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abstract: 'We study transport through a ferromagnetic single-electron transistor. The resistance is represented as a path integral, so that systems where the tunnel resistances are smaller than the quantum resistance can be investigated. Beyond the low order sequential tunneling and co-tunneling regimes, a large magnetoresistance ratio at sufficiently low temperatures is found. In the opposite limit, when the thermal energy is larger than the charging energy, the magnetoresistance ratio is only slightly enhanced.'
address:
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Department of Theoretical Physics, Lund University,\
Helgonav[ä]{}gen 5, S-223 62 Lund, Sweden
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Department of Applied Physics and Delft Institute of Microelectronics and Submicrontechnology (DIMES),\
Delft University of Technology, Lorentzweg 1, 2628 CJ Delft, The Netherlands
author:
- 'X. H. Wang'
- 'Arne Brataas$^{\dag}$'
title: 'Large Magnetoresistance Ratio in Ferromagnetic Single-Electron Transistors in the Strong Tunneling Regime'
---
Electron tunneling between two ferromagnetic metallic leads separated by an insulating layer has been studied since the seventies [@jul75]. The novel feature of these structures is that the conductance depends on the relative orientation of the magnetizations in the leads. By tuning the external magnetic field, the magnetization of a ferromagnetic metal can be controlled, and hence the conductance of the device changes. The influence of the applied magnetic field on the transport properties is characterized by the tunnel magnetoresistance ratio (TMR): $$\gamma =(R_{\text{AP}}-R_{\text{P}})/R_{\text{P}}\,, \label{tmr}$$ where $R_{\text{P}}$ ($R_{\text{AP}}$) is the resistance when the magnetizations in the leads are parallel (anti-parallel). Devices with large TMR can be used in future magneto-resistive sensors or magnetic random access memories [@moo95].
In the past years, reproducible measurements of TMR have been reported [@moo95; @Levy99; @ono97; @sche97; @san97] in structures with ultrasmall tunnel junctions. When the capacitances of the tunnel junctions are so small that the charging energy is larger than the thermal energy, the current flow is blocked when the bias voltage is smaller than a critical value determined by the junction capacitances. Such a Coulomb blockade of electron tunneling through insulating barriers has been widely investigated when the leads are nonmagnetic [@ave90; @ave91; @schon; @wang]. The single-electron transistor consists of a metallic island connected to two metallic leads via tunnel junctions and coupled to a gate-voltage $U_{g}$ via a capacitor $C_{g}$. The lowest order perturbation in terms of the dimensionless tunnel conductance $ \alpha =R_{K}/R_{T}$, where $R_{K}\equiv h/e^{2}$ is the quantum resistance and $R_{T}$ is the tunnel resistance, predicts that when the thermal energy $k_{B}T$ is much smaller than the charging energy $E_{c}=e^{2}/2C$, the conductance of the SET vanishes for $C_{g}U_{g}/e=n$, and the conductance attains a maximum for $C_{g}U_{g}/e=(n+1/2)$, where $n$ is an integral number [@ave91]. When the tunnel resistances $R_{T}$ are not much smaller than the quantum resistance, the Coulomb blockade of electron tunneling is softened due to co-tunneling and higher order tunneling processes [@ave90].
In order to describe ferromagnetic single-electron transistors (FSET’s), the spin dependence of the density of states and the transition matrix elements must be taken into account. Theoretical works on this subject based on the lowest order sequential tunnelling processes have recently been carried out [@bar98; @bra99; @kor99]. By including the co-tunneling processes in the Coulomb blockade regime, it has been shown [@tak98; @ima99] that the TMR is larger in such ultrasmall structures than in macroscopic devices, where the charging energy is negligible. When the junction resistances are smaller than the quantum resistance, the calculations based on sequential tunneling or co-tunneling become inaccurate because higher order tunneling processes determine the transport properties. It is so far unclear if the TMR of the FSET is further enhanced by the higher order tunneling processes. This question cannot be addressed from the results of the sequential tunneling or co-tunneling formulations. We will employ a non-perturbative approach to clarify this issue and show that indeed there is a further large enhancement of the TMR in the strong tunneling regime as compared with the weak tunneling case where the junction resistances are larger than the quantum resistance. This enhancement of the TMR is much larger than what calculations based on the co-tunneling processes predict.
We consider the situation when the two leads are made of the same material and have the same magnetization directions, and denote the resistance of the FSET as $R_{\eta }$, where $\eta =\text{P}$ ($\eta =\text{AP}$) for the case that the magnetization of the central grain is parallel (antiparallel) to the magnetization in the leads [@ono97; @bar98; @bra99; @kor99; @tak98]. The magnetoresistance ratio is defined by Eq. (\[tmr\]). In order to calculate the resistance of the FSET for arbitrary tunneling resistances the tunneling Hamiltonian must be treated in a nonperturbative manner. This can be realized via the path integral approach [@schon]. We use the standard single-electron tunneling Hamiltonian with spin-dependent density of states and tunneling matrix elements [@bar98; @tak98] and limit our discussions to the linear response regime.
At sufficiently high temperatures ($k_{B}T\gg E_{c}$) the system is in the classical regime, the Coulomb charging effects are negligible and the conductance of the system can be readily found. The dimensionless conductance of the FSET for the parallel magnetic alignments is $\alpha _{\text{P}}^{\text{cl}}=\alpha _{\text{M}}\left[
1+(1-P)(1-P^{\prime })/(1+P)(1+P^{\prime })\right] $, where $\alpha
_{\text{M}}$ is the dimensionless majority-band conductance of the FSET with parallel magnetic alignments, and the spin polarizations of the leads and island are denoted by $P$ and $P^{\prime }$, respectively. Similarly, the dimensionless conductance of the FSET for the anti-parallel magnetic alignments is $\alpha
_{\text{AP}}^{\text{cl}}=\alpha _{M}\left[ (1-P)/(1+P)+(1-P^{\prime
})/(1+P^{\prime })\right] $. The classical (high-temperature) magnetoresistance ratio is $$\gamma ^{\text{cl}}\equiv
\frac{R_{\text{AP}}^{\text{cl}}-R_{\text{P}}^{\text{cl}}}{R_{\text{P}}^{\text{c
l}}}=\frac{2PP^{\prime
}}{1-PP^{\prime }}\,.
\label{trc}$$
At lower temperatures the Coulomb charging effects must be taken into account. In this case, the resistance of the FSET can be calculated via Kubo’s formula [@wang], $$\frac{R_{\eta }^{\text{cl}}}{R_{\eta }}=4\pi \lim_{\omega \rightarrow
0}\omega ^{-1}{\Im }m\{\lim_{i\omega _{l}\rightarrow \omega +i\delta
}\int_{0}^{\beta }d\tau e^{i\omega _{l}\tau }r(\tau )\}\,,
\label{kubo}$$ where the Matsubara frequencies are $\omega _{l}=2\pi l/\beta $ ($l$ is an integral number) and $$\begin{aligned}
&&r(\tau )=Z_{\eta }^{-1}\sum_{k=-\infty }^{\infty }e^{2\pi ikn_{{\rm
ex}}} \nonumber \\ &&\times \int_{b_{k}}D\varphi
e^{-S_{\eta }[\varphi ]}\chi (\tau )\cos [\varphi (\tau )-\varphi
(0)]\,. \label{rt}\end{aligned}$$ where the damping kernel $\chi (\tau )$ is an even function with a period $\beta $ and Fourier components $\chi (\omega _{l})=-|\omega
_{l}|/4\pi $ and $n_{\text{ex}}=C_{g}V_{g}/e$ is controlled by the gate voltage. In Eq. (\[rt\]) the boundary condition for the path-integral is $b_{k}\rightarrow \varphi (\beta )=\varphi (0)+2\pi
k$ and $Z_{\eta }$ is the partition function of the unbiased device, namely a single-electron box (SEB)[@but]. $Z_{\eta }$ is readily represented as a path integral [@schon; @wang] $$\label{z}
Z_{\eta }=\sum_{k=-\infty }^{\infty }e^{2\pi ikn_{{\rm
ex}}}\int_{b_{k}}D\varphi e^{-S_{\eta }[\varphi ]}\,.
\label{functional}$$ The action is given by $$\begin{aligned}
&&S_{\eta }[\varphi ]=\int_{0}^{\beta }d\tau
\frac{\dot{\varphi}^{2}(\tau )}{4E_{c}} \nonumber \\
&&-\alpha _{\eta }^{\text{cl}}\int_{0}^{\beta }d\tau \int_{0}^{\beta
}d\tau ^{\prime }\chi (\tau -\tau ^{\prime })\cos [\varphi (\tau
)-\varphi (\tau ^{\prime })]\,, \label{action}\end{aligned}$$ which depends on the relative orientations of the magnetizations in the central grain and the leads through the second term in Eq.(\[action\]) describing the contribution of the tunnel junctions. If this term is neglected, the charging energy determines the action, and the sequential tunneling processes are recovered. If one expands the partition function (\[functional\]) in terms of the dimensionless conductance, a Taylor series is obtained containing sequential tunneling, co-tunneling and higher order contributions. However in practice, it is quite tedious to evaluate the path integral term by term in this way. Instead, we will evaluate the DC current nonperturbatively.
Note that the current-current correlation functions of the SET contain both sine and cosine terms in the phase differences, as described in Ref. [@wang]. However, using current conservation through the first and the second junction it can be seen that the contribution from the sine term to the conductance is proportional to the contribution from the cosine term and we obtain the simplified expression (\[rt\]).
Let us first discuss the most interesting case when the Coulomb charging energy is much larger than the thermal energy and there is a strong influence of the Coulomb charging on the current. In this case the conductance of the FSET can be varied by tuning the gate voltage [@ave91; @wang]. We use Monte Carlo simulations to compute $r(\tau
)$ (\[rt\]), and then calculate the conductances via the Kubo’s formula (\[kubo\]). Since in the Coulomb blockade regime, the TMR for integer $n_{{\rm ex}}$ is at a maximum [@ono97; @tak98], we will concentrate on this case to find the largest TMR and we set $n_{{\rm
ex}}=0$. The problem is reduced to the evaluation of expectation values, which we will treat by Monte Carlo simulations [@weg].
The computations have been carried out using the standard Metropolis algorithm. The convergence of the simulations have been checked by increasing the number of Trotter indices, which are the numbers of slices in the imaginary time interval $[0,\beta ]$ and are found to scale linear in $\beta $. Therefore at low temperatures where $\beta $ is large the simulations are time-consuming. Samplings have been done every 5 passes, and the program has been ran for a sufficiently long time to equilibrate the system before sampling. In order to guarantee a high precision of the Monte Carlo simulations, the number of samplings have been determined so that the result with doubled sampling number $2N$ is the same as the result with sampling number $N$. The typical number of samplings are found to be of order of $10^{6}$ for the parameters described below. This procedure is necessary because we calculate the Fourier components from the Monte Carlo data and then carry out the analytical continuation (\[kubo\]) using Páde approximate to get the conductance of the FSET [@man90; @cha95].
When the tunnel resistances are close to the quantum resistance, the co-tunneling result of the conductance of the SET [@ave91; @tak98; @joy97] is recovered. In order to find out if higher order tunneling processes will enhance the TMR, we show the result for the conductance of a typical value in the strong tunneling case, namely $\alpha _{M}=10$. We consider Ni leads with spin polarization $P=0.23$, and Co island with $P^{\prime }=0.35$, as in the experiment by Ono [*et al*]{} [@ono97]. For this device, the prediction of the calculations of the TMR in the classical regime (high-temperature) yields a value of 17.5 % and in the Coulomb blockade regime the co-tunneling formalism gives a value of 38 % [@tak98] . The computed results of the TMR as a function of $E_{c}/k_{B}T$ are shown in Fig. \[fig1\]. In the low temperature quantum transport regime, the TMR is denoted by $\gamma ^{\text{qu}}$. We see that the TMR is strongly enhanced by lowering the temperature and reaches the co-tunneling result [@tak98] when $E_{c}/k_{B}T\simeq 15$. In the case of co-tunneling, the TMR saturates at this temperature and remains constant at lower temperatures. We would like to emphasize that in our case by including higher order tunneling processes a further large enhancement of the TMR occurs at even lower temperatures, e.g. when $E_{c}/k_{B}T\simeq 40$ the TMR is enhanced to 90 %, which is a much larger value than in the co-tunneling regime. At even lower temperatures a further enhancement is expected. The fact that there is a large enhancement of the TMR in the strong tunneling regime can be understood as a consequence of higher order tunneling. In the co-tunneling regime the conductance of the device is proportional to the square of the tunnel conductance which roughly doubles the TMR with respect to the classical value. In our case with $\alpha _{M}=10$ the conductance increases even faster than the square of the tunnel conductance and hence the TMR is further enhanced. The numerical results of the TMR in the low temperature quantum regime can roughly be fitted by $$\gamma ^{\text{qu}}=\gamma ^{\text{cl}}+\mu ^{\text{qu}}\left(
1+\gamma ^{\text{cl}}\right) (E_{c}/k_{B}T)^{2}\,, \label{trq}$$ where we choose $\mu ^{\text{qu}}=0.0006$ so that the fitting curve is close to most of the data. Note that although $\mu ^{\text{qu}}$ is quite small, the second term on the right hand side of (\[trq\]) can be as large as unity when the temperatures are sufficiently low leading to a large enhancement of the TMR. We have thus shown that there is a large enhancement of the TMR in the strong tunneling regime at low temperatures, which is well beyond the results from the sequential tunneling and co-tunneling formalisms.
Let us now consider the semiclassical regime when the thermal energy $k_{B}T$ is larger than the Coulomb charging energy $E_{c}$. The fluctuation amplitude of the phase variable can then be estimated from the kinetic part of the action giving a value of the order of $[E_{c}/k_{B}T]^{1/2}$, and the dominating part of the action is Gaussian with some corrections from the nonquadratic parts of the tunneling term in the total action. By expanding the cosine function in the action Eq. (\[action\]) and in Eq. (\[rt\]), the resistance of the FSET to the second order in $E_{c}/k_{B}T$ is $$\frac{R_{\eta }^{\text{se}}}{R_{\eta
}^{\text{cl}}}=1+\frac{E_{c}}{3k_{B}T}+(0.04-0.02\alpha _{\eta
}^{\text{cl}})\left( \frac{E_{c}}{k_{B}T}\right) ^{2}\,.
\label{rht}$$ The calculation leading to the above equation is very similar to the one of the nonmagnetic SET carried out in Ref. [@wang]. The two first terms in Eq. (\[rht\]) have also been obtained in Ref.[@tak98]. The third term could also have been derived from the formalism used in Ref. [@tak98], but the last term is due to higher-order tunneling. It is only this term that gives a temperature-dependent contribution to the TMR at high temperatures. The TMR in the semiclassical regime is $$\gamma ^{\text{se}}=\gamma ^{\text{cl}}+\mu ^{\text{se}}\left(
1+\gamma ^{\text{cl}}\right) \left( E_{c}/k_{B}T\right) ^{2},\,
\label{trse}$$ where $$\mu ^{\text{se}}=0.04\alpha _{M}(1+PP^{\prime })/\left[ (1+P)(1+P^{\prime
})\,\right] .$$ For $P=0.23$ and $P^{\prime }=0.35$, we find $\mu ^{\text{se}}\approx
0.013\alpha _{M}$ and the TMR for intermediate Coulomb charging energy in the strong tunneling regime, e.g. for $E_{c}=k_{B}T/4$ and $\alpha _{M}=10$, increases by less than 1 % giving a TMR much smaller than the low-temperature co-tunneling result of 38 %. The above analysis shows that when the thermal energy is dominating, the Coulomb charging effect does not drastically change the TMR, and the role of the higher order tunneling processes is not very important. In order to make the TMR significantly different from the corresponding values of a macroscopic device, one has to minimize the junction capacitances so that the devices work in the low temperature regime.
It is interesting to note that the enhancement of the TMR the in low temperature quantum regime (\[trq\]) has a similar temperature-dependence as the enhancement of the TMR in the high temperature semiclassical regime (\[trse\]), with $\mu
^{\text{qu}}$ much smaller than $\mu ^{\text{se}}$. This can be understood in terms of smearing of the Coulomb blockade by the strong tunneling processes. Taking into account that at low temperatures the renormalized charging energy of the FSET, $E_{c}^{\ast }$, is one order of magnitude smaller than the bare charging energy $E_{c}$ when $\alpha _{M}=10$ [@weg], we see that $(E_{c}^{\ast }/k_{B}T)^{2}$ $\sim $ $0.01(E_{c}/k_{B}T)^{2}$, consistent with Eqs. (\[trq\]) and (\[trse\]) where $\mu ^{\text{qu}}/\mu ^{\text{se}}\sim 0.01$. The argument used here will fail at sufficiently low temperatures, where the TMR can hardly be described by a power law of $(E_{c}/k_{B}T)^{2}$. In such an extremely low temperature and strong tunneling regime, more investigations need to be done.
In conclusion, we have derived formulas for the resistance of the ferromagnetic single-electron transistor which are valid also in the regime when the junction resistances are smaller than the quantum resistance. In the strong tunneling regime we find that the magnetoresistance ratio is largely enhanced by the higher order tunneling processes at low temperatures. The magnetoresistance ratio in this regime is much larger than the value predicted by only including sequential or co-tunneling processes. We find that the TMR can at least be enhanced by a factor of 5 relative to the classical value at low temperatures. When the thermal energy is larger than the charging energy, the enhancement of the TMR is small, even when the junction resistances are much smaller than the quantum resistance.
The authors are grateful to G. E. W. Bauer, K.-A. Chao, J. Inou, A. N. Korotkov, Yu. V. Nazarov and A. A. Odintsov for stimulating discussions and R. Egger for valuable support of the Monte Carlo simulations. X. H. W. would like to thank the Swedish TFR for financial support and A. B. acknowleges financial support from the EU TMR Research Network No. FMRX-CT96-0089 (DG12-MIHT).
also at Philips Research Laboratories, Prof.Holstlaan 4, 5656 AA Eindhoven, The Netherlands.
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---
author:
- 'B.L. de Vries'
- 'M. Min'
- 'L.B.F.M. Waters'
- 'J.A.D.L. Blommaert'
- 'F. Kemper'
bibliography:
- 'references.bib'
date: 'Received 3 november, 2009; accepted 10 march 2010'
title: Determining the forsterite abundance of the dust around Asymptotic Giant Branch stars
---
[We present a diagnostic tool to determine the abundance of the crystalline silicate forsterite in AGB stars surrounded by a thick shell of silicate dust. Using six infrared spectra of high mass-loss oxygen rich AGB stars we obtain the forsterite abundance of their dust shells.]{} [We use a monte carlo radiative transfer code to calculate infrared spectra of dust enshrouded AGB stars. We vary the dust composition, mass-loss rate and outer radius. We focus on the strength of the 11.3 and the 33.6 $\mu$m forsterite bands, that probe the most recent (11.3 $\mu$m) and older (33.6 $\mu$m) mass-loss history of the star. Simple diagnostic diagrams are derived, allowing direct comparison to observed band strengths.]{} [Our analysis shows that the 11.3 $\mu$m forsterite band is a robust indicator for the forsterite abundance of the current mass-loss period for AGB stars with an optically thick dust shell. The 33.6 $\mu$m band of forsterite is sensitive to changes in the density and the geometry of the emitting dust shell, and so a less robust indicator. Applying our method to six high mass-loss rate AGB stars shows that AGB stars can have forsterite abundances of 12% by mass and higher, which is more than the previously found maximum abundance of 5%.]{}
Introduction
============
Asymptotic Giant Branch (AGB) stars represent a late phase in the evolution of low- and intermediate mass stars (0.8-8.0M$_{\odot}$). At the end of the Red Giant Branch phase when the helium in the core is depleted a star enters the AGB phase, with a degenerate carbon and oxygen core surrounded by a helium and a hydrogen burning shell. The burning shells are surrounded by a convective envelope and a dynamically active atmosphere [@HO03]. Depending on the carbon-to-oxygen ratio, either oxygen or carbon is locked in the CO molecule. This leads to two possible chemistries. Initially AGB stars exhibit an oxygen-rich chemistry reflecting the original composition of the natal molecular cloud. An AGB star becomes C-rich when sufficient carbon is dredged up from the helium-burning shell where it is formed by nucleosynthesis processes. In this study we will only consider O-rich AGB stars.
One of the most important aspects of AGB stars is their stellar wind. AGB stars can have mass-loss rates between $10^{-8}$ M$_{\odot}$ yr$^{-1}$ and about $10^{-4}$ M$_{\odot}$ yr$^{-1}$. The high mass-loss rate automatically constrains the life-time of these objects, and gives rise to dense circumstellar envelopes. Two mechanisms contribute to this mass-loss. First matter gets accelerated outwards due to shocks [@HO03]. When the material reaches distances where the temperature is low enough the material can condense into solid-state particles (from here on called dust). Then a second acceleration mechanism occurs: radiation pressure on the dust grains. The gas is coupled to the dust and dragged along. This leads to expansion speeds of 10 km s$^{-1}$ and higher.
Even though amorphous silicate dust is more abundant, crystalline silicate dust is also seen in many astronomical environments like disks around pre-main-sequence stars [@waelkens96; @meeus01; @spitzer1], comets [@wooden02], post-main-sequence stars [@waters96; @syl99; @mol02] and active galaxies [@kemper07; @spoon06]. Theoretical studies predict that the formation of crystalline silicate dust is dependent on the gas density [@tielens98; @gailsedl99; @sogawa99]. Higher densities are more favorable for the formation of crystals than low densities. @speck08 also argue that there is a correlation between crystallinity, mass-loss rate and the initial mass of the star.
These suggestions are consistent with the detection of crystalline silicates in the high density winds of OH/IR stars and the disks around post-AGB stars. Indeed, observations show a lack of crystalline silicate dust bands in the spectra of low mass-loss rate AGB stars [@waters96; @syl99]. However, an alternative explanation for this lack exists. @kemper01 have shown that a contrast effect hinders the detection of crystalline silicate bands in infrared spectra of low mass-loss rate AGB stars. This contrast effect is further discussed in sect. 3.2.2.
@kemper01 obtained an indication for the crystalline silicate abundance in the outflow of AGB stars by comparing the strength of crystalline silicate emission bands of observations to those of model spectra. For the crystalline silicates enstatite and forsterite they measured a maximum abundance of $\sim$5% by mass.
AGB stars are the main contributors of crystalline silicate material to the ISM. The creation of crystalline silicates in the ISM is not possible since the temperatures are too low. Therefore the hypothesis would be that the abundance of crystalline silicate dust in the ISM would be comparable to that found in the dust shells of AGB stars. The crystalline silicate abundance in the diffuse ISM has been investigated by @kemper04 [@kemperErr05]. They looked at the line of sight towards the Galactic Centre and found an upper limit of the degree of crystallinity in the diffuse ISM of 2.2% by mass. The abundance found in AGB stars by @kemper01 is therefore higher than that for the diffuse ISM, suggesting the crystalline material may be effectively destroyed or hidden. Possible mechanisms could be amorphization by inclusion of iron atoms in the crystalline lattice, shocks and UV radiation (see @kemper04 [@kemperErr05] for a discussion on such mechanisms).
Determining the crystalline fraction of silicates in AGB stars, as a function of mass-loss rate, will enhance our understanding in two areas: First, what are the conditions needed for the formation of crystalline silicates? Dust condensation theory suggests that the density is critical in the formation of crystalline silicates [@tielens98; @gailsedl99; @sogawa99], but this has still not been verified by observations. Secondly, how much crystalline silicate material is deposited by AGB stars to the diffuse ISM? Do the abundances in the outflow of AGB stars and the ISM match?
The most thorough way to determine the forsterite abundance in the outflows of AGB stars is to fit the full spectral range offered by ISO or Spitzer spectra. This is an exhausting endeavor to apply to multiple stars and it does not necessary deliver a good understanding on how the spectral bands depend on dust shell parameters. Therefore it is useful to follow and expand the approach of @kemper01 who derived diagnostics for the 33.6 $\mu$m forsterite band. We expand on this study by carefully studying the dependence of the spectral bands on the mass-loss rate, dust composition, thermal contact between grains and we also use two well-chosen bands, one probing the most recent mass-loss (11.3 $\mu$m) and one probing the older, colder mass-loss (33.6 $\mu$m). We derive a diagnostic diagram that is fairly easy to use and allows one to derive the forsterite abundance in the AGB star outflows.
In sect. 2 we outline how the models were computed and show several examples of computed spectra. The strength determination of the two forsterite bands is discussed in sect. 3. Also the diagnostic diagram to measure the forsterite abundance is introduced and possible limitations are discussed. In sect. 4 we apply the diagnostic diagram to six ISO-SWS spectra of AGB stars.
Modeling the AGB dust shell
===========================
In this section we present the setup used to compute model spectra and show a few examples of infrared spectra.
The model spectra were computed using the Monte Carlo radiative transport code MCMax [@min09]. Within the code we treat AGB stars as a central star enclosed in a spherically symmetric dust shell. The central star is defined by its radius and spectrum. A spectrum of an M9III giant (2667K) is used for the central star [@fluks94]. For the luminosity a typical value of 7000 $\rm{L}_{\odot}$ is used [@HO03]. Using this luminosity and integrating the spectrum one obtains a radius of $395\,\rm{R}_{\odot}$ for the central star.
An $r^{-2}$ density distribution for the dust shell gives rise to a time-independent outflow at a constant velocity. There is ample evidence for strong time variability of AGB mass-loss, see for example @decin07, which would correspond to a different density structure than the one adopted here. We will come back to this point in section 3.2.3.
The terminal wind speed (v$_{\mathrm{exp}}$) is set to the typical value of $10$ km s$^{-1}$. For the dust-to-gas ratio a value of 1/100 is assumed. For our models the dust-to-gas ratio only determines the amount of material in the dust shell that is in the form of dust. Changing this value will result in a different dust mass-loss rate and therefore it only scales the total mass-loss rates used in this paper. The temperature of the dust shell reaches a value of 1000 K at a radius of $\sim$10 AU. This radius is taken to be the radius at which the amorphous silicate condenses, and therefore defines the inner radius of the dust shell. Doing this we assume that the dust species formed before this 1000 K limit have a negligible effect on the spectrum. This is especially true for the optically thick dust shells where the infrared spectrum is not sensitive anymore to changes in the inner part of the dust shell. We note that in optically thin dust shells the maximum dust temperature detected in a study by @heras05 is around 700 K. So the calculations take somewhat high condensation temperatures for optically thin dust shells.
Dust opacities
--------------
In this paper we only consider the major components of the dust in AGB outflows. We use silicate grains, both in crystalline and amorphous form and metallic iron grains. To calculate the opacities of dust grains, the chemical composition, lattice structure and shape distribution have to be chosen. These choices will be discussed in this section.
The composition of the amorphous silicate component in AGB stars is currently not well known. In this paper we fix the amorphous silicate composition to be consistent with the spectroscopic signature of the low mass-loss rate AGB star Mira. Mira shows no clear signs of crystalline silicates making it an ideal case to study the amorphous component in AGB outflows. We have fitted the ISO-SWS infrared spectrum of Mira using amorphous silicates with different compositions and find that the best fit was made with 65.7% MgFeSiO$_{4}$ and 34.3% Mg$_{2}$SiO$_{4}$ implying an average composition of Mg$_{1.36}$Fe$_{0.64}$SiO$_{4}$. The fit was made with CDE shaped particles (see below). By adopting the composition for the amorphous silicates as found for Mira we make the implicit assumption that the composition of the amorphous silicate dust grains is not dependent on the mass-loss rate of the AGB star.
For the refractive indices we have used data measured in the laboratory for a large number of material compositions. We use here for the amorphous silicates the data from @jager2003 and @dorschner95, the forsterite data from @servoin73, the enstatite data from @jager98 and the metallic iron laboratory data from @ordal88. We always use an equal abundance of enstatite and forsterite. The opacities of forsterite are shown in fig. \[kappas\].
@min03 have shown that the use of homogeneous spheres as particle shapes introduces unrealistic effects. This is because the symmetries in these particles introduce resonances. It has been shown by several authors that the CDE (Continuous Distribution of Ellipsoids, @BH83) shape distribution provides a very good fit to both astronomical observations and laboratory measurements. Therefore, in this study we use the CDE shape distribution.
Laboratory experiments show that crystalline silicates, metallic iron and amorphous silicates have different condensation temperatures, which are also pressure dependent [@gailsedl99]. It is very well possible that high temperature condensates may act as seeds for lower temperature materials to condense onto [@gailsedl99]. Therefore we must consider chemically inhomogeneous grains as a possibility, since this will affect their temperatures and so their IR emission. However, grain condensation is likely to be a dynamical process which is difficult to model, and so it is hard to predict from chemical equilibrium condensation theory what the nature of the chemically inhomogeneous grains may be. Given these uncertainties we decided to use two choices for chemically inhomogeneous grains and investigate the effects of these two choices on the calculated spectra. These two choices should span most of parameter space. Our standard model assumes that amorphous silicate grains condense first, followed by metallic iron that forms on top of the amorphous silicates, along the lines suggested by @gailsedl99. Therefore metallic iron and amorphous silicates are in thermal contact and have the same temperature in our calculations. We assume the crystalline silicates to remain isolated and so not in thermal contact with other dust species. This minimizes the temperature of these grains, due to the low opacity at near-IR and optical wavelengths. Our second choice is to assume that all grains are in thermal contact and so have the same temperature. This obviously eliminates any temperature differences and probes the extreme case in which crystalline silicates, despite their low opacities at short wavelengths, are still efficiently heated by stellar photons.
Model spectra
-------------
In this section we show examples of computed spectra. This will illustrate the effect of the mass-loss rate, metallic iron abundance, forsterite abundance and the outer radius of the dust shell on the infrared spectra. A standard set of model parameters is used (see table \[table:1\]) and for the rest of this paper, if the parameters are not mentioned explicitly, they are set to these standard values.
For the metallic iron abundance a standard value of 4% is chosen. This is consistent with @kemper02, who have shown that the infrared spectrum of OH127.8+0.0 can be explained by including that amount of metallic iron in their models. A starting value for the forsterite and enstatite abundance has been chosen to be 4%. In most studies the outer radius of the dust shell is practically set to infinity. In this study we want to treat the outer radius as a variable, in order to investigate its effect. As a standard value we have chosen an outer radius of 500 AU. An interesting example is that @ches05 have shown that the dust shell of the current mass-loss of OH 26.5+0.6 is about 250 yr old. Using a 10 km s$^{-1}$ outflow velocity this translates to an outer radius of the dust shell of roughly 500 AU.
Parameter Standard value Minimum Maximum
------------------------- --------------------------------------- --------------------------------------- ---------------------------------------
Mass-loss $3 \cdot 10^{-5} M_{\odot} $yr$^{-1}$ $1 \cdot 10^{-6} M_{\odot} $yr$^{-1}$ $1 \cdot 10^{-4} M_{\odot} $yr$^{-1}$
Metallic iron abundance 4% 0% 9%
Forsterite abundance 4% 0% 10%
Enstatite abundance set equal to forsterite 0% 10%
Outer radius 500 AU 300 AU 3000 AU
Figure \[mdot\_compressed\] shows three spectra with different mass-loss rates. These mass-loss rates are chosen to represent objects with the 9.7 $\mu$m band in emission, partially in absorption and in absorption. The spectrum with a mass-loss rate of $10^{-6}$ M$_{\odot}$ yr$^{-1}$ shows both the 9.7 $\mu$m and the 18 $\mu$m band of amorphous silicate in emission. In this case the dust shell is optically thin and the spectrum of the central star can be observed. Increasing the mass-loss rate to $10^{-5}$ M$_{\odot}$ yr$^{-1}$ makes the blue part of the spectrum optically thick, obscuring the spectrum of the central star and causing the 9.7 $\mu$m band to go into self-absorption. The spectrum of the $10^{-4}$ M$_{\odot}$ yr$^{-1}$ case has the 9.7 $\mu$m and the 18 $\mu$m band of amorphous silicate firmly in absorption.
Spectra with different metallic iron abundances are shown in fig. \[ironCDE\]. Metallic iron absorbs short wavelength radiation more efficiently and since these grains are in thermal contact with the amorphous silicates, both heat up. In turn the amorphous silicates radiate this energy at longer wavelengths. It can be seen from the spectra that metallic iron influences the opacity in the 3 to 8 $\mu$m region and that the emission at longer wavelengths is increased when metallic iron is added.
Figure \[fo\_10zoom\] shows model results for different forsterite abundances. The forsterite band at 11.3 $\mu$m can clearly be recognized as a shoulder on the red side of the 9.7 $\mu$m band of amorphous silicate, and increases in strength with increasing abundance.
To investigate the influence of the outer radius on the spectrum, models were run with outer radii up to 3000 AU (figure \[Rout\]). From the figure it can be seen that changing the outer radius has a significant impact. By increasing the outer radius there will be more cold dust and therefore the spectrum becomes redder. The outer radius also has a profound influence on the strength of the 33.6 $\mu$m band. For an outer radius of 3000AU the band is seen to be strong and easily detectable. If the radius is decreased to 300AU the strength of the band is also seen to go down steeply. We will see this effect more quantitatively in sect. 3.2.2.
Forsterite abundance indicators
===============================
In this section we discuss the applicability of the 11.3 $\mu$m and the 33.6 $\mu$m forsterite bands as forsterite abundance indicators. We compute the strength of these bands for model spectra and study their dependence on the model parameters. For both bands the strength is measured in spectra with varying mass-loss rate, forsterite abundance, metallic iron abundance and outer radius. We have chosen to measure the equivalent width as a measure of the strength of the band because it is less sensitive to a choice in grain shapes, than for instance the ratio of the peak flux over the continuum. Although the band shape may change for different grain shapes, the total power remains conserved.
Both bands require a different method for measuring their strength. The 11.3 $\mu$m band is an absorption band superimposed on the 9.7 $\mu$m band of amorphous silicate and a careful measurement of its strength is needed. In our method to measure the strength of the 11.3 $\mu$m band we limit this study to the case where the 9.7 $\mu$m band of amorphous silicate is in absorption. As a measure of the strength of the band, the equivalent width is calculated from the optical depth profile (see sect. 3.1).
The 33.6 $\mu$m band is an emission band for the mass-loss rates that are considered. This means the band is predominantly formed in the optically thin region of the wind. Constructing a continuum under the band enables us to calculate the equivalent width. The details on how this is calculated are discussed in sect. 3.2.
Comparable to the 11.3 $\mu$m band of forsterite, enstatite has a band at 9.3 $\mu$m. We are looking into the applicability of this enstatite band, but its analysis is more complicated than for the 11.3 $\mu$m band of forsterite. The 9.3 $\mu$m band of enstatite lies closer to the absorption maximum of the 9.7 $\mu$m band of amorphous silicate. In this paper we focus on the bands of forsterite only, since these can be compared with the results of @kemper01.
The 11.3 $\mu$m forsterite band
-------------------------------
### Method
To measure the strength of the 11.3 $\mu$m band a local continuum is defined on top of the 9.7 $\mu$m amorphous silicate band. A linear interpolation between the flux densities at 8.0 $\mu $m to 8.1 $\mu$m and 12.9 $\mu$m to 13.0 $\mu$m is made to construct this local continuum.
For every wavelength in the range of the 9.7 $\mu$m band the optical depth is calculated: $F(\lambda)/F_{cont}(\lambda) = e^{-\tau}$, where $F(\lambda)$ is the flux of the model and $F_{cont}(\lambda)$ the flux of the continuum. In the optical depth profile $\tau(\lambda)$ a local “continuum” is defined under the 11.3 $\mu$m band. This is done by making a linear fit through the optical depths at the wavelength points 10.5 $\mu$m to 10.6 $\mu$m and 11.6 $\mu$m to 11.7 $\mu$m. Figure \[97tau\] shows three optical depth curves together with the continuum constructed under the 11.3 $\mu$m band. The 11.3 $\mu$m band can now be normalized: $\tau(\lambda)/\tau_{cont}(\lambda) -1$. The equivalent width in terms of optical depth is calculated by numerical integration from 10.5 $\mu$m to 11.7 $\mu$m. Figure \[113tau\] shows three normalized optical depth curves for the 11.3 $\mu$m band.
### Results
The effect of changes in the mass-loss rate on the strength of the 11.3 $\mu$m band is shown in fig. \[113\_cryst\_th\], for a range of forsterite abundances. The band strength decreases with increasing mass-loss, reaching a constant value for the highest mass-loss rates. This behavior can be understood as follows. For the highest mass-loss rates, optical depths are high for many lines of sight through the dust shell. In this case the band strength is determined by the ratios of the optical depths of the dust species, and is not sensitive any more to temperature differences. This is why the curves in fig. \[113\_cryst\_th\] reach a constant value. At lower mass-loss rates, *emission* from optically thin lines of sight through the dust shell become more important. The temperature difference between dust species plays a role and the warmer amorphous silicates and metallic iron will begin to fill in the 9.7 $\mu$m absorption band seen so prominently at high mass-loss rates. This results in a relatively smaller optical depth of the amorphous silicate band compared to that of the crystalline silicates, and so the ratio ($\tau (\lambda) / \tau_{cont}(\lambda) - 1$) will increase. The curve for 2% forsterite does not follow the general trend seen in fig. \[113\_cryst\_th\] at low mass-loss rates. This is due to the approximate way in which the continuum is estimated (linear fit), causing small errors that can be important for weak bands.
The outer radius of the dust shell has no significant effect on the strength of the 11.3 $\mu$m band and a figure has been omitted.
Adding metallic iron increases the emission of the continuum under the 9.7 $\mu$m band. This reduces the optical depth in the region of the 9.7 $\mu$m band ($\tau_{cont}(\lambda)$), causing the equivalent width of the 11.3 $\mu$m band to increase (see fig. \[113\_iron\_th\]).
In order to test the effects of our choice regarding thermal contact we also consider the case of thermal contact between all grains (section 2.1). Figure \[113\_th\_cont\] compares the strength of the 11.3 $\mu$m band in the case where only amorphous silicate and metallic iron are in thermal contact with the case where all dust species are in thermal contact. The strength of the 11.3 $\mu$m band decreases by adopting thermal contact between all the dust species. This is because part of the thermal energy is redistributed to the crystalline silicate species, which do not contribute to the continuum. This causes the flux of the continuum to drop, resulting in a weaker 11.3 $\mu$m band.
### Discussion
The 11.3 $\mu$m forsterite band is formed close to the star and so it probes the most recent mass-loss rate. We find that its strength is relatively insensitive to model parameters (other than the forsterite abundance itself) when the mass-loss rate exceeds roughly $3 \cdot 10^{-5} \,\rm{M}_{\odot}\,$yr$^{-1}$. The largest uncertainty at these high mass-loss rates is due to changes in the metallic iron abundance. The effect of thermal contact between forsterite and other dust species is minor (1-2%). In our standard model we assume no thermal contact, which leads to conservative estimates of the forsterite abundance.
The 33.6 $\mu$m forsterite band
-------------------------------
### Method
In order to measure the equivalent width of the 33.6 $\mu$m band a local continuum is defined by making a linear fit through the flux densities at 31.0 $\mu$m to 31.5 $\mu$m and 35 $\mu$m to 35.5 $\mu$m. An example of such a continuum is shown in fig. \[336\_cont\]. This continuum is used to normalize the spectrum in this region, calculating $F(\lambda)/F_{cont}(\lambda) -1$. Three such normalized bands with different forsterite abundances are shown in fig. \[336\_band\]. The equivalent width of the normalized band is calculated by numerical integration between the wavelength values of 31 $\mu$m and 35.5 $\mu$m. We note that due to the emission nature of the band its equivalent width will be sensitive to the abundance and temperature of the dust species emitting at 33.6 $\mu$m.
### Results
Figure \[336\_cryst\_th\] shows the equivalent width of the 33.6 $\mu$m band as function of mass-loss rate for different forsterite abundances. The mass-loss rates range from $10^{-6} \,\rm{M}_{\odot}\,$yr$^{-1}$ up to $2 \cdot 10^{-4} \,\rm{M}_{\odot}\,$yr$^{-1}$. The strength of the band initially increases with increasing mass-loss rate, followed by a drop and eventually the band is in absorption. This behavior is similar to that seen by @kemper01 and has been explained by these authors in terms of changes in the temperature of the different dust species as a function of mass-loss rate. For low mass-loss rates, the dust shell is optically thin and the (near-IR) stellar photons heat the dust directly. This causes large temperature differences between amorphous silicate (which has high opacity at near-IR wavelengths) and crystalline silicates (which are almost transparent in the near-IR). The band strength of forsterite will thus be weak. For higher mass-loss rates the dust shell will become more optically thick and more and more of the dust will receive longer wavelength photons emitted by the optically thick inner dust shell. The opacity of amorphous and crystalline dust at mid-IR wavelengths is very comparable and so the temperature difference between them will decrease. This results in an increase of the band strength. For even higher mass loss rates (larger than about $10^{-5} \,\rm{M}_{\odot}\,$yr$^{-1}$) some lines of sight become optically thick at 33.6 $\mu$m and this will result in a weakening of the band, eventually even leading to absorption.
The fact that the curve for 2% forsterite becomes negative at low mass-loss rates is due to the construction of the continuum, which is curved but approximated by a linear fit. This introduces small errors that can be important for weak bands, especially at low mass-loss rates.
The effect of metallic iron on the equivalent width of the 33.6 $\mu$m band is demonstrated in fig. \[336\_iron\_th\]. Metallic iron increases the flux of the continuum, but not that of forsterite. Since the equivalent width is a measure of the relative strength of forsterite to the continuum, increasing the metallic iron abundance will decrease the equivalent width. The reason for this is that forsterite is considered not to be in thermal contact with the amorphous silicates, but metallic iron is.
Increasing the outer radius of the dust shell increases the amount of cold, emitting dust and therefore increases the emission of the 33.6 $\mu$m band (figure \[336\_rout\_th\]). It is interesting that the effect of the size of the dust shell becomes smaller and even negligible for low mass-loss rates. Below $\sim 6\cdot 10^{-6} \,\rm{M}_{\odot}\,$yr$^{-1}$ the effect is already very small. The reason is that the wind is so optically thin that adding more material by increasing the outer radius adds material of such low temperatures that its emission is negligible.
Figure \[336\_th\_cont\] shows the equivalent width of the 33.6 $\mu$m band for the two different cases of thermal contact (see sect. 3.1.2) for models with 2% and 10% forsterite. It shows that the equivalent width of the band goes up when all the dust species are in thermal contact. In the case where all dust species are in thermal contact with each other, forsterite is heated up by metallic iron, creating a stronger emission band. The strength of this effect depends on the forsterite abundance.
### Discussion
The 33.6 $\mu$m band is sensitive to the mass-loss rate of the AGB star as well the outer radius of the dust shell. The size of the dust shell has an effect because the 33.6 $\mu$m band is mainly formed by the emission of cold dust in the outskirts of the wind. This tells us that other parameters that influence this cold dust will also affect the 33.6 $\mu$m band. As an example we know that AGB stars can have periods with different mass-loss rates and material of such a previous mass-loss period could also influence the 33.6 $\mu$m band. Such time variable mass-loss can be modeled as a sudden jump in the density, but can also be studied by introducing a steeper or flatter radial density gradient. Changes in the radial density gradient will affect the relative strength of the 33.6 $\mu$m emission band with respect to that of the 11.3 $\mu$m absorption band. We note however that the 11.3 $\mu$m band is most sensitive to the innermost regions of the shell since it is in absorption. Therefore the 11.3 $\mu$m band remains a robust indicator of the most recent mass-loss rate. But because of the dependence on the dust shell parameters the 33.6 $\mu$m band is not a good forsterite abundance indicator.
Another reason why the 33.6 $\mu$m band is not a practical indicator is that for very high mass-loss rates, typically bigger than $10^{-4}$ M$_{\odot}/yr$, all model curves converge and so become insensitive to the forsterite abundance.
Measuring the forsterite abundance
==================================
We apply our method to six spectra of high mass-loss rate AGB stars (see table \[tab\_obs\]) observed by ISO-SWS [@syl99]. The reduced data have been taken from @sloan03. We have used ISO-SWS spectra and not Spitzer spectra, since those are still in the pipe-line to be reduced and analyzed. Inspection of the spectra shows that the 9.7 $\mu$m amorphous silicate band is in absorption. In addition, a shoulder is seen in the observed spectra at 11.3 $\mu$m, which we attribute to forsterite. The latest detection of a very strong 11.3 $\mu$m band has been reported by @speck08. Their detection is the first detection where the 11.3 $\mu$m forsterite band does not appear as a shoulder on the 9.7 $\mu$m band of amorphous silicate, but as a separate absorption band. Given the presence of the 33.6 $\mu$m forsterite band in the ISO spectra of our sample of six stars [@kemper01], we suggest that the 11.3 $\mu$m shoulder can indeed be attributed to forsterite as well.
We have also fitted the 9.7 $\mu$m band of amorphous silicate using the opacities discussed in section 2.1 and find good agreement (not shown). This suggests that the adopted opacities that were obtained by fitting the 9.7 $\mu$m band of Mira also apply to the higher mass-loss rate AGB stars. It also means that we can use these opacities in this section to measure the forsterite abundance.
We have measured the band strength of the 11.3 $\mu$m forsterite band as outlined in sect. 3.1. The errors calculated for the strength of the bands are determined using standard error propagation formula applied to the errors in the flux obtained from @sloan03. A 10% error in the flux of the continuum is introduced to account for possible errors in the construction of the continuum. This is done for the two continua constructed in order to measure the strength of the 11.3 $\mu$m band. Mass-loss rates from the literature are listed in Table \[tab\_obs\]. No mass-loss rate was published for IRAS 17010$-$3840. The spectrum of IRAS 17010-3840 has its 9.7 $\mu$m band of amorphous silicate firmly in absorption (similar to the other sources), therefore its mass-loss rate has been assumed to lie between $5 \cdot 10^{-5}$ M$_{\odot}$ yr$^{-1}$ and $2 \cdot 10^{-4}$ M$_{\odot}$ yr$^{-1}$.
----------------- ------------------------- ------------- ------------- ------------ --
Source M Strength Fo. abun. Fo abun.
(M$_{\odot}$ yr$^{-1}$) 11.3 $\mu$m 11.3 $\mu$m Kemper et
$^{(}$[^1] $^{)}$ band band al. (2001)
IRAS 17010-3840 ? 0.14 $>$12% -
OH104.91+2.41 5.6 $10^{-5}$ (1) 0.03 $<$2% $<$5%
OH26.5+0.6 16 $10^{-5}$ (2) 0.06 $\sim$4% $<$5%
OH127.8+0.0 20 $10^{-5}$ (3) 0.04 $\sim$2% $<$5%
OH30.1-0.7 15 $10^{-5}$ (4) 0.10 $\sim$9% -
OH32.8-0.3 16 $10^{-5}$ (5) 0.14 $>$12% $<$5%
----------------- ------------------------- ------------- ------------- ------------ --
: A selection of AGB stars with their mass-loss rates and the results. It lists the strength of the 11.3 $\mu$m band together with the indicated forsterite (fo.) abundance using the 11.3 $\mu$m band. The sources have been selected using the catalog of @sloan03. The forsterite abundance obtained by @kemper01 has also been included.[]{data-label="tab_obs"}
Figure \[113\_OBS\] shows the strength of the 11.3 $\mu$m band for the six sources. The forsterite abundances indicated by fig. \[113\_OBS\] are listed in table \[tab\_obs\]. The results show that both stars with a high and low forsterite abundance are found (below 2% and up to $\sim$12%). Because of our choice of thermal contact between the dust species these values are lower limits.
@kemper01 have used the 33.6 $\mu$m band to measure the forsterite abundance in AGB stars. From their sample of six AGB stars (including OH104.91+2.41, OH127.8+0.0, OH26.5+0.6, OH32.8-0.3) they found a maximum forsterite abundance of 5% by mass. The forsterite abundances we find for OH104.91+2.41, OH127.8+0.0 and OH26.5+0.6 do not exceed this maximum of 5%. The sources OH32.8-0.3, OH30.1-0.7 and IRAS17010-3840 have significantly higher forsterite abundances.
Conclusions
===========
We have shown that for optically thick dust shells the strength of the 11.3 $\mu$m band is a robust quantity to use as an indicator of the forsterite abundance of the current mass-loss period.
Applying the 33.6 $\mu$m band is practically impossible without knowledge of the mass-loss rate, the size of the dust shell and other parameters that influence the emission of the cold dust in the outskirts of the dust shell of the AGB star. Additionally, for certain mass-loss rates (depending on the dust shell parameters) the 33.6 $\mu$m band is also in transition from an emission to an absorption band, making the use of this band impossible.
As an example and first result we used the 11.3 $\mu$m band as an indicator of the forsterite abundance for six AGB stars. This showed that these objects can have low (below 2%) but also very high forsterite abundances (12% and possibly higher), meaning that AGB stars can have a forsterite abundance higher than 5% by mass, which was the maximum found by @kemper01. It also means that the discrepancy between the amount of crystalline material in the ISM and that in AGB stars could even be higher.
If the forsterite abundances found for the six sources in this study represent all AGB stars, the forsterite abundance in AGB stars alone would be higher than the crystalline abundance (upper limit of 2.2%) found by @kemperErr05 for the ISM. To find out if the abundances found for these stars are typical for AGB stars, more spectra have to be analyzed. The number of spectra are also not enough to test any correlations. But this paper opens the way to a more elaborate study similar to the one described in sect. 4 of this paper. Determining the amount of forsterite that is produced by high mass-loss rate AGB stars will help to test the suggested correlations and see if there is really a discrepancy between the produced abundance in AGB stars and the abundance found in the ISM.
B.L. de Vries acknowledges support from the Fund for Scientific Research of Flanders (FWO) under grant number G.0470.07.
[^1]: 1: @riechers05, 2: @just96, 3: @just92, 4: @schut89, 5: @groene94
|
---
abstract: |
In this paper we consider a single-server cyclic polling system consisting of two queues. Between visits to successive queues, the server is delayed by a random switch-over time. Two types of customers arrive at the first queue: high and low priority customers. For this situation the following service disciplines are considered: gated, globally gated, and exhaustive. We study the cycle time distribution, the waiting times for each customer type, the joint queue length distribution at polling epochs, and the steady-state marginal queue length distributions for each customer type.
**Keywords:** Polling, priority levels, queue lengths, waiting times
author:
- |
M.A.A. Boon[^1]\
<marko@win.tue.nl>
- |
I.J.B.F. Adan\
<iadan@win.tue.nl>
- |
O.J. Boxma\
<boxma@win.tue.nl>
date: 'May, 2008'
title: 'A Two-Queue Polling Model with Two Priority Levels in the First Queue[^2][^3]'
---
Introduction {#intro}
============
A polling model is a single-server system in which the server visits $n$ queues $Q_1, \dots, Q_n$ in cyclic order. Customers that arrive at $Q_i$ are referred to as type $i$ customers. The special feature of the model considered in the present paper is that, within a customer type, we distinguish high and low priority customers. More specifically, we study a polling system which consists of two queues, $Q_1$ and $Q_2$. The first of these queues contains customers of two priority classes, high ($H$) and low ($L$). The exhaustive, gated and globally gated service disciplines are studied.
Our motivation to study a polling model with priorities is that the performance of a polling system can be improved through the introduction of priorities. In production environments, e.g., one could give highest priority to jobs with a service requirement below a certain threshold level. This might decrease the mean waiting time of an arbitrary customer without having to purchase additional resources [@wierman07]. Priority polling models also can be used to study traffic intersections where conflicting traffic flows face a green light simultaneously; e.g. traffic which takes a left turn may have to give right of way to conflicting traffic that moves straight on, even if the traffic light is green for both traffic flows. Another application is discussed in [@cicin2001], where a priority polling model is used to study scheduling of surgery procedures in medical emergency rooms. In the computer science community the Bluetooth and 802.11 protocols are frequently modelled as polling systems, cf. [@ieee802.11-1; @bluetooth1; @ieee802.11-2; @bluetooth2]. Many scheduling policies that have been considered or implemented in these protocols involve different priority levels in order to improve Quality-of-Service (QoS) for traffic that is very sensitive to delays or loss of data, such as Voice over Wireless IP. The 802.11e amendment defines a set of QoS enhancements for wireless LAN applications by differentiating between high priority traffic, like streaming multimedia, and low priority traffic, like web browsing and email traffic.
Although there is quite an extensive amount of literature available on polling systems, only very few papers treat priorities in polling models. Most of these papers only provide approximations or focus on pseudo-conservation laws. In [@wierman07] exact mean waiting time results are obtained using the Mean Value Analysis (MVA) framework for polling systems, developed in [@winands06]. The MVA framework can only be used to find the first moment of the waiting time distribution for each customer type, and the mean residual cycle time. The main contribution of the present paper is the derivation of Laplace Stieltjes Transforms (LSTs) of the distributions of the marginal waiting times for each customer type; in particular it turns out to be possible to obtain exact expressions for the waiting time distributions of both high and low priority customers at a queue of a polling system. Probability Generating Functions (PGFs) are derived for the joint queue length distribution at polling epochs, and for the steady-state marginal queue length distribution of the number of customers at an arbitrary epoch.
The present paper is structured as follows: Section \[general\] gathers known results of nonpriority polling models which are relevant for the present study. Sections \[gated\] (gated), \[globallygated\] (globally gated), and \[exhaustive\] (exhaustive) give new results on the priority polling model. In each of the sections we successively discuss the joint queue length distribution at polling epochs, the cycle time distribution, the marginal queue length distributions and waiting time distributions. The mean waiting times are given at the end of each section. A numerical example is presented in Section \[numericalexample\] to illustrate some of the improvements that can be obtained by introducing prioritisation in a polling system.
Notation and description of the nonpriority polling model {#general}
=========================================================
The model that is considered in this section, is a nonpriority polling model with two queues ($Q_1$ and $Q_2$). We consider three service disciplines: gated, globally gated, and exhaustive. The gated service discipline states that during a visit to $Q_i$, the server serves only those type $i$ customers who are present at the polling epoch. All type $i$ customers that arrive during this visit will be served in the next cycle. In this respect, a cycle is the time between two successive visit beginnings to a queue. The exhaustive service discipline states that when the server arrives at $Q_i$, all type $i$ customers are served until no type $i$ customer is present in the system. We also consider the globally gated service discipline, which means that during a cycle only those customers will be served that were present at the beginning of that cycle.
Customers of type $i$ arrive at $Q_i$ according to a Poisson process with arrival rate $\lambda_i$ $(i = 1,2)$. Service times can follow any distribution, and we assume that a customer’s service time is independent of other service times and independent of the arrival processes. The LST of the distribution of the generic service time $B_i$ of type $i$ customers is denoted by $\beta_i(\cdot)$. The fraction of time that the server is serving customers of type $i$ equals $\rho_i := \lambda_i E(B_i)$. Switches of the server from $Q_i$ to $Q_{i+1}$ (all indices modulo 2), require a switch-over time $S_i$. The LST of this switch-over time distribution is denoted by $\sigma_i(\cdot)$. The fraction of time that the server is working (i.e., not switching) is $\rho := \rho_1+\rho_2$. We assume that $\rho < 1$, which is a necessary and sufficient condition for the steady state distributions of cycle times, queue lengths and waiting times to exist.
[@takacs68] studied this model, but without switch-over times and only with the exhaustive service discipline. [@coopermurray69] analysed this polling system for any number of queues, and for both gated and exhaustive service disciplines. [@eisenberg72] obtained results for a polling system with switch-over times (but only exhaustive service) by relating the PGFs of the joint queue length distributions at visit beginnings, visit endings, service beginnings and service endings. [@resing93] was the first to point out the relation between polling systems and Multitype Branching Processes with immigration in each state. His results can be applied to polling models in which each queue satisfies the following property:
\[resingproperty\] If the server arrives at $Q_i$ to find $k_i$ customers there, then during the course of the server’s visit, each of these $k_i$ customers will effectively be replaced in an i.i.d. manner by a random population having probability generating function $h_i(z_1,\dots,z_n)$, which can be any $n$-dimensional probability generating function.
We use this property, and the relation to Multitype Branching Processes, to find results for our polling system with two queues, two priorities in the first queue, and gated, globally gated, and exhaustive service discipline. Notice that, unlike the gated and exhaustive service disciplines, the globally gated service discipline does not satisfy Property \[resingproperty\]. But the results obtained by Resing also hold for a more general class of polling systems, namely those which satisfy the following (weaker) property that is formulated in [@semphd]:
\[borstproperty\] If there are $k_i$ customers present at $Q_i$ at the beginning (or the end) of a visit to $Q_{\pi(i)}$, with $\pi(i) \in \{1, \dots, n\}$, then during the course of the visit to $Q_i$, each of these $k_i$ customers will effectively be replaced in an i.i.d. manner by a random population having probability generating function $h_i(z_1,\dots,z_n)$, which can be any $n$-dimensional probability generating function.
Globally gated and gated are special cases of the synchronised gated service discipline, which states that only customers in $Q_i$ will be served that were present at the moment that the server reaches the “parent queue” of $Q_i$: $Q_{\pi(i)}$. For gated service, $\pi(i) = i$, for globally gated service, $\pi(i) = 1$. The synchronised gated service discipline is discussed in [@khamisy92], but no observation is made that this discipline is a member of the class of polling systems satisfying Property \[borstproperty\] which means that results as obtained in [@resing93] can be extended to this model.
[@borst97] combined the results of [@resing93] and [@eisenberg72] to find a relation between the PGFs of the marginal queue length distribution for polling systems with and without switch-over times, expressed in the Fuhrmann-Cooper queue length decomposition form [@fuhrmanncooper85].
Joint queue length distribution at polling epochs
-------------------------------------------------
The probability generating function $h_i(z_1,\dots,z_n)$ which is mentioned in Property \[resingproperty\] depends on the service discipline. In a polling system with two queues and gated service we have $h_i(z_1, z_2) = \beta_i({\lambda_1(1-z_1)+\lambda_2(1-z_2)})$. For exhaustive service this PGF becomes $h_i(z_1, z_2) = \pi_i(\sum_{j\neq i} \lambda_j(1-z_j))$, where $\pi_i(\cdot)$ is the LST of a busy period (BP) distribution in an $M/G/1$ system with only type $i$ customers, so it is the root of the equation $\pi_i(\omega) = \beta_i(\omega + \lambda_i(1 - \pi_i(\omega)))$. We choose the beginning of a visit to $Q_1$ as start of a cycle. In order to find the joint queue length distribution at the beginning of a cycle, we relate the numbers of customers in each queue at the beginning of a cycle to those at the beginning of the previous cycle. Customers always enter the system during a switch-over time, or during a visit period. The first group is called *immigration*, whereas a customer from the second group is called *offspring* of the customer that is served at the moment of his arrival. We define the immigration PGF for each switch-over time and the offspring PGF for each visit period analogous to [@resing93]. The immigration PGFs are: $$\begin{aligned}
g^{(2)}(z_1, z_2) &= \sigma_2({\lambda_1(1-z_1)+\lambda_2(1-z_2)}), \\
g^{(1)}(z_1, z_2) &= \sigma_1(\lambda_1(1-z_1)+\lambda_2(1-h_2(z_1, z_2))).\end{aligned}$$ $g^{(2)}(z_1, z_2)$ is the PGF of the joint distribution of type $1$ and $2$ customers that arrive during $S_2$. For $S_1$ things are slightly more complicated, since type $2$ customers arriving during $S_1$ may be served before the end of the cycle, and generate offspring. $g^{(1)}(z_1, z_2)$ is the joint PGF of the type $1$ and $2$ customers present at the end of the cycle that either arrived during $S_1$, or are offspring of type 2 customers that arrived during $S_1$. The total immigration PGF is the product of these two PGFs: $$g(z_1, z_2) = \prod_{i=1}^2 g^{(i)}(z_1,z_2) = g^{(1)}(z_1,z_2)g^{(2)}(z_1,z_2).$$ We define the offspring PGFs for each visit period in a similar manner:$$\begin{aligned}
f^{(2)}(z_1, z_2) &= h_2(z_1, z_2),\\
f^{(1)}(z_1, z_2) &= h_1(z_1, h_2(z_1, z_2)).\end{aligned}$$ The term for $Q_1$ is again slightly more complicated than the term for $Q_2$, since type 2 customers arriving during a server visit to $Q_1$ may be served before the end of the cycle, and generate offspring.
[@resing93] shows that the following recursive expression holds for the joint queue length PGF at the beginning of a cycle (starting with a visit to $Q_1$): $$P_1(z_1, z_2) = g(z_1, z_2)P_1\left(f^{(1)}(z_1, z_2), f^{(2)}(z_1, z_2)\right).\label{P1recursive}$$ This expression can be used to compute moments of the joint queue length distribution. Alternatively, iteration of this expression yields the following closed form expression for $P_1(z_1, z_2)$: $$P_1(z_1, z_2) = \prod_{n=0}^\infty g(f_n(z_1, z_2)),\label{p1twoqueues}$$ where we use the following recursive definition for $f_n(z_1, z_2)$, $n=0,1,2,\dots$: $$\begin{aligned}
f_n(z_1, z_2) &= (f^{(1)}(f_{n-1}(z_1, z_2)), f^{(2)}(f_{n-1}(z_1, z_2))), \\
f_0(z_1, z_2) &= (z_1, z_2).\end{aligned}$$ [@resing93] proves that this infinite product converges if and only if $\rho < 1$.
We can relate the joint queue length distribution at other polling epochs to $P_1(z_1, z_2)$. We denote the PGF of the joint queue length distribution at a visit beginning to $Q_i$ by $V_{b_i}(\cdot)$, so $P_1(\cdot) = V_{b_1}(\cdot)$. The PGF of the joint queue length distribution at a visit completion to $Q_i$ is denoted by $V_{c_i}(\cdot)$. The following relations hold: $$\begin{aligned}
V_{b_1}(z_1, z_2) &= V_{c_2}(z_1, z_2)\sigma_2({\lambda_1(1-z_1)+\lambda_2(1-z_2)}) \nonumber\\
&= V_{b_2}(z_1, h_2(z_1, z_2)) \sigma_2({\lambda_1(1-z_1)+\lambda_2(1-z_2)}) \nonumber\\
&= V_{b_2}(z_1, f^{(2)}(z_1, z_2)) g^{(2)}(z_1, z_2), \label{vb1twoqueues}\\
V_{b_2}(z_1, z_2) &= V_{c_1}(z_1, z_2)\sigma_1({\lambda_1(1-z_1)+\lambda_2(1-z_2)}) \nonumber\\
&= V_{b_1}(h_1(z_1, z_2), z_2) \sigma_1({\lambda_1(1-z_1)+\lambda_2(1-z_2)}).\label{vb2twoqueues}\end{aligned}$$
Cycle time
----------
The cycle time, starting at a visit *beginning* to $Q_1$, is the sum of the visit times to $Q_1$ and $Q_2$, and the two switch-over times which are independent of the visit times. Since type 2 customers who arrive during the visit to $Q_1$ or the switch from $Q_1$ to $Q_2$ will be served during the visit to $Q_2$, it can be shown that the LST of the distribution of the cycle time $C_1$, $\gamma_1(\cdot)$, is related to $P_1(\cdot)$ as follows: $$\gamma_1(\omega) = \sigma_1(\omega + \lambda_2(1-\phi_2(\omega))) \, \sigma_2(\omega) \, P_1(\phi_1(\omega + \lambda_2(1-\phi_2(\omega))), \phi_2(\omega)),\label{cycletimelst}$$ where $\phi_i(\cdot)$ is the LST of the distribution of the time that the server spends at $Q_i$ due to the presence of one type $i$ customer there. For gated service $\phi_i(\cdot) = \beta_i(\cdot)$, for exhaustive service $\phi_i(\cdot) = \pi_i(\cdot)$. A proof of can be found in [@boxmafralixbruin08].
In some cases it is convenient to choose a different starting point for a cycle, for example when analysing a polling system with exhaustive service. If we define $C_1^*$ to be the time between two successive visit *completions* to $Q_1$, the LST of its distribution, $\gamma^*_1(\cdot)$, is: $$\begin{aligned}
\gamma^*_1(\omega) =& \sigma_1(\omega + \lambda_1(1-\phi_1(\omega)) + \lambda_2(1-\phi_2(\omega + \lambda_1(1-\phi_1(\omega)))))\nonumber\\
&\cdot \sigma_2(\omega + \lambda_1(1-\phi_1(\omega))) \, V_{c_1}(\phi_1(\omega), \phi_2(\omega+\lambda_1(1-\phi_1(\omega)))),\label{cycletimlstVc}\end{aligned}$$ with $V_{c_1}(z_1,z_2) = P_1(h_1(z_1, z_2), z_2)$.
Marginal queue lengths and waiting times
----------------------------------------
We denote the PGF of the steady-state marginal queue length distribution of $Q_1$ at the visit beginning by $\widetilde{V}_{b_1}(z) = V_{b_1}(z, 1)$. Analogously we define $\widetilde{V}_{b_2}(\cdot), \widetilde{V}_{c_1}(\cdot)$, and $\widetilde{V}_{c_2}(\cdot)$. It is shown in [@borst97] that the steady-state marginal queue length of $Q_i$ can be decomposed into two parts: the queue length of the corresponding $M/G/1$ queue with only type $i$ customers, and the queue length at an arbitrary epoch during the intervisit period of $Q_i$, denoted by ${N_{i|I}}$. [@borst97] show that by virtue of PASTA, ${N_{i|I}}$ has the same distribution as the number of type $i$ customers seen by an arbitrary type $i$ customer arriving during an intervisit period, which equals $$E(z^{N_{i|I}}) = \frac{E(z^{N_{i|I_{\textit{begin}}}}) - E(z^{N_{i|I_{\textit{end}}}})}{(1-z)(E(N_{i|I_{\textit{end}}}) - E(N_{i|I_{\textit{begin}}}))},$$ where $N_{i|I_{\textit{begin}}}$ is the number of type $i$ customers at the beginning of an intervisit period $I_i$, and $N_{i|I_{\textit{end}}}$ is the number of type $i$ customers at the end of $I_i$. Since the beginning of an intervisit period coincides with the completion of a visit to $Q_i$, and the end of an intervisit period coincides with the beginning of a visit, we know the PGFs for the distributions of these random variables: $\widetilde V_{c_i}(\cdot)$ and $\widetilde V_{b_i}(\cdot)$. This leads to the following expression for the PGF of the steady-state queue length distribution of $Q_i$ at an arbitrary epoch, $E[z^{N_i}]$: $$E[z^{N_i}] = \frac{(1-\rho_i)(1-z)\beta_i(\lambda_i(1-z))}{\beta_i(\lambda_i(1-z))-z}
\cdot\frac{\widetilde{V}_{c_i}(z) - \widetilde{V}_{b_i}(z)}{(1-z)(E(N_{i|I_{\textit{end}}}) - E(N_{i|I_{\textit{begin}}}))}. \label{queuelengthdecomposition}$$ [@keilsonservi90] show that the distributional form of Little’s law can be used to find the LST of the marginal waiting time distribution: $E(z^{N_i}) = E({\textrm{e}}^{-\lambda_i(1-z)(W_i + B_i)})$, hence $E({\textrm{e}}^{-\omega W_i}) = E[(1-\frac{\omega}{\lambda_i})^{N_i}]/\beta_i(\omega)$. This can be substituted into : $$\begin{aligned}
E[{\textrm{e}}^{-\omega W_i}] =& \frac{(1-\rho_i)\omega}{\omega-\lambda_i(1-\beta_i(\omega))}\cdot
\frac{\widetilde{V}_{c_i}\left(1-\frac{\omega}{\lambda_i}\right) - \widetilde{V}_{b_i}\left(1-\frac{\omega}{\lambda_i}\right)}{(E(N_{i|I_{\textit{end}}}) - E(N_{i|I_{\textit{begin}}}))\omega/\lambda_i} \nonumber\\
=& E[{\textrm{e}}^{-\omega W_{i|M/G/1}}]E\left[\left(1-\frac\omega{\lambda_i}\right)^{N_{i|I}}\right].\label{waitingtimedecomposition}\end{aligned}$$ The interpretation of this formula is that the waiting time of a type $i$ customer in a polling model is the sum of two independent random variables: the waiting time of a customer in an $M/G/1$ queue with only type $i$ customers, $W_{i|M/G/1}$, and the remaining intervisit time for a customer that arrives at an arbitrary epoch during the intervisit time of $Q_i$.
For *gated* service, the number of type $i$ customers at the beginning of a visit to $Q_i$ is exactly the number of type $i$ customers that arrived during the previous cycle, starting at $Q_i$. In terms of PGFs: $\widetilde V_{b_i}(z) = \gamma_i(\lambda_i(1-z))$. The number of type $i$ customers at the end of a visit to $Q_i$ are exactly those type $i$ customers that arrived during this visit. In terms of PGFs: $\widetilde V_{c_i}(z) = \gamma_i(\lambda_i(1-\beta_i(\lambda_i(1-z))))$. We can rewrite $E(N_{i|I_{\textit{end}}}) - E(N_{i|I_{\textit{begin}}})$ as $\lambda_i E(I_i)$, because this is the number of type $i$ customers that arrive during an intervisit time. In Section \[momentsgeneral\] we show that $\lambda_i E(I_i) = \lambda_i(1-\rho_i)E(C)$. Using these expressions we can rewrite Equation for gated service to: $$E[{\textrm{e}}^{-\omega W_i}] = \frac{(1-\rho_i)\omega}{\omega-\lambda_i(1-\beta_i(\omega))}\cdot
\frac{\gamma_i(\lambda_i(1-\beta_i(\omega))) - \gamma_i(\omega)}{(1-\rho_i)\omega E(C)}.\label{lstwgated}$$ For *exhaustive* service, $\widetilde V_{c_i}(z) = 1$, because $Q_i$ is empty at the end of a visit to $Q_i$. The number of type $i$ customers at the beginning of a visit to $Q_i$ in an exhaustive polling system is equal to the number of type $i$ customers that arrived during the previous intervisit time of $Q_i$. Hence, $\widetilde{V}_{b_i}(z) = \widetilde{I}_i(\lambda_i(1-z))$, where $\widetilde{I}_i(\cdot)$ is the LST of the intervisit time distribution for $Q_i$. Substitution of $\widetilde{I}_i(\omega) = \widetilde{V}_{b_i}(1-\frac{\omega}{\lambda_i})$ in leads to the following expression for the LST of the steady-state waiting time distribution of a type $i$ customer in an exhaustive polling system: $$E[{\textrm{e}}^{-\omega W_i}] = \frac{(1-\rho_i)\omega}{\omega-\lambda_i(1-\beta_i(\omega))}\cdot
\frac{1-\widetilde{I}_i(\omega)}{\omega E(I_i)}.\label{lstwexhaustive}$$ To the best of our knowledge, the following result is new.
Let the cycle time $C^*_i$ be the time between two successive visit *completions* to $Q_i$. The LST of the cycle time distribution is given by . An equivalent expression for $E[{\textrm{e}}^{-\omega W_i}]$ if $Q_i$ is served exhaustively, is: $$\begin{aligned}
E[{\textrm{e}}^{-\omega W_i}] &= \frac{1-\gamma^*_i(\omega - \lambda_i(1-\beta_i(\omega)))}{(\omega-\lambda_i(1-\beta_i(\omega)))E(C)}\label{lstwexhaustiveC}\\
&= E[{\textrm{e}}^{-(\omega - \lambda_i(1-\beta_i(\omega))) C^*_{i,\textit{res}}}],\nonumber\end{aligned}$$ where $C^*_{i,\textit{res}}$ is the residual length of $C^*_i$.
The cycle time is the length of an intervisit period $I_i$ plus the length of a visit $V_i$, which is the time required to serve all type $i$ customers that have arrived during $I_i$, and their type $i$ descendants. Hence, the following equation holds: $$\gamma^*_i(\omega) = \widetilde{I}_i(\omega + \lambda_i(1-\pi_i(\omega))).\label{lstCexhaustiveI}$$ We use this equation to find the inverse relation: $$\begin{aligned}
\widetilde{I}_i(\omega + \lambda_i(1-\pi_i(\omega))) &= \gamma^*_i(\omega) \\
&= \gamma^*_i(\omega + \lambda_i(1-\pi_i(\omega)) - \lambda_i(1-\pi_i(\omega)))\\
&= \gamma^*_i(\omega + \lambda_i(1-\pi_i(\omega)) - \lambda_i(1-\beta_i(\omega+\lambda_i(1-\pi_i(\omega))))).\end{aligned}$$ If we substitute $s := \omega + \lambda_i(1-\pi_i(\omega))$, we find $$\widetilde{I}_i(s) = \gamma^*_i(s - \lambda_i(1-\beta_i(s))).\label{intervisitC}$$ Substitution of into gives .
We can write and as follows: $$\gamma^*_i(\omega) = \widetilde{I}_i(\psi(\omega)), \qquad
\widetilde{I}_i(s) = \gamma^*_i(\phi(s)),$$ where $\phi(\cdot)$ equals the Laplace exponent of the Lévy process $\sum_{j=1}^{N(t)}B_{i,j}-t$, with $N(t)$ a Poisson process with intensity $\lambda_i$, and with $\psi(\omega) = \omega+\lambda_i(1-\pi_i(\omega))$, which is known to be the inverse of $\phi(\cdot)$.
Moments {#momentsgeneral}
-------
The focus of this paper is on LST and PGF of distribution functions, not on their moments. Moments can be obtained by differentiation, and are also discussed in [@wierman07]. In this subsection we will only mention some results that will be used later.
First we will derive the mean cycle time $E(C)$. Unlike higher moments of the cycle time, the mean does not depend on where the cycle starts: $E(C) = \frac{E(S_1) + E(S_2)}{1-\rho}$. This can easily be seen, because $1-\rho$ is the fraction of time that the server is not working, but switching. The total switch-over time is $E(S_1) + E(S_2)$.
The expected length of a visit to $Q_i$ is $E(V_i) = \rho_i E(C)$. The mean length of an intervisit period for $Q_i$ is $E(I_i) = (1-\rho_i)E(C)$. Notice that these expectations do not depend on the service discipline used. The expected number of type $i$ customers at polling moments does depend on the service discipline. For gated service the expected number of type $i$ customers at the beginning of a visit to $Q_i$ is $\lambda_i E(C)$. For exhaustive service this is $\lambda_i E(I_i)$. The expected number of type $i$ customers at the beginning of a visit to $Q_{i+1}$ is $\lambda_i (E(V_i) + E(S_i))$ for gated service, and $\lambda_i E(S_i)$ for exhaustive service.
Moments of the waiting time distribution for a type $i$ customer at an arbitrary epoch can be derived from the LSTs given by , and . We only present the first moment: $$\begin{aligned}
&\textrm{Gated: } & E(W_i) &= (1+\rho_i)\frac{E(C_i^2)}{2E(C)},\label{ewgated}\\
&\textrm{Exhaustive: } & E(W_i) &= \frac{E(I_i^2)}{2E(I_i)}+\frac{\rho_i}{1-\rho_i}\frac{E(B_i^2)}{2E(B_i)},\nonumber\\
& & &= (1-\rho_i)\frac{E({C^*_i}^2)}{2E(C)}.\label{ewexhaustiveC}\end{aligned}$$ Notice that the start of $C_i$ is the *beginning* of a visit to $Q_i$, whereas the start of $C^*_i$ is the *end* of a visit. Equations and are in agreement with Equations (4.1) and (4.2) in [@boxmaworkloadsandwaitingtimes89]. Although at first sight these might seem nice, closed formulas, it should be noted that the expected residual cycle time and the expected residual intervisit time are not easy to determine, requiring the solution of a large set of equations. MVA is an efficient technique to compute mean waiting times, the mean residual cycle time, and also the mean residual intervisit time. We refer to [@winands06] for an MVA framework for polling models.
Gated service {#gated}
=============
In this section we study the gated service discipline for a polling system with two queues and two priority classes in the first queue: high ($H$) and low ($L$) priority customers. All type $H$ and $L$ customers that are present at the moment when the server arrives at $Q_1$, will be served during the server’s visit to $Q_1$. First all type $H$ customers will be served, then all type $L$ customers. Type $H$ customers arrive at $Q_1$ according to a Poisson process with intensity $\lambda_H$, and have a service requirement $B_H$ with LST $\beta_H(\cdot)$. Type $L$ customers arrive at $Q_1$ with intensity $\lambda_L$, and have a service requirement $B_L$ with LST $\beta_L(\cdot)$. If we do not distinguish between high and low priority customers, we can still use the results from Section \[general\] if we regard the system as a polling system with two queues where customers in $Q_1$ arrive according to a Poisson process with intensity $\lambda_1 := \lambda_H + \lambda_L$ and have service requirement $B_1$ with LST $\beta_1(\cdot) = \frac{\lambda_H}{\lambda_1} \beta_H(\cdot) + \frac{\lambda_L}{\lambda_1} \beta_L(\cdot)$.
We follow the same approach as in Section \[general\]. First we study the joint queue length distribution at polling epochs, then the cycle time distribution, followed by the marginal queue length distribution and waiting time distribution. The last subsection provides the first moment of these distributions.
Joint queue length distribution at polling epochs
-------------------------------------------------
Equations and give the PGFs of the joint queue length distribution at visit beginnings, $V_{b_i}(z_1, z_2)$. A type 1 customer entering the system is a type $H$ customer with probability $\lambda_H/\lambda_1$, and a type $L$ customer with probability $\lambda_L/\lambda_1$. We can express the PGF of the joint queue length distribution in the polling system with priorities, $V_{b_i}(\cdot,\cdot,\cdot)$, in terms of the PGF of the joint queue length distribution in the polling system without priorities, $V_{b_i}(\cdot,\cdot)$.
\[p1\_3qs\] $$V_{b_i}(z_H, z_L, z_2) = V_{b_i}\left(\frac{\lambda_H z_H+\lambda_L z_L}{\lambda_1}, z_2\right).$$
Let $X_H$ be the number of high priority customers present in $Q_1$ at the beginning of a visit to $Q_i$, $i=1,2$. Similarly define $X_L$ to be the number of low priority customers present in $Q_1$ at the beginning of a visit to $Q_i$. Let $X_1 = X_H + X_L$. Since the type $H$/$L$ customers in $Q_1$ are exactly those $H$/$L$ customers that arrived since the previous visit beginning at $Q_i$, we know that $$P(X_H=i,X_L=k-i|X_1=k) = \binom ki \left(\frac{\lambda_H}{\lambda_1}\right)^i \left(\frac{\lambda_L}{\lambda_1}\right)^{k-i}.$$ Hence $$\begin{aligned}
E[z_H^{X_H}z_L^{X_L}|X_1 = k]
&= \sum_{i=0}^\infty\sum_{j=0}^\infty z_H^iz_L^j P(X_H=i,X_L=j|X_1=k) \\
&=\left(\frac{\lambda_H z_H + \lambda_L z_L}{\lambda_1}\right)^k.\end{aligned}$$ Finally, $$\begin{aligned}
V_{b_i}(z_H, z_L, z_2)
&= \sum_{i=0}^\infty\sum_{j=0}^\infty \left(\frac{\lambda_H z_H + \lambda_L z_L}{\lambda_1}\right)^i z_2^j P(X_1=i, X_2=j) \\
&= V_{b_i}\left(\frac{1}{\lambda_1}(\lambda_H z_H+\lambda_L z_L), z_2\right).\end{aligned}$$
Cycle time
----------
The LST of the cycle time distribution is still given by if we define $\lambda_1 := \lambda_H + \lambda_L$ and $\beta_1(\cdot) := \frac{\lambda_H}{\lambda_1} \beta_H(\cdot) + \frac{\lambda_L}{\lambda_1} \beta_L(\cdot)$, because the cycle time does not depend on the order of service.
Equation is valid for polling systems with queues having any branching type service discipline. In the present section we can derive an alternative, shorter expression for $\gamma_1(\cdot)$ by explicitly using the fact that $Q_1$ receives gated service. The type 1 (i.e. both $H$ and $L$) customers present at the visit beginning to $Q_1$ are those that arrived during the previous cycle: $P_1(z, 1) = \gamma_1(\lambda_1(1-z))$. By setting $\omega = \lambda_1(1-z)$, this leads to the following expression for the LST of the distribution of $C_1$ if service in $Q_1$ is gated: $$\gamma_1(\omega) = P_1(1-\frac{\omega}{\lambda_1}, 1).\label{lstCgatedShort}$$
Marginal queue lengths and waiting times {#gatedmarginalw}
----------------------------------------
We first determine the LST of the waiting time distribution for a type $L$ customer, using the fact that this customer will not be served until the next cycle (starting at $Q_1$). The time from the start of the cycle until the arrival will be called “past cycle time”, denoted by $C_{1P}$. The residual cycle time will be denoted by $C_{1R}$. The waiting time of a type $L$ customer is composed of $C_{1R}$, the service times of all high priority customers that arrived during $C_{1P}+C_{1R}$, and the service times of all low priority customers that have arrived during $C_{1P}$. Let $N_H(T)$ be the number of high priority customers that have arrived during time interval $T$, and equivalently define $N_L(T)$.
$$\begin{aligned}
E\left[{\textrm{e}}^{-\omega W_L}\right] =& \frac{\gamma_1(\lambda_H(1-\beta_H(\omega))+\lambda_L(1-\beta_L(\omega)))
- \gamma_1(\omega+\lambda_H(1-\beta_H(\omega)))}{(\omega-\lambda_L(1-\beta_L(\omega)))E(C)}.\end{aligned}$$
$$\begin{aligned}
E\left[{\textrm{e}}^{-\omega W_L}\right]
&= E\left[{\textrm{e}}^{-\omega (C_{1R}+\sum_{i=1}^{N_H(C_{1P}+C_{1R})}B_{H,i}+\sum_{i=1}^{N_L(C_{1P})}B_{L,i})}\right] \nonumber\\
&= \int_{t=0}^\infty \int_{u=0}^\infty \sum_{m=0}^\infty\sum_{n=0}^\infty E\left[{\textrm{e}}^{-\omega\sum_{i=1}^{m}B_{H,i}}\right]E\left[{\textrm{e}}^{-\omega\sum_{i=1}^{n}B_{L,i}}\right]
\nonumber\\
&\quad\cdot{\textrm{e}}^{-\omega u} \frac{(\lambda_H(t+u))^m}{m!}{\textrm{e}}^{-\lambda_H(t+u)}\frac{(\lambda_L t)^n}{n!}{\textrm{e}}^{-\lambda_Lt} {\,\textrm{d}}P(C_{1P}<t, C_{1R}<u)\nonumber\\
\nonumber\\
&= \int_{t=0}^\infty \int_{u=0}^\infty {\textrm{e}}^{-t(\lambda_H(1-\beta_H(\omega))+\lambda_L(1-\beta_L(\omega)))}{\textrm{e}}^{-u(\omega + \lambda_H(1-\beta_H(\omega)))} {\,\textrm{d}}P(C_{1P}<t, C_{1R}<u)\nonumber\\
&=\frac{\gamma_1(\lambda_H(1-\beta_H(\omega))+\lambda_L(1-\beta_L(\omega))) - \gamma_1(\omega+\lambda_H(1-\beta_H(\omega)))}{(\omega-\lambda_L(1-\beta_L(\omega)))E(C)}.
\label{lstwlgated}\end{aligned}$$
For the last step in the derivation of we used $$E[{\textrm{e}}^{-\omega_P C_{1P}-\omega_R C_{1R}}] = \frac{E[{\textrm{e}}^{-\omega_P C_1}]-E[{\textrm{e}}^{-\omega_R C_1}]}{(\omega_R-\omega_P)E(C)},$$ which is obtained in [@boxmalevyyechiali92].
The Fuhrmann-Cooper decomposition [@fuhrmanncooper85] still holds for the waiting time of type $L$ customers, because can be rewritten into $$\begin{aligned}
E\left[{\textrm{e}}^{-\omega W_L}\right] =& \frac{(1-\rho_L)\omega}{\omega-\lambda_L(1-\beta_L(\omega))} \nonumber\\
&\cdot
\frac{\gamma_1(\lambda_H(1-\beta_H(\omega))+\lambda_L(1-\beta_L(\omega))) - \gamma_1(\omega+\lambda_H(1-\beta_H(\omega)))}{(1-\rho_L)\omega E(C)}.\label{lstwlgateddecomposition}\end{aligned}$$ We recognise the first term on the right-hand side of as the LST of the waiting time distribution of an $M/G/1$ queue with only type $L$ customers. An interpretation of the other two terms on the right-hand side can be found when regarding the polling system as a polling system with *three* queues $(Q_H, Q_L, Q_2)$ and no switch-over time between $Q_H$ and $Q_L$. The service discipline of this equivalent system is synchronised gated, which is a more general version of gated. The gates for queues $Q_H$ and $Q_L$ are set simultaneously when the server arrives at $Q_H$, but the gate for $Q_2$ is still set when the server arrives at $Q_2$. In the following paragraphs we show that the second and third term on the right-hand side of together can be interpreted as $E[\left(1-\frac\omega{\lambda_L}\right)^{N_{L|I}}]$, where $N_{L|I}$ is the number of type $L$ customers at a random epoch during the intervisit period of $Q_L$.
The expression for the LST of the distribution of the number of type $L$ customers at an arbitrary epoch is determined by first converting the waiting time LST to sojourn time LST, i.e., multiplying expression with $\beta_L(\omega)$. Second, we apply the distributional form of Little’s law [@keilsonservi90] to . This law can be applied because the required conditions are fulfilled for each customer class (H, L, and 2): the customers enter the system in a Poisson stream, every customer enters the system and leaves the system one at a time in order of arrival, and for any time $t$ the entry process into the system of customers after time $t$ and the time spent in the system by any customer arriving before time $t$ are independent. The result is: $$E\left[z^{N_L}\right] = \frac{(1-\rho_L)(1-z)\beta_L(\lambda_L(1-z))}{\beta_L(\lambda_L(1-z))-z}
\cdot \frac{\widetilde{V}_{c_L}(z) - \widetilde{V}_{b_L}(z)}{(1-z)(E(N_{L|I_{\textit{end}}}) - E(N_{L|I_{\textit{begin}}}))}.\label{gfzlgateddecomposition}$$ In this equation $\widetilde{V}_{b_L}(z)$ denotes the PGF of the distribution of the number of type $L$ customers at the beginning of a visit to $Q_L$, and $\widetilde{V}_{c_L}(z)$ denotes the PGF at the completion of a visit to $Q_L$: $$\begin{aligned}
\widetilde{V}_{b_L}(z) &= V_{b_1}(\beta_H(\lambda_L(1-z)), z, 1)\\
&= \gamma_1(\lambda_H(1-\beta_H(\lambda_L(1-z))) + \lambda_L(1-z)), \\
\widetilde{V}_{c_L}(z) &= V_{b_1}(\beta_H(\lambda_L(1-z)), \beta_L(\lambda_L(1-z)), 1)\\
&= \gamma_1(\lambda_H(1-\beta_H(\lambda_L(1-z))) + \lambda_L(1-\beta_L(\lambda_L(1-z)))).\end{aligned}$$ The last term in is the PGF of the distribution of the number of type $L$ customers at an arbitrary epoch during the intervisit period of $Q_L$, $E[z^{N_{L|I}}]$. Substitution of $\omega := \lambda_L(1-z)$ in , and using $(E(N_{L|I_{\textit{end}}}) - E(N_{L|I_{\textit{begin}}})) = \lambda_L E(I_L)$, shows that the second and third term at the right-hand side of together indeed equal $E[\left(1-\frac\omega{\lambda_L}\right)^{N_{L|I}}]$.
The derivation of the LSTs of $W_H$ and $W_2$ is similar and leads to the following expressions: $$\begin{aligned}
E\left[{\textrm{e}}^{-\omega W_H}\right] =& \frac{(1-\rho_H)\omega}{\omega-\lambda_H(1-\beta_H(\omega))} \cdot
\frac{\gamma_1(\lambda_H(1-\beta_H(\omega))) - \gamma_1(\omega)}{(1-\rho_H)\omega E(C)},\label{lstwhgated}\\
E\left[{\textrm{e}}^{-\omega W_2}\right] =& \frac{(1-\rho_2)\omega}{\omega-\lambda_2(1-\beta_2(\omega))} \cdot
\frac{\gamma_2(\lambda_2(1-\beta_2(\omega))) - \gamma_2(\omega)}{(1-\rho_2)\omega E(C)}.\label{lstw2gated}\end{aligned}$$
Equations and are equivalent to the LST of $W_i$ in a nonpriority polling system , which illustrates that the Fuhrmann-Cooper decomposition also holds for the waiting time distributions of high priority customers in $Q_1$ and type 2 customers in a polling system with gated service.
Application of the distributional form of Little’s law to these expressions results in: $$\begin{aligned}
E\left[z^{N_H}\right] &= \frac{(1-\rho_H)(1-z)\beta_H(\lambda_H(1-z))}{\beta_H(\lambda_H(1-z))-z} \cdot
\frac{\gamma_1(\lambda_H(1-\beta_H(\lambda_H(1-z)))) - \gamma_1(\lambda_H(1-z))}{\lambda_H(1-\rho_H)(1-z)E(C)},\\
E\left[z^{N_2}\right] &= \frac{(1-\rho_2)(1-z)\beta_2(\lambda_2(1-z))}{\beta_2(\lambda_2(1-z))-z} \cdot
\frac{\gamma_2(\lambda_2(1-\beta_2(\lambda_2(1-z)))) - \gamma_2(\lambda_2(1-z))}{\lambda_2(1-\rho_2)(1-z)E(C)}.\end{aligned}$$
If the service discipline in $Q_2$ is not gated, but another branching type service discipline that satisfies Property \[resingproperty\], should be replaced by the more general expression .
Moments {#momentsgated}
-------
As mentioned in Section \[momentsgeneral\], we do not focus on moments in this paper, and we only mention the mean waiting times of type $H$ and $L$ customers. For a type $H$ customer, it is immediately clear that $E(W_H) = (1+\rho_H)E(C_{1,\textit{res}})$. The mean waiting time for a type $L$ customer can be obtained by differentiating . This results in: $$E(W_L) = (1+2\rho_H+\rho_L)E(C_{1,\textit{res}}).$$ These formulas can also be obtained using MVA, as shown in [@wierman07].
Globally gated service {#globallygated}
======================
In this section we discuss a polling model with two queues $(Q_1, Q_2)$ and two priority classes ($H$ and $L$) in $Q_1$ with globally gated service. For this service discipline, only customers that were present when the server started its visit to $Q_1$ are served. This feature makes the model exactly the same as a nonpriority polling model with three queues $(Q_H, Q_L, Q_2)$. Although this system does not satisfy Property \[resingproperty\], it does satisfy Property \[borstproperty\] which implies that we can still follow the same approach as in the previous sections.
Joint queue length distribution at polling epochs
-------------------------------------------------
We define the beginning of a visit to $Q_1$ as the start of a cycle, since this is the moment that determines which customers will be served during the next visits to the queues. Arriving customers will always be served in the next cycle, so the three $(i=H,L,2)$ offspring PGFs are: $$\begin{aligned}
f^{(i)}(z_H, z_L, z_2) &= h_i(z_H, z_L, z_2) \\
&= \beta_i(\lambda_H(1-z_H)+\lambda_L(1-z_L)+\lambda_2(1-z_2)),\end{aligned}$$ The two $(i=1,2)$ immigration functions are: $$g^{(i)}(z_H, z_L, z_2) = \sigma_i(\lambda_H(1-z_H)+\lambda_L(1-z_L)+\lambda_2(1-z_2)),$$ Using these definitions, the formula for the PGF of the joint queue length distribution at the beginning of a cycle is similar to the one found in Section \[general\]: $$P_1(z_H, z_L, z_2) = \prod_{n=0}^\infty g(f_n(z_H, z_L, z_2)).$$
Notice that in a system with globally gated service it is possible to express the joint queue length distribution at the beginning of a cycle in terms of the cycle time LST, since all customers that are present at the beginning of a cycle are exactly all of the customers that have arrived during the previous cycle:
$$P_1(z_H, z_L, z_2) = \gamma_1(\lambda_H(1-z_H)+\lambda_L(1-z_L)+\lambda_2(1-z_2)).\label{p1globallygated}$$
Cycle time
----------
Since only those customers that are present at the start of a cycle, starting at $Q_1$, will be served during this cycle, the LST of the cycle time distribution is $$\gamma_1(\omega) = \sigma_1(\omega)\sigma_2(\omega)P_1(\beta_H(\omega), \beta_L(\omega), \beta_2(\omega)).\label{lstcgloballygated}$$ Substitution of into this expression gives us the following relation: $$\begin{aligned}
&\gamma_1(\omega) = \sigma_1(\omega)\sigma_2(\omega)\\
&\cdot\gamma_1(\lambda_H(1-\beta_H(\omega))+\lambda_L(1-\beta_L(\omega))+\lambda_2(1-\beta_2(\omega))).\end{aligned}$$ [@boxmalevyyechiali92] show that this relation leads to the following expression for the cycle time LST: $$\gamma_1(\omega) = \prod_{i=0}^\infty \sigma(\delta^{(i)}(\omega)),$$ where $\sigma(\cdot) = \sigma_1(\cdot)\sigma_2(\cdot)$, and $\delta^{(i)}(\omega)$ is recursively defined as follows: $$\begin{aligned}
\delta^{(0)}(\omega) &= \omega,\\
\delta^{(i)}(\omega) &= \delta(\delta^{(i-1)}(\omega)), \qquad\qquad i=1,2,3,\dots,\\
\delta(\omega) &= \lambda_H(1-\beta_H(\omega)) + \lambda_L(1-\beta_L(\omega)) + \lambda_2(1-\beta_2(\omega)).\end{aligned}$$
Marginal queue lengths and waiting times {#marginal-queue-lengths-and-waiting-times-1}
----------------------------------------
For type $H$ and $L$ customers, the expressions for $E({\textrm{e}}^{-\omega W_H})$ and $E({\textrm{e}}^{-\omega W_L})$ are exactly the same as the ones found in Section \[gatedmarginalw\], but with $\gamma_1(\cdot)$ as defined in .
The expression for $E({\textrm{e}}^{-\omega W_2})$ can be obtained with the method used in Section \[gatedmarginalw\]: $$\begin{aligned}
E\left[{\textrm{e}}^{-\omega W_2}\right] =&\sigma_1(\omega)\cdot\frac{\gamma_1(\sum_{i=H,L,2}\lambda_i(1-\beta_i(\omega))) - \gamma_1(\omega+\sum_{i=H,L}\lambda_i(1-\beta_i(\omega)))}{(\omega-\lambda_2(1-\beta_2(\omega)))E(C)}\\
=& \sigma_1(\omega)\cdot\frac{(1-\rho_2)\omega}{\omega-\lambda_2(1-\beta_2(\omega))} \\
&\cdot
\frac{\gamma_1(\sum_{i=H,L,2}\lambda_i(1-\beta_i(\omega))) - \gamma_1(\omega+\sum_{i=H,L}\lambda_i(1-\beta_i(\omega)))}{(1-\rho_2)\omega E(C)}.\end{aligned}$$
We can use the distributional form of Little’s law to determine the LST of the marginal queue length distribution of $Q_2$: $$\begin{aligned}
E\left[z^{N_2}\right] =& \sigma_1(\lambda_2(1-z))\frac{(1-\rho_2)(1-z)\beta_2(\lambda_2(1-z))}{\beta_2(\lambda_2(1-z))-z} \\
&\cdot
\frac{\gamma_1\left(\sum_{i=H,L,2}\lambda_i(1-\beta_i(\lambda_2(1-z)))\right)
- \gamma_1\left(\lambda_2(1-z)+\sum_{i=H,L}\lambda_i(1-\beta_i(\lambda_2(1-z)))\right)}{
\lambda_2(1-\rho_2)(1-z)E(C)}.\end{aligned}$$
The Fuhrmann-Cooper queue length decomposition also holds for all customer classes in a polling system with globally gated service.
Moments {#moments}
-------
The expressions for $E(W_H)$ and $E(W_L)$ from Section \[momentsgated\] also hold in a globally gated polling system, but with a different mean residual cycle time. We only provide the mean waiting time of type 2 customers: $$E(W_2) = E(S_1) + (1 + 2\rho_H + 2\rho_L + \rho_2) E(C_{1,\textit{res}}).$$
Exhaustive service {#exhaustive}
==================
In this section we study the same polling model as in the previous two sections, but the two queues are served exhaustively. The section has the same structure as the other sections, so we start with the derivation of the LST of the joint queue length distribution at polling epochs, followed by the LST of the cycle time distribution. LSTs of the marginal queue length distributions and waiting time distributions are provided in the next subsection. In the last part of the section the mean waiting time of each customer type is studied.
It should be noted that, although we assume that both $Q_1$ and $Q_2$ are served exhaustively, a model in which $Q_2$ is served according to another branching type service discipline, requires only minor adaptations.
Joint queue length distribution at polling epochs
-------------------------------------------------
We can derive the joint queue length distribution at the beginning of a cycle for a polling system with two queues and two priority classes in $Q_1$, $P_1(z_H, z_L, z_2)$, directly from for $P_1(z_1, z_2)$. Similar to the proof of Lemma \[p1\_3qs\], we can prove that $$P_1(z_H, z_L, z_2) = P_1\left(\frac{1}{\lambda_1}(\lambda_H z_H+\lambda_L z_L), z_2\right).$$ The same holds for $V_{b_2}(\cdot, \cdot, \cdot)$ and visit completion epochs $V_{c_i}(\cdot, \cdot, \cdot)$, for $i = 1, 2$.
Cycle time
----------
For the cycle time starting with a visit to $Q_1$, is still valid. However, when studying the waiting time of a specific customer type in an exhaustively served queue, it is convenient to consider the *completion* of a visit to $Q_1$ as the start of a cycle. Hence, in this section the notation $C^*_1$, or the LST of its distribution, $\gamma^*_1(\cdot)$, refers to the cycle time starting at the completion of a visit to $Q_1$. Equation gives the LST of the distribution of $C^*_1$.
Using the fact that customers in $Q_1$ are served exhaustively, we can find an alternative, compact expression for $\gamma_1^*(\cdot)$. The type 1 (i.e. both type $H$ and $L$ customers) customers at the beginning of a visit to $Q_1$ are exactly those type 1 customers that have arrived during the previous intervisit time: $P_1(z, 1) = \widetilde{I}_1(\lambda_1(1-z))$. Hence, by setting $\omega=\lambda_1(1-z)$, we get $\widetilde{I}_1(\omega) = P_1(1-\frac{\omega}{\lambda_1}, 1)$, and thus by , $$\gamma_1^*(\omega) = P_1(\pi_1(\omega)-\frac{\omega}{\lambda_1}, 1).\label{lstCexhaustiveShort}$$
Marginal queue lengths and waiting times {#marginal-queue-lengths-and-waiting-times-2}
----------------------------------------
Analysis of the model with exhaustive service requires a different approach. The key observation, made by [@fuhrmanncooper85], is that a nonpriority polling system from the viewpoint of a type $i$ customer is an $M/G/1$ queue with multiple server vacations. This implies that the Fuhrmann-Cooper decomposition can be used, even though the intervisit times are strongly dependent on the visit times. The $M/G/1$ queue with priorities and vacations can be analysed by modelling the system as a special version of the *nonpriority* $M/G/1$ queue with multiple server vacations, and then applying the results from Fuhrmann and Cooper. This approach has been used by [@kellayechiali88] who used the concept of *delay cycles*, and also by [@Shanthikumar89] who used *level crossing analysis*; see also [@takagi91]. We apply Kella and Yechiali’s approach to the polling model under consideration to find the waiting time LST for type $H$ and $L$ customers. In [@kellayechiali88] systems with single and multiple vacations, preemptive resume and nonpreemptive service are considered. In the present paper we do not consider preemptive resume, so we only use results from the case labelled as NPMV (nonpreemptive, multiple vacations) in [@kellayechiali88]. We consider the system from the viewpoint of a type $H$ and type $L$ customer separately to derive $E[{\textrm{e}}^{-\omega W_H}]$ and $E[{\textrm{e}}^{-\omega W_L}]$.
From the viewpoint of a type $H$ customer and as far as waiting times are concerned, a polling system is a *nonpriority* single server system with multiple vacations. The vacation can either be the intervisit period $I_1$, or the service of a type $L$ customer. The LSTs of these two types of vacations are: $$\begin{aligned}
E[{\textrm{e}}^{-\omega I_1}] &= P_1(1-\omega/\lambda_1, 1), \label{intervisitexhaustive}\\
E[{\textrm{e}}^{-\omega B_L}] &= \beta_L(\omega). \nonumber\end{aligned}$$ Equation follows immediately from the fact that the number of type 1 (i.e. both H and L) customers at the beginning of a visit to $Q_1$ is the number of type 1 customers that have arrived during the previous intervisit period: $P_1(z, 1) = E[{\textrm{e}}^{-(\lambda_1(1-z)) I_1}]$.
We now use the concept of delay cycles, introduced in [@kellayechiali88], to find the waiting time LST of a type $H$ customer. The key observation is that an arrival of a tagged type $H$ customer will always take place within either an $I_H$ cycle, or an $L_H$ cycle. An $I_H$ cycle is a cycle that starts with an intervisit period for $Q_1$, followed by the service of all type $H$ customers that have arrived during the intervisit period, and ends at the moment that no type $H$ customers are left in the system. Notice that at the start of the intervisit period, no type $H$ customers were present in the system either. An $L_H$ cycle is a similar cycle, but starts with the service of a type $L$ customer. This cycle also ends at the moment that no type $H$ customers are left in the system.
The fraction of time that the system is in an $L_H$ cycle is $\frac{\rho_L}{1-\rho_H}$, because type $L$ customers arrive with intensity $\lambda_L$. Each of these customers will start an $L_H$ cycle and the length of an $L_H$ cycle equals $\frac{E(B_L)}{1-\rho_H}$: $$\begin{aligned}
E(L_H\textrm{ cycle}) &= E(B_L) + \lambda_H E(B_L) E(\textit{BP}_H) \\
&= E(B_L) + \lambda_H E(B_L) \frac{E(B_H)}{1-\rho_H} \\
&= (1+\frac{\rho_H}{1-\rho_H})E(B_L) = \frac{E(B_L)}{1-\rho_H},\end{aligned}$$ where $E(\textit{BP}_H)$ is the mean length of a busy period of type $H$ customers.
The fraction of time that the system is in an $I_H$ cycle, is $1-\frac{\rho_L}{1-\rho_H} = \frac{1-\rho_1}{1-\rho_H}$. This result can also be obtained by using the argument that the fraction of time that the system is in an intervisit period is the fraction of time that the server is not serving $Q_1$, which is equal to $1-\rho_1$. A cycle which starts with such an intervisit period and stops when all type $H$ customers that arrived during the intervisit period and their type $H$ descendants have been served, has mean length $E(I_1) + \lambda_H E(I_1) E(\textit{BP}_H) = \frac{E(I_1)}{1-\rho_H}$. This also leads to the conclusion that $\frac{1-\rho_1}{1-\rho_H}$ is the fraction of time that the system is in an $I_H$ cycle. A customer arriving during an $I_H$ cycle views the system as a nonpriority $M/G/1$ queue with multiple server vacations $I_1$; a customer arriving during an $L_H$ cycle views the system as a nonpriority $M/G/1$ queue with multiple server vacations $B_L$.
[@fuhrmanncooper85] showed that the waiting time of a customer in an $M/G/1$ queue with server vacations is the sum of two independent quantities: the waiting time of a customer in a corresponding $M/G/1$ queue without vacations, and the residual vacation time. Hence, the LST of the waiting time distribution of a type $H$ customer is: $$E[{\textrm{e}}^{-\omega W_H}] = \frac{(1-\rho_H)\omega}{\omega-\lambda_H(1-\beta_H(\omega))}\cdot
\left[\frac{1-\rho_1}{1-\rho_H} \cdot \frac{1-\widetilde{I}_1(\omega)}{\omega E(I_1)}+ \frac{\rho_L}{1-\rho_H}\cdot\frac{1-\beta_L(\omega)}{\omega E(B_L)}\right].\label{lstwhexhaustive}$$ Equation is in accordance with the more general equation in Section 4.1 in [@kellayechiali88].
The LST of the distribution of the waiting time of a high priority customer in a two priority $M/G/1$ queue without vacations is $$E[{\textrm{e}}^{-\omega W_{H|M/G/1}}] =\frac{(1-\rho_1)\omega+\lambda_L(1-\beta_L(\omega))}{\omega-\lambda_H(1-\beta_H(\omega))}\label{lstwhmg1},\\$$ see, e.g., Equation (3.85) in [@cohen82], Chapter III.3. Equation can be rewritten to , with $\frac{1-\widetilde{I}_1(\omega)}{\omega E(I_1)}$ replaced by 1. Hence, the waiting time distribution of a high priority customer in a two priority $M/G/1$ queue equals the waiting time distribution of a customer in a nonpriority $M/G/1$ queue with only type $H$ customers, where the server goes on a vacation $B_L$ with probability $\frac{\rho_L}{1-\rho_H}$.
Substitution of in expresses $E[{\textrm{e}}^{-\omega W_H}]$ in terms of the LST of the cycle time distribution starting at a visit *completion* to $Q_1$, $\gamma_1^*(\cdot)$: $$E[{\textrm{e}}^{-\omega W_H}] = \frac{1-\gamma_1^*(\omega - \lambda_H(1-\beta_H(\omega)) - \lambda_L(1-\beta_L(\omega)))
+\lambda_L (1-\beta_L(\omega))E(C)}{(\omega-\lambda_H(1-\beta_H(\omega)))E(C)}.\label{lstwhexhaustiveC}$$
The concept of cycles is not really needed to model the system from the perspective of a type $L$ customer, because for a type $L$ customer the system merely consists of $I_{HL}$ cycles. An $I_{HL}$ cycle is the same as an $I_H$ cycle, discussed in the previous paragraphs, except that it ends when no type $H$ *or $L$* customers are left in the system. So the system can be modelled as a nonpriority $M/G/1$ queue with server vacations. The vacation is the intervisit time $I_1$, plus the service times of all type $H$ customers that have arrived during that intervisit time and their type $H$ descendants. We will denote this extended intervisit time by $I_1^*$ with LST $$\widetilde{I}_1^*(\omega) = \widetilde{I}_1(\omega+\lambda_H(1-\pi_H(\omega))).$$ The mean length of $I_1^*$ equals $E(I_1^*) = \frac{E(I_1)}{1-\rho_H}$.
We also have to take into account that a busy period of type $L$ customers might be interrupted by the arrival of type $H$ customers. Therefore the alternative system that we are considering will not contain regular type $L$ customers, but customers still arriving with arrival rate $\lambda_L$, whose service time equals the service time of a type $L$ customer in the original model, plus the service times of all type $H$ customers that arrive during this service time, and all of their type $H$ descendants. The LST of the distribution of this extended service time $B_L^*$ is $$\beta_L^*(\omega) = \beta_L(\omega + \lambda_H(1-\pi_H(\omega))).$$ This extended service time is often called *completion time* in the literature. In this alternative system, the mean service time of these customers equals $E(B_L^*) = \frac{E(B_L)}{1-\rho_H}$. The fraction of time that the system is serving these customers is $\rho_L^* = \frac{\rho_L}{1-\rho_H} = 1 - \frac{1-\rho_1}{1-\rho_H}$.
Now we use the results from the $M/G/1$ queue with server vacations (starting with the Fuhrmann-Cooper decomposition) to determine the LST of the waiting time distribution for type $L$ customers: $$\begin{aligned}
E[{\textrm{e}}^{-\omega W_L}] =& \frac{(1-\rho_L^*)\omega}{\omega-\lambda_L(1-\beta_L^*(\omega))} \cdot
\frac{1-\widetilde{I}_1^*(\omega)}{\omega E(I_1^*)} \nonumber\\
=&
\frac{(1-\rho_1)(\omega+\lambda_H(1-\pi_H(\omega)))}{\omega-\lambda_L(1-\beta_L(\omega+\lambda_H(1-\pi_H(\omega))))} \cdot
\frac{1-\widetilde{I}_1(\omega+\lambda_H(1-\pi_H(\omega)))}{(\omega + \lambda_H(1-\pi_H(\omega)))E(I_1)}.
\label{lstwlexhaustive}\end{aligned}$$ The last term of is the LST of the distribution of the residual intervisit time, plus the time that it takes to serve all type $H$ customers and their type $H$ descendants that arrive during this residual intervisit time. The first term of is the LST of the waiting time distribution of a low-priority customer in an $M/G/1$ queue with two priorities, without vacations (see e.g. (3.76) in [@cohen82], Chapter III.3).
The $M/G/1$ queue with two priorities can be viewed as a nonpriority $M/G/1$ queue with vacations, if we consider the waiting time of type $L$ customers. We only need to rewrite the first term of : $$\begin{aligned}
E[{\textrm{e}}^{-\omega W_{L|M/G/1}}] =& \frac{(1-\rho_1)(\omega+\lambda_H(1-\pi_H(\omega)))}{\omega-\lambda_L(1-\beta_L(\omega+\lambda_H(1-\pi_H(\omega))))}\\
=&\frac{(1-\rho_L^*)\omega}{\omega-\lambda_L(1-\beta_L^*(\omega))} \cdot \frac{1-\rho_1}{1-\rho_L^*}
\cdot \frac{\omega+\lambda_H(1-\pi_H(\omega))}{\omega} \\
=& E[e^{-\omega W_{L|M/G/1}^*}]
\cdot\left[(1-\rho_H) + \rho_H \frac{1-\pi_H(\omega)}{\omega E(\textit{BP}_H)}\right],\end{aligned}$$ where $E[e^{-\omega W_{L|M/G/1}^*}]$ is the LST of the waiting time distribution of a customer in an $M/G/1$ queue where customers arrive at intensity $\lambda_L$ and have service requirement LST $\beta_L(\omega + \lambda_H(1-\pi_H(\omega)))$. So with probability $1-\rho_H$ the waiting time of a customer is the waiting time in an $M/G/1$ queue with no vacations, and with probability $\rho_H$ the waiting time of a customer is the sum of the waiting time in an $M/G/1$ queue and the residual length of a vacation, which is a busy period of type $H$ customers.
Substitution of in leads to a different expression for $E[{\textrm{e}}^{-\omega W_L}]$: $$\begin{aligned}
E[{\textrm{e}}^{-\omega W_L}] &=\frac{1-\gamma^*_1(\omega - \lambda_L(1-\beta_L(\omega+\lambda_H(1-\pi_H(\omega)))))}{(\omega-\lambda_L(1-\beta_L(\omega+\lambda_H(1-\pi_H(\omega)))))E(C)}\nonumber\\
&=E[{\textrm{e}}^{-(\omega - \lambda_L(1-\beta_L(\omega+\lambda_H(1-\pi_H(\omega)))))C^*_{1,\textit{res}}}].\label{lstwlexhaustiveC}\end{aligned}$$
The waiting time of type 2 customers is not affected at all by the fact that $Q_1$ contains multiple classes of customers, so is still valid for $E({\textrm{e}}^{-\omega W_2})$.
We will refrain from mentioning the PGFs of the marginal queue length distributions here, because they can be obtained by applying the distributional form of Little’s law as we have done before.
Moments {#moments-1}
-------
The mean waiting times for high and low priority customers can be found by differentiation of and : $$\begin{aligned}
E(W_H) &= \frac{\rho_H E(B_{H,\textit{res}})+\rho_L E(B_{L,\textit{res}})}{1-\rho_H}+\frac{1-\rho_1}{1-\rho_H}E(I_{1,\textit{res}}),\\
E(W_L) &= \frac{\rho_H E(B_{H,\textit{res}}) + \rho_L E(B_{L,\textit{res}})}{(1-\rho_H)(1-\rho_1)} + \frac{1}{1-\rho_H}E(I_{1,\textit{res}}).\end{aligned}$$ Differentiation of and leads to alternative expressions, that can also be found in [@wierman07]. $$\begin{aligned}
E(W_H) &= \frac{(1-\rho_1)^2}{1-\rho_H}\frac{E({C^*_1}^2)}{2E(C)},\\
E(W_L) &= \frac{(1-\rho_1)^2}{(1-\rho_H)(1-\rho_1)}\frac{E({C^*_1}^2)}{2E(C)}\\
&= \left(1-\frac{\rho_L}{1-\rho_H}\right)\frac{E({C^*_1}^2)}{2E(C)}.\end{aligned}$$
Example {#numericalexample}
=======
Consider a polling system with two queues, and assume exponential service times and switch-over times. Suppose that $\lambda_1 = \frac{6}{10}, \lambda_2 = \frac{2}{10}, E(B_1) = E(B_2) = 1, E(S_1) = E(S_2) = 1$. The workload of this polling system is $\rho = \frac{8}{10}$. This example is extensively discussed in [@winands06] where MVA was used to compute mean waiting times and mean residual cycle times for the gated and exhaustive service disciplines.
In this example we show that the performance of this system can be improved by giving higher priority to jobs with smaller service times. We define a threshold $t$ and divide the jobs into two classes: jobs with a service time less than $t$ receive high priority, the other jobs receive low priority. In Figures \[fig:gated\] and \[fig:exhaustive\] the mean waiting times of customers in $Q_1$ are shown as a function of the threshold $t$. The following four cases are distinguished:
- the mean waiting time of the low priority customers in $Q_1$ (indicated as “Type L”);
- the mean waiting time of the high priority customers in $Q_1$ (indicated as “Type H”);
- a weighted average of the above two mean waiting times: $\frac{\lambda_L}{\lambda_1}E(W_L) + \frac{\lambda_H}{\lambda_1}E(W_H)$ (indicated as “Type 1 with priorities”). This can be interpreted as the mean waiting time of an arbitrary customer in $Q_1$;
- the mean waiting time of an arbitrary customer in $Q_1$ if no priority rules would be applied to this queue (indicated as “Type 1 no priorities”). In this situation there is no such thing as high and low priority customers, so the mean waiting time does not depend on $t$, and has already been computed in [@winands06].
The figures show that a unique optimal threshold exists that minimises the mean weighted waiting time for customers in $Q_1$. This value depends on the service discipline used and is discussed in [@wierman07]. In this example the optimal threshold is 1 for gated, and 1.38 for exhaustive. Figure \[fig:gated\] confirms that the mean waiting times for type $H$ and $L$ customers in the gated model only differ by a constant value: $E(W_L) - E(W_H) = \rho_1 E(C_{1,\textit{res}})$. For globally gated service no figure is included, because we again have $E(W_L) - E(W_H) = \rho_1 E(C_{1,\textit{res}})$. The mean residual cycle time is different from the one in the gated model, but this does not affect the optimal threshold which is still $t=1$.
In the exhaustive model we have the following relation: $$E(W_L) - E(W_H) = \frac{\rho_1(1-\rho_1)}{1-\rho_H}E(C^*_{1,\textit{res}}).$$ If we increase threshold $t$, the fraction of customers in $Q_1$ that receive high priority grows, and so does their mean service time. This means that $\rho_H$ increases as $t$ increases, so $E(W_L) - E(W_H)$ gets bigger, which can be seen in Figure \[fig:exhaustive\]. Notice that $\frac{E(W_H)}{E(W_L)} = 1-\rho_1$, so it does not depend on $t$.
![Mean waiting time of customers in $Q_1$ in the gated polling system, versus threshold $t$.\[fig:gated\]](gatedMean){width="\linewidth"}
![Mean waiting time of customers in $Q_1$ in the exhaustive polling system, versus threshold $t$.\[fig:exhaustive\]](exhaustiveMean){width="\linewidth"}
It is interesting to also consider the variance, or rather the standard deviation of the waiting time. Figures \[fig:gatedvar\] and \[fig:exhaustivevar\] show the standard deviation of the type $H$ and $L$ customers versus the threshold $t$. The figures also show the standard deviation of an arbitrary customer in $Q_1$, with and without priorities. The figures indicate that the waiting times in the gated system have smaller standard deviations than in the exhaustive case. In this example, the introduction of priorities affects the standard deviation of an arbitrary type 1 customer only slightly. However, it is interesting to zoom in to investigate the influence of threshold $t$. Figure \[fig:gatedexhaustivevarzoom\] contains zoomed versions of Figures \[fig:gatedvar\] and \[fig:exhaustivevar\] and indicates that the threshold $t$ that minimises the overall mean waiting time of type $1$ customers in the priority system does not minimise the standard deviation. In fact, changing threshold $t$ affects the entire service time distributions $B_H$ and $B_L$, which results in two local minima for the standard deviation as function of threshold $t$.
![Standard deviation of the waiting time of customers in $Q_1$ in the gated polling system, versus threshold $t$.\[fig:gatedvar\]](gatedVar){width="\linewidth"}
![Standard deviation of the waiting time of customers in $Q_1$ in the exhaustive polling system, versus threshold $t$.\[fig:exhaustivevar\]](exhaustiveVar){width="\linewidth"}
![Zoomed versions of Figures \[fig:gatedvar\] (left) and \[fig:exhaustivevar\] (right).\[fig:gatedexhaustivevarzoom\]](gatedVarZoom "fig:"){width="0.45\linewidth"} ![Zoomed versions of Figures \[fig:gatedvar\] (left) and \[fig:exhaustivevar\] (right).\[fig:gatedexhaustivevarzoom\]](exhaustiveVarZoom "fig:"){width="0.45\linewidth"}
Possible extensions and future research {#extensions}
=======================================
The polling system studied in the present paper leaves many possibilities for extensions or variations. In this section we discuss some of them.
#### Multiple queues and priority levels.
Probably the most obvious extension of the model under consideration, is a polling system with any number of queues and any number of priority levels in each queue. In recent research [@boonadanboxma2008], we have discovered that such a polling model can be analysed in detail. Each queue can have its own service discipline, either exhaustive or (synchronised) gated.
#### Preemptive resume.
In the present paper, the service of low priority customers is not interrupted by the arrival of a high priority customer. If we allow for service interruptions, these would only take place in a queue with exhaustive service, since (globally) gated service forces high priority customers to wait behind the gate. We note that allowing service interruptions does not affect the joint queue length distributions at polling instants, nor the cycle time. Also the waiting time of low priority customers is unaffected (but they might have a longer *sojourn time*). It only affects the waiting time of high priority customers, because they do not have to wait for a residual service time of a low priority customer. The LST of the waiting time distribution of a high priority customer if service is preemptive resume, is: $$E[{\textrm{e}}^{-\omega W_H}] = \frac{(1-\rho_H)\omega}{\omega-\lambda_H(1-\beta_H(\omega))}\cdot
\left[\frac{1-\rho_1}{1-\rho_H} \cdot \frac{1-\widetilde{I}_1(\omega)}{\omega E(I_1)}+ \frac{\rho_L}{1-\rho_H}\right].$$
#### Mixed gated/exhaustive service.
In the present paper, customers in $Q_1$ receive either exhaustive or (globally) gated service. One may consider serving each priority level according to a different service discipline. In [@boonadan2008], high priority customers receive exhaustive service, whereas low priority customers receive gated service. This gives high priority customers an additional advantage, but it turns out that for low priority customers this strategy may be better than, e.g., gated service for all priority levels. A mixture of globally gated service for low priority customers and exhaustive service for high priority customers can be analysed similarly.
The “opposite” strategy, where low priority customers are served exhaustively and high priority customers are served according to the gated service discipline is easier to analyse, since we can model it as a nonpriority polling model with $Q_1$ replaced by two queues, $Q_{H}$ and $Q_L$, containing the type $H$ and type $L$ customers and having gated and exhaustive service respectively.
#### Partially gated.
A variant of the gated service discipline is partially gated service: every customer, type $H$ or $L$, standing in front of the gate is served during a visit with a fixed probability $p$, and is not served with probability $1-p$. The probability $p$ might even depend on the customer type. Whether a rejected customer is eligible for service in the next cycle, or leaves the system, does not matter. Both situations can be analysed.
#### Different polling sequences.
We assume that the server alternates between $Q_1$ and $Q_2$. A different way of introducing priorities to a polling system is by increasing the frequency of visits to a queue within a cycle. One can, e.g., decide to visit $Q_1$ two consecutive times if gated service is used. Or one can think of a system where the server switches to $Q_j$ after completing a visit to $Q_i$ with probability $p_{ij}$.
#### Large setup times.
[@winandsPhD] establishes fluid limits for polling systems with any branching type service discipline and *deterministic* switch-over times tending to infinity. The scaled waiting time distribution is shown to converge to a uniform distribution with bounds that can be computed explicitly. The results are relevant to applications in production systems, where large setup times are common. These fluid limits can also be computed for the polling model that is discussed in the present paper and give explicit insight in when each of the discussed service disciplines is optimal.
[27]{} \[1\][\#1]{} \[1\][`#1`]{} urlstyle \[1\][doi: \#1]{}
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[^1]: <span style="font-variant:small-caps;">Eurandom</span> and Department of Mathematics and Computer Science, Eindhoven University of Technology, P.O. Box 513, 5600MB Eindhoven, The Netherlands
[^2]: The research was done in the framework of the BSIK/BRICKS project, and of the European Network of Excellence Euro-FGI.
[^3]: The present paper is an adapted and extended version of [@boonadanboxma2queues2008].
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title: 'Wireless Quantization Index Modulation: Enabling Communication Through Existing Signals'
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[**Abstract –**]{} As the number of IoT devices continue to exponentially increase and saturate the wireless spectrum, there is a dire need for additional spectrum to support large networks of wireless devices. Over the past years, many promising solutions have been proposed but they all suffer from the drawback of new infrastructure costs, setup and maintenance, or are difficult to implement due to FCC regulations. In this paper, we propose a novel Wireless Quantization Index Modulation (QIM) technique which uses existing infrastructure to embed information into existing wireless signals to communicate with IoT devices with negligible impact on the original signal and zero spectrum overhead. We explore the design space for wireless QIM and evaluate the performance of embedding information in TV, FM and AM radio broadcast signals under different conditions. We demonstrate that we can embed messages at up to 8–200 kbps with negligible impact on the audio and video quality of the original FM, AM and TV signals respectively.
Introduction {#intro}
============
Over the last decade, we have witnessed a rapid growth in the deployment of IoT devices. By some estimates, there will be more than 26 billion connected IoT devices by the year 2020 [@iot2020]. However, as more devices connect to wireless networks, ***available spectrum is insufficient and existing wireless protocols are ill-equipped to support the growing number of devices.*** To understand the challenge, consider a home with wireless cameras, security sensor, smart watches, fitness trackers and wireless speakers. These devices use Wi-Fi/Bluetooth or proprietary wireless in the 2.4 GHz ISM band and operate alongside Wi-Fi routers, smartphones, laptops and tablets. As more devices share the wireless channel, wireless interference and packet collisions increase, negatively impacting the throughput and latency [@musaloiu2008minimising] [@gollakota2011clearing].
As wireless spectrum (such as the 2.4 GHz ISM band) becomes crowded, conventional wisdom dictates that we migrate to new protocols in less congested wireless channels. New protocols such as 802.11ah, LoRaWAN [@lorawan], SIGFOX [@sigfox] operate in 915 MHz ISM band, high speed 802.11 n/ac Wi-Fi is moving to 5.8 GHz ISM band, NB-IoT operates in the licensed cellular bands and TV white space networking [@whitespace] operates in unused channels in the TV UHF spectrum. Although these solutions are a step in the right direction, let’s discuss these approaches in terms of cost and spectrum utilization.
![[]{data-label="fig:model"}](model.eps){width="0.8\columnwidth"}
- *Infrastructure and Maintenance Costs*: Migration to new protocols such as LoRaWAN, SIGFOX or 802.11ah require setup, deployment and maintenance of dedicated expensive gateways and base stations. Instead, if we can reuse existing wireless infrastructure for communication, we could develop a far simpler and cost effective solution.
- *Spectrum Utilization*: Wireless spectrum is an extremely valuable and highly regulated resource. TV white space networking uses allocated but otherwise under-utilized TV spectrum for wireless communication. However, more often than not the availability of unused TV channels in urban areas are scarce and it is very cumbersome and expensive to deploy a TV whitespace network [@fcc]. New protocols such as LoRaWAN, SIGFOX, and 802.11ah are moving to the less crowded 915 MHz ISM band, but over time as the number of devices increase, they are going to run into familiar interference and capacity issues: as the number of devices increase, the spectrum is going to become more crowded and eventually saturate. Unless new spectrum is made available, using traditional methods, it is impossible to scale beyond a certain point.
In this paper, we propose a new cost and spectrally efficient solution for wireless communication. Consider an urban city environment as shown in Fig. \[fig:model\] with existing deployments of AM, FM, TV, and cellular base stations. These base stations have been setup with tremendous infrastructure cost, undergo periodic maintenance and pay licensing fees to transmit at pre-assigned licensed frequencies. The base stations are designed and geographically located for optimal signal coverage. For example, a typical FM tower can be received up to 100 kms.
We introduce Wireless Quantization Index Modulation (QIM), a communication technique which leverages existing infrastructure and reuses broadcast signals to provide additional communication channels for IoT devices. To understand Wireless QIM, without the loss of generality, let’s consider a broadcast TV station. A TV transmitter can use the QIM technique in its baseband to embed a message into the broadcast TV signal by introducing small perturbations while having a negligible impact on the broadcasted TV signal. Legacy TV receivers in the coverage area decode the broadcasted signal as before while IoT devices with a QIM receiver can decode the embedded message without any prior knowledge of the broadcast signal. So, in summary with a small modification to the baseband of the broadcast station, Wireless QIM reuses infrastructure, spectrum and broadcast signals to simultaneously communicate with QIM enabled IoT devices and legacy AM/FM/TV/cellular devices.
Wireless communication requires both uplink and downlink. However, more often than not, it’s asymmetric i.e. depending on the application, either uplink or downlink communication dominates. In this paper, we focus on downlink heavy applications and design a Wireless QIM system for downlink communication. Our target application is a smart city where using Wireless QIM, existing wireless infrastructure provides connectivity for real-time update of electronic bus schedule displays, billboard signs and advertisements, traffic alerts to name a few. With a minimal change in the baseband of existing broadcast towers, we can embed data to wirelessly update devices with a QIM receiver in real-time. These applications would require a minimal uplink channel to send acknowledgement messages, however such a low bandwidth and infrequent task can be accomplished using traditional LoRa, SigFox or cellular radios for the time being. In future work, we will extend the Wireless QIM technique to uplink communication and develop a bi-directional communication system which can leverage existing infrastructure and communicate with smart devices with zero spectrum overhead and minimal additional cost to target a broader set of applications.
To demonstrate the efficacy of Wireless QIM for these applications, we extensively evaluate the design space and explore various tradeoffs between performance of the message and host signal. We implement Wireless QIM on three existing infrastructure broadcast signals: AM, FM, and TV and show that information can be reliably embedded with negligible impact on audio (AM and FM) and video (TV) quality of the host signals. Our results show that using Wireless QIM, we can embed messages for IoT devices at up to 8 kbps in AM radio signals and 200 kbps in FM signals.
![[]{data-label="fig:qim_blk"}](embedding_model.eps){width="1\columnwidth"}
Quantization Index Modulation {#sec:methods}
=============================
Quantization Index Modulation (QIM) was originally introduced as a scheme for information hiding and digital watermarking [@qim]. In these applications, a message signal is embedded inside another signal called the host signal such that the embedded message is robust to common degradations, while the host signal suffers minimal degradation. QIM technique achieves an efficient tradeoff between the data rate of the message signal, distortion, and robustness of embedding.
In this work, we use QIM for wireless communication. We show how to embed messages for IoT devices inside existing wireless signals at existing wireless transmitters such that IoT devices with QIM receivers can decode the messages while the broadcasted wireless signal experiences minimal degradation. Fig. \[fig:qim\_blk\] shows the block diagram for a typical Wireless QIM system. At the AM/FM/TV/cellular broadcast, data $d$ needs to be transmitted and is passed through the host transmitter to generate a baseband host signal $s$. The host signal can be of any form, for instance, a frequency modulated signal from a FM station. Now, we also want to embed a message $m$ intended for IoT devices. The host signal is passed through a baseband QIM transmitter that uses a quantizer $Q(s)$ to embed the message inside the host signal. This generates a composite signal $x$ that propagates through the (noisy) channel and signal $y$ is received at the receiver. A legacy device would use standard compliant host receiver to decode the host signal, while an IoT device would use a QIM receiver to decode the embedded message $m$. QIM is designed to ensure that both the IoT and legacy devices can decode the desired signals with minimal degradation.
A Wireless QIM system consists of a QIM transmitter at the base station and a QIM receiver at the IoT device. To understand how Wireless QIM works, we will first describe the QIM transmitter followed by the QIM receiver.
![[]{data-label="fig:qim_tx"}](qim_tx.eps){width="50.00000%"}
QIM Transmitter {#sec:scalar}
---------------
On a high level, a QIM transmitter embeds information by using a form of quantization to introduce small perturbations in the host signal. Fig. \[fig:qim\_tx\] shows the block diagram of a QIM transmitter, where data $d$ is passed through the baseband host modulator to generate the host signal $s$. Then the host signal and message $m$ are passed through the QIM encoder to generate a composite signal $x$ in baseband. Finally, the composite signal is upconverted to carrier frequency $f_c$ and transmitted. The implementation of the baseband QIM encoder depends on the host signal. For analog systems such as AM radio, QIM encoder consists of the digital baseband followed by the ADC, a standard component. Whereas in digital FM radio and TV systems, the QIM encoder can be integrated into pre-existing digital baseband. In the following section we explain how the QIM embedding process works. First lets define a uniform quantizer $Q\left(s\right)$ as
$$\vspace{-2pt}
Q(s) = \Delta\cdot
round(\frac{s}{\Delta})
\vspace{-2pt}
\label{eq:quantize}$$
where $\Delta$ is the quantization step size and is defined as
$$\label{delta}
\vspace{-2pt}
\Delta = \frac{2\cdot max(s)}{N}
\vspace{-2pt}$$
Here $N$ is the number of quantization levels. The step size and number of levels determine the embedding resolution for the host signal. Quantizer *Q(s)* can now be used to define the QIM embedding function,
$$\label{eq:qim}
\vspace{-2pt}
Q_m(s) = Q(s - d_m) + d_m
\vspace{-2pt}$$
where $d_m$ is the dither, a function of the message $m$ that is applied to the host signal. $d_m$ can take one of the two following values to represent embedding of either a 0-bit or 1-bit.
$$\vspace{-2pt}
\label{dither} d_1=\pm\frac{\Delta}{4} \text{ and } d_0=\begin{cases} d_1 + \frac{\Delta}{2}, & \text{if } d_1 \leq 0\\ d_1 -
\frac{\Delta}{2}, & \text{otherwise} \end{cases}
\vspace{-2pt}$$
The equation shows that if the 1-bit dither is negative, $d_1$ = $\nicefrac{-\Delta}{4}$, then 0-bit dither, $d_0$ = $\nicefrac{\Delta}{4}$, will be a positive value. Similarly, for a positive 1-bit dither, the 0-bit dither would be a negative value. So, in summary we create two dithered quantizers to embed data in the host signal which is dependent on the bit value of the embedded message $m$.
Fig. \[fig:scalar\_basic\] illustrates this QIM embedding process for a positive 1-bit dither $d_1$ with $N=4$ levels, represented by solid horizontal lines. The dashed horizontal lines represent quantization levels for the 1-bit dithered quantizer and dotted lines for the 0-bit quantizer, defined by Eq. \[eq:qim\]. We can see that because the dither for a 1-bit is defined as $\nicefrac{\Delta}{4}$, the dashed lines are shifted up by $\nicefrac{\Delta}{4}$ from the original set of levels and vise versa for 0-bit quantizer.
Each sample point of the host signal (dashed green line) is perturbed to the appropriate level depending on the message $m$ (shown at the top of the figure). For instance, at the first highlighted sample point (green dot) we encode a 0-bit and the composite signal (solid blue line) goes down to $\nicefrac{-5\Delta}{4}$, the nearest 0-bit level (blue X). Similarly, at the second highlighted point we embed a 1-bit, which jumps up to $\nicefrac{5\Delta}{4}$, the nearest level for a 1-bit. We can see from the example that the distance between quantization points is uniformly distributed between \[$\nicefrac{\Delta}{2}$, $\nicefrac{-\Delta}{2}$\]. As a result, the mean error due to embedding is equal to $\nicefrac{\Delta^2}{12}$.
Finally, we characterize the impact of the QIM encoder on the host signal. We define distortion in the host signal by comparing the original host signal to the composite signal generated by embedding and can be expressed as,
$$\label{eq:distortion} \vspace{-2pt}
D_s = \frac{1}{K} \sum_{i=1}^{K}
|s_i - x_i|^2
\vspace{-2pt}$$
![[]{data-label="fig:scalar_basic"}](scalarqim_example.eps){width="50.00000%"}
![[]{data-label="fig:qim_rx"}](qim_rx.eps){width="50.00000%"}
QIM Receiver {#sec:decode}
------------
Fig. \[fig:qim\_rx\] shows the block diagram of a QIM receiver. A QIM receiver is based on standard RF architecture and is similar to a commodity receiver in terms of complexity and power consumption. The QIM decoder is implemented in digital baseband and we simply apply the same QIM embedding function to the received signal for both $0$ and $1$ bit messages to obtain two quantized signals $q_{0}$ and $q_{1}$. These are passed through a minimum distance decoder to compare with the original received signal $y$ to obtain estimated/received message,
$$\label{decode} \hat{m} = \arg\min_{m} dist(y, y_m)$$
QIM Techniques {#sec:tech}
==============
In this section we describe different QIM techniques that can be used to embed messages in a host signal.
Scalar QIM
----------
Scalar QIM is the primary QIM technique which was described in the previous section and can be applied to a real valued or scalar signal.
Distortion Compensated QIM
---------------------------
Distortion compensated QIM (DC-QIM) is an extended version of the QIM method that reduces the distortion in the host signal and significantly improves the distortion to robustness trade-off [@qim]. The robustness of QIM is a function of the distance between the quantization points. If we increase the distance, robustness of QIM would also increase. This operation is the same as scaling the QIM embedded signal by a factor, $\alpha$. For example, if a sample point $s_i$ is shifted to a point, $x$, then by scaling it by $\alpha$, the sample point $s_i$ would now be at $\nicefrac{x}{\alpha}$. However, this increases the distortion of the host signal by a factor $\nicefrac{1}{\alpha^2}$ and results in a $\nicefrac{\Delta^2}{12\alpha^2}$ mean distortion. We compensate for the distortion by adding back a fraction, $1-\alpha$ of the host signal. This operation can be represented by the following QIM embedding function,
$$\label{eq:dc} \vspace{-2pt}
Q_m(\alpha s) = Q(\alpha s -
d_m ) + (1-\alpha)s + d_m
\vspace{-2pt}$$
The parameter $\alpha$ is defined as $0 \leq \alpha \leq 1$ where 1 represents the original QIM method. Since, $\alpha$ determines the distortion, a component of noise in the composite signal due to embedding, we can determine the optimal value of $\alpha$ maximizing the signal to noise ratio $SNR$.
$$\label{eq:alpha} \vspace{-2pt}
\begin{aligned} SNR(\alpha) &=
\frac{d_1^2}{(1-\alpha)^2 + \alpha^2\sigma_n^2} \\ \\
\frac{\partial SNR(\alpha)}{\partial \alpha} &= 0 \text{ gives } \alpha^* = \frac{D_s}{D_s + \sigma_n^2}, \end{aligned}
\vspace{-2pt}$$
where, $\sigma_n^2$ is the noise power. Hence, the optimal factor $\alpha^*$ is a function of distortion and noise.
Lattice QIM
-----------
The QIM technique can be extended to higher dimensional signals by using a lattice form [@qim]. We can arrange the quantization points to be an integer lattice, $Z^N$, in an N-dimensional Euclidean space $\textbf{R}^N$. Let’s consider a simplified case of a complex or two-dimensional signal which has both in-phase and quadrature phase components. For N=2, we will have two grids to quantize to either a 0-bit or 1-bit representation. As an extension of scalar QIM, the quantization points for a 0-bit and 1-bit can be represented as co-sets of the lattice,
$$\label{l2} \vspace{-2pt}
\Lambda_0 =
\big(\frac{-\Delta}{4}-j\frac{\Delta}{4}\big) + n_i \text{ and }
\Lambda_1 = \big(\frac{\Delta}{4}+j\frac{\Delta}{4}\big) + n_i
\vspace{-2pt}$$
where $n_i = 1, 2,..., l$ and $l$ represents the total number of quantization levels. Fig. \[fig:l2\] illustrates the quantization grid for $N=2$. In lattice QIM, the quantization levels are separated by $\nicefrac{\Delta}{2}$ distance in in-phase and quadrature components resulting in an overall distance of $\nicefrac{\Delta}{\sqrt{2}}$. However, the mean error due to embedding is still $\frac{\Delta^2}{12}$ and because of larger distance between data points, lattice QIM should perform better than scalar QIM. To implement lattice QIM for complex wireless signal, we quantize both the real and imaginary parts of the signal. We use the same embedding function and modify the dither as $$\label{l2_dither} \vspace{-10pt} \small
d_1=\frac{\Delta}{4}+j\frac{\Delta}{4} \text{ and } d_0=\begin{cases} d_1+
(\frac{\Delta}{2}+j\frac{\Delta}{2}), & \text{if } d_1 \leq 0\\ d_1-
(\frac{\Delta}{2}+j\frac{\Delta}{2}), & \text{otherwise} \end{cases}
\vspace{8pt}$$
Finally, we note that the distortion compensation technique described above can be applied to the real and complex values of the signal to implement distortion compensated lattice QIM.
Evaluating the QIM Wireless Link {#sec:theory}
================================
Evaluation of an embedded communication system (such as QIM) significantly differs from conventional communication systems. Traditional systems deal with only one signal whereas embedded communication systems operate on two signals: the host signal and the embedded signal. We need to evaluate the impact of QIM on the performance of both the host and the embedded message signal: what is the rate and robustness of the embedded signal, and how much did the embedded process distort the host signal and its impact on the output of the host (legacy) receiver.
Performance of Embedded Message
-------------------------------
We start by defining the capacity for QIM embedded message in a Gaussian channel. The output of the channel is the sum of the input host signal and Gaussian white noise. Let $\sigma_s^2$ be the average power of the host signal and channel noise follows a Gaussian distribution with variance $\sigma_n^2$. We can express information capacity, i.e., the supremum of the achievable rate for a real value one dimensional signal as [@shannon]
$$\label{eq:hostcapacity}
C_{host}=\frac{1}{2}\log_2\left(1+\frac{\sigma_s^2}{\sigma_n^2}\right)$$
where $\sigma_s^2/\sigma_n^2$ denotes the signal-noise ratio (SNR) and assumes that the input also follows Gaussian distribution. Since embedded QIM signal is distortion in the host signal, we can re-write the capacity expression in terms of distortion by treating the embedded QIM signal as a form of power-limited communication over a Gaussian channel:
$$\label{eq:capacity}
C_{qim}=\frac{1}{2}\log_2\left(1+\frac{D_s}{\sigma^2_n}\right)
\mathrm{bps/sec/Hz},$$
where the distortion constraint is given by Eq. \[eq:distortion\]. The capacity equation, $C_{qim}$, shows that the performance of the embedded message is directly proportional to distortion experienced by the host signal. The QIM technique maximizes the capacity of the embedded message for a given distortion of the host signal. The capacity is also a function of the bandwidth of the host signal and a higher bandwidth host signal would enable a higher data rate embedded message. Finally, we note that the capacity in Eq. \[eq:capacity\] is the maximum achievable data rate per unit bandwidth with arbitrarily small error probability. Practical QIM implementation would require error correction coding mechanisms to achieve performance close to the limits promised by channel capacity.
Impact on the Host Signal
-------------------------
The host signal experiences distortion due to perturbations introduced by the embedded signal which was described in Eq. \[eq:distortion\]. For a fair comparison across host signals, we introduce normalized distortion which is independent of the signal strength of the host signal and is defined as follows,
$$\label{eq:dist_norm} D_{s}^{n} = \frac{\sum_{i=1}^{N}
|s_i - x_i|^2}{\sum_{i=1}^{N} |s_i|^2} * 100\%$$
Finally, in addition to computing the distortion, we will also analyze the quality of the multimedia signal at the output of the host receiver to ensure that QIM embedding operation has minimal impact on the performance of the host signal.
Simulation Results {#sec:results}
==================
The Wireless QIM technique is independent of the host signal and is universally applicable. Here we consider three host signals: AM, FM and broadcast TV which are ubiquitous in cities. We start with a short primer on the host signals. 0.05in[*[**TV.**]{}*]{} In the United States, Digital TV (DTV) operates in the UHF band from 470-614 MHz with 6 MHz wide channels and follows the Advanced Television System Committee (ATSC) standard [@atsc]. ATSC uses 8-level vestigial sideband (8-VSB) modulation to transmit data. 8-VSB is a digital modulation technique which uses eight amplitude levels to represent symbols on a 6 MHz channel. Transmissions from a TV tower can be typically received up to 50 miles.
0.05in[*[**FM Radio.**]{}*]{} FM radio operates in the 87.8-108 MHz frequency band with 200 kHz wide channels. FM uses analog frequency modulation to encode audio and data i.e. information is transmitted by varying the frequency of the transmitted RF signal. Most FM stations can be heard up to 100 miles from the transmit tower.
0.05in[*[**AM Radio.**]{}*]{} In the United States, AM radio operates in the 525-1705 kHz band with 10 kHz channel spacing. AM radio uses amplitude modulation to encode data i.e. information is represented in the amplitude of the signal. AM signals propagate long distances and have been reported to have been received 200 miles away from the station.
We evaluate wireless QIM on recorded AM, FM, and TV signals. The USRP X300 [@usrp] was used to record TV signals centered at 539 MHz (UHF channel 25) with 6.25 MHz sampling rate and FM signal centered at 106.1 MHz with 200kHz sampling rate. We use a WebSDR [@websdr] to record an AM signal centered at 1630 kHz with 8kHz sampling rate. We implement QIM methods described in Section \[sec:methods\] and simulate different channel conditions and data rates using MATLAB.
We embed pseudo-random message bits and implement scalar QIM and scalar DC-QIM for real valued AM and FM radio signals. For complex TV signals in addition to scalar QIM, we also evaluate lattice QIM and lattice DC-QIM. In DC-QIM, we set $\alpha = 0.7$, the optimal value as per Eq. \[eq:alpha\]. We introduce additive white Gaussian noise to simulate different channel conditions. Finally, a QIM receiver recovers the transmitted message bits using the algorithm described in Section \[sec:decode\]. The value of $\Delta$, the QIM embedding parameter is known at the receiver. This is a reasonable assumption since it can be either pre-set or periodically updated. We evaluate the system by measuring the impact of QIM on both the host signal and the performance of the embedded QIM signal for different QIM methods at different channel conditions and number of quantization levels (affects distortion).
Impact on the Host Signal {#sec:imp_host_signal}
-------------------------
The distortion experienced by the host signal is a function of number of levels $N$ used in the QIM embedding process. We vary the number of quantization levels from 2 to 45 for embedding random messages in TV, FM and AM host signals and measure the normalized distortion and its impact on the performance of the legacy host signal receiver.
\[fig:dist\_all\]
\[fig:distortion\]
0.05in[*[**Normalized Distortion.**]{}*]{} Fig. \[fig:distortion\] shows the percentage of distortion experienced by each host signal as a function of number of levels for scalar DC-QIM technique. The AM signal experiences the most distortion followed by TV and FM. This is expected since AM uses analog amplitude modulation to encode data and QIM introduces amplitude perturbations, which distorts the information carrying amplitude of the AM signal. Similarly, TV also uses 8 level (digital) amplitude modulation and amplitude perturbations would impact the digital TV signal but since digital amplitude modulation is more robust compared to analog modulation, QIM introduces less distortion in case of TV compared to AM. The FM signal is the most robust among evaluated signals with less than 8% for four quantization levels since amplitude perturbations introduced by QIM have minimal impact on the frequency modulated FM signal.
-0.0in[*[**Distortion in the Frequency Domain.**]{}*]{} Next, we evaluate impact of embedding data using QIM in the frequency domain. Fig. \[fig:spectrum\] shows the spectrum of the baseband FM signal before and after the QIM embedding process. The two signals are passed through pulse shaping low pass filters to comply with spectral mask requirements. Our results show that there is small distortion in the in-band spectral characteristics of the baseband signal which corroborate the time domain distortion analysis. The out of band frequency components for both before and after QIM embedded baseband FM signal are atleast 35 dB below the main lobe thereby having minimal impact on any side channels.
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\[fig:mos\]
0.05in[*[**Impact on Host Signal Multimedia.**]{}*]{} The next step is to translate the distortion to the quality of the multimedia audio and video signal carried by AM, FM and TV signals. This is the key to understanding how QIM impacts the information carried by the host signal. We use the Perceptual Evaluation of Speech Quality (PESQ) metric to quantify the quality of demodulated audio from AM and FM signals. The results model a mean opinion score (MOS) that ranks the quality of speech from 1(bad) to 5(excellent). As a reference, a PESQ $\geq$ 1 is sufficient for human hearing [@speech]. We evaluate the PESQ as function of the SNR of the host signal and distortion introduced by scalar DC-QIM technique. Fig. \[fig:mos\](a) shows the audio quality of an AM signal as a function of SNR of the host signal and the number of quantization levels. We can see that even at the lowest SNR of 6 dB, PESQ is greater than 2.8 which is sufficient for most applications. As we increase the SNR and number of quantization levels, the audio quality improves. The results confirm that QIM has minimal impact on the audio quality of AM signals. We perform similar analysis for FM signal and Fig. \[fig:mos\](b) shows the audio quality of a demodulated QIM embedded FM signal at different host signal SNR and quantization levels. Since the FM signal is more robust to QIM, we evaluate the scalar DC-QIM technique for 2-6 quantization levels (42$\%$-2$\%$ distortion) in the FM baseband. However, due to the robustness of frequency modulation, even at the worst-case SNR of 6 dB and 42$\%$ distortion, PESQ is close to 2, which is satisfactory. Finally, we evaluate the video quality of the QIM embedded TV signal, using the peak SNR (PSNR) metric which is the ratio of maximum possible power of a signal to the power of the distorting noise [@psnr]. The PSNR is computed as follows
$$PSNR = 10log_{10}(\frac{(max(s))^2}{D_s})$$
where $D_s$ is distortion defined in Eq. \[eq:distortion\]. For typical applications, PSNR values between 20-25 dB are acceptable for wireless systems [@psnr].
To evaluate the PSNR of the video signal, we first extract the video by demodulating the QIM embedded TV host signal. We use a software defined radio (USRP X300) to re-transmit the QIM embedded TV signal at different SNR to a TV tuner card by Hauppauge to recover the video. The TV signal was embedded with 20-36 quantization levels which translates to 0.9%–0.3% distortion in the TV baseband signal. In Fig. \[fig:mos\](c) we plot the PSNR of the video output of the TV tuner card as a function of SNR of the host TV signal and number of levels. The PSNR of the recovered video was around 34 for majority of the cases expect for the lowest SNR of 16 dB at the highest distortion. However, even the lowest values of 28 dB PSNR is acceptable for most applications. Our analysis considers TV signals above an SNR of 16 dB, a constraint placed by the sensitivity of the TV tuner card. The TV tuner was only able to play video from original distortion free TV signal above an SNR of 16 dB which placed the limit on the SNR of the TV signal evaluated in this work.
To give readers an intuition about the quality metric used in the evaluation, we created a composite video of audio and video clips for AM, FM and TV host signals for different SNR and distortion in the host signal which can be found at the following web link:
[ https://youtu.be/gKn09ctlFMA]{}.
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Performance of the QIM Embedded Message {#sec:results_qim_embedded}
---------------------------------------
The next step is to evaluate the performance of the QIM embedded message signal. We only consider scenarios where the distortion and impact on the host multimedia is within acceptable bounds described in Section \[sec:imp\_host\_signal\].
In Section \[sec:theory\] we showed that the information carrying capacity of QIM, like any communication system, is directly proportional to the bandwidth of the host signal. To have a fair comparison across different signals, we normalize the embedded message data rate to the bandwidth of the host signal and evaluate performance for all three host signals. Specifically, we embed at the rate of 1 bps for 10 kHz bandwidth AM signal, 20 bps for 200 kHz bandwidth FM signal and 625 bps for 6.25 MHz bandwidth TV signal. Fig. \[fig:ber\_compare\_all\] shows the BER of the embedded message using 22 level scalar DC-QIM for the three host signals which translates to 1%, 0.3% and 0.7% distortion respectively in the AM, FM and TV signals. We can see that there is a 4 dB difference between AM and TV and a 7 dB difference between AM and FM. This can be attributed to the fact that for the same number of levels, the AM signal experiences 0.35% more distortion compared to TV and 0.76% more distortion compared to FM. Since distortion is the embedded signal, a higher distortion translates to higher signal strength for the embedded message signal and better performance. We empirically note an approximate 1 dB increase in performance for every 0.1$\%$ increase in distortion.
Next, we individually analyze each of the three host signals by evaluating the BER of embedded message as a function of the SNR of the host signal and the signal strength of the embedded message. We evaluate the AM signal for 8-16 levels at 200bps embedded message rate for both scalar QIM and scalar DC QIM. Fig. \[fig:ber\](a) shows that the BER decreases with decrease in number of levels which translates to an increase in distortion of the host signal or signal strength of the embedded signal. Specifically, for every decrease in two quantization levels, there is a 2 dB increase in the performance. Finally, the distortion compensation technique improves the performance by about 2 dB and this is true for all host signals and both scalar and lattice DC-QIM methods.
For the distortion tolerant FM signals we embed message at 20 kbps and use 2–6 quantization levels which translates to a distortion of 42% to 4%. Fig. \[fig:ber\](b) shows the performance of scalar and scalar DC QIM for the host FM signal. We observe a 2 dB increase in performance for every unit decrease in number of quantization levels.
Finally, we evaluate the TV signal for both scalar and lattice DC-QIM techniques. Since TV signals are susceptible to distortion, we embed messages at 250 bps and use 22–48 quantization levels which translates to a distortion of 0.7% to 0.1% in the host TV signal. Fig. \[fig:ber\](c) plots the performance of an embedded QIM message in a host TV signal and we can see that performance increase with decrease in number of levels or increase in distortion. Additionally, lattice DC-QIM outperforms scalar DC-QIM by about 2 dB.
\[fig:bps\]
Achievable Throughput
---------------------
We analyze achievable throughput for high and low distortion in the host signal. For low distortion in a host signal, we consider the 10 kHz wide AM radio signal and embed messages using scalar DC-QIM method at different data rates using 8–16 quantization levels (8%–3.5% distortion). Fig. \[fig:bps\](a) shows the throughput as a function of SNR for different quantization levels. We can see that QIM embedded message achieves the maximum throughput of 8 kbps for SNR of 20 dB for 8 quantization levels (8% distortion). Every decrement in two quantization levels increases the throughput by a factor of 1.5 and every 2 dB increase in SNR increase the throughput by an average factor of 3. We can scale these results for FM and TV signals by respectively multiplying the AM data rates by 20 and 625. For a high distortion case, we consider the 200 kHz wide FM radio signal and embed messages using scalar DC-QIM method with 2–6 quantization levels (42%–4% distortion). We plot throughput as a function of SNR for different quantization levels in Fig. \[fig:bps\](b). We achieve a maximum throughput of 200 kbps.
Related Work
============
Our work is related to recent efforts in spread spectrum watermarking of RF signals [@ssp]. Spread spectrum is a promising technique because it is robust against interfering noise. However, it is a linear method and is susceptible to host signal interference. On the other hand, QIM is a non-linear techniques which is aware of the host signal and therefore efficiently manages host signal interference, resulting in better overall performance. Wireless QIM is also related to inter-protocol communication techniques which enable communication between IoT devices using different wireless standards. This is especially beneficial in the crowded 2.4GHz spectrum where devices with a software modification can using existing hardware to communicate between different devices employing different standards. For instance, the WiZip system uses the presence and absence of packets to encode information for transmission from a Wi-Fi device to a ZigBee device [@wizig]. FreeBee uses variance in timing of regular Wi-Fi beacons to transmit information to ZigBee devices [@freebee]. $B^2W^2$ uses presence and absence of packets to enable BLE to Wi-Fi communication while concurrently supporting existing Wi-Fi and BLE communication [@b2w2].
All of these techniques are promising, but they are limited to low data rates, are short range and spectrally inefficient since Wi-Fi and ZigBee use drastically different bandwidths. Instead, the Wireless QIM technique can achieve data rates up to 8 kbps with only a 10 kHz wide host AM signal which is 52, 470, and 5.5 order of magnitude higher when compared to WiZig, FreeBee, and $B^2W^2$ which use a significantly wider bandwidth (20 MHz) signal.
Conclusion and Future Work
==========================
We have introduced Wireless QIM technique to embed information into existing signals and communicate with smart devices while having negligible impact on the host signal. We have demonstrated communication at up to of 8 kbps for low bandwidth distortion sensitive AM signals and 200 kbps for higher bandwidth, but distortion resilient FM signals. To the best of our knowledge, this is the first work to use QIM technique to embed messages into wireless signals. We believe Wireless QIM presents a new and exciting opportunity for the radio/TV/cellular providers to enable smart cities and IoT applications by reusing their existing infrastructure and deliver additional value with connectivity at zero spectrum overhead.
In this paper, although we have evaluated data rate and reliability of the embedded message, we haven’t explored errors correcting codes. Development of error correcting codes for Wireless QIM to achieve data rates closer to the theoretical capacity of the channel is an exciting avenue for future research. Finally, this work was focused on downlink communication and in the future work we will extend the Wireless QIM technique for uplink communication as well.
Acknowledgements
================
This work is supported in part by NSF award CNS-1305072 and a Google faculty research award.
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abstract: 'It is noted that X-ray tails (XRTs) of short, hard $\gamma$-ray bursts (SHBs) are similar to X-ray flashes (XRFs). We suggest a universal central engine hypothesis, as a way of accounting for this curiosity, in which SHBs differ from long $\gamma$-ray bursts (GRBs) in prompt emission because of the differences in the host star and attendant differences in the environment they present to the compact central engine (as opposed to differences in the central engine itself). Observational constraints and implications are discussed, especially for confirming putative detections of gravitational waves from merging compact objects.'
author:
- 'David Eichler , Dafne Guetta , & Hadar Manis'
title: A Universal Central Engine Hypothesis for Short and Long GRBs
---
key words: $\gamma$-rays: bursts
Short $\gamma$- ray bursts (GRBs), originally defined to be GRBs lasting less than 2 seconds and predicted to be a separate class of phenomena (Kouveliotou et al 1993), are now widely suspected of being two merging compact objects. This somehow distinguishes them in their duration and other properties from the core collapse of a massive star. In the former case, primary emission can in principle be detected even if it comes from less than 2 lightseconds from the black hole, which would probably be impossible for long bursts, where envelopes would obscure photon emission from these scales. The central engines of each, however, are likely to be similar: a black hole of maximum angular momentum surrounded by an accretion disk of matter near nuclear densities.
A fundamental open question about short GRBs is why they are so short. Is it because the central engine (presumably a black hole fed by an accretion disk) operates on a shorter timescale than those of long GRBs? Alternatively, it may be the timescale over which the $\gamma$- rays are visible. For example, the prompt $\gamma$- rays could be scattered off slow baryons (Eichler and Manis 2007) and the short duration is a result of the baryons getting accelerated to a high enough Lorentz factor to exclude much further emission along our line of sight. What seems like a short burst to us would then appear to be a much longer burst to some other observer along a different line of sight. The hypothesis can account for the fact that short bursts tend to have harder spectra, lower luminosity, and a larger solid angle than long bursts. It also accounts for the inverse correlation between luminosity and spectral lag (Hakkila et al. 2008 and references therein) in a simple manner, assuming the acceleration is due to the primary radiation pressure.
In this letter we propose a universal central engine hypothesis for both short and long bursts. We suggest that the same compact, central engine can produce what seems to be a short, hard burst to a viewer at large viewing angle, and a long burst to a viewer at smaller viewing angle. The distinction between “central engine” - by which we mean the post-collapse compact object - and “progenitor” is emphasized. We keep the now common view that short hard bursts come from merging neutron stars or other systems without a large envelope, whereas the long bursts typically come from progenitors with large, post main-sequence envelopes.[^1]
That typical viewing angle should correlate with the progenitor and host environment is straightforward: a central engine sitting within a massive envelope is obscured by the envelope, and fireball material within the envelope can be observed directly down the hole bored by the fireball in the envelope (Figure 1). Observers outside the opening angle $\theta_o$ established by this hole can see emission from material only after it has nearly emerged from the envelope, and, from well outside the opening angle of the emerging material (including its $1/\Gamma$ emission cone), they detect kinematically softened emission that is nearly backwards in the frame of the fireball. Such an observer cannot view matter while it is well within the confines of the envelope ($R\le 10^{12}$cm). When, on the other hand, there is no massive envelope obscuring the central engine, matter can be seen from within $10^{12}$cm, and subsecond timescales become possible. The significance of the host/progenitor may then be the angles at which it allows the burst to be observed, and, therefore, the stage of the burst that is observable.
The unified model proposed here should be contrasted to that of Yamazaki et al (2004), which, somewhat presciently, was made [*before*]{} the operation of Swift (and the resulting localization of SHBs). They proposed that short bursts come from the same central engine as long bursts, and that they appear short when one emitting “minijet” of many comes close enough to the line of sight as to dominate over the contribution of all the others. Observers at large offset angle to the axis of the swarm of minijets see X-ray flashes, but little or no hard emission. In the Yamazaki et al picture, it would be hard to understand why SHBs typically come from different types of galaxies and, by inference, different types of progenitors. It would also be hard to understand why SHBs are [*underluminous*]{} (or overly hard) in the context of the Amati relation. It is not clear that the hard part of the GRB would always precede the soft part. Finally, it would be hard to understand why the small scale time structure and spectral lags in SHBs are qualitatively different from those in long GRBs. In the unified model we propose here, on the other hand, the hard photons of the SHB are seen at large viewing angles, but from an earlier stage of the fireball’s acceleration, and these observations follow naturally.
Below, we summarize the observations that motivate the universal central engine hypothesis. We then show that a particular model for GRB subpulses can produce a viable model for SHBs and the X-ray tails.
[*Observational Motivations:*]{} SHBs frequently display long X-ray tails that compare in duration to long X-ray flashes. The discovery (e.g. Donaghy et al 2006; Norris and Bonnel, 2006; Gehrels et al 2006) confirmed by Swift that short bursts have X-ray tails (XRTs) of much longer duration than the burst itself heightens the suspicion that the central engine continues to operate for longer than 2 seconds. Donaghy et al report that most SHBs observed with HETE II (which has a lower photon energy threshold than Swift) have long, soft tails, whereas the fraction of Swift SHBs is somewhat less, about half. We may interpret this as XRTs being slightly softer than XRFs and/or as Swift being more sensitive than HETE II to the short, hard phase of SHBs. We argue below that this is expected, because the larger the viewing angle, the shorter, harder the emission of the short phase, when by hypothesis $\Gamma \sim
1/\theta$, and the softer the tail emission when $\Gamma$ has reached its terminal value.
Apart from possibly being slightly softer, these XRTs are quite similar to $\gamma$-ray-silent X-ray flashes (XRFs), which have been proposed to be “off axis” GRBs. This interpretation of XRFs has also been supported by their tendency to show depressed X-ray afterglow (relative to normal GRBs) until $\sim 3 \times 10^5$s after the prompt emission (Sakamoto et al 2008), after which the afterglow appears to be of about the same intensity as that of a classical GRB. This suggests that we are seeing a kinematically suppressed, under-blue shifted signal that is predicted for an offset viewer.
There exists by now some evidence that SHBs are beamed into a small solid angle, similar to long GRBs. Fox et al. (2005) interpreted the steepening of the optical afterglow light curve of GRB 050709 and GRB 050724 in terms of a jet break, translating into a beaming factor $f_b^{-1}\sim 50$ (with $f_b$ the fraction of the $4\pi$ solid angle within which the GRB is emitted). Soderberg et al (2006) found a beaming factor of $\sim 130$ for GRB 051221A. Therefore, with the present data, the beaming angle of SHBs seems to be in a range of $\sim 0.1-0.2$ radians. The question is how this beam width compares to that of the X-ray tail.
Below, we summarize the data on SHBs, note that the XRTs (unlike the hard emission) obey the Amati relation (Amati et al. 2002), describe a particular version of a universal central engine hypothesis, and attempt a rough estimate of the beam width of the XRTs of SHBs based on the supposition that they are basically XRFs.
We have considered all the short bursts reported by Swift from its launch (November 2004) until March 2008; this constitutes a sample of 28 bursts. In Table 1 we report the observed data relative to those for which X-ray emission was detected by the X-ray telescope on board Swift. As noted by Donaghy et al (2006), prolonged soft emission in HETE II data is a rather reliable signature of SHBs, but it is clear that there is large scatter in the luminosity of the XRTs relative to the $\gamma$-ray luminosity, and, in the Swift data, XRTs are [*not*]{} a reliable indicator for a SHB.
The XRTs are quite similar to $\gamma$-ray-silent XRFs detected by HETE-II, BeppoSax and Swift (see Figure 2). We made a spectral analysis of all the XRTs that could be detected by the WFC and did not find any evidence of a spectral break in the band 0.3-10 keV. We have therefore assumed a spectral peak at $\sim 10$ keV with an uncertainty of $\sim 8 $ keV. The only burst that seems to have a higher energy break is 050724 (Campana et al. 2006) and we took $E_{\rm peak} \sim 20 $ keV for this burst. We see that the XRTs obey the Amati correlation to within the uncertainties, whereas the prompt $\gamma$- ray emission is far removed from this correlation.
[*Interpretation:*]{} That XRTs of SHBs are consistent with the Amati relation is what is expected if both the short hard emission and the X-ray tail are attributed to the offset viewing of what might be observable as a classic long GRB from a different direction. The X-rays are photons beamed backward in the frame of the classical fireball. They are reduced in frequency as the first power of the Doppler factor, and, in time-integrated fluence, by the square of the Doppler factor, when the beam is wider than the angular separation $\theta_V$ of the observer’s line of sight from the beam (Eichler and Levinson 2004). This gives the Amati correlation. (When, on the other hand, $\theta_V$ is comparable to or larger than $\theta_o$, the apparent luminosity decreases as a steeper power of peak frequency \[Yamazaki et al 2002, 2004, Eichler and Levinson 2004\]). The hard photons, on the other hand, are scattered into the line of sight by baryonic material that has not yet been accelerated beyond a Lorentz factor of $1/\theta_V$ (but soon will be). As such, the primary luminosity, as seen by observers in the beam, is diluted by the scattering, because the $1/\Gamma$ cone at low $\Gamma$ is much wider than it will end up when maximum Lorentz factor is reached. Whereas the scattering reduces only modestly the individual photon energies to observers within $1/\Gamma$ of the axis of scatterer’s motion \[$cos\theta_V \ge \beta$\] - $E^{\prime}/E$ being at least $1/\Gamma^2(1-\beta^2)(1+\beta)\ge 1/2$ - it greatly dilutes the fluence by spreading it over a much wider solid angle. So the fluence is greatly below what the Amati relation would predict for viewers that are within $1/\Gamma_a$ of the primary beam when it is finally at its terminal Lorentz factor $\Gamma_a$. The observer at wide angle (from the direction of the low $\Gamma$ scatterer) sees a much shorter hard pulse than the viewer at smaller offset angle (Eichler and Manis 2007), as shown in Figure 3, because the acceleration time as measured by the observer is proportional to $\Gamma$. In these figures we plot the light curve of the scattered emission from a single accelerating cloud with point-like geometry as seen by viewers at two different viewing angles. (We stress that this is not the same as predicting a light curve from the burst, which has a finite solid angle and time interval in which scatterers can be injected. The data superimposed in the same graph is merely for reference.) Other factors could contribute to the duration, such as the intensity of radiation pressure that causes the acceleration.
Assuming a Lorentz factor for the blast of $\Gamma(t)
\simeq 100 (t/100s)^{-3/8}$ (Sari et al. 1998), we estimate the Lorentz factor of the blast wave after $3 \times 10^5$s, the typical recovery time for XRFs (Sakamoto et al 2008), to be $\Gamma(10^{5.5}s )\sim
(10^{11/16}) \sim 5$. Attributing the afterglow recovery to the decrease of the blast’s Lorentz factor down to $1/\theta_V$, one estimates that XRFs are typically observed at an offset angle of $\theta_V \sim 0.2 \sim 10^o$ from the blast. In fact, a complete recovery requires that $\Gamma$ decline to comfortably below $1/\theta_V$, so $\theta_V$, defined to be the angular distance to the edge of the jet, is better estimated to be less than 0.2. Writing $\theta_V$ as $10^{-1}\theta_{-1}$, the typical spectral peak of the XRTs $E_{peak}$ as $30 E_{x,30}$KeV, and the expected spectral peak measured by the head-on observer as $ E_{ho} \times 1 $Mev, we estimate the Lorentz factor of the fireball that emits the prompt emission to be given by $0.03E_{x,30}/E_{ho}=1/\Gamma^2
(1+\beta)(1-\beta cos\theta) \sim (\theta_V\Gamma)^{-2}$, or $$\theta_V\Gamma \sim 6 [E_{x,30}/E_{ho}]^{1/2}.$$ For $\Gamma \sim 10^2$, and $E_{x,30}, E_{ho}$ both $ \sim 1$, this gives $\theta_V \sim 6 \times 10^{-2}[E_{x,30}/E_{ho}]^{1/2}$, which is consistent with the estimate of $\theta_V$ from the afterglow recovery time. Assuming that the jet itself has an opening angle of order $10^{-1}$ radians, this gives an opening angle for an XRF of about 0.1 to 0.2 radians, in reasonable agreement with the estimate from the flat phase of the afterglow. That the offset $\theta_V$ is comparable to the opening angle of the fireball jet $\theta_o$ suggests that for $E_{30}\ll 1$, the luminosity of the XRF should drop below that predicted by the Amati relation.
That extended soft emission is a reliable indicator for SHBs (Donaghy al. 2006) suggests that the solid angle in which the soft photons are detectable by HETE II is at least as large as that from which the hard $\gamma$-beam is detectable. On the other hand, the large variation in X-ray to $\gamma$-ray fluence suggests that we should be cautious about making simple generalizations regarding the relative characteristics of the X-ray tail and the $\gamma$-ray beams. Given our estimates of 0.1 to 0.2 radians for both the soft and hard beams, we could attribute the large variation in hard/soft emission ratio to the fact that the opening angles of the XRTs and hard $\gamma$-ray emissions are comparable, and that one can be observed near the ragged edge of the other. In our model, moreover, the fraction of hard $\gamma$- rays scattered into our line of sight can be highly variable from one burst to the next. Furthermore, the directions of the prompt $\gamma$-radiation and the accelerating baryons need not be the same (e.g. Eichler and Granot 2006, and references therein), so the respective relations of the observer’s line of sight to each of them is a somewhat free parameter. This affects the Lorentz factor of the scatterer that contributes to our line of sight, and hence the extent of solid angle dilution of the photon intensity. XRTs should therefore be considered as an important complement to (but not necessarily better than) SHBs in corroborating LIGO signals.
Long GRBs, XRFs, SHBs and XRTs each have two parameters - their cosmic event rates per unit volume and their beaming factors - for a total of eight parameters. Measuring the relative detection rates and distribution of distances of each of the four categories of events reduces this to four free parameters. The universal central engine hypothesis, in its simplest and most naive form, together with the offset viewing hypothesis for XRFs posit that a) the rate per unit volume of XRTs is the same as that of SHBs, b) the physical parameters of XRFs and XRTs are the same, c) the relative event cosmic rates of XRFs and XRTs per unit volume are the same as for long vs. SHBs, and d) the rate per unit volume of XRFs is the same as for classical long GRBs. These are only four assertions that constrain the four unknowns. So, although it may be possible to constrain the parameters of such a beaming factor within the framework of a universal central engine hypothesis via observations, the above considerations do not overconstrain the model enough to test its validity.
On the other hand, additional information could provide further tests. There may be small differences between XRTs and XRFs imposed by the different types of host stars, differences in their subsequent afterglow patterns as well as information on the host galaxies. Further into the future, a viable data set of LIGO events would allow a test of the relative beaming factors of SHBs and XRTs. We suggest that LIGO should operate together with efficient SHB detectors and wide field X-ray cameras. This would not only improve the chances for corroborating LIGO detections of mergers, but would enable these detections to teach us more about the associated high energy processes as well.
Our suggestion that some XRTs of SHBs are XRFs, [*combined with*]{} the hypothesis that they correspond to offset viewing of a long burst in some other direction, predicts that a large enough sample of XRFs, even if unbiased by any $\gamma$-ray trigger, should have a subset that correlates with SHBs. A careful analysis, however, shows that BATSE should have detected less than one SHB coincident with any X-ray flash. A larger sample of XRFs detected while a SHB detector is operating would give tighter constraints.
[*Further Consequences:*]{} Short bursts, inasmuch as they are believed to be merging compact objects (neutron star-neutron star, NS-NS, or neutron star-black hole, NS-BH), are expected to be closely connected to gravitational wave signals, and potential candidates, if close enough, for detection by LIGO. The horizon of first generation LIGO and Virgo for NS-NS, NS-BH mergers is $\sim
20$ and $43$ Mpc, respectively, while advanced LIGO/Virgo should detect them out to a distance of $\sim 300$ and $650$ Mpc (for a review see Cutler & Thorne 2002). Guetta and Stella (2008) have recently estimated that, assuming a beaming factor $\sim 100$, a sizeable fraction of gravitational wave events detectable by LIGO II is expected to be coincident with SHBs, which provides a new, interesting perspective for the Advanced LIGO/Virgo era. Here we note that a wide angle X-ray camera in addition to a Swift-type detector that triggers on hard $\gamma$-ray emission could possibly increase our ability to corroborate LIGO signals as well as learn more about merger events. As these events could be of marginal statistical significance, it would be good to verify them independently with detections of high energy emission that is believed to be associated with mergers. The complex, multicomponent nature of SHBs suggests that careful thought should be given as to the best way to corroborate putative gravitational wave events.
[*Summary:*]{} We have suggested that short, hard GRBs may have the same central engine as that of long GRBs, though having a different size host envelope, and that their duration is determined by the acceleration time of a relativistic scatterer that scatters them into our line of sight. A very rough estimate of the opening angle of XRTs, based on their hypothesized similarity to XRFs, is 0.1 to 0.2 radians, which is comparable to estimates of the opening angles for the hard emission. It is thus difficult to say which would be better for corroborating nearby compact mergers following LIGO triggers. Further information on the relative detectabilities of XRTs and the corresponding short hard $\gamma$-emission could be obtained by a wide field X-ray camera and $\gamma$-ray detectors working together. In six of 12 cases with known redshifts, the X-ray tail would exceed $10^{-10}$erg/cm$^2$s had the burst been within 300 Mpc (distance of GW detectability if the SHBs come from NS-NS mergers), so that LIGO might in such cases act as the primary trigger, which would have the valuable property of being free of any electromagnetic spectral bias.
One interesting prediction of the universal central engine hypothesis is that, while much of the X-ray fluence comes after the location and slewing of the Swift X-ray telescope, much of the fluence also can come out on a short timescale, the so called “spike”, depending on whether the source has a soft primary component. This appears to be consistent with HETE-II observations of short bursts (Donaghy et al 2006, figures 2-5,13, table 7). An important distinction should be made between the short rise of soft emission and the longer tail. The “spike” (i.e. rise and peak) consists either of a) photons that were soft at the primary source, and lost only about half their (observer frame) energy during the scattering or b) primary radiation that was softened by reprocessing at the scatterer, while the tail consists of photons that may well have been hard at the source and were drastically softened in the observer’s frame by making a near 180 degree rear end collision with the scatterer. The former should have a rise time that is nearly energy independent, as it is established by the acceleration time of the scatterer to $1/\theta $, while the latter have a much longer decay time at low energy because the collision-softened photons populate the low energy bins at large t (Eichler and Manis 2007). Thus, the relative shapes of the light curves in different energy bins, which reflect the relative contributions of these two classes of photons to the X-ray tail, can be very sensitive to the primary spectrum, which, in turn, can be sensitive to the type of progenitor/host. For example, if the prompt, soft photons have a component of thermal emission from the back side of the scatterer that has been heated by Compton recoil, we expect this component to have the same time profile as the hard $\gamma$-rays. Such a component could be more important for SHB environments, where the scatterer is closer to the central engine and sees a stronger radiation field. Moreover, at lower Lorentz factor, its back end sees a harder radiation field, so the Compton recoil heats more efficiently. This and several other matters beyond the scope of this paper bear further investigation.
A serious quantitative model of a SHB light curve within the context of the ideas sketched here should take into account the following: a) The scatterer may in fact be a swarm of individual clumps running into each other and accelerating uniformly only as a group. Figure 3 should therefore be taken, at best, as a rough envelope that characterizes the general trend of the light curve. The individual subpulses may be scattered photons from scatterers that are accelerating (as well as decelerating) on a somewhat faster timescale, and therefore have smaller positive (as well as negative) spectral lags. b) The distribution of scatterers as a function of angular separation from the observer’s line of sight is unknown but there are probably more at larger angles. The observed light curve is the sum over the individual contributions from the members of the swarm. c) While the scatterer is at modest Lorentz factor, backscattered photons may be intercepted by pair-producing collisions with primary photons. This can lower the X-ray luminosity near and just after the peak, when, for viewers at large angle, the Lorentz factor is still modest. (Eventually, as the Lorentz factor picks up, the primary photons in the frame of the scatterer are not energetic enough for pair production.) Thus, while primary, soft photons may escape immediately, the nearly 180 degree backscattered hard photons escape only after the overall Lorentz factor of the scatterers is high enough.
The universal central engine hypothesis also predicts that, occasionally, we are close to the axis of a GRB that originates from a SHB-type host. In such a case, the GRB would appear long in duration, but of the “short” variety in other ways. Indeed, there are such bursts (e.g. 060614) that confound a simple two class classification scheme (Donaghy et al 2006; Gehrels et al 2006).
We thank Luigi Piro and Matteo Perri for useful discussions. We acknowledge support from the U.S.-Israel Binational Science Foundation, the Israel Academy of Science, and the Robert and Joan Arnow Chair of Theoretical Astrophysics.
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-------------- --------- ---------------------- ------------------- --------------------------- --------------------- ---------------------------
GRB $z$ $S_{\gamma}$ $E_{\gamma, iso}$ $F_x $ $ E_{x,iso}$ $F_x $ (@ 300 Mpc)
$10^{-7}$ erg/cm$^2$ $10^{49}$ erg $10^{-11}$ erg/cm$^{2}/s$ $10^{49}$ erg $10^{-11}$ erg/cm$^{2}/s$
050709\*$^+$ 0.16 3$\pm$ 0.38 1.4 800 3.4 3.3 $\times 10^3$
050724$^+$ $0.258$ 6.3 $\pm$ 1 7.2 1200 10.1 9.9 $\times 10^3$
051210$^+ $ 1.9$\pm$ 0.3 90
051221$^+$ 0.546 22.2 $\pm$ 0.8 84 20 0.6 590
060313 $^+$ 32.1 $\pm$ 1.4 30
071227$^+$ 0.383 2.2 $\pm$ 0.3 4.0 46 0.87 854
050509B 0.225 0.23$\pm$ 0.09 0.2 0.06 4.5$\times 10^{-4}$ 0\. 44
050813 0.7 1.24 $\pm$ 0.46 5.2 0.6 0.025 25
050906 0.84 $\pm$ 0.46 $<0.007$
050925 0.92 $\pm$ 0.18 $ <0.003$
060502B 0.287 1 $\pm$ 0.13 1.15 0.1 0.001 0.98
060801 0.8 $\pm 0.1$ 0.1
061201 0.11 3.3 $\pm$ 0.3 0.7 10 0.02 24
061217 0.827 0.46 $\pm$ 0.08 2.4 0.1 0.005 4.9
070429B 0.904 0.63 $\pm 0.1$ 3.5 0.11 0.006 5.9
070724 0.45 0.3 $\pm 0.2$ 0.6 0.05 0.0012 1.2
070729 0.904 1.0 $\pm$ 0.2 5.6 0.024 0.001 0.98
070809 1.0 0.179
071112B 0.48 $<0.02$
\[t:fit\]
-------------- --------- ---------------------- ------------------- --------------------------- --------------------- ---------------------------
: Properties of SHB prompt and afterglow emission as detected by Swift X-ray telescope and HETE-2 (indicated with the \*); the + indicates that they could be detected by the WFC. The X-ray flux is estimated at 60-100 sec after the burst and is given in the 0.3-10 keV energy range. In the last column we report what the X-ray flux would be if the SHB were at a distance of LIGO (advanced version) detectability (300 Mpc if SHBs come from NS-NS mergers).
{width="1.0\columnwidth"}
{width="1.0\columnwidth"}
{width="0.7\columnwidth"} {width="0.7\columnwidth"}
[^1]: The differences between the progenitors of short and long GRB, might, [*a priori*]{}, be suspected of giving rise to different types of central engines; our point is simply that such differences do not appear to be crucial or necessary for understanding the systematic differences between short and long bursts. Similarly, the differences in the host environment of the central engines might give rise to differences in the collimation of short and long bursts, and this might indeed have some observational consequences, but neither is this the main point of this letter. Here we merely note that the observed differences can be accounted for by different viewing angle.
|
---
abstract: 'The belief revision literature has largely focussed on the issue of how to revise one’s beliefs in the light of information regarding matters of fact. Here we turn to an important but comparatively neglected issue: How might one extend a revision operator to handle [*conditionals*]{} as input? Our approach to this question of ‘conditional revision’ is distinctive insofar as it abstracts from the controversial details of how to revise by factual sentences. We introduce a ‘plug and play’ method for uniquely extending any iterated belief revision operator to the conditional case. The flexibility of our approach is achieved by having the result of a conditional revision by a Ramsey Test conditional (‘arrow’) determined by that of a plain revision by its corresponding material conditional (‘hook’). It is shown to satisfy a number of new constraints that are of independent interest.'
author:
- Jake Chandler$^1$
- |
Richard Booth$^2$\
$^1$La Trobe University\
$^2$Cardiff University\
jacob.chandler@latrobe.edu.au, boothr2@cardiff.ac.uk
bibliography:
- 'RBC.bib'
title: |
Revision by Conditionals:\
From Hook to Arrow
---
Introduction
============
The past three decades have witnessed the development of a substantial, if inconclusive, body of work devoted to the issue of [*belief revision*]{}, namely
- determining the impact of a local change in belief on both (i) the remainder of one’s prior beliefs and (ii) one’s prior [*conditional beliefs*]{} (‘Ramsey Test conditionals’).
Surprisingly, however, very little has been done to this date on the question of [*conditional belief revision*]{}, that is
- determining the impact of a local change in conditional beliefs on both (i) and (ii).
Furthermore, nearly all of the few proposals to tackle issue (B), namely [@Hansson1992-HANIDO-3], [@DBLP:conf/aaai/BoutilierG93], and [@DBLP:conf/ecai/NayakPFP96], have typically rested on somewhat contentious assumptions about how to approach (A). (A noteworthy exception to this [@DBLP:conf/ijcai/Kern-Isberner99], who introduced a number of plausible general postulates governing revision by conditionals whose impact on revision simpliciter remains fairly modest. More on these below.)
In this paper, we consider the prospects of providing a ‘plug and play’ solution to issue (B) that is independent of the details of how to address (A). Its remainder is organised as follows. First, in Section \[s:Revision\], we present some standard background on problem (A), introducing along the way the well-known notion of a Ramsey Test conditional or again conditional belief. In Section \[s:CRevision\], we outline and discuss our proposal regarding (B). Subsection \[ss:Aproposal\] presents the basic idea, according to which computing the result of a revision by a Ramsey Test conditional can be derived by minimal modification, under constraints, of the outcome of a revision by its corresponding material conditional. Our key technical contribution is presented in Subsection \[ss:RResult\], where we prove that this minimal change under constraints can be achieved by means of a simple and familiar transformation. Subsection \[ss:AltChar\] outlines some interesting general properties of the proposal. These strengthen, in a plausible manner, the aforementioned constraints presented in [@DBLP:conf/ijcai/Kern-Isberner99] and are of independent interest. Subsection \[ss:Elementary\] considers the upshot of pairing our proposal regarding (B) with some well-known suggestions regarding how to tackle (A). Finally, in Section \[s:RResearch\], we compare the suggestion made with existing work on the topic noting some important shortcomings of the latter. We close the paper in Section \[s:CComments\] with a number of questions for future research. The proofs of the various propositions and theorems have been relegated to a technical appendix.
Revision {#s:Revision}
========
The beliefs of an agent are represented by a [*belief state*]{}. Such states will be denoted by upper case Greek letters $\Psi, \Theta,\ldots$. We denote by $\mathbb{S}$ the set of all such states. Each state determines a [*belief set*]{}, a consistent and deductively closed set of sentences, drawn from a finitely generated propositional, truth-functional language $L$, equipped with the standard connectives ${\supset}$, $\wedge$, $\vee$, and $\neg$. We denote the belief set associated with state $\Psi$ by ${[\Psi]}$. Logical equivalence is denoted by $\equiv$ and the set of classical logical consequences of $\Gamma\subseteq L$ by $\mathrm{Cn}(\Gamma)$, with $\top$ denoting an arbitrary propositional tautology. The set of propositional worlds will be denoted by $W$ and the set of models of a given sentence $A$ by ${[\![A]\!]}$.
The operation of [*revision*]{} $\ast$ returns the posterior state $\Psi \ast A$ that results from an adjustment of $\Psi$ to accommodate the inclusion of the consistent input $A$ in its associated belief set, in such a way as to maintain consistency of the resulting belief set.
The beliefs resulting from single revisions are conveniently representable by a [*conditional belief set*]{} ${[\Psi]_{\mathrm{c}}}$, which can be viewed as encoding the agent’s rules of inference over $L$ in state $\Psi$. It is defined via the Ramsey Test:
BLAH:= [$(\mathrm{RT})$]{} For all $A, B \in L$, $A \shortTo B \in {[\Psi]_{\mathrm{c}}}$ iff $B \in {[\Psi \ast A]}$\
We shall call $L_c$ the minimal extension of $L$ that additionally includes all sentences of the form $A \shortTo B$, with $A, B \in L$. We shall call sentences of the form $A\shortTo B$ ‘conditionals’ and sentences of the form $A{\supset}B$ ‘material conditionals’. We shall say that a sentence of the form $A\shortTo B$ is consistent just in case $A\wedge B$ is consistent (later in the paper, we shall explicitly disallow revisions by inconsistent conditionals).
Conditional belief sets are constrained by the AGM postulates of [@alchourron1985logic; @darwiche1997logic] (henceforth ‘AGM’). Given these, ${[\Psi]_{\mathrm{c}}}$ corresponds to a consistency-preserving [*rational consequence relation*]{}, in the sense of [@lehmann1992does]. Equivalently, it is representable by a [*total preorder (TPO)*]{} $\preccurlyeq_{\Psi}$ of worlds, such that $A\shortTo B\in {[\Psi]_{\mathrm{c}}}$ iff $\min(\preccurlyeq_{\Psi}, \llbracket A\rrbracket)\subseteq \llbracket B\rrbracket$ [@grove1988two; @katsuno1991propositional]. Note that $A\in{[\Psi]}$ iff $\top\shortTo A\in{[\Psi]_{\mathrm{c}}}$ or equivalently iff $\min(\preccurlyeq_{\Psi}, W)\subseteq {[\![A]\!]}$.
Following convention, we shall call principles presented in terms of belief sets ‘syntactic’, and call ‘semantic’ those principles couched in terms of TPOs, denoting the latter by subscripting the corresponding syntactic principle with ‘$\preccurlyeq$’. Due to space considerations and for ease of exposition, we will largely restrict our focus to a semantic perspective on our problem of interest.
The AGM postulates do not entail that one’s conditional beliefs are determined by one’s beliefs—in the sense that, if ${[\Psi]}={[\Theta]}$, then ${[\Psi]_{\mathrm{c}}}={[\Theta]_{\mathrm{c}}}$—and there is widespread consensus that such determination would be unduly restrictive, with [@Hansson1992-HANIDO-3] providing supporting arguments. A fortiori, one should not identify conditional beliefs with beliefs in the corresponding material conditional. That said, there does remain a connection between $A \shortTo B \in {[\Psi]_{\mathrm{c}}}$ and $A {\supset}B \in {[\Psi]}$. The following is well known:
[prop]{}[RamseyMat]{} \[RamseyMat\] Given AGM, (a) if $A \shortTo B \in {[\Psi]_{\mathrm{c}}}$, then $A {\supset}B \in {[\Psi]}$, but (b) the converse does not hold.
Indeed, (a) is simply equivalent, given [$(\mathrm{RT})$]{}, to the AGM postulate of Inclusion, according to which ${[\Psi\ast A]}\subseteq {\mathrm{Cn}}({[\Psi]}\cup\{A\})$. This suggests the following catchline:
- ‘Conditional beliefs are beliefs in material conditionals plus’
That is, conditional beliefs are beliefs in material conditionals that satisfy certain additional constraints.
Regarding the conditional beliefs resulting from single revisions, i.e. the beliefs resulting from sequences of two revisions, we assume an ‘irrelevance of syntax’ property, which, in its semantic form, is given by:
BLAHI: = (Eq$^*_\preccurlyeq$) If $A\equiv B$, then $\preccurlyeq_{\Psi\ast A}=\preccurlyeq_{\Psi\ast B}$\
Given this principle, we take the liberty to abuse both language and notation and occasionally speak of revision by a set of worlds $S$ rather than by an arbitrary sentence whose set of models is given by $S$.
The DP postulates of [@darwiche1997logic] provide widely endorsed further constraints. We simply give them here in their semantic form:
BLAHHH:= [$(\mathrm{C}{1}^{\ast}_{\scriptscriptstyle \preccurlyeq})$]{} If $x,y \in {[\![A]\!]}$ then $x \preccurlyeq_{\Psi\ast A} y$ iff $x \preccurlyeq_{\Psi} y$\
\[0.1cm\] [$(\mathrm{C}{2}^{\ast}_{\scriptscriptstyle \preccurlyeq})$]{} If $x,y \in {[\![\neg A]\!]}$ then $x \preccurlyeq_{\Psi\ast A} y$ iff $x \preccurlyeq_{\Psi} y$\
\[0.1cm\] [$(\mathrm{C}{3}^{\ast}_{\scriptscriptstyle \preccurlyeq})$]{} If $x \in {[\![A]\!]}$, $y \in {[\![\neg A]\!]}$ and $x \prec_{\Psi} y$, then\
$x \prec_{\Psi\ast A} y$\
\[0.1cm\] [$(\mathrm{C}{4}^{\ast}_{\scriptscriptstyle \preccurlyeq})$]{} If $x \in {[\![A]\!]}$, $y \in {[\![\neg A]\!]}$ and $x \preccurlyeq_{\Psi} y$, then\
$x \preccurlyeq_{\Psi\ast A} y$\
Importantly, while there appears to be a degree of consensus that these postulates should be strengthened, there is no agreement as to how this should be done. Popular options include the principles respectively associated with the operators of natural revision ${\ast_{\mathrm{N}}}$ [@boutilier1996iterated], restrained revision ${\ast_{\mathrm{R}}}$ [@booth2006admissible] and lexicographic revision ${\ast_{\mathrm{L}}}$ [@nayak2003dynamic], semantically defined as follows:
[defo]{}[elemdef]{} \[elemdef\] The operators ${\ast_{\mathrm{N}}}$, ${\ast_{\mathrm{R}}}$ and ${\ast_{\mathrm{L}}}$ are such that:
- $x \preccurlyeq_{\Psi {\ast_{\mathrm{N}}}A} y$ iff (1) $x \in \min(\preccurlyeq_{\Psi}, {[\![A]\!]})$, or (2) $x, y \notin \min(\preccurlyeq_{\Psi}, {[\![A]\!]})$ and $x \preccurlyeq_{\Psi} y$
- $x \preccurlyeq_{\Psi {\ast_{\mathrm{R}}}A} y$ iff (1) $x \in \min(\preccurlyeq_{\Psi}, {[\![A]\!]})$, or (2) $x, y \notin \min(\preccurlyeq_{\Psi}, {[\![A]\!]})$ and either (a) $x \prec_{\Psi} y$ or (b) $x \sim_{\Psi} y$ and ($x\in{[\![A]\!]}$ or $y\in{[\![\neg A]\!]}$)
- $x \preccurlyeq_{\Psi {\ast_{\mathrm{L}}}A} y$ iff (1) $x\in{[\![A]\!]}$ and $y\in{[\![\neg A]\!]}$, or (2) ($x\in{[\![A]\!]}$ iff $y\in{[\![A]\!]}$) and $x \preccurlyeq_{\Psi} y$.
The suitability of all three operators, which we will group here under the heading of ‘[*elementary*]{} revision operators’ [@DBLP:conf/lori/Chandler019], has been called into question. Indeed, they assume that a state $\Psi$ can be identified with its corresponding TPO $\preccurlyeq_{\Psi}$ and that belief revision functions map pairs of TPOs and sentences onto TPOs. (For this reason, we will sometimes abuse language and notation and speak, for instance, of the lexicographic revision of a TPO rather than of a state.) But this assumption has been criticised as implausible, with [@DBLP:journals/jphil/BoothC17] providing a number of counterexamples.
Accordingly, [@DBLP:conf/kr/0001C18] propose a strengthening of the DP postulates that is weak enough to avoid an identification of states with TPOs and is consistent with the characteristic postulates of both ${\ast_{\mathrm{R}}}$ and ${\ast_{\mathrm{L}}}$ (albeit not of ${\ast_{\mathrm{N}}}$). They suggest associating states with structures that are richer than TPOs: ‘proper ordinal interval (POI) assignments’.
Conditional revision {#s:CRevision}
====================
We now turn to our question of interest: How might one extend a revision operator to handle conditionals as inputs? We shall call such an extended operator, which maps pairs of states and consistent sentences in $L_c$ onto states, a [*conditional revision*]{} operator.
In view of the considerable disagreement regarding revision that we noted in the previous section, it would be desirable to find a solution that abstracts from some of the details regarding how this problem is handled. In what follows, we shall propose a method that achieves just this. The idea that we will exploit is that the result of a conditional revision by a Ramsey Test conditional is determined by that of a plain revision by its corresponding material conditional. More specifically, we will be suggesting the following kind of procedure for constructing $\preccurlyeq_{\Psi\ast A \shortTo B}$:
- \(1) Determine $\preccurlyeq_{\Psi\ast A {\supset}B}$.
- \(2) Remain as ‘close’ to this TPO as possible, while:
- \(a) ensuring that $A\shortTo B\in{[\Psi\ast A\shortTo B]_{\mathrm{c}}}$, and
- \(b) retaining some of $\preccurlyeq_{\Psi\ast A {\supset}B}$’s relevant features.
Our proposal then is to derive $\preccurlyeq_{\Psi\ast A \shortTo B}$ from $\preccurlyeq_{\Psi\ast A {\supset}B}$, via distance minimisation under constraints. Importantly, this suggestion [*does not tie us to any particular revision operator*]{}, since it takes $\preccurlyeq_{\Psi\ast A {\supset}B}$ as its starting point, irrespective of how it is arrived at.
Distance-minimisation under constraints {#ss:Aproposal}
---------------------------------------
In an early paper on conditional revision, [@DBLP:conf/ecai/NayakPFP96] suggest that the task of conditional revision is no different from that of revision by the corresponding material conditional. Indeed, they note that, on their view of rational revision, whereby they identify $\ast$ with lexicographic revision ${\ast_{\mathrm{L}}}$, revision by the material conditional is sufficient to ensure that the corresponding conditional is included in the resulting conditional belief set. In other words, identifying $\ast A\shortTo B$ with ${\ast_{\mathrm{L}}}A{\supset}B$ is sufficient to secure the following desirable property of ‘[*Success*]{}’ for conditional revisions:
BLAHHHI =
[$(\mathrm{S}^{\ast}_{\scriptscriptstyle \preccurlyeq})$]{} $\min(\preccurlyeq_{\Psi\ast A\shortTo B}, {[\![A]\!]})\subseteq {[\![B]\!]}$\
$A \shortTo B \in {[\Psi\ast A \shortTo B]_{\mathrm{c}}}$\
Since we have, in Section \[s:Revision\], rejected identifying rational revision with lexicographic revision, Nayak [*et al’s*]{} proposal is not on the cards for us. But one might still wonder whether there exists a more acceptable conception of iterated revision that, like lexicographic revision, allows us to meet the requirement of Success by simply revising by the material conditional. But it is easy to find counterexamples to the inclusion $\min(\preccurlyeq_{\Psi\ast A{\supset}B}, {[\![A]\!]})\subseteq {[\![B]\!]}$ for the best known strengthenings of the DP postulates (Figure \[fig:NoSuccess\] provides a case in point for restrained revision). In fact, we can easily show that, given mild conditions, lexicographic revision is the only operator that fits the bill:
[prop]{}[OnlyLex]{} \[OnlyLex\] If $\ast$ satisfies AGM, [$(\mathrm{C}{1}^{\ast}_{\scriptscriptstyle \preccurlyeq})$]{}, [$(\mathrm{C}{2}^{\ast}_{\scriptscriptstyle \preccurlyeq})$]{}, $(\mathrm{Eq}^\ast_\preccurlyeq)$, and the principle according to which, for all $A, B\in L$ and $\Psi\in\mathcal{S}$, $A \shortTo B \in {[\Psi\ast A {\supset}B]_{\mathrm{c}}}$, then $\ast ={\ast_{\mathrm{L}}}$.
\(K) [ ]{};
(astR) [ ]{};
\(K) edge node\[anchor=north, above\] [ ]{} (astR) ;
Short of endorsing lexicographic revision, then, revision by the corresponding material conditional is not sufficient for the inclusion of a conditional in the resulting belief set. So just as conditional beliefs can be viewed as ‘beliefs in material conditionals plus’, we could say that:
- ‘Conditional revision is revision by material conditionals plus.’
How, then, might we plausibly modify $\preccurlyeq_{\Psi\ast A{\supset}B}$ so as to arrive at a TPO $\preccurlyeq_{\Psi\ast A\shortTo B}$ that satisfies [$(\mathrm{S}^{\ast}_{\scriptscriptstyle \preccurlyeq})$]{}?
Satisfaction of this principle, of course, will require some worlds in ${[\![A\wedge B]\!]}$ to be promoted in the ranking, notably in relation to certain worlds in ${[\![A\wedge \neg B]\!]}$. But we must be cautious as to how this is to take place. Plausibly, for instance, it should not occur at the expense of the worlds in ${[\![\neg A]\!]}$. In fact, it seems quite reasonable that, more broadly, the internal ordering of ${[\![A{\supset}B]\!]}$ should be left untouched. We therefore suggest supplementing [$(\mathrm{S}^{\ast}_{\scriptscriptstyle \preccurlyeq})$]{} with the following ‘retainment’ principle, which ensures the preservation of these features of $\preccurlyeq_{\Psi\ast A{\supset}B}$:
BLAHHHI = [$(\mathrm{Ret}{1}^{\ast}_{\preccurlyeq})$]{} If $x, y \in {[\![A{\supset}B]\!]}$, then $x \preccurlyeq_{\Psi\ast A\shortTo B}y$ iff\
$x \preccurlyeq_{\Psi\ast A{\supset}B}y$\
Its syntactic counterpart is given as follows:
[prop]{}[SyntacticPprincOne]{} \[SyntacticPprincOne\] Given AGM, [$(\mathrm{Ret}{1}^{\ast}_{\preccurlyeq})$]{} is equivalent to
BLAHHHI = [$(\mathrm{Ret}{1})$]{} If $A{\supset}B\in{\mathrm{Cn}}(C)$, then ${[(\Psi\ast A\shortTo B)\ast C]}=$\
${[(\Psi\ast A{\supset}B)\ast C]}$\
Given the DP postulates, this constraint obviously translates into one that connects $\preccurlyeq_{\Psi} $ and $\preccurlyeq_{\Psi\ast A\shortTo B}$ and whose syntactic counterpart is easily inferable from Proposition \[SyntacticPprincOne\]:
[prop]{}[PropertiesTwoA]{} \[PropertiesTwoA\] Given [$(\mathrm{C}{1}^{\ast}_{\scriptscriptstyle \preccurlyeq})$]{}, [$(\mathrm{Ret}{1}^{\ast}_{\preccurlyeq})$]{} is equivalent to:
BLAHHHI = [$(\mathrm{Ret}{1'}^{\ast}_{\preccurlyeq})$]{} If $x, y \in {[\![A{\supset}B]\!]}$, then $x \preccurlyeq_{\Psi\ast A\shortTo B}y$ iff\
$x \preccurlyeq_{\Psi} y$\
That conditional revision does not affect the internal ordering of ${[\![A\wedge B]\!]}$ or of ${[\![\neg A]\!]}$ is in fact required by a set of principles for conditional revision proposed in [@DBLP:conf/ijcai/Kern-Isberner99], to which we shall return later. Our principle adds to these the constraint that conditional revision by $A\shortTo B$ does not affect the relative standing of worlds in ${[\![\neg A]\!]}$ in relation to worlds in ${[\![A\wedge B]\!]}$. This further restriction yields the correct verdict in the following scenario:
\[ex:AkiraOne\] My friend and I have taken our preschoolers Akira and Bashir on holiday. They slept in bunkbeds last night. Since both beds were unmade by the morning, I initially believe that they did not choose to sleep in the same bed but suspend judgment as to which respective beds they did choose. Furthermore, in the event of coming to believe that they in fact did decide to share a bed, I would suspend judgment as to which bed they opted for. I then find out that, if Akira slept on top, then Bashir would have done so too (because he does not like people sleeping above him). What changes? Plausibly, my beliefs will change in the following respect: since I will still believe that they did not share a bed, I will now infer that Akira slept on the bottom bed and Bashir on the top. What of my conditional beliefs? Plausibly, we will have the following continuity: It will remain the case that, were to find out that they in fact decided to share a bed, I would suspend judgment as to which bed they chose.
Indeed, let $A$ and $B$ respectively stand for Akira and for Bashir’s sleeping on the top bed and $\Psi$ be my initial state. Assume for simplicity that the set of atomic propositions in $L$ is $\{A, B\}$. Let ${[\![A\wedge\neg B]\!]}=\{x\}$, ${[\![\neg A\wedge B]\!]}=\{y\}$, ${[\![A\wedge B]\!]}=\{z\}$ and ${[\![\neg A\wedge \neg B]\!]}=\{w\}$. We then have $\preccurlyeq_{\Psi}$ plausibly given by $x \sim_{\Psi} y \prec_{\Psi} z \sim_{\Psi} w$. Since $z\sim_{\Psi} w$, our principle entails the plausible result that $z\sim_{\Psi \ast A\shortTo B} w$.
But unfortunately, [$(\mathrm{S}^{\ast}_{\scriptscriptstyle \preccurlyeq})$]{} and [$(\mathrm{Ret}{1}^{\ast}_{\preccurlyeq})$]{} are not generally jointly sufficient to have the TPO $\preccurlyeq_{\Psi\ast A{\supset}B}$ determine the TPO $\preccurlyeq_{\Psi\ast A\shortTo B}$. Our suggestion is to close the gap by means of distance minimisation. More specifically, we propose to consider the closest TPO that satisfies–or, in the event of a tie, some aggregation of the closest TPOs that satisfy–our two constraints.
In terms of measuring the distance between TPOs, a natural choice is the so-called Kemeny distance $d_K$:
[defo]{}[SymDiff]{} \[SymDiff\] $d_K(\preccurlyeq, \preccurlyeq'):= \lvert (\preccurlyeq - \preccurlyeq') \cup (\preccurlyeq' - \preccurlyeq) \rvert$.
Informally, $d_K(\preccurlyeq, \preccurlyeq')$ returns the number of disagreements over relations of weak preference between the two orderings, returning the number of pairs that are in $\preccurlyeq$ but not in $\preccurlyeq'$ and vice versa. This measure is standard fare in the social choice literature. It was introduced there in [@10.2307/20026529] and received an axiomatisation in terms of a set of prima facie attractive properties in [@KemenySnell62].
In the section that follows we shall show that there exists a unique $d_K$-closest TPO that meets the requirements [$(\mathrm{S}^{\ast}_{\scriptscriptstyle \preccurlyeq})$]{} and [$(\mathrm{Ret}{1}^{\ast}_{\preccurlyeq})$]{}, which can be obtained from $\preccurlyeq_{\Psi\ast A {\supset}B}$ in a simple and familiar manner.
A construction of the posterior TPO {#ss:RResult}
-----------------------------------
To outline our main result, we first need the following item of notation (see Figure \[fig:Downset\] for illustration):
[defo]{}[Downset]{} \[Downset\] For any sentence $A\in L$ and TPO $\preccurlyeq$, we denote by $D(\preccurlyeq, A)$ the [*down-set*]{} of the members of $\min(\preccurlyeq, {[\![A]\!]})$. It is given by $D(\preccurlyeq, A):=\{x\mid x\preccurlyeq z \mathrm{,~for~some~} z\in\min(\preccurlyeq, {[\![A]\!]})\}$.
With this in hand, we propose:
[defo]{}[MyProposal]{} \[MyProposal\] Let $\ast$ be a function from $\mathbb{S}\times L$ to $\mathbb{S}$. Then we denote by $\circledast$ an arbitrary extension of $\ast$ to the domain $\mathbb{S}\times L_c$, such that $\preccurlyeq_{\Psi \circledast A\shortTo B}$ is given by the lexicographic revision of $\preccurlyeq_{\Psi \ast A{\supset}B}$ by $D(\preccurlyeq_{\Psi\ast A{\supset}B}, A\wedge B)\cap{[\![A{\supset}B]\!]}$.[^1]
The operator so-defined is illustrated in Figure \[fig:RevCon\], which depicts the resulting relation between $\preccurlyeq_{\Psi\ast A{\supset}B}$ and $\preccurlyeq_{\Psi\circledast A\shortTo B}$. Interestingly, in the special case of a Ramsey Test conditional with a tautologous antecedent, this transformation of $\preccurlyeq_{\Psi\ast A{\supset}B}$ amounts to its natural revision by the consequent.
We propose to identify $\preccurlyeq_{\Psi\ast A\shortTo B}$ with $\preccurlyeq_{\Psi\circledast A\shortTo B}$. We do so on the basis of our main technical result, which is:
[thm]{}[Representation]{} \[Representation\] The unique TPO that minimises the distance $d_K$ to $\preccurlyeq_{\Psi\ast A{\supset}B}$, given constraints [$(\mathrm{S}^{\ast}_{\scriptscriptstyle \preccurlyeq})$]{} and [$(\mathrm{Ret}{1}^{\ast}_{\preccurlyeq})$]{} is given by $\preccurlyeq_{\Psi\circledast A\shortTo B}$.
As indicated above, short of endorsing lexicographic revision, which we do not want to do, the constraint of Success prevents us from having ${[\Psi\ast A\shortTo B]_{\mathrm{c}}} = {[\Psi\ast A{\supset}B]_{\mathrm{c}}}$ for all $A, B\in L$, $\Psi\in\mathcal{S}$. Having said that, a restricted version of this equality does hold for our proposal in the form of the following plausible ‘Vacuity’ postulate, which tells us that if revision by the material conditional leads to the conditional being accepted, then it is revision enough:
BLAHHHI = $(\mathrm{V}^\ast)$ If $A\shortTo B\in{[\Psi\ast A{\supset}B]_{\mathrm{c}}}$, then\
${[\Psi\ast A\shortTo B]_{\mathrm{c}}}={[\Psi\ast A{\supset}B]_{\mathrm{c}}}$\
Furthermore, as a consequence of one of the results established in the proof of Theorem \[Representation\], we can also derive an interesting minimal change result with a more syntactic flavour:
[prop]{}[SyntacticMinimality]{} \[SyntacticMinimality\] Let $\ast$ be a function from $\mathbb{S}\times L$ to $\mathbb{S}$ and $\ast'$ an extension of $\ast$ to the domain $\mathbb{S}\times L_c$, satisfying [$(\mathrm{S}^{\ast}_{\scriptscriptstyle \preccurlyeq})$]{} and [$(\mathrm{Ret}{1}^{\ast}_{\preccurlyeq})$]{}. Then, if ${[\Psi\ast' A\shortTo B]_{\mathrm{c}}}$ agrees with ${[\Psi\ast A{\supset}B]_{\mathrm{c}}}$ on all conditionals with a given antecedent $C$, so does ${[\Psi\circledast A\shortTo B]_{\mathrm{c}}}$
Some general features {#ss:AltChar}
---------------------
We have seen that our proposal to handle conditional revision using distance minimisation under constraints yields a unique TPO that can be obtained via lexicographic revision of $\preccurlyeq_{\Psi\ast A{\supset}B}$ by a particular proposition. In this section, we discuss some of its general consequences, including three additional retainment principles that it implies.
It is easy to establish the following:
[prop]{}[RetSoundness]{} \[RetSoundness\] Let $\ast$ be a function from $\mathbb{S}\times L_c$ to $\mathbb{S}$. Then, if $\ast=\circledast$, then $\ast$ satisfies:
BLAHHHI = [$(\mathrm{Ret}{2}^{\ast}_{\preccurlyeq})$]{} If $x, y \in {[\![A\wedge \neg B]\!]}$, then $x \preccurlyeq_{\Psi\ast A\shortTo B}y$ iff\
$x \preccurlyeq_{\Psi\ast A{\supset}B}y$\
[$(\mathrm{Ret}{3}^{\ast}_{\preccurlyeq})$]{} If $x\in {[\![A{\supset}B]\!]}$, $y\in {[\![A\wedge \neg B]\!]}$, and\
$x \prec_{\Psi\ast A{\supset}B}y$, then $x \prec_{\Psi\ast A\shortTo B}y$\
[$(\mathrm{Ret}{4}^{\ast}_{\preccurlyeq})$]{} If $x\in {[\![A{\supset}B]\!]}$, $y\in {[\![A\wedge \neg B]\!]}$, and\
$x \preccurlyeq_{\Psi\ast A{\supset}B}y$, then $x \preccurlyeq_{\Psi\ast A\shortTo B}y$\
The conjunction of [$(\mathrm{Ret}{1}^{\ast}_{\preccurlyeq})$]{}, with these three principles simply tells us that the only admissible transformations, when moving from $\preccurlyeq_{\Psi\ast A{\supset}B}$ to $\preccurlyeq_{\Psi\ast A\shortTo B}$, involve a doxastic ‘demotion’ of worlds in ${[\![A\wedge \neg B]\!]}$ in relation to worlds in ${[\![A{\supset}B]\!]}$, raising, in the ordering, the position of the former in relation to the latter. They have a similar flavour to that of the DP postulates, which tell us that that the only admissible transformations, when moving from $\preccurlyeq_{\Psi}$ to $\preccurlyeq_{\Psi\ast A}$, involve a demotion of worlds in ${[\![\neg A]\!]}$ in relation to worlds in ${[\![A]\!]}$.
We note the immediate implications of these principles, in the presence of the DP postulates:
[prop]{}[PropertiesTwo]{} \[PropertiesTwo\] Given [$(\mathrm{C}{1}^{\ast}_{\scriptscriptstyle \preccurlyeq})$]{}–[$(\mathrm{C}{4}^{\ast}_{\scriptscriptstyle \preccurlyeq})$]{}, [$(\mathrm{Ret}{2}^{\ast}_{\preccurlyeq})$]{} holds iff:
BLAHHHI = [$(\mathrm{Ret}{2'}^{\ast}_{\preccurlyeq})$]{} If $x, y \in {[\![A\wedge \neg B]\!]}$, $x \preccurlyeq_{\Psi\ast A\shortTo B}y$ iff $x \preccurlyeq_{\Psi} y$\
and [$(\mathrm{Ret}{3}^{\ast}_{\preccurlyeq})$]{} and [$(\mathrm{Ret}{4}^{\ast}_{\preccurlyeq})$]{} respectively entail:
BLAHHHI = [$(\mathrm{Ret}{3'}^{\ast}_{\preccurlyeq})$]{} If $x\in {[\![A{\supset}B]\!]}$, $y\in {[\![A\wedge \neg B]\!]}$, and $x \prec_{\Psi} y$,\
then $x \prec_{\Psi\ast A\shortTo B}y$\
[$(\mathrm{Ret}{4'}^{\ast}_{\preccurlyeq})$]{} If $x\in {[\![A{\supset}B]\!]}$, $y\in {[\![A\wedge \neg B]\!]}$, and $x \preccurlyeq_{\Psi} y$,\
then $x \preccurlyeq_{\Psi\ast A\shortTo B}y$\
but the converse entailments do not hold.
The syntactic counterparts of [$(\mathrm{Ret}{2}^{\ast}_{\preccurlyeq})$]{}–[$(\mathrm{Ret}{4}^{\ast}_{\preccurlyeq})$]{} are given in the following proposition, with the counterparts of [$(\mathrm{Ret}{2'}^{\ast}_{\preccurlyeq})$]{}–[$(\mathrm{Ret}{4'}^{\ast}_{\preccurlyeq})$]{} being easily inferable from these:
[prop]{}[SyntacticPprinc]{} \[SyntacticPprinc\] Given AGM, [$(\mathrm{Ret}{2}^{\ast}_{\preccurlyeq})$]{}–[$(\mathrm{Ret}{4}^{\ast}_{\preccurlyeq})$]{} are respectively equivalent to
BLAHHHI = [$(\mathrm{Ret}{2})$]{} If $A\wedge \neg B\in{\mathrm{Cn}}(C)$, then ${[(\Psi\ast A\shortTo B)\ast C]}=$\
${[(\Psi\ast A{\supset}B)\ast C]}$\
[$(\mathrm{Ret}{3})$]{} If $A{\supset}B\in{[(\Psi\ast A{\supset}B)\ast C]}$, then $A{\supset}B\in$\
${[(\Psi\ast A\shortTo B)\ast C]}$\
[$(\mathrm{Ret}{4})$]{} If $A\wedge\neg B\notin{[(\Psi\ast A{\supset}B)\ast C]}$, then $A\wedge\neg B\notin$\
${[(\Psi\ast A\shortTo B)\ast C]}$\
In introducing [$(\mathrm{Ret}{1})$]{} above, we noted that, given [$(\mathrm{C}{1}^{\ast}_{\scriptscriptstyle \preccurlyeq})$]{}, it strengthens, in a plausible manner, part of a principle proposed in [@DBLP:conf/ijcai/Kern-Isberner99]. It turns out that, in the presence of the full set of DP postulates, [$(\mathrm{Ret}{1})$]{}–[$(\mathrm{Ret}{4})$]{} enable us to recover the trio of principles proposed by Kern-Isberner. These “KI postulates”, originally named “(CR5)” to“(CR7)”, are given semantically by:
BLAHHHI = If $x,y\in {[\![A\wedge B]\!]}$, $x,y\in {[\![\neg A]\!]}$\
or $x,y\in {[\![A\wedge \neg B]\!]}$, then $x \preccurlyeq_{\Psi} y$ iff\
$x \preccurlyeq_{\Psi\ast A\shortTo B}y$\
If $x\in {[\![A\wedge B]\!]}$, $y\in {[\![A\wedge \neg B]\!]}$ and $x \prec_{\Psi} y$,\
then $x \prec_{\Psi\ast A\shortTo B}y$\
If $x\in {[\![A\wedge B]\!]}$, $y\in {[\![A\wedge \neg B]\!]}$ and $x \preccurlyeq_{\Psi} y$,\
then $x \preccurlyeq_{\Psi\ast A\shortTo B}y$\
We can see that [$(\mathrm{KI}{1}^{\ast}_{\preccurlyeq})$]{} follows from [$(\mathrm{Ret}{1}^{\ast}_{\preccurlyeq})$]{} and [$(\mathrm{Ret}{2}^{\ast}_{\preccurlyeq})$]{}, given [$(\mathrm{C}{1}^{\ast}_{\scriptscriptstyle \preccurlyeq})$]{} and [$(\mathrm{C}{2}^{\ast}_{\scriptscriptstyle \preccurlyeq})$]{}. [$(\mathrm{KI}{2}^{\ast}_{\preccurlyeq})$]{} follows from the conjunction of [$(\mathrm{Ret}{3}^{\ast}_{\preccurlyeq})$]{} and [$(\mathrm{C}{3}^{\ast}_{\scriptscriptstyle \preccurlyeq})$]{}, while [$(\mathrm{KI}{3}^{\ast}_{\preccurlyeq})$]{} follows from the conjunction of [$(\mathrm{Ret}{4}^{\ast}_{\preccurlyeq})$]{} and [$(\mathrm{C}{4}^{\ast}_{\scriptscriptstyle \preccurlyeq})$]{}.[^2]
In view of Theorem \[Representation\] and Proposition \[RetSoundness\], it follows that, if a revision operator $\ast$ satisfies [$(\mathrm{C}{1}^{\ast}_{\scriptscriptstyle \preccurlyeq})$]{} to [$(\mathrm{C}{4}^{\ast}_{\scriptscriptstyle \preccurlyeq})$]{}, then the conditional revision $\circledast$ operator that extends it in the manner described in Definition \[MyProposal\] satisfies [$(\mathrm{KI}{1}^{\ast}_{\preccurlyeq})$]{} to [$(\mathrm{KI}{3}^{\ast}_{\preccurlyeq})$]{}.
[$(\mathrm{Ret}{3}^{\ast}_{\preccurlyeq})$]{} and [$(\mathrm{Ret}{4}^{\ast}_{\preccurlyeq})$]{} tell us that conditional revision by $A\shortTo B$ preserves any ‘good news’ for worlds in ${[\![A{\supset}B]\!]}$, compared to worlds in ${[\![A\wedge \neg B]\!]}$, that revision by $A{\supset}B$ would bring. Given [$(\mathrm{C}{3}^{\ast}_{\scriptscriptstyle \preccurlyeq})$]{} and [$(\mathrm{C}{4}^{\ast}_{\scriptscriptstyle \preccurlyeq})$]{}, they notably add to [$(\mathrm{KI}{2}^{\ast}_{\preccurlyeq})$]{} and [$(\mathrm{KI}{3}^{\ast}_{\preccurlyeq})$]{} the idea that worlds in ${[\![\neg A]\!]}$ should not be demoted with respect to worlds in ${[\![A\wedge\neg B]\!]}$ in moving from $\preccurlyeq_{\Psi}$ to $\preccurlyeq_{\Psi \ast A\shortTo B}$. The appeal of this constraint is highlighted in the following case:
I am due to visit my hometown and would like to catch up with my friends Alex and Ben. Unfortunately, both of them moved away years ago and I doubt that I will see either. If I were to find out of either of them that he was going to be around, I would still believe that the other was not. Furthermore if I were to find out that exactly one of them would be back, I would not be able to guess which one of the two that would be. A friend now tells me that if Alex will be in town, then so will Ben. Very clearly, it should not be the case that, as a result of this new information, I would now take Alex to be a more plausible candidate for being the only one of my two friends that I will see (quite the contrary).
Let $A$ and $B$ respectively stand for Alex and for Ben’s being back in town and $\Psi$ be my initial state. Assume for simplicity that the set of atomic propositions in $L$ is simply $\{A,B\}$. Let ${[\![A\wedge\neg B]\!]}=\{x\}$, ${[\![\neg A\wedge B]\!]}=\{y\}$, ${[\![A\wedge B]\!]}=\{z\}$ and ${[\![\neg A\wedge \neg B]\!]}=\{w\}$. We then have $\preccurlyeq_{\Psi}$ plausibly given by $w \prec_{\Psi} x \sim_{\Psi} y \prec_{\Psi} z$. Since $ y \preccurlyeq_{\Psi} x$, our principle entails that $ y \preccurlyeq_{\Psi\ast A\shortTo B} x$, as it intuitively should be.
Aside from entailing the three further retainment principles that we have discussed, we also note that our postulates have the happy consequence of securing the following ‘Doxastic Equivalence’ principle, according to which conditional revisions are indistinguishable from revisions by material conditionals at the level of belief sets:
BLAHHHI =
[$(\mathrm{DE}^{\ast}_{\scriptscriptstyle \preccurlyeq})$]{} $\min(\preccurlyeq_{\Psi\ast A\shortTo B}, W) = \min(\preccurlyeq_{\Psi\ast A{\supset}B}, W)$\
More precisely, it is easy to show that:
[prop]{}[NoAddedFactualBeliefs]{} \[NoAddedFactualBeliefs\] [$(\mathrm{S}^{\ast}_{\scriptscriptstyle \preccurlyeq})$]{}, [$(\mathrm{Ret}{1}^{\ast}_{\preccurlyeq})$]{}, [$(\mathrm{Ret}{3}^{\ast}_{\preccurlyeq})$]{} and [$(\mathrm{Ret}{4}^{\ast}_{\preccurlyeq})$]{} collectively entail [$(\mathrm{DE}^{\ast}_{\scriptscriptstyle \preccurlyeq})$]{}.
Elementary conditional revision operators {#ss:Elementary}
-----------------------------------------
A few interesting observations can be made regarding the more specific case in which $\ast$ is an elementary operator (i.e. belongs to the set $\{{\ast_{\mathrm{N}}}, {\ast_{\mathrm{R}}}, {\ast_{\mathrm{L}}}\}$), which we illustrate in Figure \[fig:Elementary\]. Having said that, we have noted above our significant reservations about identifying rational revision with any of these operators. This section is therefore addressed to those who are rather more optimistic.
First, we note that, in two of the special cases of interest, one of our two steps becomes superfluous.
If $\ast={\ast_{\mathrm{L}}}$, then the second step of our procedure is redundant. Indeed, for any $x$ such that $x\in{[\![A\wedge B]\!]}$ and any $y\in{[\![A\wedge \neg B]\!]}$, we have $x\prec_{\Psi{\ast_{\mathrm{L}}}A{\supset}B}y$. Hence every world that is in $D(\preccurlyeq_{\Psi\ast A{\supset}B}, A\wedge B)\cap{[\![A{\supset}B]\!]}$ is already strictly more minimal, in $\preccurlyeq_{\Psi\ast A{\supset}B}$, than any world that is not.
If $\ast={\ast_{\mathrm{N}}}$, then the first step of our procedure plays no role: we would obtain the same result by simply directly applying the second transformation to the initial TPO. This is apparent from the fact that natural revision by $A{\supset}B$ leaves unaffected the respective internal orderings of $\overline{D(\preccurlyeq_{\Psi\ast A{\supset}B}, A\wedge B)}$, $D(\preccurlyeq_{\Psi\ast A{\supset}B}, A\wedge B)\cap{[\![A{\supset}B]\!]}$ and $D(\preccurlyeq_{\Psi\ast A{\supset}B}, A\wedge B)\cap{[\![A\wedge \neg B]\!]}$, while the latter, on our proposal, jointly determine $\preccurlyeq_{\Psi\ast A \shortTo B}$.
Secondly, in the case of elementary operators more generally, it can be shown that, on our proposal for $\ast A\shortTo B$, the posterior internal ordering $\preccurlyeq_{\Psi\ast A\shortTo B}\cap{[\![A]\!]}$ of the set of $A$-worlds is recovered by revising by $B$ the restriction $\preccurlyeq\cap{[\![A]\!]}$ of the prior ordering to the $A$-worlds:[^3]
[prop]{}[PropertiesThree]{} \[PropertiesThree\] If $\ast$ is an elementary revision operator, then $\ast$ and $\circledast$ satisfy: $\preccurlyeq_{\Psi\circledast A\shortTo B}\cap{[\![A]\!]}=(\preccurlyeq_{\Psi}\cap{[\![A]\!]})\ast B$.
In other words: if one disregards the worlds in which the antecedent is false, the proposed transformation amounts to revision by the consequent. Finally, in [@DBLP:conf/lori/Chandler019 Theorem 4], it was noted that there is an interesting connection between natural revision and the rational closure operator ${\mathrm{C_{rat}}}$ [@lehmann1992does Defs 20 and 21], which minimally extends any consistent set of conditionals to a set of conditionals corresponding to a rational consequence relation. This connection was that, if $\neg A\notin{[\Psi]_{\mathrm{c}}}$, then ${[\Psi{\ast_{\mathrm{N}}}A]_{\mathrm{c}}}={\mathrm{C_{rat}}}({[\Psi]_{\mathrm{c}}}\cup\{A\})$. This connection deepens on the proposed extension of natural revision to the conditional case. The proof of Chandler & Booth’s theorem can be built upon to establish the following non-trivial result:
[prop]{}[NatAndRatClosure]{} \[NatAndRatClosure\] If $\ast={\ast_{\mathrm{N}}}$, then, if $A\shortTo \neg B\notin{[\Psi]_{\mathrm{c}}}$, then ${[\Psi\circledast A\shortTo B]_{\mathrm{c}}}={\mathrm{C_{rat}}}({[\Psi]_{\mathrm{c}}}\cup\{A\shortTo B\})$
Related research {#s:RResearch}
================
We have already presented Kern-Isberner’s trio of postulates for conditional revision and briefly discussed (and rejected) Nayak [*et al*]{}’s suggestion to treat conditional revision as lexicographic revision by a material conditional. In this section we turn to two further proposals that have been made in the literature and briefly compare them to ours. As we shall see, these both commit to identifying $\ast$ with ${\ast_{\mathrm{N}}}$–which we have argued is undesirable–and exhibit further shortcomings.[^4]
Hansson
-------
[@Hansson1992-HANIDO-3] also takes a distance based approach, albeit an unconstrained one. He proposes to use the operator $\ast_{\mathrm{H}}$:
[defo]{}[HanssonRev]{} \[HanssonRev\] ${[\Psi\ast_{\mathrm{H}} A\shortTo B]_{\mathrm{c}}}:=\bigcap {[\Theta_i]_{\mathrm{c}}}$, such that the $\Theta_i$ minimise the distance to $\Psi$, subject to the constraint that $A\shortTo B\in {[\Theta_i]_{\mathrm{c}}}$.
The fate of this suggestion, of course, hinges on (i) one’s view of the nature of states and (ii) the distance metric used. But if one equates states with TPOs and measures distance by means of $d_K$, then, first of all, rational revision coincides with natural revision: $\Psi\ast \top\shortTo B = \Psi{\ast_{\mathrm{N}}}B$. Indeed:
[prop]{}[closestNat]{} \[closestNat\] Let $\star$ be a revision operator that satisfies AGM. Then, if $\preccurlyeq_{\Psi{\ast_{\mathrm{N}}}A}\neq \preccurlyeq_{\Psi\star A}$, then $d_K(\preccurlyeq_{\Psi{\ast_{\mathrm{N}}}A}, \preccurlyeq_{\Psi})< d_K(\preccurlyeq_{\Psi\star A}, \preccurlyeq_{\Psi})$.
This, we take, is already not an appealing feature. Furthermore, Hansson’s use of the intersection of a set of rational conditional belief sets should raise concerns, since it is well known that such intersections can fail to be rational. As it turns out, this worry is substantiated, and his suggestion is in fact inconsistent with at least one of the AGM postulates:
[prop]{}[HanssonProblem]{} \[HanssonProblem\] The operator $\ast_{\mathrm{H}}$ does not satisfy
BLAHHHI = [$(\mathrm{K}{8}^{\ast})$]{} If $\neg B\notin [\Psi*A]$, then $\textrm{Cn}([\Psi*A]\cup\{B\})\subseteq$\
$ [\Psi*A\wedge B]$\
An alternative way of aggregating the closest TPOs, which would guarantee an AGM-compliant output, would be to make use of an extension to the n-ary case of the binary TPO aggregation operator ${\oplus_{\mathrm{STQ}}}$ of [@DBLP:journals/ai/BoothC19]. We leave the study of this option to those who are more enthusiastic about the prospects of natural revision.
Boutilier & Goldszmidt
----------------------
[@DBLP:conf/aaai/BoutilierG93] offer an alternative extension of ${\ast_{\mathrm{N}}}$, which makes use of two further standard belief change operators: (i) the [*contraction*]{} operator ${\div}$, which returns the posterior state $\Psi {\div}A$ that results from an adjustment of $\Psi$ to accommodate the retraction of $A$ and (ii) the [*expansion*]{} operator $+$, which is similar to revision, save that consistency of the resulting beliefs needn’t be ensured.
Like ours, their proposal involves a two-stage process, this time involving a first step of contraction by $A\shortTo \neg B$, then a step of expansion by $A\shortTo B$. In the case in which $A\shortTo B$ is not initially accepted, the contraction step involves moving the minimal ${[\![A\wedge B]\!]}$ worlds down to the rank $r$ in which the minimal ${[\![A\wedge \neg B]\!]}$ worlds sit. The expansion step then has these minimal ${[\![A\wedge \neg B]\!]}$ worlds move up to a position immediately above $r$, while preserving their relations with any worlds that were strictly above or below them. Formally:
[defo]{}[BGcontract]{} \[BGcontract\] The [*Boutilier-Goldszmidt contraction operator ${{\div}_{\mathrm{BG}}}$*]{} is such that
- - If $x, y\notin \min(\preccurlyeq_{\Psi}, {[\![A\wedge B]\!]})$, then $x\preccurlyeq_{\Psi {{\div}_\mathrm{BG}}A\shortTo \neg B} y$ iff $x\preccurlyeq_{\Psi} y$, and
- If $x\in \min(\preccurlyeq_{\Psi}, {[\![A\wedge B]\!]})$, then
- $x\preccurlyeq_{\Psi {{\div}_\mathrm{BG}}A\shortTo \neg B} y$ iff $z\preccurlyeq_{\Psi} y$, for some $z\in\min(\preccurlyeq_{\Psi}, {[\![A]\!]})$, and
- $y\preccurlyeq_{\Psi {{\div}_\mathrm{BG}}A\shortTo \neg B} x$ iff $y\preccurlyeq_{\Psi} z$, for some $z\in\min(\preccurlyeq_{\Psi}, {[\![A]\!]})$
[defo]{}[BGexpand]{} \[BGexpand\] The [*Boutilier-Goldszmidt expansion operator ${+_{\mathrm{BG}}}$*]{} is such that
- - If $x\notin \min(\preccurlyeq_{\Psi}, {[\![A\wedge \neg B]\!]})$, then $x\preccurlyeq_{\Psi {+_{\mathrm{BG}}} A\shortTo B} y$ iff $x\preccurlyeq_{\Psi} y$, and
- If $x\in \min(\preccurlyeq_{\Psi}, {[\![A\wedge \neg B]\!]})$, then
- if $y\in \min(\preccurlyeq_{\Psi}, {[\![A\wedge \neg B]\!]})$, then $y\preccurlyeq_{\Psi {+_{\mathrm{BG}}} A\shortTo B} x$, and
- if $y\notin \min(\preccurlyeq_{\Psi}, {[\![A\wedge \neg B]\!]})$, then $x\preccurlyeq_{\Psi {+_{\mathrm{BG}}} A\shortTo B} y$ iff $x\preccurlyeq_{\Psi} y$ and there is no $z\in\min(\preccurlyeq_{\Psi}, {[\![A\wedge B]\!]})$ such that $y\preccurlyeq_{\Psi} z$
The corresponding revision operator is then defined as the composition of ${\div}_{\mathrm{BG}} A\shortTo \neg B$ and $+_{\mathrm{BG}} A\shortTo B$:
[defo]{}[BGrevise]{} \[BGrevise\] The [*Boutilier-Goldszmidt revision operator ${\ast_{\mathrm{BG}}}$*]{} is given by $\Psi \ast_{\mathrm{BG}} A\shortTo B := (\Psi {\div}_{\mathrm{BG}} A\shortTo \neg B) +_{\mathrm{BG}} A\shortTo B$.
The operation ${+_{\mathrm{BG}}}A\shortTo B$ bears some striking similarities to the second step in our construction of $\ast A\shortTo B$. In fact, it [*coincides*]{} with it in the kind of circumstances under which it is supposed to operate, i.e. on the heels of ${{\div}_{\mathrm{BG}}}A\shortTo \neg B$.
Having said that, the introduction of the contraction step means that, overall, Boutilier & Goldszmidt’s proposal quite clearly departs from the proposed extension of ${\ast_{\mathrm{N}}}$ put forward in the previous section. In Figure \[fig:BG\], we see that it notably violates the requirement [$(\mathrm{Ret}{1}^{\ast}_{\preccurlyeq})$]{}, according to which the ordering internal to ${[\![A{\supset}B]\!]}$ should be preserved (since, although $4, 7\in {[\![A{\supset}B]\!]}$ and $7\prec_{\Psi} 4$, we have $4\prec_{\Psi {\ast_{\mathrm{BG}}} A\shortTo B} 7$). This particular example is also an instance of the following feature of their revision operator:
[prop]{}[ModusTollens]{} \[ModusTollens\] If $A\in {[\Psi]}$, then $A\wedge B\in {[\Psi \ast_{\mathrm{BG}} A\shortTo B]}$
But this is a rather questionable property: it essentially precludes reasoning by Modus Tollens (aka denying the consequent). The following example highlights the counterintuitive character of this proscription:
I believe that the light in the bathroom next door is on ($A$), because the light switch in this room is down ($\neg B$). The owner of the house tells me that, contrary to what one might expect, when the bathroom light is on, that means that the switch in this room is [*up*]{}. So I revise by $A\shortTo B$. In doing so, I maintain my belief about the state of the switch ($\neg B$) and conclude that the bathroom light is off ($\neg A$).
Concluding comments {#s:CComments}
===================
In what precedes we have offered a fresh approach to the problem of revision by conditionals, which imposes no constraints on the behaviour of the revision operator in relation to non-conditional inputs. This independence was achieved by deriving the result of a revision by a conditional from the result of a revision by its material counterpart. This approach, we have argued, satisfies a number of attractive new properties and enjoys a number of distinctive advantages over existing alternative proposals.
Having said that, the scope of a number of results that we have established could perhaps be broadened.
Firstly, Proposition \[PropertiesThree\] shows that, at the level of the internal ordering of ${[\![A]\!]}$, conditional revision by $A\shortTo B$ operates like revision by $B$, for the special case of extensions of [*elementary*]{} revision operators. We do not know to what extent this generalises to a broader class of revision operators, such as the POI revision operators of [@DBLP:conf/kr/0001C18].
Secondly, we establish in Proposition \[NatAndRatClosure\] that, if $\ast={\ast_{\mathrm{N}}}$ and $A\shortTo \neg B\notin{[\Psi]_{\mathrm{c}}}$, then ${[\Psi\circledast A\shortTo B]_{\mathrm{c}}}={\mathrm{C_{rat}}}({[\Psi]_{\mathrm{c}}}\cup\{A\shortTo B\})$. This raises the following question: For $\ast = {\ast_{\mathrm{L}}}$ or $\ast={\ast_{\mathrm{R}}}$, if $A\shortTo \neg B\notin{[\Psi]_{\mathrm{c}}}$, do we have ${[\Psi\ast A\shortTo B]_{\mathrm{c}}}={\mathrm{C}}({[\Psi]_{\mathrm{c}}}\cup\{A\shortTo B\})$ for some suitable closure operator ${\mathrm{C}}$?
Finally, at a number of points, we have made use of the distance metric $d_K$, noting that it was ubiquitous in the social choice literature. We are however aware of at least one alternative to this metric, proposed in [@10.2307/41681743 Sec. 3.2], which coincides with $d_K$ in the special case of linear orders. It would be interesting to assess the impact of this alternative choice on the proposal made here (another potential point of relevance concerns our assessment of Hansson’s proposal, which also made use of $d_K$).
In addition to the question of the generalisability of certain results, we note that there is an extensive literature on a related issue for models of graded, rather than categorical, belief (esp. probabilistic models): how to update one’s degrees of belief on information specifying a particular conditional degree of belief or presented in the form of a natural language indicative conditional. A natural approach here is to move to the posterior distribution that is “closest” to the prior one, on some appropriate distance measure, subject of the relevant informational constraint. However, an apparent issue with the use of the popular cross-entropy measure was presented in the classic Judy Benjamin example of [@vanFraassen1981-VANAPF], with similar observations being made in relation to two further measures in [@vanFraassen1986-VANAPF-9]. For further discussions, see [@doi:10.1111/j.1468-0017.2012.01443.x], [@Douven2011-DOUAPA], [@Douven2011-ROMANR], [@EvaForthcoming-EVALFC], and [@Grove:1997:PUC:2074226.2074251]. An examination of potential points of contact with the present work would be interesting to pursue.
Acknowledgements {#acknowledgements .unnumbered}
================
This research was supported by the Australian Government through an Australian Research Council Future Fellowship (project number FT160100092) awarded to Jake Chandler.
Appendix {#s:appendix .unnumbered}
========
Regarding (b) : Let $W=\{w,x,y,z\}$, with $w, x, y$ and $z$ respectively in ${[\![A\wedge B]\!]}$, ${[\![A\wedge \neg B]\!]}$, ${[\![\neg A\wedge B]\!]}$ and ${[\![\neg A\wedge \neg B]\!]}$. Let $\preccurlyeq_{\Psi}$ be given by $z\prec_{\Psi} \{w, x, y\}$. Then $A{\supset}B\in{[\Psi]}$, but $A\shortTo B\notin{[\Psi]_{\mathrm{c}}}$.
Given AGM, [$(\mathrm{C}{1}^{\ast}_{\scriptscriptstyle \preccurlyeq})$]{}, [$(\mathrm{C}{2}^{\ast}_{\scriptscriptstyle \preccurlyeq})$]{}, and $(\mathrm{Eq}^\ast_\preccurlyeq)$, ${\ast_{\mathrm{L}}}$ is characterised by the ‘Recalcitrance’ property
BLAHI: = (Rec$^\ast$) If $A \wedge B$ is consistent, then $A \in {[(\Psi \ast A) \ast B]}$.\
It will suffice to show that this property is entailed by:
BLAHI: = (1) If $A \wedge B$ is consistent, then $B \in$\
$ {[(\Psi \ast A{\supset}B) \ast A]}$\
So let $A\wedge B$ be consistent. Since $A\equiv (A{\supset}B){\supset}A$, it suffices, by (Eq$^\ast$) to show $A \in {[(\Psi \ast (A{\supset}B){\supset}A) \ast B]}$. We know that for any AGM operator $\ast'$ and state $\Psi'$, $A\in{[(\Psi'\ast' B)]}$ iff $A\in{[(\Psi'\ast' A{\supset}B)]}$. Hence it suffices to show $A \in {[(\Psi \ast (A{\supset}B){\supset}A) \ast A{\supset}B]}$. Since $A\wedge B \equiv ((A{\supset}B){\supset}A)\wedge ( A{\supset}B)$ is consistent, we can apply (1) to recover the required result.
- From [$(\mathrm{Ret}{1}^{\ast}_{\preccurlyeq})$]{} to [$(\mathrm{Ret}{1})$]{}: Assume that $A{\supset}B\in{\mathrm{Cn}}(C)$. We prove each inclusion in turn
- Regarding ${[(\Psi\ast A{\supset}B)\ast C]}\subseteq {[(\Psi\ast A\shortTo B)\ast C]}$: Assume that $x\in\min(\preccurlyeq_{(\Psi\ast A\shortTo B)\ast C}, W)$ but, for reductio, $x\notin\min(\preccurlyeq_{(\Psi\ast A{\supset}B)\ast C}, W)$. From the former, by Success, $x\in {[\![C]\!]}$ and so, by the fact that $A{\supset}B\in{\mathrm{Cn}}(C)$, it follows that $x\in {[\![A{\supset}B]\!]}$. Now, by Success and $A{\supset}B\in{\mathrm{Cn}}(C)$ we also have $\min(\preccurlyeq_{(\Psi\ast A{\supset}B)\ast C}, W)\subseteq{[\![A{\supset}B]\!]}$. Hence, since $x\notin\min(\preccurlyeq_{(\Psi\ast A{\supset}B)\ast C}, W)$, there exists $y\in {[\![A{\supset}B]\!]}$ such that $y\prec_{(\Psi\ast A{\supset}B)\ast C} x$. But $x\in\min(\preccurlyeq_{(\Psi\ast A\shortTo B)\ast C}, W)$ and so $x\preccurlyeq_{(\Psi\ast A\shortTo B)\ast C} y$. This contradicts [$(\mathrm{Ret}{1}^{\ast}_{\preccurlyeq})$]{}. Hence $x\in\min(\preccurlyeq_{(\Psi\ast A{\supset}B)\ast C}, W)$, as required.
- Regarding ${[(\Psi\ast A\shortTo B)\ast C]}\subseteq {[(\Psi\ast A{\supset}B)\ast C]}$: Similar to the item immediately above.
- From [$(\mathrm{Ret}{1})$]{} to [$(\mathrm{Ret}{1}^{\ast}_{\preccurlyeq})$]{}: Assume that $x, y \in {[\![A{\supset}B]\!]}$. We prove each direction of the biconditional in turn
- From $x \preccurlyeq_{\Psi\ast A\shortTo B}y$ to $x \preccurlyeq_{\Psi\ast A{\supset}B}y$: Assume $x \preccurlyeq_{\Psi\ast A\shortTo B}y$, but, for reductio, $y \prec_{\Psi\ast A{\supset}B}x$. Then, by Success, $\neg x\in{[(\Psi\ast A{\supset}B)\ast x\vee y]}$ but $\neg x\notin{[(\Psi\ast A\shortTo B)\ast x\vee y]}$, contradicting [$(\mathrm{Ret}{1})$]{}. So $x \preccurlyeq_{\Psi\ast A{\supset}B}y$, as required.
- From $x \preccurlyeq_{\Psi\ast A{\supset}B}y$ to $x \preccurlyeq_{\Psi\ast A\shortTo B}y$: Similar to the item immediately above.
We prove the result in three lemmas. The first of these is the following:
[lem]{}[TheBasics]{} \[lem:TheBasics\] For any function $\ast$ from $\mathbb{S} \times L$ to $\mathbb{S}$, $\circledast$ satisfies [$(\mathrm{S}^{\ast}_{\scriptscriptstyle \preccurlyeq})$]{} and [$(\mathrm{Ret}{1}^{\ast}_{\preccurlyeq})$]{}.
We note that, given the definition of ${\ast_{\mathrm{L}}}$ in Definition \[elemdef\], Definition \[MyProposal\] is equivalent to:
- $x \preccurlyeq_{\Psi \circledast A\shortTo B} y$ iff
- $x\in D(\preccurlyeq_{\Psi\ast A{\supset}B}, A\wedge B)\cap{[\![A{\supset}B]\!]}$ and $y\notin D(\preccurlyeq_{\Psi\ast A{\supset}B}, A\wedge B)\cap{[\![A{\supset}B]\!]}$, or
- ($x\in D(\preccurlyeq_{\Psi\ast A{\supset}B}, A\wedge B)\cap{[\![A{\supset}B]\!]}$ iff $y\in D(\preccurlyeq_{\Psi\ast A{\supset}B}, A\wedge B)\cap{[\![A{\supset}B]\!]}$) and $x \preccurlyeq_{\Psi\ast A{\supset}B} y$
We now show that [$(\mathrm{S}^{\ast}_{\scriptscriptstyle \preccurlyeq})$]{} is satisfied. By definition, $D(\preccurlyeq_{\Psi\ast A{\supset}B}, A\wedge B)\cap{[\![A\wedge B]\!]}\neq \varnothing$. So let $x$ be such that $x\in D(\preccurlyeq_{\Psi\ast A{\supset}B}, A\wedge B)\cap{[\![A\wedge B]\!]}$. Now consider $y\in {[\![A\wedge\neg B]\!]}$. By Definition \[MyProposal\], $x\prec_{\Psi\circledast A\shortTo B}y$, and hence $\min(\preccurlyeq_{\Psi\circledast A\shortTo B}, {[\![A]\!]})\subseteq {[\![B]\!]}$, as required.
Regarding [$(\mathrm{Ret}{1}^{\ast}_{\preccurlyeq})$]{}: Assume $x, y \in {[\![A{\supset}B]\!]}$. For the left to right direction, assume that $x\preccurlyeq_{\Psi\circledast A\shortTo B}y$. From this one of either (1) or (2) holds. (2) immediately entails that $x\preccurlyeq_{\Psi\ast A{\supset}B}y$. (1) gives us: $x\in D(\preccurlyeq_{\Psi\ast A{\supset}B}, A\wedge B)$ and $y \notin D(\preccurlyeq_{\Psi\ast A{\supset}B}, A\wedge B)$. $x\preccurlyeq_{\Psi\ast A{\supset}B}y$ then follows from the definition of $D(\preccurlyeq_{\Psi\ast A{\supset}B}, A\wedge B)$. For the right to left direction, assume that $x\preccurlyeq_{\Psi\ast A{\supset}B}y$. If $x, y\in D(\preccurlyeq_{\Psi\ast A{\supset}B}, A\wedge B)$ or $x, y\notin D(\preccurlyeq_{\Psi\ast A{\supset}B}, A\wedge B)$, then we obtain $x\preccurlyeq_{\Psi\circledast A\shortTo B}y$ by (1). So assume that one of either $x$ or $y$ is in $D(\preccurlyeq_{\Psi\ast A{\supset}B}, A\wedge B)$, while the other is not. From $x\preccurlyeq_{\Psi\ast A{\supset}B}y$, it must be the case that $x\in D(\preccurlyeq_{\Psi\ast A{\supset}B}, A\wedge B)$ and $y \notin D(\preccurlyeq_{\Psi\ast A{\supset}B}, A\wedge B)$. From (2), we then again recover $x\preccurlyeq_{\Psi\circledast A\shortTo B}y$.
Our next lemma states a general fact about lexicographic combinations of ordered pairs of TPOs, defined by
[defo]{}[lexcombo]{} \[lexcombo\] The [*lexicographic combination*]{} $\mathrm{lex}(\preccurlyeq_1, \preccurlyeq_2)$ of two TPOs $\preccurlyeq_1$ and $\preccurlyeq_2$ is given by the TPO $\preccurlyeq$ such that $x\preccurlyeq y$ iff (i) $x \prec_1 y$ or (ii) $x \sim_1 y$ and $x \prec_2 y$
It is given as follows:
[lem]{}[Lexresult]{} \[lem:Lexresult\] Let $\preccurlyeq_{1}$, $\preccurlyeq_{2}$ be two given TPOs and let $X(\preccurlyeq_{2}) =\{\preccurlyeq\mid \preccurlyeq\mbox{ is a TPO s.t.~}\preccurlyeq\subseteq \preccurlyeq_{2}\}$. Then the TPO in $X(\preccurlyeq_{2})$ that minimises the distance $d_K$ to $\preccurlyeq_{1}$ is $\mathrm{lex}(\preccurlyeq_1, \preccurlyeq_2)$.
Let $\preccurlyeq'= \mathrm{lex}(\preccurlyeq_1, \preccurlyeq_2)$. First we need to check that $\preccurlyeq'\in X(\preccurlyeq_2)$, i.e. $\preccurlyeq'\subseteq\preccurlyeq_2$. But this is clear from Definition \[lexcombo\]. It remains to be shown that $d_K(\preccurlyeq_1, \preccurlyeq') \leq d_K(\preccurlyeq_1, \preccurlyeq'')$ for all $\preccurlyeq''\in X(\preccurlyeq_2)$. To see this, we first reformulate $d_K$.
[defo]{}[hardsoft]{} \[hardsoft\] A [*hard conflict*]{} between $\preccurlyeq_1, \preccurlyeq'$ is a 2-element set $\{x, y\}$ s.t. $x\prec' y$ and $y\prec_{1} x$. Let $\mathrm{Hard}(\preccurlyeq_1, \preccurlyeq')$ denote the set of such hard conflicts.
A [*soft conflict*]{} between $\preccurlyeq_1, \preccurlyeq'$ is a 2-element set $\{x, y\}$ s.t. either (i) $x\prec' y$ and $x\sim_{1} y$ or (ii) $x\prec_1 y$ and $x\sim' y$. Let $\mathrm{Soft}(\preccurlyeq_1, \preccurlyeq')$ denote the set of such soft conflicts.
So $d_K(\preccurlyeq_1, \preccurlyeq') = 2\times \lvert \mathrm{Hard}(\preccurlyeq_1, \preccurlyeq')\rvert + \lvert \mathrm{Soft}(\preccurlyeq_1, \preccurlyeq')\rvert$ and similarly for $d_K(\preccurlyeq_1, \preccurlyeq'')$. Hence, to show that $d_K(\preccurlyeq_1, \preccurlyeq') < d_K(\preccurlyeq_1, \preccurlyeq'')$ when $\preccurlyeq'\neq \preccurlyeq''$, it suffices to prove
- \(1) $\mathrm{Hard}(\preccurlyeq_1, \preccurlyeq')\subseteq \mathrm{Hard}(\preccurlyeq_1, \preccurlyeq'')$
- \(2) $\mathrm{Soft}(\preccurlyeq_1, \preccurlyeq')\subseteq \mathrm{Soft}(\preccurlyeq_1, \preccurlyeq'')$
Regarding (1): Let $\{x, y\}\in \mathrm{Hard}(\preccurlyeq_1, \preccurlyeq')$, i.e. $x\prec_1 y$ and $y\prec' x$. We must show $y\prec'' x$. By definition of $\preccurlyeq'= \mathrm{lex}(\preccurlyeq_1, \preccurlyeq_2)$, we have, from $y\prec' x$, (i) $y\prec_2 x$ or (ii) $y\sim_2 x$ and $y\prec_1 x$. We cannot have $y\prec_1 x$, since we already have $x\prec_1 y$. So $y\prec_2 x$. Hence, since $\preccurlyeq''\in X(\preccurlyeq_2)$, i.e. $\preccurlyeq''\subseteq\preccurlyeq_2$, we have $y\prec' x$ as well, as required.
Regarding (2): Let $\{x, y\}\in \mathrm{Soft}(\preccurlyeq_1, \preccurlyeq')$. We then have two cases to consider:
- $x\sim_1 y$ and $x\prec' y$: From $x\prec' y$, we get either (i) $x\prec_2 y$ or (ii) $x\sim_2 y$ and $x\prec_1 y$. The latter cannot occur, since we assume $x\sim_1 y$. Hence $x\prec_2 y$. Since $\preccurlyeq''\subseteq \preccurlyeq_2$, we also then have $x\prec'' y$, so $\{x,y\}\in\mathrm{Soft}(\preccurlyeq_1, \preccurlyeq'')$, as required.
- $x\sim' y$ and $x\prec_1 y$: Impossible, since $x\sim' y$ entails that both $x\sim_1 y$ and $x\sim_2 y$ but we assume $x\prec_1 y$.
We now show that:
[lem]{}[Subsetresult]{} \[lem:Subsetresult\] For any $\preccurlyeq'$ satisfying [$(\mathrm{S}^{\ast}_{\scriptscriptstyle \preccurlyeq})$]{} and [$(\mathrm{Ret}{1}^{\ast}_{\preccurlyeq})$]{}, we must have $\preccurlyeq'\subseteq \preccurlyeq_{D}$, where $\preccurlyeq_D$ is defined as follows:
- $x \preccurlyeq_D y$ iff $x\in D(\preccurlyeq_{\Psi\ast A{\supset}B}, A\wedge B)\cap{[\![A{\supset}B]\!]}$ or $y\notin D(\preccurlyeq_{\Psi\ast A{\supset}B}, A\wedge B)\cap{[\![A{\supset}B]\!]}$
Let $\preccurlyeq'$ satisfy [$(\mathrm{S}^{\ast}_{\scriptscriptstyle \preccurlyeq})$]{} and [$(\mathrm{Ret}{1}^{\ast}_{\preccurlyeq})$]{}. Suppose $y\preccurlyeq_D x$. We must show that $y\prec' x$. From $y\preccurlyeq_D x$, by definition of $\preccurlyeq_D$, we have $x\notin D(\preccurlyeq_{\Psi\ast A{\supset}B}, A\wedge B)\cap{[\![A{\supset}B]\!]}$ and $y\in D(\preccurlyeq_{\Psi\ast A{\supset}B}, A\wedge B)\cap{[\![A{\supset}B]\!]}$. From $y\in D(\preccurlyeq_{\Psi\ast A{\supset}B}, A\wedge B)$, we have $y\preccurlyeq_{\Psi\ast A{\supset}B}z$, where $z\in\min(\preccurlyeq_{\Psi\ast A{\supset}B}, {[\![A\wedge B]\!]})$. We now consider two cases:
- $x\in{[\![A{\supset}B]\!]}$: Then, since $x\notin D(\preccurlyeq_{\Psi\ast A{\supset}B}, A\wedge B)\cap{[\![A{\supset}B]\!]}$, we have $x\notin D(\preccurlyeq_{\Psi\ast A{\supset}B}, A\wedge B)$ and so, form this and $y\in D(\preccurlyeq_{\Psi\ast A{\supset}B}, A\wedge B)$, we get $y\preccurlyeq_{\Psi\ast A{\supset}B}x$. Since $x, y \in{[\![A{\supset}B]\!]}$, we get from this $y\prec' x$ by [$(\mathrm{Ret}{1}^{\ast}_{\preccurlyeq})$]{}, as required.
- $x\in{[\![A\wedge\neg B]\!]}$: By [$(\mathrm{S}^{\ast}_{\scriptscriptstyle \preccurlyeq})$]{}, we know that $u\prec' x$ for some $u\in{[\![A\wedge B]\!]}$. By the minimality of $z$, we know $z\preccurlyeq_{\Psi\ast A{\supset}B} u$. Hence $y\preccurlyeq_{\Psi\ast A{\supset}B} u$. Then, by [$(\mathrm{Ret}{1}^{\ast}_{\preccurlyeq})$]{}, we recover $y\preccurlyeq' u$ and so $y\prec' x$, as required.
Putting together Lemmas \[lem:TheBasics\], \[lem:Lexresult\] and \[lem:Subsetresult\] yields the proof of the theorem. Lemma \[lem:TheBasics\] tells us that $\preccurlyeq_{\Psi\circledast A\shortTo B}$ satisfies [$(\mathrm{S}^{\ast}_{\scriptscriptstyle \preccurlyeq})$]{} and [$(\mathrm{Ret}{1}^{\ast}_{\preccurlyeq})$]{}. Next, note that Definition \[MyProposal\] can be equivalently presented in terms of a lexicographic combination, so that $\preccurlyeq_{\Psi\circledast A\shortTo B} = \mathrm{lex}(\preccurlyeq_D, \preccurlyeq_{\Psi\ast A{\supset}B})$.
In view of this, we can see that, by Lemma \[lem:Lexresult\], $\preccurlyeq_{\Psi\circledast A\shortTo B}$ minimises $d_K$ to $\preccurlyeq_{\Psi\ast A{\supset}B}$ among all TPOs $\preccurlyeq$ s.t. $\preccurlyeq\subseteq\preccurlyeq_D$. Finally, since all $\preccurlyeq$ that satisfy [$(\mathrm{S}^{\ast}_{\scriptscriptstyle \preccurlyeq})$]{} and [$(\mathrm{Ret}{1}^{\ast}_{\preccurlyeq})$]{} are such that $\preccurlyeq\subseteq\preccurlyeq_D$, by Lemma \[lem:Subsetresult\], $\preccurlyeq_{\Psi\circledast A\shortTo B}$ must also minimise $d_K$ among all TPOs satisfying [$(\mathrm{S}^{\ast}_{\scriptscriptstyle \preccurlyeq})$]{} and [$(\mathrm{Ret}{1}^{\ast}_{\preccurlyeq})$]{} .
We will prove the equivalent statement: if ${[(\Psi\circledast A\shortTo B)\circledast C]}\neq {[(\Psi\circledast A{\supset}B)\circledast C]}$, then ${[(\Psi\ast' A\shortTo B)\ast' C]}\neq {[(\Psi\ast' A{\supset}B)\ast' C]}$.
Suppose ${[(\Psi\circledast A\shortTo B)\circledast C]}\neq {[(\Psi\circledast A{\supset}B)\circledast C]}$. Then $\min(\preccurlyeq_{\Psi\circledast A\shortTo B}, {[\![C]\!]})\neq \min(\preccurlyeq_{\Psi\circledast A{\supset}B}, {[\![C]\!]})$. So we have two cases to consider, corresponding to the failures of each direction of subset inclusion.
Assume $\min(\preccurlyeq_{\Psi\circledast A\shortTo B}, {[\![C]\!]})\nsubseteq \min(\preccurlyeq_{\Psi\ast A{\supset}B}, {[\![C]\!]})$. Let $x\in \min(\preccurlyeq_{\Psi\circledast A\shortTo B}, {[\![C]\!]}) - \min(\preccurlyeq_{\Psi\ast A{\supset}B}, {[\![C]\!]})$. Let $y\in \min(\preccurlyeq_{\Psi\ast A{\supset}B}, {[\![C]\!]})$. Then $x \preccurlyeq_{\Psi\circledast A\shortTo B} y$ (by minimality of $x$) and $y \preccurlyeq_{\Psi\ast A{\supset}B} x$. Since $x \preccurlyeq_{\Psi\circledast A\shortTo B} y$ , we know that either $x \sim_{\Psi\circledast A\shortTo B} y$ or $x \prec_{\Psi\circledast A\shortTo B} y$. By our construction of $\circledast$, it is not possible to have both $x \sim_{\Psi\circledast A\shortTo B} y$ and $y \prec_{\Psi\ast A{\supset}B} x$. So we must have $x \prec_{\Psi\circledast A\shortTo B} y$ and hence $\{x, y \}\in \mathrm{Hard}(\preccurlyeq_{\Psi\circledast A\shortTo B}, \preccurlyeq_{\Psi\ast A{\supset}B})$. Now, since we assume $\ast'$ to satisfy [$(\mathrm{S}^{\ast}_{\scriptscriptstyle \preccurlyeq})$]{} and [$(\mathrm{Ret}{1}^{\ast}_{\preccurlyeq})$]{}, we know from our proof of Theorem \[Representation\] that $\mathrm{Hard}(\preccurlyeq_{\Psi\circledast A\shortTo B}, \preccurlyeq_{\Psi\ast A{\supset}B})\subseteq \mathrm{Hard}(\preccurlyeq_{\Psi\ast' A\shortTo B}, \preccurlyeq_{\Psi\ast A{\supset}B})$. So $\{x, y \}\in \mathrm{Hard}(\preccurlyeq_{\Psi\ast' A\shortTo B}, \preccurlyeq_{\Psi\ast A{\supset}B})$ and hence $x \prec_{\Psi\ast' A\shortTo B} y$. So we have $y\in \min(\preccurlyeq_{\Psi\ast' A{\supset}B}, {[\![C]\!]})=\min(\preccurlyeq_{\Psi\ast A{\supset}B}, {[\![C]\!]})$ (since $\ast'$ extends $\ast$, i.e. agrees with it on unconditional revisions) but $y\notin \min(\preccurlyeq_{\Psi\ast' A{\supset}B}, {[\![C]\!]})$. So $\min(\preccurlyeq_{\Psi\ast' A{\supset}B}, {[\![C]\!]})\neq \min(\preccurlyeq_{\Psi\ast' A\shortTo B}, {[\![C]\!]})$, i.e. ${[(\Psi\ast' A\shortTo B)\ast' C]}\neq {[(\Psi\ast' A{\supset}B)\ast' C]}$, as required
Assume $\min(\preccurlyeq_{\Psi\ast A{\supset}B}, {[\![C]\!]})\nsubseteq \min(\preccurlyeq_{\Psi\circledast A\shortTo B}, {[\![C]\!]})$. Let $x\in \min(\preccurlyeq_{\Psi\ast A{\supset}B}, {[\![C]\!]}) - \min(\preccurlyeq_{\Psi\circledast A\shortTo B}, {[\![C]\!]})$. Let $y\in \min(\preccurlyeq_{\Psi\circledast A\shortTo B}, {[\![C]\!]})$. Then $y\prec_{\Psi\circledast A\shortTo B} x$ and $x \preccurlyeq_{\Psi\ast A{\supset}B} y$, from which $x \prec_{\Psi\ast A{\supset}B} y$ or $x \sim_{\Psi\ast A{\supset}B} y$.
If $x \prec_{\Psi\ast A{\supset}B} y$, then $\{x, y \}\in \mathrm{Hard}(\preccurlyeq_{\Psi\circledast A\shortTo B}, \preccurlyeq_{\Psi\ast A{\supset}B})$. Since, as noted above, $\mathrm{Hard}(\preccurlyeq_{\Psi\circledast A\shortTo B}, \preccurlyeq_{\Psi\ast A{\supset}B})\subseteq \mathrm{Hard}(\preccurlyeq_{\Psi\ast' A\shortTo B}, \preccurlyeq_{\Psi\ast A{\supset}B})$, we get $y \prec_{\Psi\ast' A\shortTo B} x$ and so $x\in \min(\preccurlyeq_{\Psi\ast' A{\supset}B}, {[\![C]\!]}) - \min(\preccurlyeq_{\Psi\ast' A\shortTo B}, {[\![C]\!]})$, i.e. ${[(\Psi\ast' A\shortTo B)\ast' C]}\neq {[(\Psi\ast' A{\supset}B)\ast' C]}$, as required.
If $x \sim_{\Psi\ast A{\supset}B} y$, then $\{x, y \}\in \mathrm{Soft}(\preccurlyeq_{\Psi\circledast A\shortTo B}, \preccurlyeq_{\Psi\ast A{\supset}B})$. Since, $\mathrm{Soft}(\preccurlyeq_{\Psi\circledast A\shortTo B}, \preccurlyeq_{\Psi\ast A{\supset}B})\subseteq \mathrm{Soft}(\preccurlyeq_{\Psi\ast' A\shortTo B}, \preccurlyeq_{\Psi\ast A{\supset}B})$ (again, see proof of Theorem \[Representation\]), we have $\{x, y \}\in \mathrm{Soft}(\preccurlyeq_{\Psi\ast' A\shortTo B}, \preccurlyeq_{\Psi\ast A{\supset}B})$, so either $x \prec_{\Psi\ast' A\shortTo B} y$ or $y \prec_{\Psi\ast' A\shortTo B} x$. In the latter case, we get ${[(\Psi\ast' A\shortTo B)\ast' C]}\neq {[(\Psi\ast' A{\supset}B)\ast' C]}$ as above. In the former case, we deduce $y\notin \min(\preccurlyeq_{\Psi\ast' A\shortTo B}, {[\![C]\!]})$. But from $x\sim_{\Psi\ast A{\supset}B} y$ and $x\in \min(\preccurlyeq_{\Psi\ast A{\supset}B}, {[\![C]\!]})$ we must have $y\in \min(\preccurlyeq_{\Psi\ast A{\supset}B}, {[\![C]\!]})$. Hence $y\in \min(\preccurlyeq_{\Psi\ast A{\supset}B}, {[\![C]\!]})-\min(\preccurlyeq_{\Psi\ast' A\shortTo B}, {[\![C]\!]})$, so $\min(\preccurlyeq_{\Psi\ast A{\supset}B}, {[\![C]\!]})\neq \min(\preccurlyeq_{\Psi\ast' A\shortTo B}, {[\![C]\!]})$, which gives again ${[(\Psi\ast' A\shortTo B)\ast' C]}\neq {[(\Psi\ast' A{\supset}B)\ast' C]}$, as required.
Given the definition of ${\ast_{\mathrm{L}}}$ in Definition \[elemdef\], if we assume $\ast=\circledast$, then Definition \[MyProposal\] tells us that:
- $x \preccurlyeq_{\Psi \ast A\shortTo B} y$ iff
- \(i) $x\in D(\preccurlyeq_{\Psi\ast A{\supset}B}, A\wedge B)\cap{[\![A{\supset}B]\!]}$ and
\(ii) $y\notin D(\preccurlyeq_{\Psi\ast A{\supset}B}, A\wedge B)\cap{[\![A{\supset}B]\!]}$,
or
- \(i) ($x\in D(\preccurlyeq_{\Psi\ast A{\supset}B}, A\wedge B)\cap{[\![A{\supset}B]\!]}$ iff $y\in D(\preccurlyeq_{\Psi\ast A{\supset}B}, A\wedge B)\cap{[\![A{\supset}B]\!]}$) and
\(ii) $x \preccurlyeq_{\Psi\ast A{\supset}B} y$
Regarding [$(\mathrm{Ret}{2}^{\ast}_{\preccurlyeq})$]{}: Assume $x, y \in {[\![A\wedge \neg B]\!]}$. For the left to right direction, assume that $x\preccurlyeq_{\Psi\ast A\shortTo B}y$. From this one of either (1) or (2) holds. Since, $x, y \in {[\![A\wedge \neg B]\!]}$, it must be the case that (2). But this immediately entails that $x\preccurlyeq_{\Psi\ast A{\supset}B}y$. The proof of the right to left direction is entirely analogous to the one given immediately above.
Regarding [$(\mathrm{Ret}{3}^{\ast}_{\preccurlyeq})$]{}: Assume $x\in {[\![A{\supset}B]\!]}$, $y\in {[\![A\wedge \neg B]\!]}$, and $x \prec_{\Psi\ast A{\supset}B} y$. We want to show that $x \prec_{\Psi\ast A\shortTo B} y$, i.e. that both
- \(i) $y\notin D(\preccurlyeq_{\Psi\ast A{\supset}B}, A\wedge B)$ or
\(ii) $y\in{[\![A\wedge \neg B]\!]}$ or
\(iii) $x\in D(\preccurlyeq_{\Psi\ast A{\supset}B}, A\wedge B)\cap{[\![A{\supset}B]\!]}$,
and
- \(i) $y\in D(\preccurlyeq_{\Psi\ast A{\supset}B}, A\wedge B)\cap{[\![A{\supset}B]\!]}$ iff $x\notin D(\preccurlyeq_{\Psi\ast A{\supset}B}, A\wedge B)\cap{[\![A{\supset}B]\!]}$ or
\(ii) $x \prec_{\Psi\ast A{\supset}B} y$
Our assumption that $y\in {[\![A\wedge \neg B]\!]}$ gets us (3), while $x \prec_{\Psi\ast A{\supset}B} y$ gets us (4).
Regarding [$(\mathrm{Ret}{4}^{\ast}_{\preccurlyeq})$]{}: Assume $x\in {[\![A{\supset}B]\!]}$, $y\in {[\![A\wedge \neg B]\!]}$, and $x \preccurlyeq_{\Psi\ast A{\supset}B} y$. We want to establish the disjunction of (1) and (2). If $x\in D(\preccurlyeq_{\Psi\ast A{\supset}B}, A\wedge B)$, then, since $x\in {[\![A{\supset}B]\!]}$ and $y\in {[\![A\wedge \neg B]\!]}$, we have (1) and we are done. So assume $x\notin D(\preccurlyeq_{\Psi\ast A{\supset}B}, A\wedge B)$ and hence, by $x \preccurlyeq_{\Psi\ast A{\supset}B} y$, $x, y\notin D(\preccurlyeq_{\Psi\ast A{\supset}B}, A\wedge B)$. (2) then holds by virtue of this and the fact that $x \preccurlyeq_{\Psi\ast A{\supset}B} y$.
The equivalence and entailments are obvious. Regarding the failure of entailment from [$(\mathrm{Ret}{3'}^{\ast}_{\preccurlyeq})$]{} and [$(\mathrm{Ret}{4'}^{\ast}_{\preccurlyeq})$]{} to [$(\mathrm{Ret}{3}^{\ast}_{\preccurlyeq})$]{} and [$(\mathrm{Ret}{4}^{\ast}_{\preccurlyeq})$]{}, see the countermodel in Figure \[fig:PrimeCM\].
There, we see that the DP postulates, as well as [$(\mathrm{S}^{\ast}_{\scriptscriptstyle \preccurlyeq})$]{}, [$(\mathrm{Ret}{1}^{\ast}_{\preccurlyeq})$]{}, [$(\mathrm{Ret}{2}^{\ast}_{\preccurlyeq})$]{}, [$(\mathrm{Ret}{3'}^{\ast}_{\preccurlyeq})$]{} and [$(\mathrm{Ret}{4'}^{\ast}_{\preccurlyeq})$]{} are satisfied. However, neither [$(\mathrm{Ret}{3}^{\ast}_{\preccurlyeq})$]{} nor [$(\mathrm{Ret}{4}^{\ast}_{\preccurlyeq})$]{} are satisfied because, notably, $1 \prec_{\Psi\ast A{\supset}B} 4$, but $4 \prec_{\Psi\ast A\shortTo B} 1$.
The derivation of the equivalences is straightforward and resembles the well known derivation of the equivalences between [$(\mathrm{C}{1}^{\ast}_{\scriptscriptstyle \preccurlyeq})$]{}–[$(\mathrm{C}{4}^{\ast}_{\scriptscriptstyle \preccurlyeq})$]{} and their syntactic counterparts:
- [$(\mathrm{Ret}{2}^{\ast}_{\preccurlyeq})$]{} and [$(\mathrm{Ret}{2})$]{}: Similar to (a) above.
- [$(\mathrm{Ret}{3}^{\ast}_{\preccurlyeq})$]{} and [$(\mathrm{Ret}{3})$]{}:
- From [$(\mathrm{Ret}{3}^{\ast}_{\preccurlyeq})$]{} to [$(\mathrm{Ret}{3})$]{}: Assume that $A\rightarrow B\in{[(\Psi\ast A\rightarrow B)\ast C]}$, but, for reductio, $A\rightarrow B\notin{[(\Psi\ast A\Rightarrow B)\ast C]}$. Then there exists $y\in\min(\preccurlyeq_{(\Psi\ast A\shortTo B)\ast C}, W)\cap {[\![A\wedge \neg B]\!]}$. Let $x$ be in $\min(\preccurlyeq_{(\Psi\ast A{\supset}B)\ast C}, W)\cap {[\![A{\supset}B]\!]}$ (we know that such an $x$ exists, since $\min(\preccurlyeq_{(\Psi\ast A{\supset}B)\ast C}, W)\subseteq {[\![A{\supset}B]\!]}$). Since $y\in {[\![A\wedge \neg B]\!]}$ and $\min(\preccurlyeq_{(\Psi\ast A{\supset}B)\ast C}, W)\subseteq {[\![A{\supset}B]\!]}$, we have $x \prec_{\Psi\ast A{\supset}B}y$. As $x\in {[\![A{\supset}B]\!]}$ and $y\in {[\![A\wedge \neg B]\!]}$, we can then apply [$(\mathrm{Ret}{3}^{\ast}_{\preccurlyeq})$]{} to recover $x \prec_{\Psi\ast A\shortTo B}y$. But this contradicts $y\in\min(\preccurlyeq_{(\Psi\ast A\shortTo B)\ast C}, W)$. Hence $A\rightarrow B\in{[(\Psi\ast A\Rightarrow B)\ast C]}$, as required.
- From [$(\mathrm{Ret}{3})$]{} to [$(\mathrm{Ret}{3}^{\ast}_{\preccurlyeq})$]{}: Assume $x\in {[\![A{\supset}B]\!]}$, $y\in {[\![A\wedge \neg B]\!]}$, and $x \prec_{\Psi\ast A{\supset}B}y$. Then, since $y\in {[\![A\wedge \neg B]\!]}$, by Success, $A\rightarrow B\in{[(\Psi\ast A{\supset}B)\ast C]}$. Hence, by [$(\mathrm{Ret}{3})$]{}, $A\rightarrow B\in{[(\Psi\ast A\shortTo B)\ast C]}$. Once again, by Success and the fact that $y\in {[\![A\wedge \neg B]\!]}$, $x \prec_{\Psi\ast A\shortTo B}y$, as required.
- [$(\mathrm{Ret}{4}^{\ast}_{\preccurlyeq})$]{} and [$(\mathrm{Ret}{4})$]{}: Similar to (c) above.
We need to show that that $\min(\preccurlyeq_{\Psi\ast A\shortTo B}, W) = \min(\preccurlyeq_{\Psi\ast A{\supset}B}, W)$.
We first prove $\min(\preccurlyeq_{\Psi\ast A\shortTo B}, W) \subseteq \min(\preccurlyeq_{\Psi\ast A{\supset}B}, W)$. Assume that $y\in \min(\preccurlyeq_{\Psi\ast A\shortTo B}, W)$ but, for reductio, that $y\notin \min(\preccurlyeq_{\Psi\ast A{\supset}B}, W)$. From the latter and [$(\mathrm{S}^{\ast}_{\scriptscriptstyle \preccurlyeq})$]{}, there exists $x\in {[\![A{\supset}B]\!]}$ such that $x\prec_{\Psi\ast A{\supset}B}y$. If $y\in {[\![A{\supset}B]\!]}$, then $x \prec_{\Psi\ast A\shortTo B} y$ by [$(\mathrm{Ret}{1}^{\ast}_{\preccurlyeq})$]{}. If $y\in {[\![A\wedge \neg B]\!]}$, then $x \prec_{\Psi\ast A\shortTo B} y$ by [$(\mathrm{Ret}{3}^{\ast}_{\preccurlyeq})$]{}. Either way, $y\notin \min(\preccurlyeq_{\Psi\ast A\shortTo B}, W)$. Contradiction. Hence $y\in \min(\preccurlyeq_{\Psi\ast A{\supset}B}, W)$ and so $\min(\preccurlyeq_{\Psi\ast A\shortTo B}, W) \subseteq \min(\preccurlyeq_{\Psi\ast A{\supset}B}, W)$, as required.
We now show that $\min(\preccurlyeq_{\Psi\ast A{\supset}B}, W) \subseteq \min(\preccurlyeq_{\Psi\ast A\shortTo B}, W)$. Assume that $y\in \min(\preccurlyeq_{\Psi\ast A{\supset}B}, W)$ but, for reductio, that $y\notin \min(\preccurlyeq_{\Psi\ast A\shortTo B}, W)$. From the former and [$(\mathrm{S}^{\ast}_{\scriptscriptstyle \preccurlyeq})$]{}, we have $y\in {[\![A{\supset}B]\!]}$. From the latter, there exists $x\in W$ such that $x \prec_{\Psi\ast A\shortTo B} y$. If $x\in {[\![A{\supset}B]\!]}$, then $x \prec_{\Psi\ast A{\supset}B} y$ by [$(\mathrm{Ret}{1}^{\ast}_{\preccurlyeq})$]{}. If $x\in {[\![A\wedge \neg B]\!]}$, then $x \prec_{\Psi\ast A{\supset}B} y$ by [$(\mathrm{Ret}{4}^{\ast}_{\preccurlyeq})$]{}. Either way, $y\notin \min(\preccurlyeq_{\Psi\ast A{\supset}B}, W)$. Contradiction. Hence $y\in \min(\preccurlyeq_{\Psi\ast A\shortTo B}, W)$ and so $\min(\preccurlyeq_{\Psi\ast A{\supset}B}, W) \subseteq \min(\preccurlyeq_{\Psi\ast A\shortTo B}, W)$, as required.
For this proof, we will make use of the fact that elementary operators satisfy the following principle from [@DBLP:conf/kr/0001C18]:
BLAHI: =
[$(\beta{2}^{\ast}_{\scriptscriptstyle \preccurlyeq})$]{} If $x \not\in \min(\preccurlyeq_{\Psi}, {[\![C]\!]})$, $x \in {[\![A]\!]}$, $y\in {[\![\neg A]\!]}$, and\
$y \prec_{\Psi \ast A} x$, then $y \prec_{\Psi\ast C} x$\
We will also make use of the concept of a [*state restriction*]{}. Informally, the restriction $\Psi\,\mid\, A$ of a state $\Psi$ to a sentence $A$ corresponds to the agent’s categorically ruling out as impossible any world inconsistent with $A$. In terms of the TPO associated with a restricted state, we define $\preccurlyeq_{\Psi\mid A}$ as $\preccurlyeq_{\Psi}\cap ({[\![A]\!]}\times {[\![A]\!]})$. This notion allows us to articulate the following property, which is also satisfied by elementary operators:
BLAHI: =
(SR) (1) If $x\in\min(\preccurlyeq_{\Psi\mid A}, {[\![B]\!]})$ and $y\in{[\![\neg B]\!]}$, then\
$x\prec_{(\Psi\mid A) \ast B}y$\
(2) Otherwise $x\preccurlyeq_{(\Psi\mid A) \ast B}y$ iff $x \preccurlyeq_{\Psi\ast B}y$\
Assume that $x, y\in{[\![A]\!]}$. We want to show that $x\preccurlyeq_{\Psi\ast A\shortTo B}y$ iff $x\preccurlyeq_{(\Psi\mid A) \ast B}y$.
- $x, y\in{[\![B]\!]}$: Then $x, y \in{[\![A{\supset}B]\!]}$, so, by [$(\mathrm{Ret}{1}^{\ast}_{\preccurlyeq})$]{}, $x\preccurlyeq_{\Psi\ast A\shortTo B}y$ iff $x\preccurlyeq_{\Psi\ast A{\supset}B}y$ and, by [$(\mathrm{C}{1}^{\ast}_{\scriptscriptstyle \preccurlyeq})$]{}, $x\preccurlyeq_{\Psi\ast A{\supset}B}y$ iff $x\preccurlyeq_{\Psi} y$. Since $x, y\in{[\![A]\!]}$ and $x, y\in{[\![B]\!]}$, by [$(\mathrm{C}{1}^{\ast}_{\scriptscriptstyle \preccurlyeq})$]{} and (SR), $x\preccurlyeq_{\Psi} y$ iff $x\preccurlyeq_{(\Psi\mid A) \ast B}y$.
- $x, y\in{[\![\neg B]\!]}$: Similar to (a), using [$(\mathrm{Ret}{2}^{\ast}_{\preccurlyeq})$]{} and [$(\mathrm{C}{2}^{\ast}_{\scriptscriptstyle \preccurlyeq})$]{}.
- $x\in{[\![B]\!]}$, $y\in{[\![\neg B]\!]}$: If $x\in\min(\preccurlyeq_{\Psi\mid A}, {[\![B]\!]})$, then, by (SR), we have $x\prec_{(\Psi\mid A) \ast B} y$. Furthermore, since $x\in{[\![A]\!]}$, it follows from $x\in\min(\preccurlyeq_{\Psi\mid A}, {[\![B]\!]})$ that $x\in\min(\preccurlyeq_{\Psi}, {[\![A\wedge B]\!]})$. By Success and the fact that $y\in{[\![A\wedge\neg B]\!]}$, we then recover $x\prec_{\Psi\ast A\shortTo B}y$. Hence $x\preccurlyeq_{\Psi\ast A\shortTo B}y$ iff $x\preccurlyeq_{(\Psi\mid A) \ast B}y$, as required.
So assume that $x\notin\min(\preccurlyeq_{\Psi\mid A}, {[\![B]\!]})$, then (SR) gives us $x\preccurlyeq_{(\Psi\mid A) \ast B}y$ iff $x\preccurlyeq_{\Psi\ast B}y$. We now show that $x \preccurlyeq_{\Psi\ast A{\supset}B} y$ iff $x \preccurlyeq_{\Psi\ast B}y$.
We know from $x\in{[\![A]\!]}$ that $x\in\min(\preccurlyeq,{[\![A{\supset}B]\!]})$ iff $x\in\min(\preccurlyeq,{[\![B]\!]})$. So we consider two cases:
- $x\in\min(\preccurlyeq_{\Psi},{[\![A{\supset}B]\!]})$ and $x\in\min(\preccurlyeq_{\Psi},{[\![B]\!]})$: Since $y\in{[\![A\wedge\neg B]\!]}$, by Success, we have $x \prec_{\Psi\ast A{\supset}B} y$ and $x \prec_{\Psi\ast B}y$. Hence $x \preccurlyeq_{\Psi\ast A{\supset}B} y$ iff $x \preccurlyeq_{\Psi\ast B}y$.
- $x\notin\min(\preccurlyeq_{\Psi},{[\![A{\supset}B]\!]})$ and $x\notin\min(\preccurlyeq_{\Psi},{[\![B]\!]})$: We’ll establish our biconditional by showing that $y \prec_{\Psi\ast A{\supset}B} x$ iff $y \prec_{\Psi\ast B}x$. We recover the implication from $y \prec_{\Psi\ast A{\supset}B} x$ to $y \prec_{\Psi\ast B}x$, by [$(\beta{2}^{\ast}_{\scriptscriptstyle \preccurlyeq})$]{}, since $x\notin\min(\preccurlyeq_{\Psi},{[\![B]\!]})$, $x\in{[\![A{\supset}B]\!]}$ and $y\in{[\![A\wedge\neg B]\!]}$. We recover the converse implication by the same principle, this time because $x\notin\min(\preccurlyeq_{\Psi},{[\![A{\supset}B]\!]})$, $x\in{[\![ B]\!]}$ and $y\in{[\![\neg B]\!]}$.
So we have established that $x \preccurlyeq_{\Psi\ast A{\supset}B} y$ iff $x \preccurlyeq_{\Psi\ast B}y$. Since we already have $x\preccurlyeq_{(\Psi\mid A) \ast B}y$ iff $x\preccurlyeq_{\Psi\ast B}y$, it then follows that $x\preccurlyeq_{(\Psi\mid A) \ast B}y$ iff $x \preccurlyeq_{\Psi\ast A{\supset}B} y$.
We now note the following consequence of Definition \[MyProposal\]:
- If $x, y\in{[\![A]\!]}$, then
- If $x\in\min(\preccurlyeq_{\Psi}, {[\![A\wedge B]\!]})$ and $y\in{[\![A\wedge \neg B]\!]}$, then $x\prec_{\Psi\ast A\shortTo B}y$
- Otherwise $x\preccurlyeq_{\Psi\ast A\shortTo B}y$ iff $x\preccurlyeq_{\Psi\ast A{\supset}B} y$
So, if $x\notin\min(\preccurlyeq_{\Psi}, {[\![A\wedge B]\!]})$, it follows from $x\in{[\![B]\!]}$ and $y\in{[\![\neg B]\!]}$, by (1’), that $x\preccurlyeq_{\Psi\ast A\shortTo B}y$ iff $x\preccurlyeq_{\Psi\ast A{\supset}B} y$ and so $x\preccurlyeq_{\Psi\ast A\shortTo B}y$ iff $x\preccurlyeq_{(\Psi\mid A) \ast B}y$, as required.
If $x\in\min(\preccurlyeq_{\Psi}, {[\![A\wedge B]\!]})$, then $x\prec_{\Psi\ast A\shortTo B}y$, by (2’). But we will also have $x\in\min(\preccurlyeq_{\Psi\mid A}, {[\![B]\!]})$ and so, by (SR), $x\prec_{(\Psi\mid A) \ast B}y$. Hence $x\preccurlyeq_{\Psi\ast A\shortTo B}y$ iff $x\preccurlyeq_{(\Psi\mid A) \ast B}y$, as required.
- $x\in{[\![\neg B]\!]}$, $y\in{[\![B]\!]}$: Similar to (c).
We want to show that our proposal has the result that $\preccurlyeq_{\Psi{\ast_{\mathrm{N}}}A\shortTo B}$ is the ‘flattest’ TPO–in a technical sense to be defined below–such that the following lower bound constraint is satisfied, given $A\shortTo \neg B\notin{[\Psi]_{\mathrm{c}}}$:
BLAHBLI: = $ {[\Psi]_{\mathrm{c}}} \cup \{A\shortTo B\}\subseteq {[\Psi {\ast_{\mathrm{N}}}A\shortTo B]_{\mathrm{c}}}$\
In view of Definitions 20 and 21 of [@lehmann1992does], the upshot of this is then that $\preccurlyeq_{\Psi{\ast_{\mathrm{N}}}A\shortTo B}$ is the unique TPO corresponding to the rational closure of ${[\Psi]_{\mathrm{c}}}\cup\{A\shortTo B\}$.
We first note that, given AGM, the syntactic condition $A\shortTo \neg B\notin{[\Psi]_{\mathrm{c}}}$ corresponds to the semantic condition $\min(\preccurlyeq_{\Psi}, {[\![A]\!]})\nsubseteq{[\![\neg B]\!]}$, while the lower bound condition can be expressed as follows:
BLAHBLI: = (a) If $x \prec_{\Psi} y$, then $x \prec_{\Psi {\ast_{\mathrm{N}}}A\shortTo B} y$ and\
(b) $\min(\preccurlyeq_{\Psi{\ast_{\mathrm{N}}}A\shortTo B},{[\![A]\!]})\subseteq {[\![B]\!]}$\
Indeed, the lower bound constraint simply amounts to the conjunction of Success, which is equivalent to (b), with the claim that $ {[\Psi]_{\mathrm{c}}} \subseteq {[\Psi {\ast_{\mathrm{N}}}A\shortTo B]_{\mathrm{c}}}$, which is equivalent to (a). With this in hand, we now prove two lemmas. First:
[lem]{}[SPP]{} \[SPP\] When the extension of ${\ast_{\mathrm{N}}}$ to $L_c$ is defined as in Definition \[MyProposal\], if $\min(\preccurlyeq_{\Psi}, {[\![A]\!]})\nsubseteq{[\![\neg B]\!]}$, then $\preccurlyeq_{\Psi {\ast_{\mathrm{N}}}A{\supset}B}$ satisfies the lower bound condition.
We have already established (unconditional) satisfaction of (b) in Theorem \[Representation\]. So we just need to establish satisfaction of (a).
Assume $\min(\preccurlyeq_{\Psi}, {[\![A]\!]})\nsubseteq{[\![\neg B]\!]}$ and $x\prec_{\Psi}y$. We need to establish that $x \prec_{\Psi {\ast_{\mathrm{N}}}A\shortTo B} y$. To do this, we will first establish that $x~\prec_{\Psi {\ast_{\mathrm{N}}}A{\supset}B}~y$. Given the definition of ${\ast_{\mathrm{N}}}$, we will have $x~\prec_{\Psi {\ast_{\mathrm{N}}}A{\supset}B}~y$ iff either
- $x \in \min(\preccurlyeq_{\Psi}, {[\![A{\supset}B]\!]})$ and $y \notin \min(\preccurlyeq_{\Psi}, {[\![A{\supset}B]\!]})$, or
- $x, y \notin \min(\preccurlyeq_{\Psi}, {[\![A{\supset}B]\!]})$ and $x\prec_{\Psi}y$
Assume for reductio that $y \in \min(\preccurlyeq_{\Psi}, {[\![A{\supset}B]\!]})$. Then, since $x\prec_{\Psi}y$, we must have $\min(\preccurlyeq_{\Psi}, W)\subseteq {[\![A\wedge\neg B]\!]}$. But we have assumed $\min(\preccurlyeq_{\Psi}, {[\![A]\!]})\nsubseteq{[\![\neg B]\!]}$. Contradiction. Hence $y \notin \min(\preccurlyeq_{\Psi}, {[\![A{\supset}B]\!]})$. This leaves us with two possibilities. The first is that $x, y \notin \min(\preccurlyeq_{\Psi }, {[\![A{\supset}B]\!]})$, which, given $x\prec_{\Psi}y$, places us in case (2). The second is that $x \in \min(\preccurlyeq_{\Psi}, {[\![A{\supset}B]\!]})$ and $y \notin \min(\preccurlyeq_{\Psi}, {[\![A{\supset}B]\!]})$, which places us in case (1). Either way, then, $x~\prec_{\Psi {\ast_{\mathrm{N}}}A{\supset}B}~y$, as required.
We now need to show, from this, that, given Definition \[MyProposal\], it follows that $x~\prec_{\Psi {\ast_{\mathrm{N}}}A\shortTo B}~y$. If (i) $x, y\in{[\![A{\supset}B]\!]}$, (ii) $x, y\in{[\![A\wedge\neg B]\!]}$ or (iii) $x\in{[\![A{\supset}B]\!]}$ and $y\in{[\![A\wedge\neg B]\!]}$, then the required result follows from [$(\mathrm{Ret}{1}^{\ast}_{\preccurlyeq})$]{}, [$(\mathrm{Ret}{2}^{\ast}_{\preccurlyeq})$]{} and [$(\mathrm{Ret}{3}^{\ast}_{\preccurlyeq})$]{}, respectively. So assume $x\in{[\![A\wedge\neg B]\!]}$ and $y\in{[\![A{\supset}B]\!]}$. We will now show that $y\notin D(\preccurlyeq_{\Psi\ast A{\supset}B}, A\wedge B)$. Given this, the required conclusion will follow by [$(\mathrm{Ret}{5}^{\ast}_{\preccurlyeq})$]{}. So assume for reductio that $y\in D(\preccurlyeq_{\Psi\ast A{\supset}B}, A\wedge B)$, i.e. that $y\preccurlyeq_{\Psi{\ast_{\mathrm{N}}}A{\supset}B} z$, for all $z\in \min(\preccurlyeq_{\Psi{\ast_{\mathrm{N}}}A{\supset}B}, {[\![A\wedge B]\!]})$. Since $x~\prec_{\Psi {\ast_{\mathrm{N}}}A{\supset}B}~y$, we will therefore have $x~\prec_{\Psi {\ast_{\mathrm{N}}}A{\supset}B}~z$, for all $z\in \min(\preccurlyeq_{\Psi{\ast_{\mathrm{N}}}A{\supset}B}, {[\![A\wedge B]\!]})$. Since $x\in{[\![A\wedge \neg B]\!]}$ and $z\in{[\![A{\supset}B]\!]}$, by [$(\mathrm{C}{4}^{\ast}_{\scriptscriptstyle \preccurlyeq})$]{}, we recover $x\prec_{\Psi} z$, for all $z\in \min(\preccurlyeq_{\Psi{\ast_{\mathrm{N}}}A{\supset}B}, {[\![A\wedge B]\!]})$. However, by [$(\mathrm{C}{1}^{\ast}_{\scriptscriptstyle \preccurlyeq})$]{}, we have the result that $\min(\preccurlyeq_{\Psi}, {[\![A\wedge B]\!]})= \min(\preccurlyeq_{\Psi{\ast_{\mathrm{N}}}A{\supset}B}, {[\![A\wedge B]\!]})$. So $x\prec_{\Psi} z$ for all $z\in \min(\preccurlyeq_{\Psi}, {[\![A\wedge B]\!]})$. But, since $x\in{[\![A\wedge\neg B]\!]}$, this contradicts our assumption that $\min(\preccurlyeq_{\Psi}, {[\![A]\!]})\nsubseteq{[\![\neg B]\!]}$. Hence $y\notin D(\preccurlyeq_{\Psi\ast A{\supset}B}, A\wedge B)$, as required. This completes the proof of Lemma \[SPP\].
For our second lemma, we will make use of the convenient representation of TPOs by their corresponding [*ordered partitions*]{} of $W$. The ordered partition $\langle S_1, S_2, \ldots S_m\rangle$ of $W$ corresponding to a TPO $\preccurlyeq$ is such that $x \preccurlyeq y$ iff $r(x, \preccurlyeq) \leq$ where $r(x, \preccurlyeq)$ denotes the ‘rank’ of $x$ with respect to $\preccurlyeq$ and is defined by taking $S_{r(x, \preccurlyeq)}$ to be the cell in the partition that contains $x$.
This lemma is given as follows:
[lem]{}[FlattestLI]{} \[FlattestLI\] If $\min(\preccurlyeq_{\Psi}, {[\![A]\!]})\nsubseteq{[\![\neg B]\!]}$, then $\preccurlyeq_{\Psi {\ast_{\mathrm{N}}}A\shortTo B}~\sqsupseteq~~\preccurlyeq$, for any TPO $\preccurlyeq$ satisfying the lower bound condition.
where:
\[dfn:Flatter\] $\sqsupseteq$ is a binary relation on the set of TPOs over $W$ such such that, for any TPOs $\preccurlyeq_1$ and $\preccurlyeq_2$, whose corresponding ordered partitions are given by $\langle S_1, S_2, \ldots, S_m \rangle$ and $\langle T_1, T_2, \ldots, T_m \rangle$ respectively, $
\preccurlyeq_1~
\sqsupseteq~
\preccurlyeq_2
$ iff either (i) $S_i = T_i$ for all $i = 1, \ldots, m$, or (ii) $S_i \supset T_i$ for the first $i$ such that $S_i \neq T_i$.
$\sqsupseteq$ partially orders the set of TPOs over $W$ according to comparative ‘flatness’, with the flatter TPOs appearing ‘greater’ in the ordering, so that $\preccurlyeq_1~\sqsupseteq~\preccurlyeq_2$ iff $\preccurlyeq_1 $ is at least as as flat as $\preccurlyeq_2$.
Let $\langle T_1,\ldots, T_m\rangle$ be the ordered partition corresponding to the TPO $\preccurlyeq_{\Psi {\ast_{\mathrm{N}}}A\shortTo B}$. Let $\preccurlyeq $ be any TPO satisfying the lower bound condition:
BLAHBLI: = (a) If $x \prec_{\Psi} y$, then $x \prec y$ and\
(b) $\min(\preccurlyeq,{[\![A]\!]})\subseteq {[\![B]\!]}$\
Let $\langle S_1,\ldots, S_n\rangle$ be its corresponding ordered partition. We must show that the following relation holds: $\preccurlyeq_{\Psi {\ast_{\mathrm{N}}}A\shortTo B}~\sqsupseteq~\preccurlyeq$.
If $T_i=S_i$ for all $i$, then we are done. So let $i$ be minimal such that $T_i\neq S_i$. We must show $S_i\subset T_i$. So let $y\in S_i$ and assume, for contradiction, that $y\notin T_i$. We know that $T_i\neq \varnothing$, since, otherwise, $\bigcup_{j < i} T_j = W$, hence $\bigcup_{j < i} S_j = W$ and so $S_i=\varnothing$, contradicting $S_i\neq T_i$. So let $x\in T_i$. Then, since $y\notin T_i$, we have $x \prec_{\Psi {\ast_{\mathrm{N}}}A\shortTo B} y$. We are going to show that this entails that $\exists z$ such that
- $z\sim_{\Psi {\ast_{\mathrm{N}}}A\shortTo B} x$, i.e. $z\in T_i$, and
- $z\prec y$.
But if it were the case that $z\prec y$, then, since $y\in S_i$, $z\in \bigcup_{j < i} S_j = \bigcup_{j < i} T_j$, contradicting $z\in T_i$. Hence $y\in T_i$ and so we can conclude that $S_i\subset T_i$, as required.
So assume $\min(\preccurlyeq_{\Psi}, {[\![A]\!]})\nsubseteq{[\![\neg B]\!]}$. From $x \prec_{\Psi {\ast_{\mathrm{N}}}A\shortTo B} y$ we know that both
- \(a) $y\notin D(\preccurlyeq_{\Psi\ast A{\supset}B}, A\wedge B)$ or
\(b) $y\in{[\![A\wedge \neg B]\!]}$ or
\(c) $x\in D(\preccurlyeq_{\Psi\ast A{\supset}B}, A\wedge B)\cap{[\![A{\supset}B]\!]}$,
and
- \(a) $y\in D(\preccurlyeq_{\Psi\ast A{\supset}B}, A\wedge B)\cap{[\![A{\supset}B]\!]}$ iff $x\notin D(\preccurlyeq_{\Psi\ast A{\supset}B}, A\wedge B)\cap{[\![A{\supset}B]\!]}$ or
\(b) $x \prec_{\Psi\ast A{\supset}B} y$
Assume (2) (b), i.e. $x \prec_{\Psi\ast A{\supset}B} y$. By the definition of ${\ast_{\mathrm{N}}}$, either:
- $x \in \min(\preccurlyeq_{\Psi}, {[\![A{\supset}B]\!]})$ and $y \notin \min(\preccurlyeq_{\Psi}, {[\![A{\supset}B]\!]})$, or
- $x, y \notin \min(\preccurlyeq_{\Psi}, {[\![A{\supset}B]\!]})$ and $x\prec_{\Psi}y$
If (4), then from $x\prec_{\Psi} y$ and (a) of the lower bound condition, it follows that $x\prec y$ and we are done, with $x$ playing the role of $z$ in (i) and (ii) above. So assume (3). We now split into two cases:
- Assume $y\in{[\![A{\supset}B]\!]}$. Then from $x \in {[\![A{\supset}B]\!]}$ (which follows from (3)), [$(\mathrm{C}{1}^{\ast}_{\scriptscriptstyle \preccurlyeq})$]{} and $x \prec_{\Psi\ast A{\supset}B} y$, it follows that $x\prec_{\Psi} y$. From this and (a) of the lower bound condition, we then again have $x\prec y$ and we are done, with $x$ playing the role of $z$ in (i) and (ii) above.
- Assume $y\in{[\![A\wedge \neg B]\!]}$. If $x\in\min(\preccurlyeq, W)$, then, since, by (b) of the lower bound condition, $\min(\preccurlyeq,{[\![A]\!]})\subseteq {[\![B]\!]}$, we have $x \prec y$ and we are done, with $x$ playing the role of $z$ in (i) and (ii) above. So assume $x\notin\min(\preccurlyeq, W)$. Let $z\in\min(\preccurlyeq, W)$. Then $z\preccurlyeq x$. By the contrapositive of (a) of the lower bound condition, $z\preccurlyeq_{\Psi} x$. If $z\in{[\![A]\!]}$, then, by $\min(\preccurlyeq, {[\![A]\!]})\subseteq {[\![B]\!]}$, we have $z\in{[\![A{\supset}B]\!]}$. Obviously, if $z\in{[\![\neg A]\!]}$, then again $z\in{[\![A{\supset}B]\!]}$. So $z\in{[\![A{\supset}B]\!]}$. Since, furthermore, $x \in \min(\preccurlyeq_{\Psi}, {[\![A{\supset}B]\!]})$ (by (3)) and $z\preccurlyeq_{\Psi} x$, it follows from this that $z \in \min(\preccurlyeq_{\Psi}, {[\![A{\supset}B]\!]})$. By [$(\mathrm{C}{1}^{\ast}_{\scriptscriptstyle \preccurlyeq})$]{}, we then have $z\sim_{\Psi{\ast_{\mathrm{N}}}A{\supset}B} x$. From $\min(\preccurlyeq, {[\![A]\!]})\subseteq {[\![B]\!]}$ and $y\in{[\![A\wedge \neg B]\!]}$, $y\notin\min(\preccurlyeq, W)$. Hence, since $z\in\min(\preccurlyeq, W)$, it follows that $z\prec y$. So $z$ satisfies conditions (i) and (ii) above and we are done.
Assume (2)(a), i.e. $y\in D(\preccurlyeq_{\Psi\ast A{\supset}B}, A\wedge B)\cap{[\![A{\supset}B]\!]}$ iff $x\notin D(\preccurlyeq_{\Psi\ast A{\supset}B}, A\wedge B)\cap{[\![A{\supset}B]\!]}$. We now split into three cases:
- Assume (1)(a), i.e. $y\notin D(\preccurlyeq_{\Psi\ast A{\supset}B}, A\wedge B)$. Then, by (2)(a), $x\in D(\preccurlyeq_{\Psi\ast A{\supset}B}, A\wedge B)$ and $x\in{[\![A{\supset}B]\!]}$. From the first two facts, we have $x\prec_{\Psi\ast A{\supset}B}y$. Since we can see from the item immediately below that, if $y\in{[\![A\wedge \neg B]\!]}$, then we recover the required result, we can assume $y\in{[\![A{\supset}B]\!]}$. From this, the fact that $x\in{[\![A{\supset}B]\!]}$, and $x\prec_{\Psi\ast A{\supset}B}y$, we recover $x\prec_{\Psi}y$ by [$(\mathrm{C}{1}^{\ast}_{\scriptscriptstyle \preccurlyeq})$]{}. From this, by (a) of the lower bound principle, we obtain $x\prec y$ and we are done, with $x$ playing the role of $z$ in (i) and (ii) above.
- Assume (1)(b), i.e. $y\in{[\![A\wedge \neg B]\!]}$. If $x\prec y$, then we are done, with $x$ playing the role of $z$ in (i) and (ii) above. So assume $y\preccurlyeq x$, from which it follows by the contrapositive of (a) of the lower bound condition that $y\preccurlyeq_{\Psi} x$. By the definition of ${\ast_{\mathrm{N}}}$, we then have $y\preccurlyeq_{\Psi{\ast_{\mathrm{N}}}A{\supset}B} x$ iff $x\notin\min(\preccurlyeq_{\Psi},{[\![A{\supset}B]\!]})$. Now, as we show above, if $x\in\min(\preccurlyeq_{\Psi},{[\![A{\supset}B]\!]})$ and $y\in{[\![A\wedge \neg B]\!]}$, then we are done. So assume $x\notin\min(\preccurlyeq_{\Psi},{[\![A{\supset}B]\!]})$ and hence $y\preccurlyeq_{\Psi{\ast_{\mathrm{N}}}A{\supset}B} x$. Let $z\in\min(\preccurlyeq_{\Psi}, {[\![A]\!]})$. Since $\min(\preccurlyeq_{\Psi}, {[\![A]\!]})\nsubseteq{[\![\neg B]\!]}$ and $y\in{[\![A\wedge \neg B]\!]}$, it follows that $z\preccurlyeq_{\Psi} y$. By [$(\mathrm{C}{4}^{\ast}_{\scriptscriptstyle \preccurlyeq})$]{}, since $x\in {[\![A{\supset}B]\!]}$ and $y\in{[\![A\wedge \neg B]\!]}$, we then have $z\preccurlyeq_{\Psi{\ast_{\mathrm{N}}}A{\supset}B} y$. Since, furthermore, $y\preccurlyeq_{\Psi{\ast_{\mathrm{N}}}A{\supset}B} x$, we then have $z\preccurlyeq_{\Psi{\ast_{\mathrm{N}}}A{\supset}B} x$. But from $x\in D(\preccurlyeq_{\Psi\ast A{\supset}B}, A\wedge B)$, it follows, by definition, that $x\preccurlyeq_{\Psi{\ast_{\mathrm{N}}}A{\supset}B} z$ and hence $z\sim_{\Psi{\ast_{\mathrm{N}}}A{\supset}B} x$. Therefore $z$ satisfies condition (i) above.
It now remains to be shown that $z$ satisfies condition (ii) above, i.e. that $z\prec y$. We first show that it follows from (a) of the lower bound condition that $\min(\preccurlyeq_{\Psi}, {[\![A]\!]})\cap \min(\preccurlyeq, {[\![A]\!]})\neq \varnothing$. Indeed, we can show that, given (a), if $x\in \min(\preccurlyeq_{\Psi}, {[\![A]\!]})$ but $x\notin \min(\preccurlyeq, {[\![A]\!]})$, then there exists a distinct $z\in \min(\preccurlyeq_{\Psi}, {[\![A]\!]})$, such that $z\prec x$. Since we have assumed that $W$ is finite and hence so is $\min(\preccurlyeq_{\Psi}, {[\![A]\!]})$, it cannot be the case that such a $z$ exists for all $x\in \min(\preccurlyeq_{\Psi}, {[\![A]\!]})$. Hence $\min(\preccurlyeq_{\Psi}, {[\![A]\!]})\cap \min(\preccurlyeq, {[\![A]\!]})\neq \varnothing$. In view of this, we are free to assume that $z\in\min(\preccurlyeq, {[\![A]\!]})$. Since $\min(\preccurlyeq,{[\![A]\!]})\subseteq {[\![B]\!]}$ and $y\in{[\![A\wedge \neg B]\!]}$, we then have $z\prec y$, as required.
- Assume (1)(c), i.e. $x\in D(\preccurlyeq_{\Psi\ast A{\supset}B}, A\wedge B)\cap{[\![A{\supset}B]\!]}$. Then, by (2)(a), we are in one of the two cases immediately above and the required result follows.
This completes the proof of Lemma \[FlattestLI\] and hence the proof of Proposition \[NatAndRatClosure\].
Note first that the distance between TPOs over $W$ is simply the sum of the distances between the restrictions of these TPOs to the different pairs drawn from $W$.
We assume $\preccurlyeq_{\Psi{\ast_{\mathrm{N}}}A}\neq \preccurlyeq_{\Psi\star A}$. We have three cases to consider:
- $x, y\in \min(\preccurlyeq_{\Psi}, {[\![A]\!]})$: Since both ${\ast_{\mathrm{N}}}$ and $\star$ satisfy AGM, we have $\min(\preccurlyeq_{\Psi\star A}, W) = \min(\preccurlyeq_{\Psi}, {[\![A]\!]}) = \min(\preccurlyeq_{\Psi{\ast_{\mathrm{N}}}A}, W) $. Hence $d_K(\preccurlyeq_{\Psi{\ast_{\mathrm{N}}}A}\cap \{x, y\}, \preccurlyeq_{\Psi}\cap \{x, y\}) = d_K(\preccurlyeq_{\Psi\star A}\cap \{x, y\}, \preccurlyeq_{\Psi}\cap \{x, y\})$.
- $x, y\notin \min(\preccurlyeq_{\Psi}, {[\![A]\!]})$: We know that, by definition, for all $x, y \notin \min(\preccurlyeq_{\Psi}, {[\![A]\!]})$, $x \preccurlyeq_{\Psi {\ast_{\mathrm{N}}}A} y$ iff $x \preccurlyeq_{\Psi} y$, so that $\preccurlyeq_{\Psi{\ast_{\mathrm{N}}}A}\cap \{x, y\} = \preccurlyeq_{\Psi}\cap \{x, y\}$. Hence $d_K(\preccurlyeq_{\Psi{\ast_{\mathrm{N}}}A}\cap \{x, y\}, \preccurlyeq_{\Psi}\cap \{x, y\}) = 0$. Since $\preccurlyeq_{\Psi{\ast_{\mathrm{N}}}A}\neq \preccurlyeq_{\Psi\star A}$, but $\min(\preccurlyeq_{\Psi\star A}, W) = \min(\preccurlyeq_{\Psi{\ast_{\mathrm{N}}}A}, W) $, we have $\preccurlyeq_{\Psi\star A}\cap \{x, y\} \neq \preccurlyeq_{\Psi{\ast_{\mathrm{N}}}A}\cap \{x, y\} = \preccurlyeq_{\Psi}\cap \{x, y\}$. So $d_K(\preccurlyeq_{\Psi\star A}\cap \{x, y\}, \preccurlyeq_{\Psi}\cap \{x, y\}) > 0$ and hence $d_K(\preccurlyeq_{\Psi{\ast_{\mathrm{N}}}A}\cap \{x, y\}, \preccurlyeq_{\Psi}\cap \{x, y\}) < d_K(\preccurlyeq_{\Psi\star A}\cap \{x, y\}, \preccurlyeq_{\Psi}\cap \{x, y\})$.
- $x\in \min(\preccurlyeq_{\Psi}, {[\![A]\!]})$ and $y\notin \min(\preccurlyeq_{\Psi}, {[\![A]\!]})$: Since both ${\ast_{\mathrm{N}}}$ and $\star$ satisfy AGM, we have $\min(\preccurlyeq_{\Psi\star A}, W) = \min(\preccurlyeq_{\Psi}, {[\![A]\!]}) = \min(\preccurlyeq_{\Psi{\ast_{\mathrm{N}}}A}, W) $. So $x \prec_{\Psi{\ast_{\mathrm{N}}}A} y$ and $x \prec_{\Psi\star A} y$ and hence $d_K(\preccurlyeq_{\Psi{\ast_{\mathrm{N}}}A}\cap \{x, y\}, \preccurlyeq_{\Psi}\cap \{x, y\}) = d_K(\preccurlyeq_{\Psi\star A}\cap \{x, y\}, \preccurlyeq_{\Psi}\cap \{x, y\})$.
From the above, it then follows that $d_K(\preccurlyeq_{\Psi{\ast_{\mathrm{N}}}A}, \preccurlyeq_{\Psi})< d_K(\preccurlyeq_{\Psi\star A}, \preccurlyeq_{\Psi})$, as required.
We first recall the fact that [$(\mathrm{K}{8}^{\ast})$]{} is equivalent, given the remainder of the AGM postulates, to the principle of Disjunctive Inclusion:
BLAHHI: =
$\mathrm{(DI^\ast)}$ If $\neg A\notin {[\Psi\ast A\vee C ]}$, then ${[\Psi\ast A\vee C ]}\subseteq$\
${[\Psi\ast A]}$\
Consider now the initial TPO $\preccurlyeq_{\Psi}$ depicted in Figure \[fig:HanssonInit\].
It is easily verified that the two closest TPOs to this one, subject to the constraint that $A\shortTo B$ is in the associated conditional belief set, are given by, the TPOs $\preccurlyeq_{\Theta 1}$, pictured in Figure \[fig:HanssonClosest\] on the left, and $\preccurlyeq_{\Theta 2}$, pictured to its right:
Let ${[\![X]\!]} = \{2, 3, 4\}$ and ${[\![Y]\!]} = \{3, 4\}$. From the diagram, we can see that the following hold:
- \(i) $(X\vee Y)\shortTo \neg Y\notin {[\Theta_1]_{\mathrm{c}}}\cap {[\Theta_2]_{\mathrm{c}}}$
- \(ii) $(X\vee Y)\shortTo \neg 3\in {[\Theta_1]_{\mathrm{c}}}\cap {[\Theta_2]_{\mathrm{c}}}$
- \(iii) $Y\shortTo \neg 3\notin {[\Theta_1]_{\mathrm{c}}}\cap {[\Theta_2]_{\mathrm{c}}}$
But this contradicts $\mathrm{(DI^\ast)}$, since from (i), it would follow that
- For all $Z$, if $(X\vee Y)\shortTo Z\notin {[\Theta_1]_{\mathrm{c}}}\cap {[\Theta_2]_{\mathrm{c}}}$, then $Y\shortTo Z\in {[\Theta_1]_{\mathrm{c}}}\cap {[\Theta_2]_{\mathrm{c}}}$
which is inconsistent with (ii) and (iii).
Semantically, the claim translates into: If $\min(\preccurlyeq_{\Psi}, W)\subseteq {[\![A]\!]}$, then $\min(\preccurlyeq_{\Psi {\ast_{\mathrm{BG}}} A\shortTo B}, W)\subseteq{[\![A\wedge B]\!]}$. So assume $\min(\preccurlyeq_{\Psi}, W)\subseteq {[\![A]\!]}$. We will first show that
- $\min(\preccurlyeq_{\Psi {{\div}_{\mathrm{BG}}} A\shortTo \neg B}, W)\subseteq {[\![A]\!]}$.
To do so, we assume that $y\in{[\![\neg A]\!]}$ and show that there exists $x$ such that $x\prec_{\Psi {{\div}_{\mathrm{BG}}} A\shortTo \neg B}y$ and hence that $y\notin\min(\preccurlyeq_{\Psi {{\div}_{\mathrm{BG}}} A\shortTo \neg B}, W)$. From $y\in{[\![\neg A]\!]}$ and $\min(\preccurlyeq_{\Psi}, W)\subseteq {[\![A]\!]}$, we have $y\notin\min(\preccurlyeq_{\Psi}, W)$. Let $x\in \min(\preccurlyeq_{\Psi}, W)$ and so, since $y\notin\min(\preccurlyeq_{\Psi}, W)$, $x\prec_{\Psi} y$. Since $\min(\preccurlyeq_{\Psi}, W)\subseteq {[\![A]\!]}$, we have $x\in{[\![A]\!]}$. So we consider two cases:
- $x\in{[\![A\wedge B]\!]}$: Then $x\in \min(\preccurlyeq_{\Psi}, {[\![A\wedge B]\!]})$. Since $ \min(\preccurlyeq_{\Psi}, {[\![A]\!]})=\min(\preccurlyeq_{\Psi}, W)$ and $y\in{[\![\neg A]\!]}$, there is no $z\in\min(\preccurlyeq_{\Psi}, {[\![A]\!]})$ such that $y\preccurlyeq_{\Psi} z$. Hence we can apply (2)(b) of Definition \[BGcontract\] to recover $x\prec_{\Psi {{\div}_{\mathrm{BG}}} A\shortTo \neg B}y$, as required.
- $x\in{[\![A\wedge \neg B]\!]}$: Then $x\notin \min(\preccurlyeq_{\Psi}, {[\![A\wedge B]\!]})$. Furthermore, since $y\in{[\![\neg A]\!]}$, $y\notin \min(\preccurlyeq_{\Psi}, {[\![A\wedge B]\!]})$. Hence we can apply (1) of Definition \[BGcontract\] to recover $x\preccurlyeq_{\Psi{{\div}_{\mathrm{BG}}} A\shortTo \neg B} y$ iff $x\preccurlyeq_{\Psi} y$. Since $x\prec_{\Psi} y$, we have $x\prec_{\Psi {{\div}_{\mathrm{BG}}} A\shortTo \neg B}y$, as required.
We now show that, given what we have just established,
- $\min(\preccurlyeq_{(\Psi {{\div}_{\mathrm{BG}}} A\shortTo \neg B) {+_{\mathrm{BG}}}A\shortTo B}, W)\subseteq{[\![A]\!]}$.
We follow the same kind of strategy as above: we assume that $y\in{[\![\neg A]\!]}$ and show that there exists $x$ such that $x \prec_{(\Psi {{\div}_{\mathrm{BG}}} A\shortTo \neg B) {+_{\mathrm{BG}}}A\shortTo B}y$ and hence that $y\notin\min(\preccurlyeq_{(\Psi {{\div}_{\mathrm{BG}}} A\shortTo \neg B) {+_{\mathrm{BG}}}A\shortTo B} , W)$. From $y\in{[\![\neg A]\!]}$ and $\min(\preccurlyeq_{\Psi {{\div}_{\mathrm{BG}}} A\shortTo \neg B}, W)\subseteq {[\![A]\!]}$, we have $y\notin \min(\preccurlyeq_{\Psi {{\div}_{\mathrm{BG}}} A\shortTo \neg B}, W)$. Let $x\in \min(\preccurlyeq_{\Psi {{\div}_{\mathrm{BG}}} A\shortTo \neg B}, W)$ and so, since $y\notin\min(\preccurlyeq_{\Psi {{\div}_{\mathrm{BG}}} A\shortTo \neg B}, W)$, $x\prec_{\Psi {{\div}_{\mathrm{BG}}} A\shortTo \neg B} y$. Since $\min(\preccurlyeq_{\Psi {{\div}_{\mathrm{BG}}} A\shortTo \neg B}, W)\subseteq {[\![A]\!]}$, we have $x\in{[\![A]\!]}$. So we consider two cases:
- $x\in{[\![A\wedge B]\!]}$: Then $x\notin \min(\preccurlyeq_{\Psi {{\div}_{\mathrm{BG}}} A\shortTo \neg B}, {[\![A\wedge \neg B]\!]})$, and so we can infer, from condition (1) of Definition \[BGexpand\], that $x\preccurlyeq_{(\Psi {{\div}_{\mathrm{BG}}} A\shortTo \neg B) {+_{\mathrm{BG}}}A\shortTo B} y$ iff $x\preccurlyeq_{\Psi {{\div}_{\mathrm{BG}}} A\shortTo \neg B} y$. But we know that $x\prec_{\Psi {{\div}_{\mathrm{BG}}} A\shortTo \neg B} y$. So $x \prec_{(\Psi {{\div}_{\mathrm{BG}}} A\shortTo \neg B) {+_{\mathrm{BG}}}A\shortTo B} y$, as required.
- $x\in{[\![A\wedge \neg B]\!]}$: Since $y\in{[\![\neg A]\!]}$, $y\notin\min(\preccurlyeq_{\Psi {{\div}_{\mathrm{BG}}} A\shortTo \neg B}, {[\![A\wedge \neg B]\!]})$. Furthermore, we know that $x\prec_{\Psi {{\div}_{\mathrm{BG}}} A\shortTo \neg B} y$. Hence, by condition (2)(b) of Definition \[BGexpand\], we have $x \prec_{(\Psi {{\div}_{\mathrm{BG}}} A\shortTo \neg B) {+_{\mathrm{BG}}}A\shortTo B} y$, as required.
From $\min(\preccurlyeq_{(\Psi {{\div}_{\mathrm{BG}}} A\shortTo \neg B) {+_{\mathrm{BG}}}A\shortTo B}, W)\subseteq{[\![A]\!]}$, it follows that $\min(\preccurlyeq_{\Psi {\ast_{\mathrm{BG}}} A\shortTo B}, W)\subseteq{[\![A]\!]}$. In other words: $A\in{[\Psi {\ast_{\mathrm{BG}}} A\shortTo B]}$. By Success, which ${\ast_{\mathrm{BG}}}$ satisfies [@DBLP:conf/aaai/BoutilierG93 Proposition 4] , $A\shortTo B\in{[\Psi {\ast_{\mathrm{BG}}} A\shortTo B]_{\mathrm{c}}}$. By the Ramsey Test, $B\in{[ \Psi {\ast_{\mathrm{BG}}} A\shortTo B]}$. So $A\wedge B\in{[\Psi {\ast_{\mathrm{BG}}} A\shortTo B]}$, i.e. $\min(\preccurlyeq_{\Psi {\ast_{\mathrm{BG}}} A\shortTo B}, W)\subseteq{[\![A\wedge B]\!]}$, as required.
[^1]: In case we identify states with TPOs, there will exist only one such extension.
[^2]: We note that the KI postulates, in turn, subsume the DP postulates, which correspond to the special cases in which $A=\top$. Indeed, [$(\mathrm{KI}{1}^{\ast}_{\preccurlyeq})$]{} yields the conjunction of [$(\mathrm{C}{1}^{\ast}_{\scriptscriptstyle \preccurlyeq})$]{} and [$(\mathrm{C}{2}^{\ast}_{\scriptscriptstyle \preccurlyeq})$]{}, while [$(\mathrm{KI}{2}^{\ast}_{\preccurlyeq})$]{} and [$(\mathrm{KI}{3}^{\ast}_{\preccurlyeq})$]{} give us [$(\mathrm{C}{3}^{\ast}_{\scriptscriptstyle \preccurlyeq})$]{} and [$(\mathrm{C}{4}^{\ast}_{\scriptscriptstyle \preccurlyeq})$]{}, respectively.
[^3]: If, that is, we extend in the obvious manner the domain of $\ast$ to cover any TPO over some [*subset*]{} of $W$.
[^4]: We have left for a future occasion the comparison of our approach with the somewhat complex “c-revision” framework of [@DBLP:journals/amai/Kern-Isberner04], defined in terms of transformations of “conditional valuation functions”, which include probability, possibility and ranking functions as special cases.
|
---
author:
- 'S. Nakata'
- 'M. Horio'
- 'K. Koshiishi'
- 'K. Hagiwara'
- 'C. Lin'
- 'M. Suzuki'
- 'S. Ideta'
- 'K. Tanaka'
- 'D. Song'
- 'Y. Yoshida'
- 'H. Eisaki'
- 'A. Fujimori'
- Charlie Author
- Delta Author
title: |
Supplementary information for\
Nematicity in the pseudogap state of cuprate superconductors revealed by angle-resolved photoemission spectroscopy
---
0
Supplementary Note {#supplementary-note .unnumbered}
==================
0
Suppression of the superstructure in the Bi-O layer
---------------------------------------------------
We performed the X-ray diffraction (XRD) measurement on (Pb-Bi2212) which is the very sample measured in the ARPES experiment in order to check whether the superstructural modulation of (Bi2212) was suppressed by substituting Pb for Bi [@MUSOLINO20031]. We used a High Luminance In-plane Type X-ray Diffractometer SmartLab 9kW (Rigaku) at room temperature. The absence of the superstructure modulation was confirmed by the absence of the satellite peaks which indicate the superstructure modulation around the main (020)${}_\text{o}$ and (040)${}_\text{o}$ peaks. Here, the orthorhombic notation for unit cell size of $\sqrt{2}a \times \sqrt{2}a$ is employed.
Strain estimated from the lattice parameters
---------------------------------------------
After the ARPES measurements, we carried out XRD measurements in order to estimate the magnitude of the tensile strain applied to the sample by the screw shown in Fig. \[fig:S2\](c). We estimated the strain from the anisotropy of the $a$ and $b$ lattice parameters because it is difficult to measure the strain directly because of the plasticity of the device. As shown in Fig. \[fig:S2\](a), we focused on the (200) and (020) Bragg peaks of Pb-Bi2212 and each peak was fitted using a single Gaussian. Figure \[fig:S2\](b) shows the $a$ and $b$ lattice parameters estimated as $a = 3.814$ Å and $b = 3.812$ Å. The anisotropy of the crystal structure is $(a-b)/(a+b) < 0.03$ %. The anisotropy of the electronic structure induced by the distortion $\delta_\text{crystal} = 7 \times (a-b)/(a+b) < 0.2$ % expected from Eq. (S4.3) is much smaller than the asymmetry parameter $\delta$ of order 1% which we observed in the ARPES measurements shown in Fig. 3. Therefore, the enhancement of the nematicity in the pseudogap state cannot be explained due merely to the anisotropy of the crystal structure. A picture and a schematic figures of the strain device are shown in Fig \[fig:S2\](c).
Matrix-element effect
---------------------
In vacuum, the vector potential $\vec{A}(\vec{r},t)$ satisfies the wave equation, $$\nabla ^2 \vec{A}(\vec{r},t) = \frac{1}{c^2} \frac{\partial ^2 \vec{A}(\vec{r},t)}{\partial t^2}. \label{eq:2.5.1}$$ One of the solutions of this equation is $$\vec{A}(\vec{r},t) = \vec{\varepsilon}e^{i(\vec{k}\cdot\vec{r}-\omega t)}, \label{eq:2.5.2}$$ where $\vec{\varepsilon}$ denotes the polarization (the oscillating direction of the electric field), and $\vec{k}$ and $\omega$ are the wave vector and the frequency of light, respectively ($\omega = c k$). The probability of photoemission is given by $$w_{\text{i} \rightarrow \text{f}} \propto \left|\braket{f|H'|i}\right|^2,\label{eq:2.5.3}$$ where $$H' = \frac{e}{m}\vec{A}(\vec{r})\cdot \vec{p} = \frac{e}{m}\vec{\varepsilon}e^{i\vec{k}\cdot\vec{r}}\cdot \vec{p},\label{eq:2.5.4}$$ according to Fermi’s golden rule. In ARPES measurement, we often use vacuum-ultra violet, whose photon energy is about 100 eV i.e., the wave length is about 100 Å[ ]{} which is much longer than the distance between neighboring atoms in the material, so that one obtains $$e^{i\vec{k}\cdot\vec{r}} = 1 + i\vec{k}\cdot\vec{r} + \cdots \simeq 1 .\label{eq:2.5.5}$$ Then the matrix element in eq (\[eq:2.5.3\]) can be expressed as $$\begin{aligned}
\braket{f|H'|i} &\simeq& \braket{f|\frac{e}{m}\vec{\varepsilon}\cdot \vec{p}|i}\\ \label{eq:2.5.6}
&=& \frac{e}{i\hbar} \braket{f|\vec{\varepsilon}\cdot [\vec{x},H_0]|i}\\
&=& \frac{e}{i\hbar} \braket{f|\vec{\varepsilon}\cdot \vec{x}H_0^\dagger - H_0\vec{\varepsilon}\cdot \vec{x}|i}\\
&=& \frac{e}{i\hbar}(E_\text{f}-E_\text{i}) \braket{f|\vec{\varepsilon}\cdot \vec{x}|i},\label{eq:2.5.9}\end{aligned}$$ using the Hermite property of $H_0$ and the commutation relationship $$[\vec{x},H_0]=\frac{i\hbar \vec{p}}{m}.\label{eq:2.5.10}$$ In summary, the transition probability can be approximated by $$w_{\text{i} \rightarrow \text{f}} \propto \left|\braket{f|\vec{\varepsilon}\cdot \vec{x}|i}\right|^2,\label{eq:2.5.11}$$ and the matrix element can be described by that of the dipole transition.
The $k$-dependence of the intensity in $k$-space obtained by ARPES can be estimated by calculating the matrix element (eq. (\[eq:2.5.11\])) in the case that $\ket{i} = \ket{d_{x^2-y^2}}$, $\vec{E} \propto (+1,-1,0)$, which represents our experimental condition, and $\ket{f} = e^{i\vec{k}\cdot\vec{r}}$ was assumed for simplicity. Thus, we obtain $$\begin{aligned}
|\braket{f|\vec{E}\cdot \vec{r}|i}|^2 &=& \left|\braket{e^{i\vec{k}\cdot\vec{r}}|\vec{E}\cdot \vec{r}|d_{x^2-y^2}}\right|^2\\
&\propto& \left|\int d^2\vec{r} (x^2-y^2)(x-y)\cos(k_xx+k_yy)\right|^2\\
&& +\left|\int d^2\vec{r} (x^2-y^2)(x-y)\sin(k_xx+k_yy)\right|^2.\end{aligned}$$ Figure \[fig:2.5.1\] is the result of the $k$-dependence of the transition probability in the first Brillouin zone.
Energy dependence of the FWHM of the momentum distribution curves in the normal, pseudogap, and superconducting states
----------------------------------------------------------------------------------------------------------------------
Figures S3 - S5 show the MDC for the off-nodal cut and the FWHM of the two MDC peaks as functions of binding energy for the superconducting, pseudogap, and normal states, respectively. The figures show that the FWHM at $E-E_\text{F}$ = -30 meV is small compared to other binding energies, because around $E-E_\text{F}$ = -10 meV the superconducting gap opens, and in the higher binding energy region, where the quasi-particle scattering rate increases.
Effect of strain on hopping parameters
--------------------------------------
One may suspect that the anisotropy arises only from the effect of the uniaxial strain and that the temperature dependence of the anisotropy is caused by thermal expansion. Such a possibility can be excluded by the following consideration: Here, we consider a transfer integral between atomic orbitals of different atoms [*i*]{} and [*j*]{}. When the axis connecting both atoms is taken as the $z$-axis, the transfer integral between the i and j atoms $-t_{ij}(0,0,1) = \braket{i|h|j}$ is given by the Slater-Koster parameter $(l_il_j\mu)$, where $l_A$ is the azimuthal quantum number of atom $A$ $(A=a, b)$ and $\mu$ is the magnetic quantum number of atoms $i$ and $j$. In the following, $l_A = 0, 1, 2,...$ and $\mu = 0, \pm1, \pm2,...$ are noted as [*s, p, d*]{},... and $\sigma, \pi, \delta,...$, respectively.
As for the transfer integral between the $2p$ orbital of O atom and the $3d$ orbital of Cu atom, there is approximate relationship between the Slater-Koster parameter and the distance between two atoms $d$ [@Harrison] $$(pd \mu ) \propto d^{-3.5}. \label{eq:3.2.3}$$ Hopping between nearest-neighbor Cu atoms can be regarded as consisting of two hopping processes from Cu to O and from O to another Cu and, therefore, the nearest-neighbor hopping parameter $t$ in eq (1) as a function of $d$ is given by $$t { \propto (pd \mu )^2} \propto d^{-7}. \label{eq:3.2.4}$$ If we assume that $t$ depends only on $d$ for simplicity, one can obtain $$\frac{\Delta t}{t} \simeq -7\frac{\Delta d}{d}. \label{eq:3.2.5}$$ If the anisotropy of $t$ which we observed is only due to the uniaxial strain, the anisotropy parameter $\delta \left(= \frac{\Delta t}{t} \right)$ should change monotonically because the atomic distance changes monotonically with temperature by thermal expansion. However, the anisotropy parameter $\delta$ observed in our measurement was enhanced only in the pseudogap state. This means that the stress works to align the nematic domain properly and ensures that the anisotropy does not come only from the distortion of the sample.
|
---
abstract: |
Cooperation and data sharing among national networks and International Meteor Organization Video Meteor Database (IMO VMDB) resulted in European viDeo MeteOr Network Database (EDMOND). The current version of the database (EDMOND 5.0) contains 144 751 orbits collected from 2001 to 2014. In our survey we used EDMOND database in order to identify existing and new meteor showers in the database.
In the first step of the survey, using $D_{SH}$ criterion we found groups around each meteor within similarity threshold. Mean parameters of the groups were calculated and compared using a new function based on geocentric parameters ($\lambda$, $\alpha$, $\delta$, and $V_g$). Similar groups were merged into final clusters (representing meteor showers), and compared with IAU Meteor Data Center list of meteor showers. This paper presents the results obtained by the proposed methodology.
**Keywords:** Meteor showers, meteoroid stream identification methods, databases.
author:
- 'R. Rudawska$^1$, P. Matlovič$^1$, J. Tóth$^1$, L. Kornoš$^1$'
bibliography:
- 'indepEdmond.bib'
date: 'Dated: '
nocite: '[@*]'
title: |
Independent identification of meteor showers\
in EDMOND database
---
Introduction
============
Nowadays, due to the international cooperation, meteor activity is monitored over almost the entire Europe. Consequently, in recent years, multi-national networks of video meteor observers have contributed many new data. As a result, the latest version of EDMOND database contains 144 751 orbits collected from 2001 to 2014.\
In this paper, we focus on determining an independent method to associate an individual meteor in the EDMOND database with a given meteor shower. The outcome of this method is confirmation of some of the previously reported meteoroid streams listed in the IAU Meteor Data Center (IAU MDC), and finding potentially new ones.\
In Section \[sec:Methodology\] we provide necessary mathematical tools used in the independent identification procedure described in Section \[sec:IdentProc\]. Section \[sec:DataPrep\] focuses on the EDMOND data preparation used in the analysis. While in Section \[sec:Conclusion\] we present our conclusions and perspectives for future work.
Methodology {#sec:Methodology}
===========
Our cluster identification procedure links two types of meteor parameters: orbital elements ($e$, $q$, $i$, $\omega$, and $\Omega$) and geocentric parameters ($\lambda$, $\alpha$, $\delta$, and $V_g$). The first set of parameters is applied by so called D-criteria that determine similarity between orbits of meteoroids. While the second set of parameters measure similarity between meteors on the sky in a given meteor shower activity period.
Orbital similarity functions {#subsec:Func}
----------------------------
The similarity between two orbits is established by measuring the distance between them with D-criterion (a similarity function). Depending on the number of parameters that defines the similarity function, the distance between two orbits might be measured in a five- [@Southworth_1963; @Drummond_1981; @Jopek_1993], seven- [@Jopek_2008], or other dimensional phase.
In our survey, we use two functions. @Southworth_1963 criterion defined as $$\begin{aligned}
D_{SH}^{2} & = & [e_{B}-e_{A}]^2+[q_{B}-q_{A}]^{2}+\left[2\cdot\sin\frac{I_{AB}}{2}\right]^{2} \nonumber \\
& + & \left[\frac{e_{B}+e_{A}}{2}\right]^{2}\left[2\cdot\sin\frac{\pi_{AB}}{2}\right]^{2},
\label{eq:dsh}\end{aligned}$$ where $e_A$ and $e_B$ is the eccentricity, and $q_A$ and $q_B$ is the perihelion distance of two orbits, $I_{AB}$ is the angle between two orbital planes, and $\pi_{AB}$ is the distance of the longitudes of perihelia measured from the intersection of the orbits.
Geocentric similarity function
------------------------------
The second criterion we propose a new distance function $D_x$ involving geocentric parameters, defined as $$\begin{aligned}
D_x^{2} & = & w_\lambda \left(2\cdot\sin\frac{(\lambda_A-\lambda_B)}{2}\right)^{2}\nonumber \\
& + & w_\alpha\, (|V_{g_A}-V_{g_B}|+1) \left(2\cdot\sin\left(\frac{\alpha_A-\alpha_B}{2}\cdot\cos\delta_A \right)\right)^{2} \nonumber \\
& + & w_\delta\, (|V_{g_A}-V_{g_B}|+1) \left(2\cdot\sin\left(\frac{\delta_A-\delta_B}{2}\right)\right)^{2} \nonumber \\
& + & w_V\, \left(\frac{|V_{g_A}-V_{g_B}|}{V_{g_A}}\right)^2,
\label{eq:dx}\end{aligned}$$ where $\lambda_A$ and $\lambda_B$ is the solar longitude, $\alpha_A$ and $\alpha_B$ is the right ascension, $\delta_A$ and $\delta_B$ is the declination, and $V{g_A}$ and $V{g_B}$ is the geocentric velocity of two meteors. The $w_\lambda$, $w_\alpha$, $w_\delta$, and $w_V$ are suitably defined weighting factors. To normalize contribution of each term in $D_x$, we used values: $w_\lambda = \mathrm{0.17}$, $w_\alpha = \mathrm{1.20}$, $w_\delta = \mathrm{1.20}$, and $w_v = \mathrm{0.20}$. Moreover, the values of weighting factors fulfil assumption that compared geocentric parameters differ only 20$^\circ$, 3.5$^\circ$, 3.5$^\circ$, and 3.5 km/s in solar longitude, right ascension, declination and velocity, respectively.
Mean parameters {#subsec:MeanCalc}
---------------
The mean values of the orbital elements and other parameters of each found cluster were obtained as a weighted arithmetic mean, where the weights were determined by @Welch_2001 $$w_i = 1 - \frac{D_{SH}^2}{D_c^2},
\label{eq:weights}$$ and where $D_c$ is the threshold of the dynamical similarity.
The mean and standard deviation of angular elements were calculated according to @Mardia_1972. The mean value of the angular element $\epsilon$ is taken as the solution of the system of equations $$\begin{array}{l}
S = r\, \sin \epsilon\, ,\\
C = r\, \cos \epsilon\, .
\end{array}$$ Here $$\begin{array}{lll}
S = \frac{\sum\limits_{i=1}^N w_i\, \sin\epsilon_i}{\sum\limits_{i=1}^N w_i}\, ,\qquad &
C = \frac{\sum\limits_{i=1}^N w_i\, \cos\epsilon_i}{\sum\limits_{i=1}^N w_i}\, ,\qquad &
r = \sqrt{S^2+C^2}.
\end{array}$$ where $N$ is the number of members in a group/cluster, and the values of the weights $w_i$ are given by Eq. \[eq:weights\].
Data preparation {#sec:DataPrep}
================
The current version of the database, EDMOND 5.0, which contains 144 751 orbits collected from 2001 to 2014, has been split between particular years of observations. At first, we pre-ordered dataset of each year in a way that the starting orbit is with the highest orbits concentrations in the phase space of orbital elements. For this purpose we calculated $\rho$, as defined by Eq. \[eq:rho\]. As a result, for each year our input data is ordered from the highest to the lowest density $\rho$ (Figure \[fig:rho\]).
Identification procedure {#sec:IdentProc}
========================
Our method may be summarised by following steps:
Step 1:
: We probe database using $D_{SH}$ with a low threshold value $D_c=\mathrm{0.05}$. Around a meteoroid orbit is created a sphere of orbital parameters and radius $D_c$. A set of orbits within the sphere creates a group, which members are excluded from following search around another meteoroid orbit. In this way, we have independent groups around each reference meteoroid orbit. Next, for each group a weighted mean of parameters is calculated (Eq. \[eq:weights\]).
Step 2:
: Using $D_x$ we are merging groups into clusters of similar weighted means of geocentric parameters found in Step 1. Groups are associated if $D_x \leq D_c'$, where $D_c' = \mathrm{0.15}$.
To calculate mean of parameters of a new cluster, first we search for an orbit within the cluster with the highest density at a point in orbital elements space [@Welch_2001] $$\rho= \sum\limits_{i=1}^{N} \left(1 - \frac{D_{i}^2}{D_c^2}\right),
\label{eq:rho}$$ where $D_{i}$ is the value of $D_{SH}$ obtained for the $i$-th meteor in the cluster by comparing its orbit with orbits of each member of the cluster, and $D_c$ is the threshold value adjust to a studied cluster. The orbit with the highest $\rho$ is a reference to calculate the new weighted mean of parameters for cluster found in Step 2.
We repeat Step 2 using new means till groups are no longer linked into clusters.
Step 3:
: We compare parameters of known meteor showers in the IAU MDC with the final mean values of the same parameters of found clusters. For this purpose we use $D_{SH}$ criterion with $D_c = \mathrm{0.15}$. We merge clusters of the same identified meteor shower. Although, a cluster must have 5 or more members to be considered as a representation of a meteor shower.
Results {#sec:Results}
=======
The results of our survey are given in Table \[tab:meanOrbRad\] and Figure \[fig:GEM\_PER\_ORI\]-\[fig:winter\]. It contains 257 meteoroid streams identified by described earlier procedure. It summarizes the mean geocentric parameters and mean orbital parameters of the detected showers (clusters), ranked according to the IAU MDC coding. In addition, in the last column, is given value of $D_{SH}$ that determines the similarity between the mean orbit of a cluster and the orbit of a given meteoroid stream from the IAU MDC. Additionally, we visualise results separately for each seasons in Figures \[fig:spring\]-\[fig:winter\], where colours represent amount of meteors within an identified meteor shower, while the size of points corresponds to $D_{SH}$ value.\
To show efficiency of the procedure we present here results for selected cases. Figure \[fig:GEM\_PER\_ORI\] shows meteor concentrations of Geminids, Perseids, and Orionids on the sky. Those meteor showers are the most prominent showers in the EDMOND database, including over 5 000 members. Those showers have been correctly identified by our procedure. Their activity period lasts about 25-35 days. As should be expected, Geminids and Orionids are more compact in comparison to Perseids meteor shower. But in contrast, Perseids is more prominent than the other two showers.\
A given identification method may fail in separation of branches of the same meteor shower. Moreover, if two meteor showers are located in close distance to each other on the sky, an identification method based on geocentric parameters may fail and link those two showers into one. However, our identification procedure succeeds in correct separation of meteor showers in such cases. Figure \[fig:pairs\] presents example of such pairs as: Southern & Northern Taurids, December Monocerotids & November Orionids, and Northern & Southern October $\delta$ Arietids. The second listed shower of a given pair is marked in blue. As shown in Figure \[fig:pairs\] our identification procedure correctly separates meteor showers.
Conclusion {#sec:Conclusion}
==========
In order to apply $D_{SH}$ criterion in Step 3 for identification of clusters, we selected meteor showers for which their orbital elements are provided by the IAU MDC (as of June 2014). In total we used 488 meteor showers.\
We identified 257 meteor showers. The list includes 42 already established streams, 152 from the working list and 63 *pro-tempore* meteor showers. For a higher threshold ($D_{c}'' = \mathrm{0.20}$), we found 284 meteor shower in total (44, 173, and 67, respectively). However, with such threshold value some of the showers are more contaminated by the sporadic background.\
There are several clusters that require further investigation. Some of them are those meteor showers for which orbital elements are not given at the IAU MDC. We plan to identify them using $D_x$ criterion, calculate their orbital elements which will be provided to the IAU MDC subsequently. Of course, not identified yet clusters may represents also possible new meteor showers, which need our additional, detailed analysis before they will be submitted to the IAU MDC as well.
Acknowledgement
===============
The work is supported by the Slovak grant APVV-0517-12, APVV-0516-10 and VEGA 1/0225/14.
|
---
abstract: 'Labeling of sentence boundaries is a necessary prerequisite for many natural language processing tasks, including part-of-speech tagging and sentence alignment. End-of-sentence punctuation marks are ambiguous; to disambiguate them most systems use brittle, special-purpose regular expression grammars and exception rules. As an alternative, we have developed an efficient, trainable algorithm that uses a lexicon with part-of-speech probabilities and a feed-forward neural network. This work demonstrates the feasibility of using prior probabilities of part-of-speech assignments, as opposed to words or definite part-of-speech assignments, as contextual information. After training for less than one minute, the method correctly labels over 98.5% of sentence boundaries in a corpus of over 27,000 sentence-boundary marks. We show the method to be efficient and easily adaptable to different text genres, including single-case texts.'
author:
- |
David D. Palmer\
CS Division, 387 Soda Hall \#1776\
University of California, Berkeley\
Berkeley, CA 94720-1776\
[*dpalmer@cs.berkeley.edu*]{}\
Marti A. Hearst\
Xerox PARC\
3333 Coyote Hill Rd\
Palo Alto, CA 94304\
[*hearst@parc.xerox.com*]{}\
title: Adaptive Sentence Boundary Disambiguation
---
0.0pt 0.0pt
Introduction {#intro}
============
Labeling of sentence boundaries is a necessary prerequisite for many natural language processing (NLP) tasks, including part-of-speech tagging [@church88], [@cutting91], and sentence alignment [@gale93], [@kay93]. End-of-sentence punctuation marks are ambiguous; for example, a period can denote an abbreviation, the end of a sentence, or both, as shown in the examples below:
- [*The group included Dr. J.M. Freeman and T. Boone Pickens Jr.*]{}
- [*“This issue crosses party lines and crosses philosophical lines!” said Rep. John Rowland (R., Conn.).*]{}
Riley determined that in the Tagged Brown corpus [@francis82] about 90% of periods occur at the end of sentences, 10% at the end of abbreviations, and about 0.5% as both abbreviations and sentence delimiters. Note from example (2) that exclamation points and question marks are also ambiguous, since they too can appear at locations other than sentence boundaries.
Most robust NLP systems, e.g., Cutting et al. , find sentence delimiters by tokenizing the text stream and applying a regular expression grammar with some amount of look-ahead, an abbreviation list, and perhaps a list of exception rules. These approaches are usually hand-tailored to the particular text and rely on brittle cues such as capitalization and the number of spaces following a sentence delimiter. Typically these approaches use only the tokens immediately preceding and following the punctuation mark to be disambiguated. However, more context can be necessary, such as when an abbreviation appears at the end of a sentence, as seen in (3a-b):
- [*It was due Friday by 5 p.m. Saturday would be too late.*]{}
- [*She has an appointment at 5 p.m. Saturday to get her car fixed.*]{}
or when punctuation occurs in a subsentence within quotation marks or parentheses, as seen in Example (2).
Some systems have achieved accurate boundary determination by applying very large manual effort. For example, at Mead Data Central, Mark Wasson and colleagues, over a period of 9 staff months, developed a system that recognizes special tokens (e.g., non-dictionary terms such as proper names, legal statute citations, etc.) as well as sentence boundaries. From this, Wasson built a stand-alone boundary recognizer in the form of a grammar converted into finite automata with 1419 states and 18002 transitions (excluding the lexicon). The resulting system, when tested on 20 megabytes of news and case law text, achieved an accuracy of 99.7% at speeds of 80,000 characters per CPU second on a mainframe computer. When tested against upper-case legal text the algorithm still performed very well, achieving accuracies of 99.71% and 98.24% on test data of 5305 and 9396 periods, respectively. It is not likely, however, that the results would be this strong on lower-case data.[^1]
Humphrey and Zhou report using a feed-forward neural network to disambiguate periods, although they use a regular grammar to tokenize the text before training the neural nets, and achieve an accuracy averaging 93%.[^2]
Riley describes an approach that uses regression trees [@breiman84] to classify sentence boundaries according to the following features:
- Probability\[word preceding “.” occurs at end of sentence\]
- Probability\[word following “.” occurs at beginning of sentence\]
- Length of word preceeding “.”
- Length of word after “.”
- Case of word preceeding “.”: Upper, Lower, Cap, Numbers
- Case of word following “.”: Upper, Lower Cap, Numbers
- Punctuation after “.” (if any)
- Abbreviation class of words with “.”
The method uses information about one word of context on either side of the punctuation mark and thus must record, for every word in the lexicon, the probability that it occurs next to a sentence boundary. Probabilities were compiled from 25 million words of pre-labeled training data from a corpus of AP newswire. The results were tested on the Brown corpus achieving an accuracy of 99.8%.[^3]
Müller provides an exhaustive analysis of sentence boundary disambiguation as it relates to lexical endings and the identification of words surrounding a punctuation mark, focusing on text written in English. This approach makes multiple passes through the data and uses large word lists to determine the positions of full stops. Accuracy rates of 95-98% are reported for this method tested on over 75,000 scientific abstracts. (In contrast to Riley’s Brown corpus statistics, Müller reports sentence-ending to abbreviation ratios ranging from 92.8%/7.2% to 54.7%/45.3%. This implies a need for an approach that can adapt flexibly to the characteristics of different text collections.)
Each of these approaches has disadvantages to overcome. We propose that a sentence-boundary disambiguation algorithm have the following characteristics:
- The approach should be robust, and should not require a hand-built grammar or specialized rules that depend on capitalization, multiple spaces between sentences, etc. Thus, the approach should adapt easily to new text genres and new languages.
- The approach should train quickly on a small training set and should not require excessive storage overhead.
- The approach should be very accurate and efficient enough that it does not noticeably slow down text preprocessing.
- The approach should be able to specify “no opinion” on cases that are too difficult to disambiguate, rather than making underinformed guesses.
In the following sections we present an approach that meets each of these criteria, achieving performance close to solutions that require manually designed rules, and behaving more robustly. Section \[algorithm\] describes the algorithm, Section \[results\] describes some experiments that evaluate the algorithm, and Section \[summary\] summarizes the paper and describes future directions.
Our Solution {#algorithm}
============
We have developed an efficient and accurate automatic sentence boundary labeling algorithm which overcomes the limitations of previous solutions. The method is easily trainable and adapts to new text types without requiring rewriting of recognition rules. The core of the algorithm can be stated concisely as follows: the part-of-speech probabilities of the tokens surrounding a punctuation mark are used as input to a feed-forward neural network, and the network’s output activation value determines what label to assign to the punctuation mark.
The straightforward approach to using contextual information is to record for each word the likelihood that it appears before or after a sentence boundary. However, it is expensive to obtain probabilities for likelihood of occurrence of all individual tokens in the positions surrounding the punctuation mark, and most likely such information would not be useful to any subsequent processing steps in an NLP system. Instead, we use probabilities for the part-of-speech categories of the surrounding tokens, thus making training faster and storage costs negligible for a system that must in any case record these probabilities for use in its part-of-speech tagger.
This approach appears to incur a cycle: because most part-of-speech taggers require pre-determined sentence boundaries, sentence labeling must be done before tagging. But if sentence labeling is done before tagging, no part-of-speech assignments are available for the boundary-determination algorithm. Instead of assigning a single part-of-speech to each word, our algorithm uses [*the prior probabilities*]{} of all parts-of-speech for that word. This is in contrast to Riley’s method [@riley89] which requires probabilities to be found for every lexical item (since it records the number of times every token has been seen before and after a period). Instead, we suggest making use of the unchanging prior probabilities for each word already stored in the system’s lexicon.
The rest of this section describes the algorithm in more detail.
Assignment of Descriptors
-------------------------
The first stage of the process is lexical analysis, which breaks the input text (a stream of characters) into tokens. Our implementation uses a slightly-modified version of the tokenizer from the PARTS part-of-speech tagger [@church88] for this task. A token can be a sequence of alphabetic characters, a sequence of digits (numbers containing periods acting as decimal points are considered a single token), or a single non-alphanumeric character. A lookup module then uses a lexicon with part-of-speech tags for each token. This lexicon includes information about the frequency with which each word occurs as each possible part-of-speech. The lexicon and the frequency counts were also taken from the PARTS tagger, which derived the counts from the Brown corpus [@francis82]. For the word [*adult*]{}, for example, the lookup module would return the tags “JJ/2 NN/24,” signifying that the word occurred 26 times in the Brown corpus – twice as an adjective and 24 times as a singular noun.
The lexicon contains 77 part-of-speech tags, which we map into 18 more general categories (see Figure 1). For example, the tags for present tense verb, past participle, and modal verb all map into the more general “verb” category. For a given word and category, the frequency of the category is the sum of the frequencies of all the tags that are mapped to the category for that word. The 18 category frequencies for the word are then converted to probabilities by dividing the frequencies for each category by the total number of occurrences of the word.
For each token that appears in the input stream, a descriptor array is created consisting of the 18 probabilities as well as two additional flags that indicate if the word begins with a capital letter and if it follows a punctuation mark.
[l l l l]{} noun & verb & article & modifier\
conjunction & pronoun & preposition & proper noun\
number & comma or semicolon & left parentheses & right parentheses\
non-punctuation character & possessive & colon or dash & abbreviation\
sentence-ending punctuation & others & &\
The Role of the Neural Network
------------------------------
We accomplish the disambiguation of punctuation marks using a feed-forward neural network trained with the back propagation algorithm [@hertz91]. The network accepts as input $k * 20$ input units, where $k$ is the number of words of context surrounding an instance of an end-of-sentence punctuation mark (referred to in this paper as “k-context”), and $20$ is the number of elements in the descriptor array described in the previous subsection. The input layer is fully connected to a hidden layer consisting of $j$ hidden units with a sigmoidal squashing activation function. The hidden units in turn feed into one output unit which indicates the results of the function.[^4]
The output of the network, a single value between 0 and 1, represents the strength of the evidence that a punctuation mark occurring in its context is indeed the end of the sentence. We define two adjustable sensitivity thresholds $t_0$ and $t_1$, which are used to classify the results of the disambiguation. If the output is less than $t_0$, the punctuation mark is not a sentence boundary; if the output is greater than or equal to $t_1$, it is a sentence boundary. Outputs which fall between the thresholds cannot be disambiguated by the network and are marked accordingly, so they can be treated specially in later processing. When $t_0 = t_1$, every punctuation mark is labeled as either a boundary or a non-boundary.
To disambiguate a punctuation mark in a k-context, a window of $k+1$ tokens and their descriptor arrays is maintained as the input text is read. The first $k/2$ and final $k/2$ tokens of this sequence represent the context in which the middle token appears. If the middle token is a potential end-of-sentence punctuation mark, the descriptor arrays for the context tokens are input to the network and the output result indicates the appropriate label, subject to the thresholds $t_0$ and $t_1$.
Section \[results\] describes experiments which vary the size of $k$ and the number of hidden units.
Heuristics
----------
A connectionist network can discover patterns in the input data without using explicit rules, but the input must be structured to allow the net to recognize these patterns. Important factors in the effectiveness of these arrays include the mapping of part-of-speech tags into categories, and assignment of parts-of-speech to words not explicitly contained in the lexicon.
As previously described, we map the part-of-speech tags in the lexicon to more general categories. This mapping is, to an extent, dependent on the range of tags and on the language being analyzed. In our experiments, when all verb forms in English are placed in a single category, the results are strong (although we did not try alternative mappings). We speculate, however, that for languages like German, the verb forms will need to be separated from each other, as certain forms occur much more frequently at the end of a sentence than others do. Similar issuse may arise in other languages.
Another important consideration is classification of words not present in the lexicon, since most texts contain infrequent words. Particularly important is the ability to recognize tokens that are likely to be abbreviations or proper nouns. Müller gives an argument for the futility of trying to compile an exhaustive list of abbreviations in a language, thus implying the need to recognize unfamiliar abbreviations. We implement several techniques to accomplish this. For example, we attempt to identify initials by assigning an “abbreviation” tag to all sequences of letters containing internal periods and no spaces. This finds abbreviations like “J.R.” and “Ph.D.” Note that the final period is a punctuation mark which needs to be disambiguated, and is therefore not considered part of the word.
A capitalized word is not necessarily a proper noun, even when it appears somewhere other than in a sentence’s initial position (e.g., the word “American” is often used as an adjective). We require a way to assign probabilities to capitalized words that appear in the lexicon but are not registered as proper nouns. We use a simple heuristic: we split the word’s probabilities, assigning a $0.5$ probability that the word is a proper noun, and dividing the remaining $0.5$ according to the proportions of the probabilities of the parts of speech indicated in the lexicon for that word.
Capitalized words that do not appear in the lexicon at all are generally very likely to be proper nouns; therefore, they are assigned a proper noun probability of $0.9$, with the remaining $0.1$ probability distributed equally among all the other parts-of-speech. These simple assignment rules are effective for English, but would need to be slightly modified for other languages with different capitalization rules (e.g., in German all nouns are capitalized).
Experiments and Results {#results}
=======================
We tested the boundary labeler on a large body of text containing 27,294 potential sentence-ending punctuation marks taken from the Wall Street Journal portion of the ACL/DCI collection [@church91c]. No preprocessing was performed on the test text, aside from removing unnecessary headers and correcting existing errors. (The sentence boundaries in the WSJ text had been previously labeled using a method similar to that used in PARTS and is described in more detail in [@liberman92]; we found and corrected several hundred errors.) We trained the weights in the neural network with a back-propagation algorithm on a training set of 573 items from the same corpus. To increase generalization of training, a separate cross-validation set (containing 258 items also from the same corpus) was also fed through the network, but the weights were not trained on this set. When the cumulative error of the items in the cross-validation set reached a minimum, training was stopped. Training was done in batch mode with a learning rate of $0.08$. The entire training procedure required less than one minute on a Hewlett Packard 9000/750 Workstation. This should be contrasted with Riley’s algorithm which required 25 million words of training data in order to compile probabilities.
If we use Riley’s statistics presented in Section \[intro\], we can determine a lower bound for a sentence boundary disambiguation algorithm: an algorithm that always labels a period as a sentence boundary would be correct 90% of the time; therefore, any method must perform better than 90%. In our experiments, performance was very strong: with both sensitivity thresholds set to $0.5$, the network method was successful in disambiguating $98.5\%$ of the punctuation marks, mislabeling only 409 of 27,294. These errors fall into two major categories: (i)“false positive”: the method erroneously labeled a punctuation mark as a sentence boundary, and (ii) “false negative”: the method did not label a sentence boundary as such. See Table \[errors\] for details.
--------------------------------------
224 (54.8%) false positives
185 (45.2%) false negatives
409 total errors out of 27,294 items
--------------------------------------
: \[errors\]
The 409 errors from this testing run can be decomposed into the following groups:
- - false positive at an abbreviation within a title or name, usually because the word following the period exists in the lexicon with other parts-of-speech ([*Mr. Gray, Col. North, Mr. Major, Dr. Carpenter, Mr. Sharp*]{}). Also included in this group are items such as [*U.S. Supreme Court*]{} or [*U.S. Army*]{}, which are sometimes mislabeled because [*U.S.*]{} occurs very frequently at the end of a sentence as well.
- false negative due to an abbreviation at the end of a sentence, most frequently [*Inc., Co., Corp.*]{}, or [*U.S.*]{}, which all occur within sentences as well.
- false positive or negative due to a sequence of characters including a punctuation mark and quotation marks, as this sequence can occur both within and at the end of sentences.
- false negative resulting from an abbreviation followed by quotation marks; related to the previous two types.
- false positive or false negative resulting from presence of ellipsis (...), which can occur at the end of or within a sentence.
- miscellaneous errors, including extraneous characters (dashes, asterisks, etc.), ungrammatical sentences, misspellings, and parenthetical sentences.
The results presented above (409 errors) are obtained when both $t_0$ and $t_1$ are set at $0.5$. Adjusting the sensitivity thresholds decreases the number of punctuation marks which are mislabeled by the method. For example, when the upper threshold is set at $0.8$ and the lower threshold at $0.2$, the network places 164 items between the two. Thus when the algorithm does not have enough evidence to classify the items, some mislabeling can be avoided.[^5]
We also experimented with different context sizes and numbers of hidden units, obtaining the results shown in Tables \[contexts\] and \[hidden\]. All results were found using the same training set of 573 items, cross-validation set of 258 items, and mixed-case test set of 27,294 items. The “Training Error” is one-half the sum of all the errors for all 573 items in the training set, where the “error” is the difference between the desired output and the actual output of the neural net. The “Cross Error” is the equivalent value for the cross-validation set. These two error figures give an indication of how well the network learned the training data before stopping.
----------- ---------- ---------- ------- --------- -----------
Context Training Training Cross Testing Testing
Size Epochs Error Error Errors Error (%)
4-context 1731 1.52 2.36 1424 5.22%
6-context 218 0.75 2.01 409 1.50%
8-context 831 0.043 1.88 877 3.21%
----------- ---------- ---------- ------- --------- -----------
----------- ---------- ---------- ------- --------- -----------
\# Hidden Training Training Cross Testing Testing
Units Epochs Error Error Errors Error (%)
1 623 1.05 1.61 721 2.64%
2 216 1.08 2.18 409 1.50%
3 239 0.39 2.27 435 1.59%
4 350 0.27 1.42 1343 4.92%
----------- ---------- ---------- ------- --------- -----------
We observed that a net with fewer hidden units results in a drastic decrease in the number of false positives and a corresponding increase in the number of false negatives. Conversely, increasing the number of hidden units results in a decrease of false negatives (to zero) and an increase in false positives. A network with 2 hidden units produces the best overall error rate, with false negatives and false positives nearly equal.
From these data we concluded that a context of six surrounding tokens and a hidden layer with two units worked best for our test set.
After converting the training, cross-validation and test texts to a lower-case-only format and retraining, the network was able to successfully disambiguate $96.2\%$ of the boundaries in a lower-case-only test text. Repeating the procedure with an upper-case-only format produced a $97.4\%$ success rate. Unlike most existing methods which rely heavily on capitalization information, the network method is reasonably successful at disambiguating single-case texts.
Discussion and Future Work {#summary}
==========================
We have presented an automatic sentence boundary labeler which uses probabilistic part-of-speech information and a simple neural network to correctly disambiguate over $98.5\%$ of sentence-boundary punctuation marks. A novel aspect of the approach is its use of prior part-of-speech probabilities, rather than word tokens, to represent the context surrounding the punctuation mark to be disambiguated. This leads to savings in parameter estimation and thus training time. The stochastic nature of the input, combined with the inherent robustness of the connectionist network, produces robust results. The algorithm is to be used in conjunction with a part-of-speech tagger, and so assumes the availability of a lexicon containing prior probabilities of parts-of-speech. The network is rapidly trainable and thus should be easily adaptable to new text genres, and is very efficient when used in its labeling capacity. Although the systems of Wasson and Riley report slightly better error rates, our approach has the advantage of flexibility for application to new text genres, small training sets (and hence fast training times), (relatively) small storage requirements, and little manual effort. Futhermore, additional experimentation may lower the error rate.
Although our results were obtained using an English lexicon and text, we designed the boundary labeler to be equally applicable to other languages, assuming the accessibility of lexical part-of-speech frequency data (which can be obtained by running a part-of-speech tagger over a large corpus of text, if it is not available in the tagger itself) and an abbreviation list. The input to the neural network is a language-independent set of descriptor arrays, so training and labeling would not require recoding for a new language. The heuristics described in Section \[algorithm\] may need to be adjusted for other languages in order to maximize the efficacy of these descriptor arrays.
Many variations remain to be tested. We plan to: (i) test the approach on French and perhaps German, (ii) perform systematic studies on the effects of asymmetric context sizes, different part-of-speech categorizations, different thresholds, and larger descriptor arrays, (iii) apply the approach to texts with unusual or very loosely constrained markup formats, and perhaps even to other markup recognition problems, and (iv) compare the use of the neural net with more conventional tools such as decision trees and Hidden Markov Models.
[**Acknowledgements**]{} The authors would like to acknowledge valuable advice, assistance, and encouragement provided by Manuel Fähndrich, Haym Hirsh, Dan Jurafsky, Terry Regier, and Jeanette Figueroa. We would also like to thank Ken Church for making the PARTS data available, and Ido Dagan, Christiane Hoffmann, Mark Liberman, Jan Pedersen, Martin Röscheisen, Mark Wasson, and Joe Zhou for assistance in finding references and determining the status of related work. Special thanks to Prof. Franz Guenthner for introducing us to the problem.
The first author was sponsored by a GAANN fellowship; the second author was sponsored in part by the Advanced Research Projects Agency under Grant No. MDA972-92-J-1029 with the Corporation for National Research Initiatives (CNRI) and in part by the Xerox Palo Alto Research Center (PARC).
Leo Breiman, Jerome H. Friedman, Richard Olshen, and Charles J. Stone. 1984. . Wadsworth International Group, Belmont, CA.
Kenneth W. Church and Mark Y. Liberman. 1991. A status report on the [ACL/DCI]{}. In [*The Proceedings of the 7th Annual Conference of the UW Centre for the New OED and Text Research: Using Corpora*]{}, pages 84–91, Oxford.
Kenneth W. Church. 1988. A stochastic parts program and noun phrase parser for unrestricted text. In [*Second Conference on Applied Natural Language Processing*]{}, pages 136–143, Austin, TX.
Doug Cutting, Julian Kupiec, Jan Pedersen, and Penelope Sibun. 1991. A practical part-of-speech tagger. In [*The 3rd Conference on Applied Natural Language Processing*]{}, Trento, Italy.
W. Francis and H. Kucera. 1982. . Houghton Mifflin Co., New York.
William A. Gale and Kenneth W. Church. 1993. A program for aligning sentences in bilingual corpora. , 19(1):75–102.
John Hertz, Anders Krogh, and Richard G. Palmer. 1991. . Santa Fe Institute studies in the sciences of complexity. Addison-Wesley Pub. Co., Redwood City, CA.
Susanne M. Humphrey. 1989. Research on interactive knowledge-based indexing: The medindex prototype. In [*Symposium on Computer Applications in Medical Care*]{}, pages 527–533.
Martin Kay and Martin Röscheisen. 1993. Text-translation alignment. , 19(1):121–142.
Mark Y. Liberman and Kenneth W. Church. 1992. Text analysis and word pronunciation in text-to-speech synthesis. In Sadaoki Furui and Man Mohan Sondhi, editors, [*Advances in Speech Signal Processing*]{}, pages 791–831. Marcel Dekker, Inc.
Hans Müller, V. Amerl, and G. Natalis. 1980. Worterkennungsverfahren als [G]{}rundlage einer [U]{}niversalmethode zur automatischen [S]{}egmentierung von [T]{}exten in [S]{}ätze. [E]{}in [V]{}erfahren zur maschinellen [S]{}atzgrenzenbestimmung im [E]{}nglischen. , 1.
Michael D. Riley. 1989. Some applications of tree-based modelling to speech and language indexing. In [*Proceedings of the DARPA Speech and Natural Language Workshop*]{}, pages 339–352. Morgan Kaufmann.
[^1]: All information about Mead’s system is courtesy of a personal communication with Mark Wasson.
[^2]: Accuracy results were obtained courtesy of a personal communication with Joe Zhou.
[^3]: Time for training was not reported, nor was the amount of the Brown corpus against which testing was performed; we assume the entire Brown corpus was used.
[^4]: The context of a punctuation mark can be thought of as the sequence of tokens preceding and following it. Thus this network can be thought of roughly as a Time-Delay Neural Network (TDNN) [@hertz91], since it accepts a sequence of inputs and is sensitive to positional information within the sequence. However, since the input information is not really shifted with each time step, but rather only presented to the neural net when a punctuation mark is in the center of the input stream, this is not technically a TDNN.
[^5]: We will report on results of varying the thresholds in future work.
|
---
author:
- |
Yonghui Wu, Mike Schuster, Zhifeng Chen, Quoc V. Le, Mohammad Norouzi\
`yonghui,schuster,zhifengc,qvl,mnorouzi@google.com`
- |
Wolfgang Macherey, Maxim Krikun, Yuan Cao, Qin Gao, Klaus Macherey,\
Jeff Klingner, Apurva Shah, Melvin Johnson, Xiaobing Liu, Łukasz Kaiser,\
Stephan Gouws, Yoshikiyo Kato, Taku Kudo, Hideto Kazawa, Keith Stevens,\
George Kurian, Nishant Patil, Wei Wang, Cliff Young, Jason Smith, Jason Riesa,\
Alex Rudnick, Oriol Vinyals, Greg Corrado, Macduff Hughes, Jeffrey Dean\
bibliography:
- 'bnmt.bib'
title: 'Google’s Neural Machine Translation System: Bridging the Gap between Human and Machine Translation'
---
Introduction {#intro}
============
Related Work {#relwork}
============
Model Architecture {#model architecture}
==================
Segmentation Approaches
=======================
Training Criteria {#sec:RL}
=================
Quantizable Model and Quantized Inference
=========================================
Decoder
=======
Experiments and Results {#Experimental Setup}
=======================
Conclusion
==========
Acknowledgements {#acknowledgements .unnumbered}
================
We would like to thank the entire Google Brain Team and Google Translate Team for their foundational contributions to this project.
|
---
abstract: |
Test beam measurements at the test beam facilities of DESY have been conducted to characterise the performance of the EUDET-type beam telescopes originally developed within the ${\ensuremath{\textrm{EUDET}}}$ project. The beam telescopes are equipped with six sensor planes using ${\ensuremath{\textrm{MIMOSA\,26}}}$ monolithic active pixel devices. A programmable Trigger Logic Unit provides trigger logic and time stamp information on particle passage. Both data acquisition framework and offline reconstruction software packages are available. User devices are easily integrable into the data acquisition framework via predefined interfaces.
The biased residual distribution is studied as a function of the beam energy, plane spacing and sensor threshold. Its standard deviation at the two centre pixel planes using all six planes for tracking in a 6GeV electron/positron-beam is measured to be $(2.88\,\pm\,0.08)\,\upmu\meter$. Iterative track fits using the formalism of General Broken Lines are performed to estimate the intrinsic resolution of the individual pixel planes. The mean intrinsic resolution over the six sensors used is found to be $(3.24\,\pm\,0.09)\,\upmu\meter$. With a 5GeV electron/positron beam, the track resolution halfway between the two inner pixel planes using an equidistant plane spacing of 20mm is estimated to $(1.83\,\pm\,0.03)\,\upmu\meter$ assuming the measured intrinsic resolution. Towards lower beam energies the track resolution deteriorates due to increasing multiple scattering. Threshold studies show an optimal working point of the ${\ensuremath{\textrm{MIMOSA\,26}}}$ sensors at a sensor threshold of between five and six times their RMS noise. Measurements at different plane spacings are used to calibrate the amount of multiple scattering in the material traversed and allow for corrections to the predicted angular scattering for electron beams.
author:
- |
H. Jansen${}^{\textrm{a,}}$, S. Spannagel${}^{\textrm{a}}$, J. Behr${}^{\textrm{a,}}$[^1], A. Bulgheroni${}^{\textrm{b,}}$[^2], G. Claus${}^{\textrm{c}}$, E. Corrin${}^{\textrm{d,}}$[^3], D. G. Cussans${}^{\textrm{e}}$, J. Dreyling-Eschweiler${}^{\textrm{a}}$, D. Eckstein${}^{\textrm{a}}$, T. Eichhorn${}^{\textrm{a}}$, M. Goffe${}^{\textrm{c}}$, I. M. Gregor${}^{\textrm{a}}$, D. Haas${}^{\textrm{d,}}$[^4], C. Muhl${}^{\textrm{a}}$, H. Perrey${}^{\textrm{a,}}$[^5], R. Peschke${}^{\textrm{a}}$, P. Roloff${}^{\textrm{a,}}$[^6], I. Rubinskiy${}^{\textrm{a,}}$[^7], M. Winter${}^{\textrm{c}}$\
${}^{\textrm{a}}$ Deutsches Elektronen-Synchrotron DESY, Hamburg, Germany\
${}^{\textrm{b}}$ INFN Como, Italy\
${}^{\textrm{c}}$ IPHC, Strasbourg, France\
${}^{\textrm{d}}$ DPNC, University of Geneva, Switzerland\
${}^{\textrm{e}}$ University of Bristol, UK
bibliography:
- 'bibtex/refs.bib'
- 'refs.bib'
title: |
Performance of the EUDET-type\
beam telescopes
---
=1
Introduction {#sec:intro}
============
Beamlines {#sec:beamlines}
=========
Components of the EUDET-type beam telescopes {#sec:tscope}
============================================
The EUDAQ data acquisition framework {#sec:eudaq}
====================================
Offline analysis and reconstruction using EUTelescope {#sec:offline}
=====================================================
Track resolution studies {#sec:trackres}
========================
Considerations for DUT integrations {#sec:dutintegration}
===================================
Conclusion {#sec:conclusion}
==========
Data and materials {#data-and-materials .unnumbered}
==================
The datasets supporting the conclusions of this article are available from reference [@jansen_data]. The software used is available from the github repositories: 1) <https://github.com/eutelescope/eutelescope>, 2) <https://github.com/simonspa/eutelescope/>, branch *scattering* and 3) <https://github.com/simonspa/resolution-simulator>. For the presented analysis, these specific tags have been used: [@jansen_2016_49065] and [@spannagel_2016_48795].
Competing interests {#competing-interests .unnumbered}
===================
The authors declare that they have no competing interests.
Acknowledgements {#acknowledgements .unnumbered}
================
We are indebted to Claus Kleinwort for his counsel and numerous discussions. Also, we would like to thank Ulrich Kötz. The test beam support at DESY is highly appreciated. This work was supported by the Commission of the European Communities under the FP7 Structuring the European Research Area, contract number RII3-026126 (EUDET). Furthermore, strong support from the Helmholtz Association and the BMBF is acknowledged.
[^1]: Now at Institut für Unfallanalysen, Hamburg, Germany
[^2]: Now at KIT, Karlsruhe, Germany
[^3]: Now at SwiftKey, London, UK
[^4]: Now at SRON, Utrecht, Netherlands
[^5]: Now at Lund University, Sweden
[^6]: Now at CERN, Geneva, Switzerland
[^7]: Now at CFEL, Hamburg, Germany
|
---
abstract: 'In this paper, we analyze the fundamental tradeoff of diversity and multiplexing in multi-input multi-output (MIMO) channels with imperfect channel state information at the transmitter (CSIT). We show that with imperfect CSIT, a higher diversity gain as well as a more efficient diversity-multiplexing tradeoff (DMT) can be achieved. In the case of multi-input single-output (MISO)/single-input multi-output (SIMO) channels with $K$ transmit/receive antennas, one can achieve a diversity gain of $d(r)=K(1-r+K\alpha)$ at spatial multiplexing gain $r$, where $\alpha$ is the *CSIT quality* defined in this paper. For general MIMO channels with $M$ ($M>1$) transmit and $N$ ($N>1$) receive antennas, we show that depending on the value of $\alpha$, different DMT can be derived and the value of $\alpha$ has a great impact on the achievable diversity, especially at high multiplexing gains. Specifically, when $\alpha$ is above a certain threshold, one can achieve a diversity gain of $d(r)=MN(1+MN\alpha)-(M+N-1)r$; otherwise, the achievable DMT is much lower and can be described as a collection of discontinuous line segments depending on $M$, $N$, $r$ and $\alpha$. Our analysis reveals that imperfect CSIT significantly improves the achievable diversity gain while enjoying high spatial multiplexing gains.'
author:
- 'Xiao Juan Zhang and Yi Gong [^1]'
title: 'On the Achievable Diversity-Multiplexing Tradeoff in MIMO Fading Channels With Imperfect CSIT'
---
Diversity-multiplexing tradeoff, MIMO, channel state information, channel estimation.
Introduction
============
The performance of wireless communications is severely degraded by channel fading caused by multipath propagation and interference from other users. Fortunately, multiple antennas can be used to increase diversity to combat channel fading. Antenna diversity where sufficiently separated or different polarized multiple antennas are put at either the receiver, the transmitter, or both, has been widely considered [@Paulraj], [@Lek]. On the other hand, multi-antenna channel fading can be beneficial since it can increase the degrees of freedom of the channel and thus can provide spatial multiplexing gain. It is shown in [@FoschiniGans] that the spatial multiplexing gain in a multi-input and multi-output (MIMO) Rayleigh fading channel with $M$ transmit and $N$ receive antennas increases linearly with $\min(M,N)$ if the channel knowledge is known at the receiver. As MIMO channels are able to provide much higher spectral efficiency and diversity gain than conventional single-antenna channels, many MIMO schemes have been proposed, which can be classified into two major categories: spatial multiplexing oriented (e.g., Layered space-time architecture [@Foschini]), and diversity oriented (e.g., space-time trellis coding [@Tarokh], [@Yi], and space-time block coding [@Alamouti], [@Orthogonal]).
For a MIMO scheme realized by a family of codes {$C(\rho)$} with signal-to-noise ratio (SNR) $\rho$, rate $R(\rho)$ (bits per channel use), and maximum-likelihood (ML) error probability $\mathcal{P}_e(\rho)$, Zheng and Tse defined in [@IEEErelay:DMT] the spatial multiplexing gain $r$ as $r\triangleq\lim_{\rho\rightarrow\infty} \frac{R(\rho)}{\log \rho}$ and the diversity gain $d$ as $d\triangleq-\lim_{\rho\rightarrow\infty}
\frac{\mathcal{P}_e(\rho)}{\log \rho}$. Under the assumption of independent and identically distributed (i.i.d.) quasi-static flat Rayleigh fading channels where the channel state information (CSI) is known at the receiver but not at the transmitter, for any integer $r\leq \min(M,N)$, the optimal diversity gain $d^*(r)$ (the supremum of the diversity gain over all coding schemes) is given by [@IEEErelay:DMT] $$\label{DMT} d^*(r)=(M-r)(N-r)$$ provided that the code length $L\geq M+N-1$. The diversity-multiplexing tradeoff (DMT) in (\[DMT\]) provides a theoretical framework to analyze many existing diversity-oriented and multiplexing-oriented MIMO schemes. It indicates that the diversity gain cannot be increased without penalizing the spatial multiplexing gain and vice versa. This pioneering work has generated a lot of research activities in finding DMT for other important channel models [@IEEErelay:MAC]-[@IEEErelay:FDTCR] and designing space-time codes that achieve the desired tradeoff of diversity and multiplexing gain [@IEEErelay:ODASC]-[@IEEErelay:RDTST]. The DMT analysis was extended to multiple-access channels in [@IEEErelay:MAC]. The automatic retransmission request (ARQ) scheme is shown to be able to significantly increase the diversity gain by allowing retransmissions with the aid of decision feedback and power control in block-fading channels [@IEEErelay:ARQ]. The work in [@IEEErelay:FDPRM] investigated the diversity performance of rate-adaptive MIMO channels at finite SNRs and showed that the achievable diversity gains at realistic SNRs are significantly lower than those at asymptotically high SNRs. The impact of spatial correlation on the DMT at finite SNRs was further studied in [@IEEErelay:FDTCR].
It is natural to expect that the DMT can be further enhanced through power and/or rate adaptation if the transmitter has channel knowledge. If the CSI at the transmitter (CSIT) is perfectly known, there will be no outage even in slow fading channels since it is always able to adjust its power or rate adaptively according to the instantaneous channel conditions. For example, it can transmit with a higher power or lower rate when the channel is poor and a lower power or higher rate when the channel is good. However, in practice the CSIT is almost always imperfect due to imperfect CSI feedback from the receiver or imperfect channel estimation at the transmitter through pilots. The work in [@IEEErelay:TCTT] showed that the transmitter training through pilots significantly increases the achievable diversity gain in a single-input multi-output (SIMO) link. In [@IEEErelay:OFT], the authors quantified the CSIT quality as $\alpha=-\log \sigma_{e}^2/\log \rho$, where $\sigma_{e}^2$ is the variance of the CSIT error, and showed that using rate adaptation, one can achieve an average diversity gain of $\bar{d}(\alpha,\bar{r})=(1+\alpha-\bar{r})K$ in SIMO/MISO links, where $K=\max(M,N)$ and $\bar{r}$ is the average multiplexing gain. Note that setting $\alpha=1$ and ignoring the multiplexing gain loss due to training symbols directly yields the result in [@IEEErelay:TCTT]. For general MIMO channels, the achievable DMT with partial CSIT is characterized in [@IEEErelay:TT], where the partial CSIT is obtained using quantized channel feedback.
In this paper, we analyze the fundamental DMT in MIMO channels and show that with power adaptation, imperfect CSIT provides significant additional diversity gain over (\[DMT\]). The imperfect CSIT considered in this paper is due to channel estimation error at the transmitter side. In the case of MISO/SIMO channels, we show that with power adaptation (under an average sum power constraint), one can achieve a higher diversity gain than that with rate adaptation in [@IEEErelay:OFT], where the authors assumed peak power transmission and thus no temporal power adaptation is considered therein. Specifically, we prove that with a CSIT quality $\alpha$, the achievable diversity gain is $d(r)=K\left(1-r+K\alpha\right)$. It has been shown in our earlier work [@SIMOMISO] that this is actually the *optimal* DMT in SIMO/MISO channels with CSIT quality $\alpha$. For general MIMO channels ($M>1$, $N>1$), we show in this paper that depending on the value of $\alpha$, different DMT can be derived and the value of $\alpha$ has a great impact on the achievable diversity, especially at high multiplexing gains. Specifically, when $\alpha$ is above a certain threshold, one can achieve a diversity gain of $d(r)=MN(1+MN\alpha)-(M+N-1)r$; otherwise, the achievable DMT is much lower and can be described as a collection of discontinuous line segments depending on $M$, $N$, $r$ and $\alpha$. It is noted that an independent and concurrent work recently reported in [@NoisyCSIT] shares some similar results. However, we wish to emphasize that our CSIT model and the involved analysis towards the achievable DMT are different from those in [@NoisyCSIT]. The noisy CSIT therein is based on the *channel mean feedback* model in [@MeanFeedback] and an example of obtaining CSIT through delayed feedback is provided, whereas the CSIT in our work is estimated from reverse channel pilots using ML estimation at the transmitter. As the variance of the channel estimation error is inversely proportional to the pilots’ SNR [@IEEErelay:SDICE], the CSIT quality $\alpha$ is naturally connected to the reverse channel power consumption and any value of $\alpha$ can be achieved by scaling the reverse channel transmit power. In addition, our paper provides detailed closed-form solutions to the achievable DMT, which offers great insight and depicts directly what the DMT curve with imperfect CSIT looks like.
*Notations*: $\mathcal{R}^N$ denotes the the set of real $N$-tuples, and $\mathcal{R}^{N+}$ denotes the set of non-negative $N$-tuples. Likewise, $\mathcal{C}^{N\times M}$ denotes the set of complex $N\times M$ matrices. For a real number $x$, $(x)^+$ denotes $\max(x,0)$, while for a set $\mathcal{O}\subseteq \mathcal{R}^N$, $\mathcal{O}^+$ denotes $\mathcal{O}\cap\mathcal{R}^{N+}$. ${\mathcal{O}}^c$ denotes the complementary set of $\mathcal{O}$ and $\emptyset$ denotes the empty set. $|\mathcal{O}|$ denotes the cardinality of set $\mathcal{O}$. $x\in (a, b]$ denotes that the scalar $x$ belongs to the interval $a < x \leq b$. Likewise, $x\in
[a, b]$ is similarly defined. $\mathcal{CN}(0,\sigma^2)$ denotes the complex Gaussian distribution with mean 0 and variance $\sigma^2$. The superscripts $^*$ and $^{\dag}$ denote the complex conjugate and conjugate transpose, respectively. $\|\cdot\|_F^2$ denotes the matrix Frobenius norm and $\textbf{I}_N$ denotes the $N\times N$ identity matrix. $E\{\cdot\}$ denotes the expectation operator and $\log(\cdot)$ denotes the base-2 logarithm. $f(\rho)\doteq \rho^b$ denotes that $b$ is the exponential order of $f(\rho)$, i.e., $\lim_{\rho\rightarrow\infty} {\log(f(\rho))}/{\log(\rho)}=b$. Likewise, ${\buildrel \textstyle .\over \leq}$ is similarly defined. Finally, for matrix $\mathbf{A}$, $\mathbf{A} \succeq 0$ denotes that $\mathbf{A}$ is positive semidefinite; if $\succeq$ is used with a vector, it denotes the componentwise inequality.
The rest of this paper is organized as follows. In section II, we describe the channel model. In section III, we propose a power adaptation scheme based on imperfect CSIT and present the main result on the achievable DMT. The achievability proof of the presented DMT is given in Section IV. Section V provides some discussions. Finally, Section VI concludes this paper.
Channel Model
=============
We consider a point-to-point TDD wireless link with $M$ transmit and $N$ receive antennas, where the downlink and uplink channels are reciprocal. Without loss of generality, we assume $M\geq N$ in this paper. As shown in [@IEEErelay:DMT], this assumption does not affect the DMT result. We also consider quasi-static Rayleigh fading channels, where the channel gains are constant within one transmission block of $L$ symbols, but change independently from one block to another. We assume that the channel gains are independently complex circular symmetric Gaussian with zero mean and unit variance. The channel model, within one block, can be written as $$\label{SIMO_channel}
\textbf{Y}=\sqrt{{P}/M}\textbf{H}\textbf{X}+\textbf{W}$$ where $\textbf{H}=\{h_{n,m}\}\in\mathcal{C}^{N\times M }$ with $h_{n,m}$, $m=1,2,\ldots,M$, $n=1,2,\ldots,N$, being the channel gain from the $m$-th transmit antenna to the $n$-th receive antenna; $\textbf{X}=\{X_{m,l}\} \in \mathcal{C}^{M\times L}$ with $X_{m,l}$, $m=1,2,\ldots,M$, $l=1,2,\ldots,L$, being the symbol transmitted from antenna $m$ at time $l$; $\textbf{Y}=\{Y_{n,l}\}\in
\mathcal{C}^{N\times L}$ with $Y_{n,l}$, $n=1,2,\ldots,N$, $l=1,2,\ldots,L$, being the received signal at antenna $n$ and time $l$; the additive noise $\textbf{W}\in \mathcal{C}^{N\times L}$ has i.i.d. entries $W_{n,l}\sim \mathcal{CN}(0,\sigma^2)$; $P$ is the instantaneous transmit power while the average energy of $X_{m,l}$ is normalized to be 1. Letting $\bar{P}$ denote the average sum power constraint, we have $E\{P\}=\bar{P}$. So, the average SNR at the receive antenna is given by $\rho=\bar{P}/\sigma^2$.
We assume that the receiver has perfect CSI $\textbf{H}\in
\mathcal{C}^{N\times M}$, but the transmitter has imperfect CSIT $\hat{\textbf{H}}\in \mathcal{C}^{N\times M}$, which is estimated from reverse channel pilots using ML estimation. Thus, $\hat{\textbf{H}}$ can be modeled as [@IEEErelay:SDICE]-[@IEEErelay:SDVA] $$\hat{\textbf{H}}=\textbf{H}+\textbf{E}$$ where the channel estimation error $\textbf{E}\in
\mathcal{C}^{N\times M}$ has i.i.d. entries $E_{n,m}\sim
\mathcal{CN}(0,\sigma_{e}^2)$, $n=1,2,\ldots,N$, $m=1,2,\ldots,M$, and is independent of $\textbf{H}$. The quality of $\hat{\textbf{H}}$ is thus characterized by $\sigma_{e}^2$. If $\sigma_{e}^2 = 0$, the transmitter has perfect channel knowledge; if $\sigma_{e}^2$ increases, the transmitter has less reliable channel knowledge. We follow [@IEEErelay:OFT] to quantify the channel quality at the transmitter. The transmitter is said to have a *CSIT quality* $\alpha$, if $\sigma_{e}^2\doteq
\rho^{-\alpha}$. The definition of $\alpha$ builds up a connection between the imperfect channel knowledge at transmitters and the forward channel SNR, $\rho$. Since the variance of the channel estimation error is inversely proportional to the pilots’ SNR, i.e., $\sigma_{e}^2\propto (SNR_{pilot})^{-1}$ [@IEEErelay:SDICE], any value of $\alpha$ can be achieved by scaling the reverse channel power such that $SNR_{pilot}\doteq\rho^{\alpha}$. One can see that the selection of $\alpha$ value actually determines the cost of obtaining CSIT in terms of the reverse channel power consumption. When $\alpha=0$, the reverse channel SNR does not scale with $\rho$, which means that the pilot power is fixed or limited; when $0<\alpha<1$, the reverse channel SNR relative to $\rho$ is asymptotically zero; when $\alpha=1$, the reverse channel SNR scales with $\rho$ at the same rate; when $\alpha>1$, the reverse channel SNR as compared to the forward channel SNR, $\rho$, is asymptotically unbounded [@IEEErelay:OFT]. In the sequel, we will study how the pilot power, or equivalently the CSIT quality $\alpha$, affects the fundamental tradeoff of diversity and multiplexing in the considered channel. Before presenting our main results, we give the following probability density function (pdf) expressions and some preliminary results that will be used later.
For an $N\times M$ ($N\leq M$) random matrix $\textbf{A}$ with i.i.d. entries $\sim \mathcal{CN}(0,1)$, let $0<\lambda_1 \leq
\lambda_2 \leq ... \leq \lambda_N$ denote the ordered nonzero eigenvalues of $\textbf{A}\textbf{A}^{\dag}$. Letting $v_n$ denote the exponential order of $1/\lambda_n$ for all $n$, the pdf of the random vector $\textbf{v}=[v_1,...,v_N]$ is given by [@IEEErelay:Eigen] $$\label{eqnpv2}
\begin{split}
p(\textbf{v})&=\lim_{\rho\rightarrow\infty}\xi^{-1}(\log \rho)^N\prod_{n=1}^N \rho^{-(M-N+1)v_n} \prod_{j>n}^N (\rho^{-v_n}-\rho^{-v_j})^2\exp\left(-\sum_{n=1}^N \rho^{-v_n}\right)\\
&\doteq
\begin{cases}
0,\ \text{for any $v_n<0$}\\
\prod_{n=1}^N \rho^{-(2n-1+M-N)v_n},\ \text{for all $v_n\geq0$}
\end{cases}
\end{split}$$where $\xi$ is a normalizing constant. Hence, the probability $\mathcal{P}_\mathcal{O}$ that $(v_1,...,v_N)$ belongs to set $\mathcal{O}$ can be characterized by $$\label{bb} \mathcal{P}_\mathcal{O}\doteq \rho^{-d_{\mathcal{O}}},\
\text{for}\ d_{\mathcal{O}}= \inf_{(v_1,...,v_N)\in \mathcal{O}^+}
\sum_{n=1}^N (2n-1+M-N)v_n$$provided that $\mathcal{O}^+$ is not empty.
Letting $\textbf{a}=[a_1,a_2...,a_N]$, $0<a_1\leq a_2\leq...\leq
a_N$, $\textbf{b}=[b_1,b_2...,b_N]$, $0<b_1\leq b_2\leq...\leq b_N$, and $\textbf{c}=[c_1,c_2...,c_N]$, $0<c_1\leq c_2\leq...\leq c_N$, denote the eigenvalue vectors of $\textbf{H}\textbf{H}^{\dag}$, $\hat{\textbf{H}}\hat{\textbf{H}}^{\dag}$ and $
\textbf{E}\textbf{E}^{\dag}$, respectively, the pdfs of $\textbf{a}$, $\textbf{b}$, and $\textbf{c}$ can be shown to be $$\begin{aligned}
\label{eqn_9}
&&p(\textbf{a})=\xi^{-1}\prod_{n=1}^N a_n^{M-N} \prod_{n<j}^N (a_n-a_j)^2\exp\left(-\sum_{n=1}^N a_n\right)\\
&&p(\textbf{b})=\hat{\xi}^{-1}
\prod_{n=1}^N b_n^{M-N} \prod_{n<j}^N (b_n-b_j)^2\exp\left(-\frac{1}{1+\sigma_e^2}\sum_{n=1}^N b_n\right) \\
\label{eqn_11} &&p(\textbf{c})=\tilde{\xi}^{-1} \prod_{n=1}^N
c_n^{M-N} \prod_{n<j}^N
(c_n-c_j)^2\exp\left(-\frac{1}{\sigma_e^2}\sum_{n=1}^N c_n\right)\end{aligned}$$ where $\hat{\xi}^{-1}=\xi^{-1} (1+\sigma_e^2)^{-MN}$ and $\tilde{\xi}^{-1}=\xi^{-1} (\sigma_e^2)^{-MN}$.
Main Result on DMT
==================
The ML error probability $\mathcal{P}_e(\rho)$ of the channel described in (\[SIMO\_channel\]) is closely related to the associated packet outage probability $\mathcal{P}_{out}$, which is defined as the probability that the instantaneous channel capacity falls below the target data rate $R(\rho)$. In fact, the error probability of an ML decoder which utilizes a fraction of the codeword such that the mutual information between the received and transmitted signals exceeds $LR(\rho)$ (no outage), averaged over the ensemble of random Gaussian codes, can be made arbitrarily small provided that the codeword length $L$ is sufficiently large [@IEEErelay:OAD]. We will thus leverage on the outage probability to examine the achievable diversity gain. If the transmitter has perfect CSIT, it may adopt the optimal power adaptation according to the actual instantaneous channel gain such that no outage will occur. With only the imperfect CSIT, in order to mitigate the channel uncertainty, we propose the following power adaptation scheme.
Given $\hat{\textbf{H}}$, the transmitter transmits with power $$\label{PA}
P(\hat{\textbf{H}})=\frac{ \kappa\bar{P}}{\left(\prod_{n=1}^N
b_n^{2n-1+M-N}\right)^t}$$where $\kappa=\hat{\xi}\prod_{n=1}^N \left[(2n-1+M-N)(1-t)\right]$ and $t$ ($0\leq t<1$) can be chosen arbitrarily close to $1$.
It is shown in Appendix A that the above power adaptation scheme satisfies the sum power constraint $E\{P(\hat{\textbf{H}})\}=
\bar{P}$. We believe that given the CSIT quality of $\alpha$, this power adaptation scheme is the optimal power adaptation scheme that maximizes the achievable diversity gain of a MIMO fading channel.
Consider a MIMO channel with $M$ transmit and $N$ receive antennas ($M\geq N$) and CSIT quality of $\alpha$. If the block length $L\geq
M+N-1$, the achievable DMT using the power adaptation scheme in Proposition 1 is characterized by
Case 1: If $N=1$ or $\alpha \geq \frac{1}{M-1}$, then $$\begin{split}
d(r)=MN(1+MN\alpha)-(M+N-1)r.
\end{split}
\label{eq14}$$
Case 2: Otherwise, the achievable DMT is a collection of discontinuous line segments, with the two end points of line segment $d_k(r)$ ($k\in \mathcal{B}$) given by $$\label{eqn_446}
\begin{split}
\text{Left end: }&d_k(r)=k(M-N+k){\tau}(k),\ \text{for}\ r=(N-k){\tau}(k)\\
\text{Right end: }&d_k(r)=((N-k)(k-N-1)+MN){\tau}(k)-(2k-1+M-N)(N-{\mathcal{I}}(k)){\tau}({\mathcal{I}}(k)),\\
&\ \ \ \ \ \ \ \ \ \ \text{for}\
r=(N-{\mathcal{I}}(k)){\tau}({\mathcal{I}}(k))
\end{split}$$where $$\mathcal{B}=\left\{k\middle | (M-N+k)(N-k)<1/\alpha,
(N-k){\tau}(k)<(N-\bar{k}){\tau}(\bar{k}), \forall \bar{k}<k,
k=1,...,N\right\}, \nonumber$$ ${\tau}(k)=1+k\alpha(M-N+k)$ and ${\mathcal{I}}(k)= \max_{\bar{k}\in
{\mathcal{B}},\bar{k}<k} \bar{k}$.
For example, when $M=N=2$ and $\alpha<1$, the DMT curve consists of two discontinuous line segments which are $(0,16\alpha+4)$—$(1+\alpha,13\alpha+1)$ and $(1+\alpha,1+\alpha)$—$(2,2\alpha)$. When $r=1+\alpha$, the achievable diversity gain is $d(r)=1+\alpha$ instead of $13\alpha+1$. From Theorem 1, we can get $d(0)=MN(1+MN\alpha)$ and $d(N)=p\alpha(M-N+p)(MN+(p-N)(N-p+1)))-p^2+p$ where $p=\min_{k\in
{\mathcal{B}}} k$. If $\alpha <\frac{1}{(N-1)(M-N+1)}$, which indicates $1\in {\mathcal{B}}$, we will have $d(N)=\alpha
N(M-N+1)^2$.
Proof of Theorem 1
==================
The proof involves the computation of the asymptotic ML error probability at high SNRs. We will first derive a lower bound of the SNR exponent of the outage probability, denoted as $d_\mathcal{O}(r)$, and then show that using a random coding argument the SNR exponent of the error probability is no less than $d_\mathcal{O}(r)$ if $L\geq M+N-1$.
Derivation of $d_{\mathcal{O}}(r)$
----------------------------------
Optimizing over all input distributions, which can be assumed to be Gaussian with a covariance matrix $\textbf{Q}$ without loss of optimality, the outage probability of a MIMO channel with transmit power $P(\hat{\textbf{H}})$ is given by $$\label{OutageProb} \mathcal{P}_{out}=\inf_{\textbf{Q}\succeq0,
tr(\textbf{Q})\leq M}\mathcal{P}\left(\log
\text{det}\left(\textbf{I}_N+\frac{P(\hat{\textbf{H}})}{M\sigma^2}\textbf{H}\textbf{Q}\textbf{H}^{\dag}\right)<R(\rho)
\right)$$ where $\mathcal{P}(\cdot)$ denotes the probability of an event. It is shown in [@IEEErelay:DMT] that one can get an upper bound and a lower bound on the outage probability by picking $\textbf{Q}=\textbf{I}_{M}$ and $\textbf{Q}=M\textbf{I}_M$, respectively, and the two bounds converge in the high SNR regime. Therefore, without loss of generality, we consider $\textbf{Q}=\textbf{I}_M$. Substituting (\[PA\]) in (\[OutageProb\]), we have $$\begin{split}
\mathcal{P}_{out}&=\mathcal{P}\left(\log \text{det}\left(\textbf{I}_N+\frac{\rho \kappa}{M\prod_{n=1}^N b_n^{(2n-1+M-N)t}}\textbf{H}\textbf{H}^{\dag}\right)<R(\rho) \right)\\
&=\mathcal{P}\left(\log\prod_{n=1}^N\left(1+\frac{\rho \kappa
a_n}{M\prod_{n=1}^N b_n^{(2n-1+M-N)t}}\right)<R(\rho) \right).
\end{split}$$
The eigenvalues of $\hat{\textbf{H}}\hat{\textbf{H}}^{\dag}$, $\textbf{H}\textbf{H}^{\dag}$ and $\textbf{E} \textbf{E}^{\dag}$ have the following relationship $$\label{bac} b_n \leq 2 (a_n+c_N), \ n=1,2,...,N.$$
We obviously have the following equality $$(\textbf{H}+\textbf{E})(\textbf{H}+\textbf{E})^{\dag}+(\textbf{H}-\textbf{E})(\textbf{H}-\textbf{E})^{\dag}=2 (\textbf{H}\textbf{H}^{\dag}+\textbf{E}\textbf{E}^{\dag})$$where both $(\textbf{H}+\textbf{E})(\textbf{H}+\textbf{E})^{\dag}$ and $(\textbf{H}-\textbf{E})(\textbf{H}-\textbf{E})^{\dag}$ are positive semidefinite matrices. We denote the vector of eigenvalues of $(\textbf{H}\textbf{H}^{\dag}+\textbf{E}\textbf{E}^{\dag})$ as $\textbf{d}=[d_1,...,d_N]$ with $d_1\leq d_2\leq...\leq d_N$. Since the eigenvalues of the sum of two positive semidefinite matrices are at least as large as the eigenvalues of any one of the positive semidefinite matrices [@IEEErelay:Math], we have $ b_n\leq 2
d_n,\ n=1,2,...,N$. Further, using the relationship of the eigenvalues of the sum of Hermitian matrices, we get $a_n + c_1 \leq
d_n \leq a_n +c_N,\ n=1,2,...,N$. It thus directly leads to (\[bac\]).
With Lemma 1, the outage probability is upper bounded by $$\label{out} \mathcal{P}_{out}\leq
\mathcal{P}\left[\log\prod_{n=1}^N \left(1+ \frac{\rho \kappa
a_n}{M\prod_{n=1}^N (2a_n+2c_N)^{(2n-1+M-N)t}}
\right)<R(\rho)\right] .
$$
Let $v_n$ and $u_n$ denote the exponential orders of $1/{a_n}$ and $1/{c_n}$, respectively, i.e., $v_n=-\lim_{\rho\rightarrow\infty}\frac{\log(a_n)}{\log(\rho)}$, $u_n=-\lim_{\rho\rightarrow\infty}\frac{\log(c_n)}{\log(\rho)}$. Using (\[eqnpv2\]), (\[eqn\_9\]) and (\[eqn\_11\]), the pdfs of the random vector $\textbf{v}=[v_1,...,v_N]$ and $\textbf{u}=[u_1,...,u_N]$ can be shown to be $$\begin{aligned}
&&p(\textbf{v})\doteq
\begin{cases}
0,\ \text{for any $v_n<0$}\\
\prod_{n=1}^N \rho^{-(2n-1+M-N)v_n},\ \text{for all $v_n\geq0$}
\end{cases}\label{pvn1}\\
&&p(\textbf{u})\doteq
\begin{cases}
0,\ \text{for any $u_n<\alpha$}\\
\prod_{n=1}^N \rho^{-(2n-1+M-N)(u_n-\alpha)},\ \text{for all
$u_n\geq \alpha$}.
\end{cases}\label{pun1}\end{aligned}$$
At high SNRs, with (\[pvn1\]) and (\[pun1\]), (\[out\]) becomes $$\label{eqn29} \mathcal{P}_{out}\leq \mathcal{P}\left[\sum_{n=1}^N
\left(1-v_n+\sum_{n=1}^N t(2n-1+M-N)\min(v_n,u_N)\right)^+
<r\right].$$
So, the outage event $\mathcal{O}$ in (\[eqn29\]) is the set of $\{v_1,\ldots,v_N,u_1,\ldots,u_N\}$ that satisfies $$\label{eqn28} \sum_{n=1}^N \left(1-v_n+\sum_{n=1}^N
t(2n-1+M-N)\min(v_n,u_N)\right)^+ <r$$ where $ v_n\geq 0, u_n\geq \alpha\geq0, n=1,2,...,N$.
According to (\[bb\]), we have $$\mathcal{P}_{out}\leq \mathcal{P}_\mathcal{O}\doteq
\rho^{-d_{\mathcal{O}}(r)}
$$ where $d_\mathcal{O}(r)$ serves as a lower bound of the SNR exponent of $\mathcal{P}_{out}$ and is given by $$\label{DMTNEW} d_\mathcal{O}(r)= \inf_{(v_1,...,v_N,u_1,...,u_N)\in
\mathcal{O}} \sum_{n=1}^N (2n-1+M-N) \left(v_n+u_n-\alpha\right).$$
Next, we work on the explicit expression of $d_\mathcal{O}(r)$. Since the left hand side (LHS) of (\[eqn28\]) is a non-decreasing function of $u_N$, decreasing $u_N$ will not violate the outage condition in (\[eqn28\]) while enjoying a reduced SNR exponent $\sum_{n=1}^N (2n-1+M-N) \left(v_n+u_n-\alpha\right)$. Combining with the fact $u_n\geq \alpha, \ n=1,2,...,N$, the solution of $\textbf{u}$ is found to be $u_1^*=...=u_N^*=\alpha$. Therefore, (\[eqn28\]) can be rewritten as $$\mathcal{O}=\left\{v_n\middle|\sum_{n=1}^N \left(1-v_n+\sum_{n=1}^N t(2n-1+M-N)\min(v_n,\alpha)\right)^+ <r, v_n\geq 0 \right\}.$$
To solve the optimization problem of (\[DMTNEW\]), we need to solve the subproblems $$\label{eqn33} d_k(r)\triangleq \inf_{(v_1,...,v_N)\in \mathcal{O}_k}
\sum_{n=1}^N (2n-1+M-N) v_n,\ k=0,1,...,N$$ where subset $\mathcal{O}_k$ $(0\leq k \leq N)$ is defined as $$\begin{split}
\mathcal{O}_k =&\left\{v_n\middle|\sum_{n=1}^N
\left(1-v_n+\sum_{n=1}^k t(2n-1+M-N)\alpha + \sum_{n=k+1}^N
t(2n-1+M-N)v_n\right)^+ <r,\right.\nonumber\\ &\left.v_1\geq...\geq
v_k\geq \alpha \geq v_{k+1}\geq...\geq v_N \right\} .
\end{split}$$
So, $d_\mathcal{O}(r)$ is given by $$\label{eqn_29} d_\mathcal{O}(r)=\min\left( d_0(r),
d_1(r),...,d_N(r)\right).$$ In other words, among all the DMT curves $d_0(r)$,...,$d_N(r)$, corresponding to the outage subsets $\mathcal{O}_1$,...,$\mathcal{O}_N$, the lowest one will be the DMT curve for the entire outage event. Since $t$ can be made arbitrarily close to 1, it is without loss of accuracy to set $t=1$ in the rest of this paper.
Firstly, we derive $d_0(r)$. It is easy to show $$\label{eqn_30}
\sum_{n=1}^N \left(1-v_n+\sum_{n=1}^N (2n-1+M-N)v_n\right)^+\geq
N-\sum_{n=1}^N v_n +N\sum_{n=1}^N (2n-1+M-N)v_n \geq N$$ which suggests that it is possible to operate at spatial multiplexing gain $r\in[0,N]$ reliably without any outage, i.e., $d_0(r)=\infty$. So we can exclude $d_0(r)$ from the optimization problem in (\[eqn\_29\]).
Secondly, we derive $d_k(r)$ ($1\leq k \leq N$). Note that the function $\sum_{n=1}^N \left(1-v_n+\sum_{n=1}^k (2n-1\right.$ $\left.+M-N)\alpha + \sum_{n=k+1}^N (2n-1+M-N)v_n\right)^+$ is an increasing function of $v_{k+1},v_{k+2},...,v_N$. That is, decreasing $v_{k+1},v_{k+2}...,v_N$ does not violate the outage condition for $\mathcal{O}_k$, while reducing the SNR exponent $\sum_{n=1}^N (2n-1+M-N)v_n$. Therefore, the optimal solutions of $v_{k+1},v_{k+2},...,v_N$ are $v_{k+1}^*=v_{k+2}^*=...=v_N^*=0$. Consequently, the optimization problem in (\[eqn33\]) can be reformulated as $$d_k(r)=\inf_{(v_1,...,v_k)\in \tilde{\mathcal{O}}_k} \sum_{n=1}^k (2n-1+M-N) v_n,\
k=1,...,N.$$ Here the modified outage subset $\tilde{\mathcal{O}}_k$ is defined as $$\label{eqn35}
\begin{split}
\tilde{\mathcal{O}}_k
=\left\{v_1,...,v_k\middle|N\tau(k)-\sum_{n=1}^k v_n<r, \alpha\leq
v_k\leq...\leq v_1 \leq \tau(k) \right\}.
\end{split}$$ where $\tau(k)=1+k\alpha(M-N+k)$. Careful observation of (\[eqn35\]) reveals that $$\label{rr}
\begin{split}
N\tau(k)-\sum_{n=1}^k v_n \geq (N-k)\tau(k).
\end{split}$$ It implies that there will be no outage ($d_k(r)=\infty$), if $r
\leq (N-k)\tau(k)$ or $(N-k)\tau(k) \geq N$. Note that $(N-k)\tau(k)
\geq N \Rightarrow (M-N+k)(N-k)\geq 1/\alpha$. So, if $(M-N+k)(N-k)<
1/\alpha$, there will be nonzero outage ($d_k(r)<\infty$), for $r\in
\Omega_k$, where $\Omega_k$ is defined as $$\label{ome} \Omega_k\triangleq \left((N-k)\tau(k),\ N\right].$$
For any $r\in \Omega_k$, we are able to explicitly calculate the optimal solutions of $v_1,...,v_k$ that minimize the SNR exponent $\sum_{n=1}^k (2n-1+M-N)v_n$. The results are summarized in the following.
1. If $r=(N-k')\tau(k)-(k-k')\alpha,\
k'=1,2,...,k$, then the achievable diversity for outage event $\tilde{\mathcal{O}}_k$ is $$\label{eqn_35} d_k(r)=k'(M-N+k')\tau(k)+(k-k')(k+k'+M-N)\alpha.$$ The corresponding optimal solutions of $v_1,...v_k$ are $v_1^*=...=v_{k'}^*=\tau(k)$, $v_{k'+1}^*=...=v_k^*=\alpha$. Specifically, $d_k(r)=k(M-N+k)\tau(k)$ for $r=(N-k)\tau(k)$.
2. If $(N-k')\tau(k)-(k-k')\alpha<r<(N-k'+1)\tau(k)-(k-k'+1)\alpha,\
k'=1,2,...,k$, the achievable diversity for outage event $\tilde{\mathcal{O}}_k$ is $$\label{eqn_38}
d_k(r)=((N-k')(k'-N-1)+MN)\tau(k)+(k-k'+1)(k-k')\alpha-(2k'-1+M-N)r.$$ The corresponding optimal solutions of $v_1,...,v_k$ are $v_1^*=...=v_{{k'-1}}=\tau(k)$, $v_{k'}^*=(N-k'+1)\tau(k)-(k-k')\alpha-r$, $v_{k'+1}^*=...=v_k^*=\alpha$.
For a particular $k'$, when spatial multiplexing gain $r$ is between $(N-k')\tau(k)-(k-k')\alpha$ and $(N-k'+1)\tau(k)-(k-k'+1)\alpha$, only one singular value of $\textbf{H}$, corresponding to the typical outage event, needs to be adjusted to be barely large enough to support the data rate. (\[eqn\_38\]) further shows that $d_k(r)$ is a continuous line segment between these two points. It is thus obvious that curve $d_k(r)$ is piecewise-linear with $(r,d_k(r))$ specified in (\[eqn\_35\]) being its corner points.
After calculating $d_1(r),...,d_N(r)$, we remain to solve $d_{\mathcal{O}}(r)=\min_k d_k(r),\ k=1,...,N$. Since $d_k(r)=\infty$ if $(M-N+k)(N-k)\geq 1/\alpha $, we only need to consider $k\in \mathcal{A}$, where set $\mathcal{A}$ is defined as $$\mathcal{A}=\left\{k\middle | (M-N+k)(N-k)< 1/\alpha,
k=1,...,N\right\}.$$
Note that we always have $k=N \in \mathcal{A}$. We consider the following two cases.
Case 1: $\mathcal{A}$ has only one element, i.e., $\mathcal{A}=\{k=N\}$. In this case, we have $d_{\mathcal{O}}(r)=d_N(r)$. If $N=1$, this condition is naturally satisfied, since there is only one element in $\mathcal{A}$ that is $k=1$. If $N>1$, we must require $(M-N+k)(N-k) \geq 1/\alpha$ for $k=1,...,N-1$, which leads to $$\label{eqn42} \alpha \geq \frac{1}{M-1},\ \ N>1.$$
We now examine the corner points of $d_N(r)$. From (\[eqn\_35\]), we have $r=(N-k')(1+\alpha MN-\alpha)> 1+MN\alpha-\alpha$ for corner point $k'$ ($k'=1,2,...,N-1$). Since $1+MN\alpha-\alpha$ is a non-decreasing function of $\alpha$, we easily get $r>
1+\frac{MN}{M-1}-\frac{1}{M-1} >N$. Thus we conclude that there is only one corner point ($0,d_N(0)$) on curve $d_N(r)$ over region $r\in \Omega_{N}$. Therefore, $d_{\mathcal{O}}(r)=d_N(r)$ is a straight line between corner points ($0,d_N(0)$) and ($N,d_N(N)$). From (\[eqn\_38\]), we have $d_N(N)=MN(1+MN\alpha)-(M+N-1)N$, so $d_{\mathcal{O}}(r)$ can be described as $$d_{\mathcal{O}}(r)=MN(1+MN\alpha)-(M+N-1)r \ \ \text{for}\ 0\leq
r\leq N.$$
Case 2: $\mathcal{A}$ has more than one element. Since $N\in
\mathcal{A}$ and $\Omega_N=[0, N]$, $\Omega_k$ ($k\neq N,k\in
\mathcal{A}$) overlaps with $\Omega_N$. That is, there are some regions of spatial multiplexing gain $r$, leading to finite diversity gains on different DMT curves. A straightforward method to find $d_{\mathcal{O}}(r)$ is to numerically calculate $d_k(r)$ for all $k\in \mathcal{A}$, and choose the minimum value among them. However, this makes $d_{\mathcal{O}}(r)$ implicit and hardly insightful. To find the closed-form solution of $d_{\mathcal{O}}(r)$, we wish to find out if there is any relationship among $d_1(r),...,d_N(r)$. This motivates the birth of the following Lemma, the proof of which is given in Appendix B.
For any spatial multiplexing gain $r\in \Omega_{k_1} \bigcap \Omega_{k_2}$ ($1\leq k_1, k_2\leq N$), if $k_1<k_2$, we have $d_{k_1}(r)<d_{k_2}(r)$.
This Lemma tells us if a spatial multiplexing gain $r$ leads to finite diversity gains on two DMT curves, we only need to select the curve with lower diversity gain. For example, if $r\in
\Omega_1\bigcap\Omega_2\bigcap...\bigcap\Omega_N$, then $d_{\mathcal{O}}(r)=d_1(r)$ since $d_1(r)<d_2(r)<...<d_N(r)<\infty$. Therefore, we can further expurgate bad $k$ (s.t. $\Omega_k\subseteq
\Omega_{\bar{k}}$, for $\bar{k}<k \in \mathcal{A} $) from $\mathcal{A}$ and only take into account $k\in {\mathcal{B}}$ for the optimization problem, where $${\mathcal{B}}=\left\{k\middle |
(N-k)\tau(k)<(N-\bar{k})\tau(\bar{k}),\ \forall \bar{k}<k,
\bar{k},k \in \mathcal{A}\right\}.$$
Letting $|{\mathcal{B}}|$ denote the cardinality of ${\mathcal{B}}$, we further divide $r\in[0,N]$ into $|{\mathcal{B}}|$ non-overlapping regions with region $\tilde{\Omega}_k$ ($k\in {\mathcal{B}}$) defined as $$\begin{split}
\tilde{\Omega}_k&=\Omega_k\bigcap \tilde{\Omega}_{\bar{k}}^c,\ \forall \bar{k}<k\&\bar{k}\in {\mathcal{B}}\\
&=
[(N-k)\tau(k), (N-\mathcal{I}(k))\tau(\mathcal{I}(k)))\\
\end{split}$$ where $\mathcal{I}(k)$ indicates the immediately preceding element of $k$ in ${\mathcal{B}}$, i.e., $\mathcal{I}(k)=\max_{\bar{k}<k,\bar{k}\in {\mathcal{B}}} \bar{k}$. From Fig. 1, which illustrates the relationship between $\Omega_{k}$ and $\tilde{\Omega}_k$, we get $d_{\mathcal{O}}(r)=d_k(r)$ for any $
r\in \tilde{\Omega}_k$.
Next we examine the corner points on curve $d_k(r)$ over $r\in
\tilde{\Omega}_k$ and give the following Lemma, the proof of which is given in Appendix C.
For $k\in {\mathcal{B}}$, there is only one corner point, $\left(
(N-k)\tau(k), k(M-N+k)\tau(k)\right)$, making $r\in
\tilde{\Omega}_k$.
As a result, $d_k(r)$ over $r\in \tilde{\Omega}_k$ is just a single line segment connecting the following two end points $$\label{eqn_44}
\begin{split}
\text{Left end: }&d_k(r)=k(M-N+k)\tau(k),\ \text{for}\ r=(N-k)\tau(k)\\
\text{Right end: }&d_k(r)=((N-k)(k-N-1)+MN)\tau(k)-(2k-1+M-N)(N-\mathcal{I}(k))\tau(\mathcal{I}(k)),\\
&\ \ \ \ \ \ \ \ \ \text{for}\
r=(N-\mathcal{I}(k))\tau(\mathcal{I}(k)).
\end{split}$$
Finally, since $d_{\mathcal{O}}(r)$ is the union of $d_k(r)$ over $r\in \tilde{\Omega}_k$ for all $k\in {\mathcal{B}}$, the DMT curve over the entire outage event is the collection of all the involved line segments and the two end points of line segment $d_k(r)$ ($k\in
{\mathcal{B}}$) are described in (\[eqn\_44\]). It should be noted that these line segments are discontinuous though $r$ is continuous between $0$ and $N$. Combining the above Cases 1 and 2 directly leads to (\[eq14\]) and (\[eqn\_446\]) in Theorem 1.
Achievability Proof
-------------------
To complete the proof of the Theorem 1, we need to show that $\mathcal{P}_e(\rho){\buildrel \textstyle .\over \leq}\rho^{-d_{\mathcal{O}}(r)}$ if $L\geq
M+N-1$. With the ensemble of i.i.d. complex Gaussian random codes at the input, the ML error probability is given by [@IEEErelay:DMT] $$\label{eqn41}
\mathcal{P}_e(\rho)=\mathcal{P}_{\mathcal{O}}\mathcal{P}(\text{error}
|\mathcal{O})+\mathcal{P}(\text{error},\mathcal{O}^c)\leq
\mathcal{P}_{\mathcal{O}}+\mathcal{P}(\text{error},\mathcal{O}^c)$$where $\mathcal{O}$ and $\mathcal{P}_{\mathcal{O}}$ are given by (\[eqn28\]) and (\[DMTNEW\]), respectively.
$\mathcal{P}(\text{error},\mathcal{{O}}^c)$ can be upper-bounded by a union bound. Assume that $\textbf{X}(0)$, $\textbf{X}(1)$ are two possible transmitted codewords, and that $\Delta
\textbf{X}=\textbf{X}(1)-\textbf{X}(0)$. Suppose $\textbf{X}(0)$ is transmitted, the probability that an ML receiver will make a detection error in favor of $\textbf{X}(1)$, conditioned on a certain realization of the channel, is $$\mathcal{P}\left(\textbf{X}(0)\rightarrow \textbf{X}(1)\middle |
\textbf{H},\hat{\textbf{H}}\right)=\mathcal{P}\left(\frac{P(\hat{\textbf{H}})}{M
\sigma^2}\left\| \frac{1}{2}\textbf{H}(\Delta \textbf{X})\right
\|_F^2\leq \|\textbf{w}\|^2\right)$$where $\textbf{w}$ is the additive noise on the direction of $\textbf{H}(\Delta \textbf{X})$, with variance $1/2$. With the standard approximation of the Gaussian tail function, $Q(x)\leq 1/2 \exp(-x^2/2)$, we have $$\mathcal{P}\left(\textbf{X}(0)\rightarrow \textbf{X}(1)\middle |
\textbf{H},\hat{\textbf{H}}\right)\leq \exp
\left(-\frac{P(\hat{\textbf{H}})}{4M \sigma^2}\|\textbf{H}(\Delta
\textbf{X})\|^2\right).$$
Averaging over the ensemble of random codes, we have the average pairwise error probability conditioned on the channel realization $$\mathcal{P}\left(\textbf{X}(0)\rightarrow \textbf{X}(1)\middle |
\textbf{H},\hat{\textbf{H}}\right)\leq \det
\left(\textbf{I}_N+\frac{P(\hat{\textbf{H}})}{2M
\sigma^2}\textbf{H}\textbf{H}^{\dag}\right)^{-L}.$$
With a data rate $R=r\log(\rho)$ (bits per channel use), we have in total $\rho^{Lr}$ codewords. Applying the union bound, we have $$\begin{split}
\mathcal{P}(\text{error} | \textbf{H},\hat{\textbf{H}})&\leq\rho^{Lr} \det \left(\textbf{I}_N+\frac{P(\hat{\textbf{H}})}{2M \sigma^2}\textbf{H}\textbf{H}^{\dag}\right)^{-L}\\
&= \rho^{Lr} \prod_{n=1}^N \left(1+ \frac{\rho \kappa
a_n}{2M\prod_{n=1}^N b_n^{2n-1+M-N}} \right)^{-L}
\\
&\leq \rho^{Lr} \prod_{n=1}^N \left(1+ \frac{\rho \kappa
a_n}{M\prod_{n=1}^N (2a_n+2c_N)^{2n-1+M-N}}
\right)^{-L}\\
&\doteq \rho^{-L\left(\sum_{n=1}^N \left(1-v_n+\sum_{n=1}^N
(2n-1+M-N)\min(v_n,u_N)\right)^+-r\right)}.
\end{split}$$
Averaging over the distributions of $\textbf{H}$ and $ \hat{\textbf{H}}$, or equivalently $\textbf{v}$ and $\textbf{u}$, we have $$\begin{split}
&\mathcal{P}(\text{error}, \mathcal{{O}}^c)=\int_{\mathcal{O}^c}p(\textbf{u})p(\textbf{v}) \mathcal{P}(\text{error} | \textbf{H},\hat{\textbf{H}})d\textbf{u} d\textbf{v}\\
&{\buildrel \textstyle .\over \leq}\int_{\mathcal{{O}}^c}\rho^{-\sum_{n=1}^N(2n-1+M-N)(v_n+u_n-\alpha)} \rho^{-L\left(\sum_{n=1}^N \left(1-v_n+\sum_{n=1}^N (2n-1+M-N)\min(v_n,u_N)\right)^+-r\right)} d\textbf{u} d\textbf{v}\\
&\doteq \rho^{-d_G(r)}
\end{split}$$ where $$\label{eqn_48}
\begin{split}
d_G(r)
=&\inf_{\textbf{u}, \textbf{v} \in \mathcal{O}^c}\sum_{n=1}^N(2n-1+M-N)(v_n+u_n-\alpha)\\
&+L\left(\sum_{n=1}^N \left(1-v_n+\sum_{n=1}^N
(2n-1+M-N)\min(v_n,u_N)\right)^+-r\right).
\end{split}$$
When $L\geq M+N-1$, $d_G(r)$ has the same monotonicity as $\sum_{n=1}^N \left(1-v_n+\sum_{n=1}^k (2n-1+M\right.$ $\left.-N)\alpha + \sum_{n=k+1}^N (2n-1+M-N)v_n\right)^+$ with respect to $v_n$ or $u_n$, $n=1,...,N$. Therefore, the minimum always occurs when $\sum_{n=1}^N \left(1-v_n+\sum_{n=1}^N
(2n-1+M-N)\min(v_n,u_N)\right)^+=r$. Hence $$\begin{split}
d_G(r)&=\inf_{\sum_{n=1}^N \left(1-v_n+\sum_{n=1}^N (2n-1+M-N)\min(v_n,u_N)\right)^+=r} \sum_{n=1}^N(2n-1+M-N)(v_n+u_n-\alpha)\\
&=d_{\mathcal{O}}(r).
\end{split}$$
Therefore, the overall error probability can be written as $$\begin{split}
\mathcal{P}_e(\rho)&\leq \mathcal{P}_{\mathcal{O}}+\mathcal{P}(\text{error},\mathcal{{O}}^c)\\
&\doteq \rho^{-d_{\mathcal{O}}(r)}+\mathcal{P}(\text{error},\mathcal{{O}}^c)\\
& {\buildrel \textstyle .\over \leq}\rho^{-d_{\mathcal{O}}(r)}+\rho^{-d_G(r)} \doteq
\rho^{-d_{\mathcal{O}}(r)}
\end{split}$$
Since the MIMO channel with the proposed power adaptation scheme leads to an error probability lower than or equal to $\rho^{-d_{\mathcal{O}}(r)}$, we can say that the MIMO channel is able to achieve the diversity gain of $d_{\mathcal{O}}(r)$. Theorem 1 is thus obtained.
Discussions
===========
In this section, we discuss the additional diversity gain $\Delta_{
d}(r)$ brought by the imperfect CSIT through power adaptation.
Case A) $N=1$ (MISO/SIMO): According to (\[eq14\]), the imperfect CSIT provides an additional diversity gain of $\Delta_{d}(r)=M^2\alpha $ at any spatial multiplexing gain in the considered MISO/SIMO channel. Most remarkably, when $\alpha = 1/M$, one can achieve both full diversity gain (i.e., $M$) and full spatial multiplexing gain (i.e., $1$) at the same time, while $\alpha$ has to be equal to or greater than $1$ to achieve the same performance in [@IEEErelay:OFT]. Note that the case of $\alpha<1$ is much more practical than the case of $\alpha\geq1$ as one usually allocates much less power to the reverse (feedback pilot) channel than the forward transmission channel.
Case B) $\alpha \geq \frac{1}{M-1},\ N>1$: For such MIMO channel, according to (\[eq14\]), the additional diversity gain is $\Delta
_{d}(r)=(M^2N^2\alpha+r-r^2)$, for $r=0,1,...,N$. Specifically, $\Delta _{d}(0)=M^2N^2\alpha$ and $\Delta _{d}(N)=\alpha
M^2N^2+N-N^2> MN^2+N$. If $0<r<N$, the additional diversity gain is between the two extreme values $\Delta_{d}(0)$ and $\Delta_{d}(N)$.
Case C) $\alpha < \frac{1}{M-1},\ N>1$: When $r=N$, $\Delta_{d}(N)=d(N)\geq d_1(N)=\alpha N(M-N+1)^2$. When $r<N$, for the convenience of comparison with [@IEEErelay:DMT], we consider integer spatial multiplexing gains, i.e., $r=N-k,\ k=1,2,...,N$. Since $r=N-k\leq (N-k){\tau}(k)$, from Theorem 1, the achievable diversity gain is $d(r)\geq k(M-N+k){\tau}(k)=
(M-r)(N-r)\left(1+\alpha (M-r)(N-r)\right)$. Recall that the optimal diversity gain without CSIT is $d^*(r)=(M-r)(N-r)$. Therefore, the additional achievable diversity gain with our scheme is $\Delta_{d}(r)\geq \alpha (M-r)^2(N-r)^2=\alpha
\left(d^*(r)\right)^2$. It indicates that even a very small $\alpha$ leads to a significant diversity gain improvement.
We use numerical results to show the additional diversity gain achieved with imperfect CSIT. We compare the following two scenarios: 1) No CSIT [@IEEErelay:DMT]; and 2) Imperfect CSIT with power adaptation. Figs. 2 and 3 plot the DMT curves for $3\times 3$ and $4\times 2$ MIMO fading channels, respectively. It is obvious that imperfect CSIT provides significant additional diversity gain improvement and offers non-zero diversity gain at any possible spatial multiplexing gain. Fig. 2 also shows the impact of $\alpha$ value. When $\alpha\geq\frac{1}{M-1}=\frac{1}{2}$, we only have $d_N(r)<\infty$ and thus $\mathcal{B}=\{3\}$. Therefore, the DMT curve is a single line segment. However, when $\alpha=\frac{1}{3}<\frac{1}{M-1}$, $\mathcal{B}=\{1,2,3\}$. Therefore, the DMT curve is made up of three discontinuous line segments. Fig. 3 shows how $d_{\mathcal{O}}(r)$ depends on $d_1(r)$ and $d_2(r)$ in a $4\times
2$ MIMO channel with $\alpha=0.1$. We observe that $d_2(r)\geq
d_1(r)$ and there is only one corner point on $d_1(r)$ (or $d_2(r)$) over spatial multiplexing gain region $r\in \tilde{\Omega}_1$ (or $r\in \tilde{\Omega}_2$).
Next we illustrate the impact of $\alpha$ on DMT. Fig. 4 plots the relationship between the achievable diversity gain and the channel quality $\alpha$ in a MISO/SIMO channel. It clearly shows that power adaptation makes better use of the imperfect CSIT than rate adaptation. In other words, to achieve the same performance our scheme saves a great amount of pilot power and thus is more applicable. Specifically, the diversity gain improvements over [@IEEErelay:DMT] and [@IEEErelay:OFT] are $M^2\alpha$ and $(M-1)M\alpha$, respectively, at any spatial multiplexing gain. It is no doubt that the achievable DMT increases with CSIT quality $\alpha$. Fig. 5 plots how the achievable diversity gain with power adaptation improves with the channel quality $\alpha$ in a $5\times
3$ MIMO channel at the full multiplexing gain. We observe that there are fast increases of diversity gain at $\alpha=0.25$ and $\alpha=0.1667$. These two values of $\alpha$ are actually thresholds for $d_k(r)<\infty, k=1,2,3$. When $\alpha\geq
\frac{1}{M-1}=0.25$, $\mathcal{B}=\{3\}$. Therefore, we have $d_{\mathcal{O}}(N)=d_3(N)$. When $0.1667 \leq \alpha<0.25$, we have $\mathcal{B}=\{2,3\}$ and $d_{\mathcal{O}}(N)=d_2(N)$. When $\alpha<0.1667$, we have $\mathcal{B}=\{1,2,3\}$ and $d_{\mathcal{O}}(r)=d_1(N)$. Combining with the fact that $d_1(r)<d_2(r)<d_3(r)$ for any fixed $\alpha$, it is not difficult to understand the cliffs on this curve.
Note that the additional diversity gain comes at the price of reverse channel pilot power to obtain the CSIT. As long as the reverse channel SNR does not scale with $\rho$, i.e., $\alpha=0$, even with some partial CSIT at the transmitter, there will be no improvement on the fundamental DMT. However, when the reverse channel SNR relative to $\rho$ becomes asymptotically zero, i.e., $\alpha<1$, there will be a significant improvement of the diversity gain. When the reverse channel SNR as compared to the forward SNR is asymptotically unbounded, i.e., $\alpha>1$, one can achieve the full spatial multiplexing gain while enjoying a even more remarkable diversity.
Conclusion
==========
In this paper, we investigated the impact of CSIT on the tradeoff of diversity and spatial multiplexing in MIMO fading channels. For MISO/SIMO channels, we showed that using power adaptation, one can achieve a diversity gain of $d(r)=K(1-r+K\alpha)$, where $K$ is the number of transmit antennas in the MISO case or the number of receive antennas in the SIMO case. This is not only higher but also more efficient than the available results in literature. For general MIMO channels with $M>1$ transmit and $N>1$ receive antennas, when $\alpha$ is above some certain threshold, one can achieve a diversity gain of $d(r)=MN(1+MN\alpha)-(M+N-1)r$; otherwise, the achievable DMT is a collection of discontinuous line segments depending on $M$, $N$, $r$ and $\alpha$. The presented DMT shows that exploiting imperfect CSIT through power adaptation significantly increases the achievable diversity gain in MIMO channels.
Letting $q_n\triangleq-\log(b_n)/\log (\rho)$ for all $n$ and $\mathbf{q}\triangleq[q_1,...,q_N]$, we have $$\begin{split}
&E\{P(\hat{\textbf{H}})\} =\int_{\textbf{b}\succeq
0}\frac{\kappa\bar{P} }{\left(\prod_{n=1}^N b_n^{2n-1+M-N}\right)^t}
\hat{\xi}^{-1}\prod_{n=1}^N b_n^{M-N} \prod_{n<j}^N
(b_n-b_j)^2\exp\left(-\frac{1}{1+\sigma_e^2}\sum_{n=1}^N b_n\right)
d \textbf{b}\\
&=\int_{\textbf{q}\succeq 0} \frac{\kappa\bar{P} \hat{\xi}^{-1}(\log
\rho)^N}{\left(\prod_{n=1}^N \rho^{-(2n-1+M-N)q_n}\right)^t}
\prod_{n=1}^N \rho^{-(M-N+1)q_n} \prod_{n<j}^N (\rho^{-q_n}-\rho^{-q_j})^2\exp\left(-\frac{1}{1+\sigma_e^2}\sum_{n=1}^N \rho^{-q_n}\right)
d \textbf{q}.
\end{split}
\label{eq13}$$
At high SNRs, it is easy to show that $$E\{P(\hat{\textbf{H}})\}= \lim_{\rho\rightarrow\infty}
\int_{\textbf{q}\succeq 0}\kappa \bar{P} \hat{\xi}^{-1}(\log
\rho)^N \left(\prod_{n=1}^N \rho^{-(2n-1+M-N)q_n}\right)^{(1-t)} d
\textbf{q}=\bar{P}.$$
Let $v_{1,k_1},...,v_{k_1,k_1}$ denote the solutions of $v_1,...,v_{k_1}$ that minimize $d_{k_1}(r)$, and let $v_{1,k_2},...,v_{k_2,k_2}$ denote the solutions of $v_1,...,v_{k_2}$ that minimize $d_{k_2}(r)$. Without loss of generality, we assume $$\begin{aligned}
&&v_{1,k_1}=...=v_{i-1,k_1}=\tau(k_1),\tau(k_1)>v_{i,k_1}\geq \alpha, v_{i+1,k_1}=...=v_{k_1,k_1}=\alpha\\
&&v_{1,k_2}=...=v_{j-1,k_2}=\tau(k_2),\tau(k_2)>v_{j,k_2}\geq \alpha, v_{j+1,k_2}=...=v_{k_2,k_2}= \alpha.\end{aligned}$$
It follows that the corresponding spatial multiplexing gain $r$ satisfies $$\begin{aligned}
&&r=(N-i+1)\tau(k_1)-(k_1-i)\alpha-v_{i,k_1}\\ &&r=(N-j+1)\tau(k_2)-(k_2-j)\alpha-v_{j,k_2}
\end{aligned}$$which leads to $$\label{eqnv2v1}
\left\{(N-j+1)\tau(k_2)-(k_2-j)\alpha-v_{j,k_2}\right\}
-\left\{(N-i+1)\tau(k_1)-(k_1-i)\alpha-v_{i,k_1}\right\}=0.
$$
We consider the following three cases.\
Case 1) $j<i$: Letting $B$ denote the LHS of (\[eqnv2v1\]), we have $$\begin{split}
B>&
(N-j)\tau(k_2)-(k_2-j)\alpha -(N-i+1)\tau(k_1)-(k_1-i+1)\alpha\\
\geq &(N-i+1)\tau(k_2)-(k_2-i+1)\alpha -(N-i+1)\tau(k_1)-(k_1-i+1)\alpha\\
\geq &\alpha(k_2-k_1)\left((N-i+1)(M-N+k_2+k_1)-1\right)>0.
\end{split}$$ This contradicts with $B=0$. Therefore, $j<i$ is not possible.
Case 2) $j>i$: It is easy to observe that $$\begin{aligned}
&&v_{1,k_2}-v_{1,k_1}=...=v_{i-1,k_2}-v_{i-1,k_1}
=k_2\alpha(M-N+k_2)-k_1\alpha(M-N+k_1)>0\\
&&v_{i,k_2}-v_{i,k_1}>k_2\alpha(M-N+k_2)-k_1\alpha(M-N+k_1)>0\\
&&v_{i+1,k_2}-v_{i+1,k_1},...,v_{k_1,k_2}-v_{k_1,k_1}\geq \alpha-\alpha= 0\\
&&v_{k_1+1,k_2},...,v_{k_2,k_2}\geq \alpha.\end{aligned}$$
Then, it follows that $$d_{k_2}(r)-d_{k_1}(r)=\sum_{i=1}^{k_2}v_{i,k_2}-\sum_{i=1}^{k_1}v_{i,k_1}>0.$$
Case 3) $j=i$: Similarly, we have $$\begin{aligned}
\label{eqn_65}
&&v_{1,k_2}-v_{1,k_1}=...=v_{i-1,k_2}-v_{i-1,k_1}=k_2\alpha(M-N+k_2)-k_1\alpha(M-N+k_1)>0\\
\label{eqn_66}
&&v_{i+1,k_2}=...=v_{k_2,k_2}=v_{i+1,k_1}=...=v_{k-1,k_1}=\alpha.\end{aligned}$$
From (\[eqnv2v1\]), we get $$\label{eqn_67}
v_{i,k_2}-v_{i,k_1}=\alpha(k_2-k_1)\left((N-i+1)(M-N+k_2+k_1)-1\right)>0.$$ Combining (\[eqn\_65\]), (\[eqn\_66\]) and (\[eqn\_67\]), we get $d_{k_2}(r)>d_{k_1}(r)$. The proof of Lemma 2 is complete.
We compare the spatial multiplexing gain $r$ of the corner point $k'$ ($k'=1,...,k-1$) on the DMT curve $d_k(r)$, i.e., $r=(N-k')\tau(k)-(k-k')\alpha$, with the lower boundary of $\Omega_{k-1}$, i.e., $(N-k+1)\tau(k-1)$, and get $$(N-k')\tau(k)-(k-k')\alpha - (N-k+1)\tau(k-1) \geq
\left((N-k+1)(M-N+2k-1)-1\right)\alpha > 0.
$$
If $(N-k')\tau(k)-(k-k')\alpha<N$, it suffices to have $(N-k')\tau(k)-(k-k')\alpha\in {\Omega}_{k-1}$. Otherwise, we get $(N-k')\tau(k)-(k-k')\alpha \notin \Omega_k$. Since ${\Omega}_{k-1}\bigcap \tilde{\Omega}_{k}=\emptyset$ and $\tilde{\Omega}_{k}\subseteq \Omega_k$, both cases lead to $r\notin\tilde{\Omega}_k$. This completes the proof of Lemma 3.
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![Relationship of $\Omega_k$ and $\tilde{\Omega}_k$.[]{data-label="fig_1"}](Fig1.eps){width="3.5in"}
![DMT in a $3 \times 3$ MIMO channel. Note that $\mathsf{d_o(r)}$ in the legend denotes $d_{\mathcal{O}}(r)$.[]{data-label="fig_3"}](Fig2.eps){width="5in"}
![DMT in a $4\times 2$ MIMO channel with $\alpha=0.1$. Note that $\mathsf{d_o(r)}$ in the legend denotes $d_{\mathcal{O}}(r)$.[]{data-label="fig_4"}](Fig3.eps){width="5in"}
![Diversity gain versus channel quality $\alpha$ in a SIMO/MISO channel.[]{data-label="fig_5"}](Fig4.eps){width="5in"}
![Diversity gain at $r=N$ versus channel quality $\alpha$ in a $5 \times 3$ MIMO channel.[]{data-label="fig_6"}](Fig5.eps){width="5in"}
[^1]: The authors are with the School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore, 639798 (e-mail: zh0012an@ntu.edu.sg, eygong@ntu.edu.sg).
|
---
author:
- 'NikolaiV.Ivanov'
title: 'Thede Bruijn–Erdös–Hananitheorem'
---
=1
plus 0pt minus 0pt plus 0pt minus 0pt plus 0pt minus 0pt
[***Contents***]{}\
[***1.***]{} N.Bourbaki and the*“Kvant”*magazine\
[***2.***]{} A solution of theN.Bourbaki exercise\
[***3.***]{} The de Bruijn–Erdös proof\
[***4.***]{} From de Bruijn–Erdös to systems of distinct representatives\
[***5.***]{} Linear algebra and the inequality$m\qff \geqslant\qff n$\
[***6.***]{} Hanani’s theorem\
[***7.***]{} Another proofofHanani’s theorem\
[***8.***]{} All the de Bruijn–Erdös inequalities\
[***References***]{}
$\phantom{1.}$[[Preface]{}]{.nodecor} {#phantom1.preface .unnumbered}
=====================================
[[.]{}]{.nodecor} {#kvant .unnumbered}
==================
[**Problem.**]{} *Let$E$be a set of $n$ elements.Suppose that $m$ different subsets of$E$ (not equal to$E$ itself)are selected in such a way that for every two elements of$E$there is exactly one selected subset containing both these elements.Prove that$m\qff \geqslant\qff n$.*
*When an equality is possible?*
In1970this problem was included as the ProblemM5in the very first issue of the Soviet*“Kvant”*magazine and attributed toN.Bourbaki[@b-70].The intended audience of the*“Kvant”*magazine(its name means*“Quantum”*)was the school students in the USSR of the last two-three grades.Nowadays the audacity of the editorial board inspires awe:ProblemM5was offered to this audience exactly as it is cited above,as an abstract problem about finite sets without any motivation and any hints.The readers were expected to be interested in this problem and to appreciate its beauty without any crutches.
In1970I was among the intended audience of*“Kvant”*,but I was more interested in the foundations of mathematics and in the set theory than in the combinatorics of finite sets.I easily found this problem in the Russian translation[@b-56]of the*“Théorie des ensembles”*byN.Bourbaki.It turned out to be the Exercise12 to the section*“Calcul sur les entiers”*.In all editions this exercise is marked as one of the most difficult.
The editors of*“Kvant”*were faithful toN.Bourbaki in not offering any motivation.But,in contrast with*“Kvant”*,N.Bourbaki split the result into few steps,offered a hint to the key one,and stated the expected result in the case of the equality.The first two steps were rather easy,but the hint to key step turned out to be incomprehensible for me.
According to the authors of the solution[@q-solution] published in*“Kvant”*a few months later,they followed *“the hints of the author of the problem,N. Bourbakihimself”*and referred to[@b-56].The habit ofN. Bourbaki to include in his tract recent results without attribution as exercises is well known,and was well known in1970in the Soviet Union.But it seems that neither the editors of the*“Kvant”*magazine,nor the authors of the solution[@q-solution]were aware that this result is due to N.G.de Bruijn andP.Erdös[@db-e]andH.Hanani[@h1; @h2].
Neither was I before by an accident I returned to this problem in 2016. By this time I was able to immediately recognize that this exercise from[@b-56]is about points and lines in a geometry,and this realization quickly lead me to the de Bruijn–Erdös paper[@db-e].The exercise turned out to be a quite faithful summary of the de Bruijn–Erdös proof,and the key part of the proof,summarized byN.Bourbaki as a hint,turned out to be nearly as obscure as this hint. Here is my translation of this exercise based on the reprint[@b-06]of the1970edition(where it appears as the Exercise14to§5).It is slightly different from the translation in[@b-04].
[**Exercise.**]{} *Let$E$be a finite set of $n$ elements, $(a_j)_{1\dff \leqslant\dff j\dff \leqslant\dff n}$ be the sequence of elements of $E$ arranged in some order, $(A_i)_{1\dff \leqslant\dff i\dff \leqslant\dff m}$ be a sequence of parts of$E$.*
(a)*For each index $j$, let $k_j$ be the number of indices $i$ such that $a_j\qff \in\qff A_i$; for each index $i$ let $s_i\qff =\qff {\mathop{\mbox{\textup{Card}\fff}}}\dff (\fff A_i\fff)$. Show that* $$\quad
\sum_{j\qff =\qff 1}^n\qff k_j
\off =\off
\sum_{i\qff =\qff 1}^m\qff s_i\off.$$
(b)*Suppose that for each subset$\{\dff x\fff,\pff y\dff\}$of two elements of$E$,there exists one and only one index $i$ such that $x$ and $y$ are contained in $A_i$.Show that if $a_j\qff \not\in\qff A_i$,then$s_i\qff \leqslant\qff k_j$.*
(c)*Under the assumptions of, show that$m\qff \geqslant\qff n$.* (Let $k_n$ be the least of the numbers $k_j$; show that one may assume that,whenever $i\qff \leqslant\qff k_n\dff,\off j\qff \leqslant\qff k_n$ and$i\qff \neq\qff j$, one has$a_j\qff \not\in\qff A_i$,and$a_n\qff \not\in\qff A_j$ for all$j\qff \geqslant\qff k_n$.)
(d)*Under the assumptions of, show that in order for$m\qff =\qff n$to hold,it is necessary and sufficient that one of the following two cases occurs:*
\(i) $A_1
\off =\off
\{\qff a_1\fff,\off a_2\fff,\off \ldots\fff,\off a_{n\qff -\qff 1} \qff\}$,${A_i
\qff =\qff
\{\qff a_{i\qff -\qff 1}\fff,\off a_n \qff\}}$*for* $i
\off =\off
2\fff,\pff \ldots\fff,\pff n$;
\(ii) $n\off =\off k\dff(k\qff -\qff 1)\qff +\qff 1$,*each$A_i$is a set of$k$elements, and each element of$E$belongs to exactly $k$ sets $A_i$.*
[**Remarks.**]{} Two aspects of this exercise need to be clarified.First,the parts $A_i$ are implicitly assumed to be different from $E$.Second,the case(ii)of the part(d)is expected to hold only up to renumbering of elements $a_i$ and parts $A_j$.
[**The troubles with the hint.**]{} The parts(a)and(b)of this exercise are rather easy,and there is a hint for the part(c).But for me this hint turned out to be more of a riddle than of a help.
It would be quite easy to accept and follow the suggestion to consider the least of the numbers $k_j$.But why it should be $k_n$?The phrase *“Let$k_n$be the least of the numbers$k_j$”*is fairly hard to interpret(the expressions used in the French original and in the Russian translation have the same meaning).The standard usage of*“Let”*(and of*“Soit”*in French)in mathematics is to introduce new notations.But $k_n$ is already defined.
The authors of the solution[@q-solution]found a clever way out.They introduce the number $k_n$*before*introducing other numbers $k_j$!This trick helps only partially:the question*“Why $k_n$?”*remains. The de Bruijn–Erdös exposition[@db-e]is better.They write*“Assume now that $k_n$ is the smallest $k_i$ …”*.Thisisless obscure, and amounts to renumbering elements of $E$,butleaves the question*“Why $k_n$?”* unanswered.
If one manages to put this question aside,there is another riddle:how the subscripts$i\fff,\pff j$,which are merely marking the points(and do not even need to be numbers)may be compared with $k_n$,which is a genuine characteristic of the point marked by $n$?Perhaps,this difficulty is encountered only by the categorically minded mathematicians;analysts appear to be quite comfortable with using the values of a function in its domain of definition.
Here de Bruijn and Erdös[@db-e]are again doing better.They write*“Assume…that $A_1$,$A_2$,…,$A_{\dff k_n}$are lines through $a_n$”*(they call the parts $A_i$ lines).This amounts to renumbering the parts $A_i$, and one may wonder why renumbering is treated as an assumption.The trick of the authors of[@q-solution]saves the day here for them.They simply denote the $k_n$lines through $a_n$ by$A_1$,$A_2$,…,$A_{\dff k_n}$and other lines by$A_{\dff k_n\dff +\dff 1}$,$A_{\dff k_n\dff +\dff 2}$,…,$A_{m}$.
There is one more riddle in the store.How one uses the assumption that $k_n$ is the least of the numbers $k_j$ in the proof of the claim in the hint?One does not,this claim is true without it.
[**Partially decrypting the hint.**]{} Even if one encounters all these troubles and is not aware of the de Bruijn–Erdös paper(like me in 1970),the hint still may be of some help.The first message is that it is important to know when an element $a_j$ is not in the part $A_i$.Together with the part(b)this suggest that the inequalities$s_{\fff i}\qff \leqslant\qff k_{\fff j}$,which hold for$a_j\qff \not\in\qff A_i$,should play a key role.
Another message is that the least of the numbers $k_{\fff j}$ should play some role. After wasting some time assuming that for a given $u$ the number $k_{\fff u}$ is minimal among all numbers $k_{\fff j}$ andtrying to use this minimality to prove something like stated in the hint,it is only natural to abandon this assumption and consider an arbitrary subscript $u$ such that$1\qff \leqslant\qff u\qff \leqslant\qff n$.
[**The1970proofof$m\qff \geqslant\qff n$.**]{} With no more than this limited help from this exercise(in1970I definitely understood less than in 2016)I managed to prove in the early1970the inequality$m\qff \geqslant\qff n$.Among my schoolmates this qualified as a solution of the ProblemM5.This solution was lost long time ago.In April of2016and another time one year later I attempted to reconstruct this proof.In these attempts I encountered the same difficulties as in1970,and it is likely that I dealt with them in the same manner.At the very least,the resulting proof does not use any tools not known to me at the time,and does not involve any tricks(such as the cyclic ordering of some parts $A_i$ by de Bruijn–Erdös)which I was unlikely to discover at the time.It is presented in Section\[solution\]below.
The question*“When an equality is possible?”*was considered by my classmates as too vague to be addressed seriously, and this was indirectly admitted by the authors of the solution[@q-solution].If$m\qff =\qss n$,then(d)easily implies that$A_i\dff \cap\dff A_j\qff \neq\qff \emptyset$if$i\qff \neq\qff j$.In fact,proving this property is an almost inevitable part of the proof of(d).This property means that the set $E$ together with the parts $A_i$ is a*finite projective plane*,possibly degenerate in the case(i)of the part(d).Therefore,this question amounts to the classification of finite projective planes and,to the best of my knowledge,it remains largely open.See the paper by Ch.Weibel[@w]for a survey of the state of the art as of2007,and[@i-planes]for an introduction(not focusing on the finite case).
[***“Kvant”*publishes a solution.**]{} *“Kvant”*published a solution[@q-solution]of the ProblemM5in the August or September of 1970,close to the beginning of the school year in the USSR(always September 1).The editors of the problem section wrote(see[@q-solution],p.49):
> The letters to editors indicate that this problem is extremely difficult,but interesting.As a matter of fact,here we have two problems:1)prove that$m\qff \geqslant\qff n$,2)when an equality is possible?
>
> The first problem was completely solved only by*A.Suslin*from the city of Leningrad.His proof is based on a basic theorem of the linear algebra:if the number of$n$-vectors is greater than $n$,then they are linearly dependent.
>
> Looking for such a proof will be interesting for whose who are familiar with these notions.Nobody solved completely the second problem.Of course,this is not surprising,since,as it will be explained below,it can be reduced to a well known, but unsolved problem in mathematics.
Among my schoolmates,these remarks stirred a renewed interest in the problem.A.Suslinwas known as a very strong problem solver and as a winner of the gold medal at 1967 International Mathematical Olympiad.Since only he submitted a complete solution,the problem had to be really difficult.Since he used tools going beyond the school level,the problem had to be even more difficult.And this caused a real interest in my unsubmitted to the*“Kvant”*solution.I had an outline as a sparsely filled with formulas sheet of paper.One of my schoolmates borrowed this sheet for few days,and I have not seen it anymore.
But I am not aware of any serious attempt to study the published solution[@q-solution].For me it was almost as condensed and obscure as the N.Bourbaki hint.The role of the numbering of elements and parts is overemphasized:
> Let us pay attention once again to the way we numbered elements and sets.
>
> First of all,$k_{\fff n}$is the least of the numbers$k_{\fff 1}\fff,\pff k_{\fff 2}\fff,\pff \ldots\fff,\pff k_{\fff n\dff -\dff 1}$(sic!–*N.I.*).…
See[@q-solution],p.51.And I always disliked random numerical examples,which are supposed to help the reader and are extensively used in[@q-solution].I must admit that I did not even look at the last two pages of[@q-solution]before writing these comments,and,in particular,before writing down the proofin the next section.Surprisingly,it turned out that the proof[@q-solution]contains a gap:it is mentioned that$k_{\fff n}\qff =\qff 2$in the situation described in the case(i)of the Bourbaki exercise,but no proof that this is the only possibility is even attempted.
[[.]{}]{.nodecor} {#solution .unnumbered}
==================
[**The terminology and notations.**]{} In contrast withN.Bourbakiand with the*“Kvant”*,Ihave no reasons to hide the geometric content of this result.Following de Bruijn andErdös,I will call the elements of $E$*points*andthe sets $A_i$*lines*.Since the lines are assumed to be proper subsets of $E$,every point is contained in at least $2$ lines.Indeed,if a point is contained in only one line,then all points are contained in this line,i.e.it is not a proper subset.
It is convenient to explicitly introduce a counterpart to the set $E$ of points,namely the set of lines$\mathcal{L}
\off =\off
\{\dff A_1\fff,\pff A_2\fff,\pff \ldots\fff,\pff A_m \dff\}$.Ifthe case(i)of the part(d)of the Bourbaki exercise occurs,up to renumbering of points and lines,then the pair$(\dff E\fff,\pff \mathcal{L}\dff)$ is called a*near-pencil*.If the case(ii)of the part(d)occurs,then$(\dff E\fff,\pff \mathcal{L}\dff)$is called a*projective plane*.
I also do not see any reason to follow the outdated fashion of using*numerical*indices(i.e.subscripts),which amounts to ordering objects even when their order is irrelevant.Instead of this,for every point $z$ we will denote by $k_{\fff z}$ the number of lines containing $z$,and for every line $l$ we will denote by $s_{\fff l}$ the number of points in $l$,i.e. the number of elements of the set $l$.
[**The part(a)of the Bourbaki exercise.**]{} With the above notations the part(a)takes the form $$\label{sums}
\quad
\sum_{l\dff \in\dff \mathcal{L}}\qff s_{\fff l}
\off\off =\off\off
\sum_{\qff z\dff \in\dff E}\qff k_{\fff z}\dff.$$
after interchanging the sides.This immediately follows from counting in two different ways the pairs$(z\fff,\pff l\dff)\qff \in\qff E\times \mathcal{L}$such that$z\qff \in\qff l$.
[**The part(b)of the Bourbaki exercise.**]{} For the rest of the paper we will assume that the assumption of the part (b) holds,i.e.that for every pair of distinct points there is exactly one line containing both of them.If a line $l$ contains$\leqslant\qff 1$points, then removing $l$ from the set of lines does not affects this assumption, and at the same time decreases number of lines by $1$.Hence we may assume for the rest of the paper that every line contains at least $2$ points.
With the above notations the part (b) takes the form $$\quad
\mbox{ If }\quad
z\qff \not\in\qff l\fff,\quad
\mbox{ then }\quad
s_{\fff l}\qff \leqslant\qff k_{\fff z}\qff.$$
We will call these inequalities the*[de Bruijn–Erdös ]{}inequalities*.
In order to prove the [de Bruijn–Erdös ]{}inequalities,suppose that$z\qff \not\in\qff l$.Then for every$y\qff \in\qff l$there is a unique line containing$\{\dff z\fff,\pff y \dff\}$and it is different from $l$ because$z\qff \not\in\qff l$.These lines are pairwise distinct because if$y\fff,\pff y'\qff \in\qff l$and$y\qff \neq\qff y'$,then $l$ is the only line containing$\{\dff y\fff,\pff y' \dff\}$.There areis $s_{\fff l}$ such lines and all of them contain $z$;therefore$s_{\fff l}\qff \leqslant\qff k_{\fff z}$.
[**Lines through an arbitrary point.**]{} Let$u\qff \in\qff E$be an arbitrary point,let$p\qff =\qff k_{\fff u}$be the number of lines containing $u$,and let $\mathcal{U}$ be the set of these lines.
By the definition of $\mathcal{U}$,if a line $l$is not in$\mathcal{U}$,then$u\qff \not\in\qff l$.For every $l\qff \not\in\qff \mathcal{U}$we have the [de Bruijn–Erdös ]{}inequality$s_{\fff l}\qff \leqslant\qff k_{\fff u}$.By summing all these inequalities and taking into account that there are$m\qff -\qff p$lines notbelongingto $\mathcal{U}$,we see that $$\label{s-upper-estimate}
\quad
\sum_{l\qff \not\in\qff \mathcal{U}} s_{\fff l}
\off\off \leqslant\off\off
(m\qff -\qff p)\dff k_u\qff.$$
Since every set of the form $\{\dff u\fff,\pff y \trf\}$with$y\qff \neq\qff u$ is contained in one and only one line,the sets$l\dff \smallsetminus\dff \{\dff u\trf\}$with$l\qff \in\qff \mathcal{U}$are pairwise disjoint and form a partition of$E\dff \smallsetminus\dff \{\dff u\trf\}$.Since we assumed that$s_l\qff \geqslant\qff 2$for all lines $l$,all these sets are non-empty.Let $U$ be a set of representatives of these sets.In other terms,$U$ is containedin$E\dff \smallsetminus\dff \{\dff u\trf\}$and intersects every set$l\dff \smallsetminus\dff \{\dff u\trf\}$with$l\qff \in\qff \mathcal{U}$in exactly $1$ point.In particular,$U$ consists of exactly $p$ points.
If $(\fff l\fff,\pff z\fff)\qff \in\qff \mathcal{U}\dff \times\dff U$ and $z\qff \not\in\qff l$,then the [de Bruijn–Erdös ]{}inequality$s_{\fff l}\qff \leqslant\qff k_{\fff z}$holds.There are $p\fff(p\qff -\qff 1)$ of such pairs$(\fff l\fff,\pff z\fff)$and hence$p\fff(p\qff -\qff 1)$of such inequalities.For each$l\qff \in\qff \mathcal{U}$the number $s_{\fff l}$ occurs$p\qff -\qff 1$times in the left hand sides of them,and for each$z\qff \in\qff U$the number $k_{\fff z}$ occurs$p\qff -\qff 1$times in the right hand sides.Hence the sum of all these inequalities is $$\label{sum-all-pairs}
\quad
\sum_{l\qff \in\qff \mathcal{U}} (p\qff -\qff 1)\dff s_{\fff l}
\off\off \leqslant\off\off
\sum_{z\qff \in\qff U} (p\qff -\qff 1)\dff k_{\fff z}\qff.$$
After dividing(\[sum-all-pairs\]) by$p\qff -\qff 1$we get $$\label{sum-divided}
\quad
\sum_{l\qff \in\qff \mathcal{U}} s_{\fff l}
\off\off \leqslant\off\off
\sum_{z\qff \in\qff U} k_{\fff z}\qff.$$
Now it is only natural to take the sum of the inequalities (\[s-upper-estimate\])and(\[sum-divided\]) and conclude that $$\label{sum-all-s-estimate}
\quad
\sum_{l\qff \in\qff \mathcal{L}} s_{\fff l}
\off\off \leqslant\off\off
(m\qff -\qff p)\dff k_{\fff u}
\off +\off
\sum_{z\qff \in\qff U} k_{\fff z}\qff.$$
The left hand side of the inequality(\[sum-all-s-estimate\]) is the same as the left hand side of the equality(\[sums\]).The right hand side of(\[sum-all-s-estimate\]) can be compared with the right hand side of the equality(\[sums\]) if$k_{\fff u}$isthe least among the numbers$k_{\fff z}$and$m\qff \leqslant\qff n$.
[**Proofoftheinequality$m\qff \geqslant\qff n$.**]{} Now we are ready to do the part(c)of the Bourbaki exercise.Let$u\qff \in\qff E$be a point such that $k_{\fff u}$ is the least of the numbers $k_{\fff z}$ over all points$z\qff \in\qff E$.Then $$\label{k-lower-estimate}
\quad
(m\qff -\qff p)\dff k_{\fff u}
\off\off \leqslant\off\off
\sum_{z\qff \in\qff Y} k_{\fff z}\qff.$$
for every subset$Y\qff \subset\qff E$consisting of$m\qff -\qff p$points.
Suppose that$m\qff \leqslant\qff n$.Then the subset $Y$ can be chosen to be disjoint from $U$(because $U$ consists of $p$ points).Let us choose an arbitrary $Y$ disjoint from $U$ andlet$Z\off =\off Y\qff \cup\qff U$.Then $Z$ is a subset of $E$ consisting of $m$ points and the inequalities (\[sum-all-s-estimate\])and(\[k-lower-estimate\]) imply that $$\label{final-estimate}
\quad
\sum_{l\qff \in\qff \mathcal{L}} s_{\fff l}
\off\off \leqslant\off\off
\sum_{z\qff \in\qff Z} k_{\fff z}
\off\off \leqslant\off\off
\sum_{z\qff \in\qff E} k_{\fff z}\qff,$$
where the last inequality is strict unless$Z\off =\off E$.In view of(\[sums\])this inequality cannot be strict and hence$Z\off =\off E$and$m\qff =\qff n$.Since$m\qff \leqslant\qff n$implies that$m\qff =\qff n$,we see that$m\qff \geqslant\qff n$.
[**The case$m\qff =\qff n$.**]{} After the work done in the proofs of(a),(b),and(c),the part(d)nearly proves itself.As we will see,in this case all inequalities (\[s-upper-estimate\])–(\[final-estimate\])are,in fact,equalities.
By(\[sums\])the leftmost and the rightmost sums in(\[final-estimate\])are equal.It follows that$Z\off =\off E$and hence$Y\off =\off E\dff \smallsetminus\dff U$.Moreover, the sides of each of the inequalities (\[sum-all-s-estimate\])and(\[k-lower-estimate\])are equal. Since $k_{\fff u}$ is the least of the numbers $k_{\fff z}$,the equality of the sides of(\[k-lower-estimate\])implies that $$\label{ku-k-not-u}
\quad
k_{\fff u}
\off =\off
k_{\fff z}
\quad\mbox{ for\qss all }\quad
z\qff \in\qff Y\off =\off E\dff \smallsetminus\dff U\qff.$$
The fact that the sides of(\[sum-all-s-estimate\])are equal implies that the sides of each of the inequalities(\[s-upper-estimate\])and(\[sum-divided\])are equal also.The equality of the sides of(\[s-upper-estimate\]) implies that $$\label{s-not-ku}
\quad
s_{\fff l}
\off =\off
k_{\fff u}
\quad\mbox{ for\qss all }\quad
l\qff \not\in\qff \mathcal{U}\qff.$$
Since the sides of(\[sum-divided\])are equal,the sides of(\[sum-all-pairs\])are also equal.Since(\[sum-all-pairs\])is the sum of the inequalities$s_{\fff l}\qff \leqslant\qff k_{\fff z}$ over all pairs $(\fff l\fff,\pff z\fff)\qff \in\qff \mathcal{U}\dff \times\dff U$ such that$z\qff \not\in\qff l$,the equality of the sides of(\[sum-all-pairs\])implies that$s_{\fff l}\qff =\qff k_{\fff z}$for all $(\fff l\fff,\pff z\fff)\qff \in\qff \mathcal{U}\dff \times\dff U$ such that$z\qff \not\in\qff l$.Equivalently, $$\label{s-k-uu}
\quad
s_{\fff l}
\off =\off
k_{\fff z}
\quad\mbox{ if }\quad
l\qff \in\qff \mathcal{U}
\quad\mbox{ and }\quad
z\qff \in\qff U\qff \smallsetminus\qff l\qff.$$
The rest of the proof splits into two subcases depending onif $p\qff =\qff 2$or$p\qff \geqslant\qff 3$.
[**The subcase$p\qff =\qff 2$.**]{} In this case$\mathcal{U}\off =\off \{\trf l\fff,\pff l' \qff\}$for some$l\fff,\pff l'$andhence$E\qff =\qff l\qff \cup\qff l'$.It follows that every line different from$l\fff,\pff l'$contains only $2$ points,namely the points of its intersection with the lines $l\fff,\pff l'$.If$s_{\fff l}\fff,\off s_{\fff l'}\pff \geqslant\pff 3$,then there are at least $4$ points$z\qff \neq\qff u$ and the part (b)implies that $k_{\fff z}\qff \geqslant\qff 3$for every$z\qff \neq\qff u$.On the other hand, (\[ku-k-not-u\])implies that $$\quad
k_{\fff z}
\off =\off
k_{\fff u}
\off =\off
p
\off =\off
2$$
for every$z\qff \not\in\qss U$.But $U$ consists of only two points and hence$k_{\fff z}\qff \geqslant\qff 3$for no more than two points $z$.The contradiction shows that either$s_{\fff l}\qff =\qff 2$or$s_{\fff l'}\qff =\qff 2$.We may assume that$s_{\fff l}\qff =\qff 2$.Then$l\qff =\qff \{\dff u\fff,\pff a \trf\}$for some$a\qff \in\qff E$and every line different from$l\fff,\pff l'$has the form$\{\dff a\fff,\pff z \dff\}$with$z\qff \in\qff l'\dff \smallsetminus\dff \{\dff u\trf\}$.Itfollows that$(\fff E\fff,\pff \mathcal{L}\fff)$ is a near-pencil.
[**The subcase$p\qff \geqslant\qff 3$.**]{} The set $U$ is a set of representatives of the sets$l\dff \smallsetminus\dff \{\fff u\trf\}$with$l\qff \in\qff \mathcal{U}$.For any two lines$l\fff,\pff l'\qff \in\qff \mathcal{U}$the assumption$p\qff \geqslant\qff 3$implies that there exists a point$z\qff \in\qff U$such that$z\qff \not\in\qff l\fff,\pff l'$.If$z$is such a point,then(\[s-k-uu\])implies that $$\quad
s_{\fff l}\off =\off k_{\fff z}\off =\off s_{\fff l'}\qff.$$ Similarly,if$z\fff,\pff z'\qff \in\qff U$,then there exists a line$l\qff \in\qff \mathcal{U}$such that$z\fff,\pff z'\qff \not\in\qff l$and hence $$\quad
k_{\fff z}\off =\off s_{\fff l}\off =\off k_{\fff z'}\qff.$$ It follows that in the subcase$p\qff \geqslant\qff 3$all numbers$s_{\fff l}$,$k_{\fff z}$with$l\qff \in\qff \mathcal{U}$and$z\qff \in\qff U$are equal.Since$k_{\fff u}\qff =\qff p\qff \geqslant\qff 3$is the smallest of the numbers $k_{\fff z}$ over all$z\qff \in\qff E$,it follows that$$\quad
s_{\fff l}
\off =\off
k_{\fff z}
\off \geqslant\off 3
\quad\mbox{ for\qss all }\quad
l\qff \in\qff \mathcal{U}\fff,\quad
z\qff \in\qff U.$$ Let$l\qff \in\qff \mathcal{U}$,and let $y$ be the unique element of $U$ contained in $l$. Since$s_{\fff l}\qff \geqslant\qff 3$, there exists a point$x\qff \in\qff l$not equal to$u\fff,\pff y$.We can replace in $U$ the point $y$ by the point $x$ and get a new set of representatives $U'$.Then all previous results apply to $U'$ in the role of $U$.In particular,$k_{\fff x}
\off =\off
k_{\fff z}$forall $z\qff \in\qff U\dff \smallsetminus\dff l
\off =\off
U'\dff \smallsetminus\dff l$and hence $$\quad
k_{\fff x}
\off =\off
k_{\fff z}
\quad\mbox{ for\qss all }\quad
z\qff \in\qff U.$$ On the other hand,$x\qff \not\in\qff U$and hence $k_{\fff x}\qff =\qff k_{\fff u}$by(\[ku-k-not-u\])applied to the original set $U$.At the same time(\[ku-k-not-u\])implies that$k_{\fff u}\qff =\qff k_{\fff z}$for all$z\qff \not\in\qff U$and hence$$\quad
k_{\fff x}
\off =\off
k_{\fff z}
\quad\mbox{ for\qss all }\quad
z\qff \not\in\qff U.$$ Itfollows that all numbers $k_{\fff z}$ are equal.At the same time by(\[s-not-ku\])and(\[s-k-uu\])every $s_{\fff l}$is equal to some $k_{\fff z}$.It follows that all numbers$s_{\fff l}$,$k_{\fff z}$with$l\qff \in\qff \mathcal{L}$and$z\qff \in\qff E$are equal.It remains to apply the following lemma.
[**Lemma1.**]{} *Ifallthe numbers$s_{\fff l}$,$k_{\fff z}$are equal,then$(\dff E\fff,\pff \mathcal{L}\dff)$is a projective plane.*
[**Proof.**]{} Let $k$ be the common value of the numbers$s_{\fff l}$,$k_{\fff z}$,andlet$y\qff \in\qff E$.The sets$l\dff \smallsetminus\dff \{\dff y\trf\}$with$y\qff \in\qff l$are pairwise disjoint and form a partition of$E\dff \smallsetminus\dff \{\dff y\trf\}$.Each of them consists of $$\quad
s_{\fff l}\qff -\qff 1\off =\off k\qff -\qff 1$$ points,and there are$k_{\fff y}\qff =\qff k$such sets.It follows that the number $n$ of elements of $E$ is equal to$k\fff(k\qff -\qff 1)\qff +\qff 1$.Therefore,$(\dff E\fff,\pff \mathcal{L}\dff)$ is a projective plane.[ $\blacksquare$]{}
[**Remarks.**]{} A key step of this solution and the solution[@q-solution]differ from the de-Bruijn–Erdös paper in the same way:the cyclic order argument of [de Bruijn–Erdös ]{} (see Section\[dbe-proof\])is replaced by the inequalities(\[sum-all-pairs\])and(\[sum-divided\]).
[[.]{}]{.nodecor} {#dbe-proof .unnumbered}
==================
Since the proof presented in Section\[solution\]grow out of a summary of the de Bruijn–Erdös proof,albeit not quite understood,it is not surprising that the two proofs have a lot in common.In the following exposition of the de Bruijn–Erdös proof we will use the notations of Section\[solution\]and will refer to Section\[solution\]for the arguments which differ from[@db-e]only in the notations and the amount of details.The Bruijn–Erdös paper is concise on the border of being cryptic.
The de Bruijn–Erdös proof begins with the parts(a)and(b)of the Bourbaki exercise.After this [de Bruijn–Erdös ]{}introduce $k_{\fff u}$ as the smallest among all numbers $k_{\fff z}$(and denote it by $k_{\fff n}$).Then [de Bruijn–Erdös ]{}observe that it can assumed that every line contains at least two points.Following the notations of Section\[solution\],let us denote by $\mathcal{U}$ the set of all lines containing $u$.By the [de Bruijn–Erdös ]{}inequalities$s_{\fff l}\qff \leqslant\qff k_{\fff u}$ for every $l\qff \not\in\qff \mathcal{U}$.The inequalities(\[s-upper-estimate\])and(\[k-lower-estimate\])follows.The following argument plays a role similar to the role of the inequality(\[sum-all-pairs\]).
[**The cyclic order argument.**]{} Let$l_{\fff 1}\fff,\pff l_{\fff 2}\fff,\pff \ldots\fff,\pff l_{\fff p}$be a cyclically ordered list of elements of $\mathcal{U}$.We treat the subscripts$1\fff,\pff 2\fff,\pff \ldots\fff,\pff p$as integers $\mod\dss p$.For each $i
\off =\off
1\fff,\pff 2\fff,\pff \ldots\fff,\pff p$let us choose some point$a_{\fff j}\qff \in\qff l_{\fff j}\dff \smallsetminus\dff \{\dff u \dff\}$andlet $U$ be the set of these points.Let $$\quad
s_{\fff i}
\off =\off
s_{\fff l_{\fff i}}
\hspace*{1.5em}\mbox{ and }\hspace*{1.5em}
k_j
\off =\off
k_{\fff a_{\fff j}}\qff.$$ Since$a_{\fff i\dff +\dff 1}\qff \not\in\qff l_{\fff i}$, by the [de Bruijn–Erdös ]{}inequalities $s_{\fff i}\qff \leqslant\qff k_{\fff i\dff +\dff 1}$for all $i\off =\off
1\fff,\pff 2\fff,\pff \ldots\fff,\pff p$,i.e. $$\label{s-k-cycle}
\quad
s_{\fff 1}\off \leqslant\off k_{\fff 2}\dff,\quad
s_{\fff 2}\off \leqslant\off k_{\fff 3}\dff,\quad
\ldots\dff,\quad
s_{\fff p}\off \leqslant\off k_{\fff 1}\dff.$$
By summing the inequalities(\[s-k-cycle\])one concludes that$$\label{sum-to-p}
\quad
\sum_{j\qff =\qff 1}^p s_j
\off \leqslant\off
\sum_{j\qff =\qff 1}^p k_j\qff.$$ The inequality(\[sum-to-p\])is nothing else but another form of (\[sum-divided\]). The arguments of Section\[solution\]show that(\[sum-to-p\]) implies that$m\qff \geqslant\qff n$. In fact,de Bruijn and Erdös do not bother to write down even the inequality(\[sum-to-p\]),to say nothing about other details presented in Section\[solution\].
[**The case$m\qff =\qff n$.**]{} In view of the equality(\[sums\])in this case the left hand and the right hand sides of the inequality(\[sum-to-p\]) are equal.Together with(\[s-k-cycle\])this implies that $$\label{cycle-of-equalities}
\quad
s_{\fff 1}\off =\off k_{\fff 2}\dff,\quad
s_{\fff 2}\off =\off k_{\fff 3}\dff,\quad
\ldots\dff,\quad
s_{\fff p}\off =\off k_{\fff 1}\qff.$$ Similarly,in this case the left hand and the right hand sides of the inequality(\[k-lower-estimate\]) are equal.Since$m\qff =\qff n$,one can take$Y\qff =\qff E\dff \smallsetminus\dff U$in(\[k-lower-estimate\]).It follows that$k_{\fff u}
\qff =\qff
k_{\fff z}$for all$z\qff \in\qff E\dff \smallsetminus\dff U$.Finally,the left hand and the right hand sides of the inequality(\[s-upper-estimate\])are equal.It follows that$s_{\fff l}
\qff =\qff
k_{\fff u}$for all$l\qff \not\in\qff \mathcal{U}$. By combining the last two observations,we see that $$\quad
s_{\fff l}
\off =\off
k_{\fff z}
\quad\mbox{ for\qss all }\quad
l\qff \in\qff \mathcal{L}\dff \smallsetminus\dff \mathcal{U}\fff,\quad
z\qff \in\qff E\dff \smallsetminus\dff U\dff.$$
Since$m\qff =\qff n$,both sets$\mathcal{L}\dff \smallsetminus\dff \mathcal{U}$and$E\dff \smallsetminus\dff U$consist of$n\qff -\qff p$elements.It follows that one can number the points and lines in such a way that(in the notation of the Bourbaki exercise) $$\quad
s_{\fff 1}\off =\off k_{\fff 1}\fff,\quad
s_{\fff 2}\off =\off k_{\fff 2}\fff,\quad
\ldots\fff,\quad
s_{\fff n}\off =\off k_{\fff n}\qff.$$ As the next step,let us renumber the points and lines once more and assume that $$\label{k-order}
\quad
k_{\fff 1}\off \geqslant\off
k_{\fff 2}\off \geqslant\off
\ldots\off
\geqslant\off k_{\fff n}\qff.$$ The rest of the proof splits into two subcases depending onif $k_{\fff 1}\qff >\qff k_{\fff 2}$or not.
[**The subcase$k_{\fff 1}\qff >\qff k_{\fff 2}$.**]{} In this case$s_{\fff 1}\qff =\qff k_{\fff 1}\qff >\qff k_{\fff i}$for all$i\qff \geqslant\qff 2$.By the [de Bruijn–Erdös ]{}inequalities this implies that $a_{\fff i}\qff \in\qff A_{\fff 1}$for all$i\qff \geqslant\qff 2$.Itfollows that$(\dff E\fff,\pff \mathcal{L}\dff)$is a near-pencil.
[**The subcase$k_{\fff 1}\off =\off k_{\fff 2}$.**]{} Suppose that$k_{\fff j}\qff <\qff k_{\fff 1}\qff =\qff k_{\fff 2}$for some $j$.By the [de Bruijn–Erdös ]{}inequalities $a_{\fff j}$ belongs to the both lines $A_{\fff 1}$ and $A_{\fff 2}$.This is possible for only one point,namely the point of the intersection of the lines $A_{\fff 1}$ and $A_{\fff 2}$.In view of(\[k-order\]),this may happen only if$$\quad
k_{\fff 1}
\off =\off
k_{\fff 2}
\off =\off
\ldots
\off =\off
k_{\fff n\dff -\dff 1}
\off >\off
k_{\fff n}$$ and hence$
s_{\fff j}
\off =\off
k_{\fff j}
\off >\off
k_{\fff n}
\off \geqslant\off
2$for all$j\qff \neq\qff n$.It follows that$s_{\fff j}\off \geqslant\off 3$if$j\qff <\qff n$.In particular,all $k_{\fff n}$ lines containing $a_{\fff n}$ consist of$\geqslant\qff 2$points and all except,perhaps,the line $A_{\fff n}$,consist of$\geqslant\qff 3$points.Therefore one can choose $2$ points$x\fff,\pff y\qff \neq\qff a_{\fff n}$on one of these lines,and a point$z\qff \neq\qff a_{\fff n}$on some other line.Let$l_{j}\fff,\pff l_{j'}$be the lines containing the pairs$\{\dff x\fff,\fff z \dff\}$and$\{\dff y\fff,\fff z \dff\}$respectively.Then$j\qff \neq\qff j'$and $a_{\fff n}\off \not\in\off l_j\fff,\pff l_{j'}$.Hence the [de Bruijn–Erdös ]{}inequalities imply that$s_{\fff j}\fff,\pff s_{\fff j'}\off \leqslant\off k_{\fff n}$,contrary to the fact that $s_{\fff j}\off >\off k_{\fff n}$if$j\off \neq\off n$.The contradiction shows that all numbers $k_{\fff j}$ are equal,and hence all numbers$s_{\fff i}\fff,\pff k_{\fff j}$are equal.Now the observation at the end of Section\[solution\]implies that$(\dff E\fff,\pff \mathcal{L}\dff)$is a projective plane.
[**Intersection of lines.**]{} After the proof is completed,[de Bruijn–Erdös ]{}point out that in the subcase$k_{\fff 1}\qff =\qff k_{\fff 2}$of the case$m\qff =\qff n$every two lines intersect.Indeed,if$l'\fff,\pff l''$are two disjoint lines and$a\qff \in\qff l''$,then there are $s_{\fff l'}$ lines containing $a$ and intersecting $l'$,and still one more line,namely $l''$,containing $a$.Therefore$k_{\fff a}\qff \geqslant\qff s_{\fff l'}\qff +\qff 1$,contrary to the fact all numbers$k_{\fff z}\fff,\pff s_{\fff l}$are equal.In fact,every two lines obviously intersect in the subcase$k_{\fff 1}\qff >\qff k_{\fff 2}$also.
[**Why $k_{\fff n}$?**]{} Now it is clear why the smallest of the numbers $k_{\fff z}$ is denoted by $k_{\fff n}$.The number $k_{\fff n}$ is indeed the smallest if the points are ordered in such a way that(\[k-order\])holds.At the same time(\[k-order\]) plays almost no role in the proof.One may speculate that(\[k-order\]) and notation $k_{\fff n}$ for the smallest of the numbers $k_{\fff z}$ are remnants of an earlier approach to the theorem.
[[.]{}]{.nodecor} {#reps .unnumbered}
==================
[**The cyclic order argument and systems of distinct representatives.**]{} The key step of the de Bruijn–Erdös proof is the cyclic order argument used to prove the inequality(\[sum-to-p\]) and the equalities(\[cycle-of-equalities\])in the case$m\qff =\qff n$.Ultimately, the cyclic order argument is based on the fact that$a_{\fff i\dff +\dff 1}\qff \not\in\qff l_{\fff i}$for all $i\off =\off
1\fff,\pff 2\fff,\pff \ldots\fff,\pff p$, i.e.on the fact that$i\qff \longmapsto\qff a_{\fff i\dff +\dff 1}$is a system of distinct representatives for the family$i\off \longmapsto\off E\qff \smallsetminus\qff l_{\fff i}$of subsets of $E$,where$i\off =\off
1\fff,\pff 2\fff,\pff \ldots\fff,\pff p$.
Once this is realized,it is only natural to look for a system of distinct representatives of the full family ${\displaystyle}l\qff \longmapsto\qff E\qff \smallsetminus\qff l$of the complements of lines,i.e.for an injective map$l\qff \longmapsto\qff a\fff({\fff}l\fff)$from$\mathcal{L}$to$E$ such that$a\fff({\fff}l\fff)\off \in\off E\qff \smallsetminus\qff l$for all$l\qff \in\qff \mathcal{L}$.
By the well knownPh.Hall’s marriage theorem,such a system of distinct representatives exists if and only if for every subset$\mathcal{K}\qff \subset\qff \mathcal{L}$the union $$\label{union}
\quad
\bigcup_{\dff l\dff \in\dff \mathcal{K}}\qff E\qff \smallsetminus\qff l
\off\off =\off\off
E\off \smallsetminus\off \bigcap_{\dff l\dff \in\dff \mathcal{K}}\qff l$$
contains$\geqslant\qff |\fff \mathcal{K}\fff |$elements,where$|\dff X\dff|$denotes the number of elements of a set $X$.But the intersection of$\geqslant\qff 2$lines consists of$\leqslant\qff 1$points,and,almost obviously,this condition holds.
[**The message.**]{} All this emerged in my mind in one instant as an irreducible revelation.My first thought after this revelation was that it cannot be true,because if it is true,then everybody writing about this topic would use systems of distinct representatives.Perhaps,the right question is not how I came up with this idea,but why experts missed it. The rest of this section is devoted to the proof[@i-db-e]based on this revelation.
[**Proofof$m\qff \geqslant\qff n$.**]{} We may assume that $m\qff \leqslant\qff n$.Let $\mathcal{K}$ be a subset of $\mathcal{L}$.If$|\dff \mathcal{K}\dff|\qff =\qff 1$,then(\[union\])is the complement of a line and hence contains$\geqslant\qff 1$elements. If$2\qff \leqslant\qff |\dff \mathcal{K}\dff |\qff \leqslant\qff m\qff -\qff 1$,then(\[union\]) is a complement in $E$ of$\leqslant\qff 1$point and hence contains$$\quad
\geqslant\qff n\qff -\qff 1
\off \geqslant\off
m\qff -\qff 1
\off \geqslant\off
|\dff \mathcal{K}\dff |$$ elements.If$|\dff \mathcal{K}\dff|\qff =\qff m$,then(\[union\])contains$n\qff \geqslant\qff m\qff =\qff |\dff \mathcal{K}\dff|$elements.Therefore there exists a system of distinct representativesfor the family $l\qff \longmapsto\qff E\qff \smallsetminus\qff l$,i.e. there exists an injective map$l\qff \longmapsto\qff a\fff({\fff}l\fff)$such that$a\fff({\fff}l\fff)\qff \not\in\qff l$for every $l$.By the [de Bruijn–Erdös ]{}inequalities $$\label{basic-reps-ineq}
\quad
s_{\fff l}\off \leqslant\off k_{\fff a\fff({\fff}l\fff)}
\quad\mbox{ for\qss every }\quad
l\qff \in\qff \mathcal{L}.$$ By summing all these inequalities and using the injectivity of$l\qff \longmapsto\qff a\fff({\fff}l\fff)$ we see that $$\label{three-sums}
\quad
\sum_{l\dff \in\dff \mathcal{L}}\qff s_{\fff l}
\off\off \leqslant\off\off
\sum_{l\dff \in\dff \mathcal{L}}\qff k_{\fff a\fff({\fff}l\fff)}
\off\off \leqslant\off\off
\sum_{\qff z\dff \in\dff E}\qff k_{\fff z}\qff.$$ Moreover,the second inequality is strict unless$m\qff =\qff n$(otherwise the last sum has more positive summands than the previous one).But(\[sums\])implies that both inequalities in(\[three-sums\])should be actually equalities.It follows that$m\qff =\qff n$.Moreover,in view of the inequalities(\[basic-reps-ineq\]),it follows that $s_{\fff l}\off =\off k_{\fff a\fff({\fff}l\fff)}$for every$l\qff \in\qff \mathcal{L}$(under the assumption$m\qff \leqslant\qff n$).
[**The case$m\qff =\qff n$.**]{} Suppose that a point $z$ is contained in$\geqslant\qff m\qff -\qff 1$lines.Each of these lines contains at least one point in addition to $z$.Since$m\qff =\qff n$,there are no other points and $z$ is contained in exactly$m\qff -\qff 1$lines.Since there are exactly $m$ lines,only one line does not contain $z$.This line should contain all points$\neq\qff z$.It follows that$(\dff E\fff,\pff \mathcal{L}\dff)$is a near-pencil.
Suppose now that no point is contained in$\geqslant\qff m\qff -\qff 1$lines.Let $\mathcal{K}$ be a proper subset of $\mathcal{L}$.If$|\dff \mathcal{K}\dff|\qff =\qff 1$, then(\[union\])is equal to$E\dff \smallsetminus\dff l$for some line $l$.If$E\dff \smallsetminus\dff l$consists of only one point $z$,then by the [de Bruijn–Erdös ]{}inequalities $z$ is contained in$\geqslant\qff s_{\fff l}\qff =\qff m\qff -\qff 1$lines,contrary to the assumption.Therefore,(\[union\])contains$\geqslant\qff 2\qff =\qff |\dff \mathcal{K}\dff|\qff +\qff 1$points.If$|\dff \mathcal{K}\dff|\qff \leqslant\qff m\qff -\qff 2$,then(\[union\])contains$\geqslant\qff n\qff -\qff 1\qff =\qff m\qff -\qff 1\qff \geqslant\qff
|\dff \mathcal{K}\dff|\qff +\qff 1$points.Finally,if$|\dff \mathcal{K}\dff|\qff =\qff m\qff -\qff 1$,then(\[union\])contains all $n\qff =\qff m\qff =\qff |\dff \mathcal{K}\dff|\qff +\qff 1$ points because no point is contained in$\geqslant\qff m\qff -\qff 1$lines.
We see that(\[union\])contains$\geqslant\qff |\dff \mathcal{K}\dff|\qff +\qff 1$elements for every proper subset$\mathcal{K}\qff \subset\qff \mathcal{L}$. This allows to get from the marriage theorem more than just the existence of a system of distinct representatives.Let$\lambda\qff \in\qff \mathcal{L}$and $z\qff \in\qff E\dff \smallsetminus\qff \lambda$.Then there exists a system of distinct representatives$l\qff \longmapsto\qff a\fff({\fff}l\fff)$such that$a\fff(\fff\lambda\fff)\qff =\qff z$.This immediately follows from an application of the marriage theorem to the family of sets$(\dff E\qff \smallsetminus\qff \{\dff z \qff\}\dff)\qff \smallsetminus\qff l$with$l\qff \in\qff \mathcal{L}\qff \smallsetminus\qff \{\dff \lambda \qff\}$.
Since$m\qff \leqslant\qff n$,the existence of a system of distinct representatives$l\qff \longmapsto\qff a\fff({\fff}l\fff)$such that$a\fff(\fff\lambda\fff)\qff =\qff z$implies that$s_{\fff \lambda}\off =\off k_{\fff a\fff({\fff}\lambda\fff)}\off =\off k_{\fff z}$.Therefore,$z\qff \not\in\qff l$implies that$s_{\fff l}\qff =\qff k_{\fff z}$and hence every line containing $z$ intersects $l$.It follows that every two lines intersect.
If $E$ cannot be obtained as the union of two lines,then for every two lines$l\fff,\pff l'$there exists a point $z$ such that$z\qff \not\in\qff l\fff,\pff l'$and hence$s_{\fff l}\qff =\qff k_{\fff z}\qff =\qff s_{\fff l'}$.In this case all the numbers $s_{\fff l}\fff,\pff k_{\fff z}$are equal and hence$(\dff E\fff,\pff \mathcal{L}\dff)$is a projective plane by Lemma1at the end of Section\[solution\].If there exist two lines$l\fff,\pff l'$such that$E\qff =\qff l\dff \cup\dff l'$,then$k_{\fff y}\qff =\qff 2$,where $y$ is the point of intersection of $l$ and $l'$,and the proofis completed by applying the following lemma.
[**Lemma2.**]{} *If$m\qff =\qff n$and$k_{\fff y}\qff =\qff 2$for some point $y$,then$(\dff E\fff,\pff \mathcal{L}\dff)$is a near-pencil.*
[**Proof.**]{} Let$l\fff,\pff l'$be the lines containing $y$.Then$E\qff =\qff l\dff \cup\dff l'$and there are$n\qff =\qff s_{\fff l}\qff +\qff s_{\fff l'}\qff -\qff 1$points.In addition to the lines$l\fff,\pff l'$thereare$(s_l\qff -\qff 1)(s_{l'}\qff -\qff 1)$lines consisting of a point in $l\qff \smallsetminus\dss \{\dff y \trf\}$and a point in $l'\qff \smallsetminus\dss \{\dff y \trf\}$.If$s_{\fff l}\off \geqslant\off s_{\fff l'}\off \geqslant\off 3$,then the number $m$oflinesis $$\quad
\geqslant\off
2\qff +\qff (s_l\qff -\qff 1)(s_{l'}\qff -\qff 1)
\off \geqslant\off
2\qff +\qff 2\dff(s_l\qff -\qff 1)
\off =\off
2\fff s_l
\off \geqslant\off
s_l\qff +\qff s_{l'}
\off =\off
n\qff +\qff 1\dff,$$
contrary to the assumption$m\qff =\qff n$.Therefore one of the lines$l\fff,\pff l'$consists of $2$ points and hence$(\dff E\fff,\pff \mathcal{L}\dff)$ is a near-pencil.[ $\blacksquare$]{}
[[.]{}]{.nodecor} {#linear-algebra .unnumbered}
==================
[**A proof of the inequality$m\qff \geqslant\qff n$based on the linear independence.**]{} This proof was communicated to me byF.Petrov[@p].Ibelieve that this is essentially the proof found by A.Suslin.
Let${ \mathbf{R}^{\dff\mathcal{L}} }$be the vector space of maps$\mathcal{L}\dff {\longrightarrow}\dff \mathbf{R}$with the scalar product $$\quad
(\dff v\fff,\pff w \dff)
\off\off =\off\off
\sum_{l\qff \in\qff \mathcal{L}}\qff v(\fff l\fff)\dff w(\fff l\fff).$$ Every$z\qff \in\qff E$ defines a map $v_{\fff z}\dff \colon\dff \mathcal{L}\dff {\longrightarrow}\dff \mathbf{R}$by the rule$v_{\fff z}\fff(\fff l\fff)\qff =\qff 1$if$z\qff \in\qff l$and$v_{\fff z}\fff(\fff l\fff)\qff =\qff 0$otherwise.There are $n$ maps $v_{\fff z}$.Since the dimension of ${ \mathbf{R}^{\dff\mathcal{L}} }$ is equal to $m$,it is sufficient to prove that the maps $v_{\fff z}$ are independent as vectors of${ \mathbf{R}^{\dff\mathcal{L}} }$.
The scalar product$(\dff v_{\fff z}\fff,\pff v_{\fff z} \dff)$is equal to the number of lines containing the point $z$,and hence$(\dff v_{\fff z}\fff,\pff v_{\fff z} \dff)
\qff \geqslant\qff 2$for all$z\qff \in\qff E$.If$z\qff \neq\qff y$,then $(\dff v_{\fff z}\fff,\pff v_{\fff y} \dff)$is equal to the number of lines containing both $z$ and $y$,and hence$(\dff v_{\fff z}\fff,\pff v_{\fff y} \dff)
\qff =\qff 1$.If the vectors $v_{\fff z}$ are linearly dependent,then $$\quad
\sum_{z\qff \in\qff E}\qff c_{\fff z}\dff v_{\fff z}
\off\off =\off\off
0$$ for some real numbers$c_{\fff z}$,$z\qff \in\qff E$,such that not all $c_{\fff z}$ are equal to $0$.For every$y\qff \in\qff E$taking the scalar product of this equality with the vector$v_{\fff y}$results in the equality $$\quad
\sum_{z\qff \in\qff E}\qff c_{\fff z}\dff (\fff v_{\fff z}\fff,\pff v_{\fff y} \fff)
\off\off =\off\off
0.$$ Since$(\fff v_{\fff z}\fff,\pff v_{\fff y} \fff)\qff =\qff 1$for all$z\qff \neq\qff y$,this equality implies that $$\quad
c_{\fff y}\dff ((\fff v_{\fff y}\fff,\pff v_{\fff y} \fff)\qff -\qff 1)
\off +\off
\sum_{z\qff \in\qff E}\qff c_{\fff z}
\off\off =\off\off
0.$$ Since$(\fff v_{\fff y}\fff,\pff v_{\fff y} \fff)\qff \geqslant\qff 2$,it follows that the coefficient $c_{\fff y}$ and the sum$$\quad
\sum_{z\qff \in\qff E}\qff c_{\fff z}$$ have opposite signs.But since not all $c_{\fff y}$ are equal to $0$, this cannot be true for all$y\qff \in\qff E$.The contradiction shows that vectors $v_{\fff z}$ are linearly independent and hence$m\qff \geqslant\qff n$.[ $\blacksquare$]{}
[**Standard linear algebra proofs of the inequality$m\qff \geqslant\qff n$.**]{} In order to present standard proofs it is convenient to return to the notations of theN.Bourbaki exercise.Let $M$ be the*incidencematrix*of the points $a_j$ and sets $A_i$.Namely,$M$isan $n \times m$ matrix with entries$m_{j i}\qff =\qff 1$if$a_j\qff \in\qff A_i$and$m_{j i}\qff =\qff 0$otherwise.Let us consider the product $M\fff M^{\fff T}$,where $M^{\fff T}$ is the matrix transposed to $M$.It is an $n \times n$ matrix with all non-diagonal entries equal to $1$ and with diagonal entries$k_{\fff 1}\fff,\pff k_{\fff 2}\fff,\pff \ldots\fff,\pff k_{\fff n}$.The most classical linear algebra proofs,going back to the paper[@bo]byR.C.Bose, proceed with the computation of the determinant of $M\fff M^{\fff T}$.It is rarely presented in details;apparently,it is expected that the readers enjoy computations of determinants.Curious readers may find a computation of$\det\qff M\fff M^{\fff T}$at the end of this section;in particular,the computation shows that this determinant is non-zero.The non-vanishing of$\det\qff M\fff M^{\fff T}$means that the rank of the matrix $M\fff M^{\fff T}$ is equal to $n$,and this implies that the rank of $M$ is$\geqslant\qff n$.Since $M$ isan $n \times m$ matrix,this may happen only if$m\qff \geqslant\qff n$.
More modern expositions avoid computation of the determinant$\det\qff M\fff M^{\fff T}$by observing that $M\fff M^{\fff T}$ is equal to the sum of the diagonal matrix with the diagonal entries $$\quad
k_{\fff 1}\qff -\qff 1\fff,\off k_{\fff 2}\qff -\qff 1\fff,\off \ldots\fff,\off k_{\fff n}\qff -\qff 1$$
and the $n \times n$ matrix $J$ with all entries equal to $1$.Since$k_{\fff j}\qff \geqslant\qff 2$and hence$k_{\fff j}\qff -\qff 1\qff \geqslant\qff 1$for all $j$,the first matrix is positive definite.The matrix $J$ is positive semi-definite,although is not definite.In fact,the associated quadratic form $\mathbf{x}\qff J\trf \mathbf{x}^{\fff T}$,where$\mathbf{x}
\off =\off
(\fff x_{\fff 1}\fff,\pff x_{\fff 2}\fff,\pff \ldots,\fff x_{\fff n} \fff)$is a row vector, is equal to$(\fff x_{\fff 1}\qff +\qff x_{\fff 2}\qff +\qff \ldots\qff +\qff x_{\fff n} \fff)^{\fff 2}$.It follows that the sum $M\fff M^{\fff T}$ of these matrices is positive definite and hence has the rank $n$.As above,this implies that$m\qff \geqslant\qff n$.
[**Comparing the proofs.**]{} The standard proofs do not fit the *“Kvant”*description of the proof by A.Suslin:they use more advanced tools than the theorem about the linear dependence of more than $n$ vectors in an $n$-dimensional vector space.One can find a proof based only on this theorem in the unpublished book draft[@bf]byL.Babai andP.Frankl.But even in this remarkable book it is hidden in the exercises.See Exercise4.1.5and its solution on p.184.The preference for using the matrix $M\fff M^{\fff T}$ seems to be a part of a dominating culture.On the other hand,all proofs based on the linear algebra more or less explicitly reduce the inequality$m\qff \geqslant\qff n$to the following lemma and then prove it.
[**Lemma.**]{} *Let$V$ be an $m$-dimensional vector space over${\mathbf{R}}$ equipped with a scalar product$(\fff \bullet\fff,\pff \bullet \fff)$.Let$P$ be a set of $n$ vectors in$V$.Suppose that there exists $\lambda\qff \in\qff {\mathbf{R}}$,$\lambda\qff >\qff 0$,such that* $$\quad
(\fff u\fff,\pff u \fff)\off >\off \lambda
\hspace*{1.5em}\mbox{ \emph{and} }\hspace*{1.5em}
(\fff v\fff,\pff w \fff)\off =\off \lambda$$
*for every$u\qff \in\qff P$and every two distinct vectors$v\fff,\pff w\qff \in\qff P$.Then$m\qff \geqslant\qff n$.* [ $\blacksquare$]{}
[**A generalization.**]{} The linear algebra proofs apply with only trivial changes to a more general situation.Namely,it is sufficient to assume that there exist a natural number$\lambda\qff \geqslant\qff 1$such that every two distinct points are contained in exactly $\lambda$ lines and every point is contained in$>\qff \lambda$lines.Then the conclusion$m\qff \geqslant\qff n$still holds.This is due to H.J.Ryser[@r].Apparently,no combinatorial proof ofRyser’s theorem is known.Ryser[@r]also used linear algebra to provide a description of the case$m\qff =\qff n$similar to de Bruijn–Erdös description in the case$\lambda\qff =\qff 1$.
[**The determinant of$M\fff M^{\fff T}$.**]{} For the benefit of the readers who do not like to compute the determinants themselves, here is a computation of $\det\qff M\fff M^{\fff T}$ following the textbook[@hp].
Let$m_{j}\qff =\qff k_{\fff j}\qff -\qff 1$for all $j$.Then $$\quad
M\fff M^{\fff T}
\off =\off
\begin{bmatrix}\off
m_{\fff 1}\dff +\qff 1 & 1 & 1 & \hdotsfor{4} & 1 & 1\\
1 & m_{\fff 2}\dff +\qff 1 & 1 & \hdotsfor{4} & 1 & 1\\
1 & 1 & m_{\fff 3}\dff +\qff 1 & \hdotsfor{4} & 1 & 1\\
& & & & & & & & \\
\hdotsfor{9} \\
& & & & & & & & \\
1 & 1 & 1 & \hdotsfor{4} & 1 & m_{\fff n}\dff +\qff 1\off
\end{bmatrix}.$$
Let us subtract the first row from every other one and get the matrix $$\quad
\phantom{M\fff M^{\fff T}
\off =\off}
\begin{bmatrix}\off
m_{\fff 1}\dff +\qff 1 & 1 & 1 & \hdotsfor{4} & 1 & 1\\
-\qff m_{\fff 1} & m_{\fff 2} & 0 & \hdotsfor{4} & 0 & 0\\
-\qff m_{\fff 1} & 0 & m_{\fff 3} & \hdotsfor{4} & 0 & 0\\
& & & & & & & & \\
\hdotsfor{9} \\
& & & & & & & & \\
-\qff m_{\fff 1} & 0 & 0 & \hdotsfor{4} & 0 & m_{\fff n}\off
\end{bmatrix}.$$
For$j\off =\off 2\fff,\pff 3\fff,\pff \ldots\fff,\pff n$,let us multiply the $j$-th column by$m_{\fff 1}/m_{\fff j}$(recall that$k_{\fff j}\qff \geqslant\qff 2$ and hence$m_j\qff \geqslant\qff 1\qff >\qff 0$)and add the result to the first column.We get the matrix $$\quad
\phantom{M\fff M^{\fff T}
\off =\off}
\begin{bmatrix}\off
D & 1 & 1 & \hdotsfor{4} & 1 & 1\\
0 & m_{\fff 2} & 0 & \hdotsfor{4} & 0 & 0\\
0 & 0 & m_{\fff 3} & \hdotsfor{4} & 0 & 0\\
& & & & & & & & \\
\hdotsfor{9} \\
& & & & & & & & \\
0 & 0 & 0 & \hdotsfor{4} & 0 & m_{\fff n}\off
\end{bmatrix},$$
where${\displaystyle}D
\off\off =\off\off
m_{\fff 1}
\qff +\qff
1
\off +\off
\sum_{j\qff =\qff 2}^n\qff \frac{\fff m_{\fff 1}\fff}{m_j}
\off\off =\off\off
m_{\fff 1}
\off +\off
\sum_{j\qff =\qff 1}^n\qff \frac{\fff m_{\fff 1}\fff}{m_j}$.
It follows that ${\displaystyle}\det\qff M\dff M^{\fff T}
\off\off =\off\off
D
\qff\cdot
\prod_{j\qff =\qff 2}^n m_j
\off\off =\off\off
\prod_{j\qff =\qff 1}^n m_j
\qff \cdot\qff
\left(\qff
1
\qff +\qff
\sum_{j\qff =\qff 1}^n\qff \frac{\fff 1\fff}{m_j}
\qff\right)
\off\off \neq\off\off 0$.
[[.]{}]{.nodecor} {#h-proof .unnumbered}
==================
[**Two papers ofH.Hanani.**]{} According to the Th.Motzkin[@m],the first proof of the inequality$m\qff \geqslant\qff n$and,it seems,of the full de Bruijn–Erdös theorem,was given in1938by H.Hanani.He published an outline of his proof[@h1] only in1951. Later on H.Hanani published a detailed exposition[@h2]of a simplified proof. In fact,in[@h2]he proved(at no extra cost)a stronger version of the de Bruijn–Erdös theorem.He also used his methods to prove a$3$-dimensional analogue dealing with points,lines,and planes.
[**Hanani’sTheorem.**]{} *Under the previous assumptions,let$L\qff \in\qff \mathcal{L}$be a line containing the maximal number of points among all lines,let$\mathcal{P}$be the set of all lines intersecting$L$(in particular,$L\qff \in\qff \mathcal{P}$),and let$p$be the number of elements of$\mathcal{P}$.Then$p\qff \geqslant\qff n$,andif$p\qff =\qff n$,then$\mathcal{P}\qff =\qff \mathcal{L}$and$(\dff E\fff,\pff \mathcal{L}\dff)$ is either a near-pencil,or a projective plane.*
Suppose that$n\qff \geqslant\qff p$and$(\dff E\fff,\pff \mathcal{L}\dff)$is not a near-pencil.As usual,we assume that every line contains$\geqslant\qff 2$points.Let$a\qff =\qff s_{\dff L}$.Let $K$ be the line with the maximal number of points among the lines different from $L$,andlet$b\qff =\qff s_{\dff K}$.Then$a\qff \geqslant\qff b$.The strategy is to estimate $n$,or,what is the same,$n\qff -\qff 1$in terms of $a$ and $b$ both from the below and from the above.
[**Hanani’sLemma.**]{} *If$x\qff \in\qff L$,then* $$\label{lemma}
k_{\fff x}\qff -\qff 1\off \geqslant\off \frac{n\qff -\qff a}{b\qff -\qff 1}\qff.$$
Let us consider pairs$(\fff l\fff,\pff y\fff)$such that $l$ is a line containing $x$ and $y$ is a point in$l\dff \smallsetminus\dff L$.Such a pair is uniquely determined by the point $y$ and hence there are $n\qff -\qff a$ such pairs.A line $l$ occurs in such a pair if and only if$x\qff \in\qff l$and$l\qff \neq\qff L$.It follows that there are$k_{\fff x}\qff -\qff 1$choices of $l$.Given a line $l$, there are$\leqslant\qff b\qff -\qff 1$choices for the point $y$.Therefore the number$n\qff -\qff a$of such pairs is $\leqslant\qff (k_{\fff x}\qff -\qff 1)(b\qff -\qff 1)$.The lemma follows.[ $\blacksquare$]{}
[**An upper estimate of$n\qff -\qff 1$.**]{} By summing the inequalities(\[lemma\])over all$x\qff \in\qff L$and adding $1$ in order to account for the line $L$ itself,we can estimate $p$ from below and conclude that $$\label{p-lower-1}
\quad
n
\off \geqslant\off
p
\off \geqslant\off
1\qff +\qff a\qff \frac{n\qff -\qff a}{b\qff -\qff 1}
\off =\off
1\qff +\qff a\qff \frac{(n\qff -\qff 1)\qff -\qff (a\qff -\qff 1)}{b\qff -\qff 1}$$
or,what is the same, $$\label{n-upper}
\quad
a\fff(a\qff -\qff 1)
\off \geqslant\off
(n\qff -\qff 1)(a\qff -\qff b\qff +\qff 1)\dff.$$
The inequality(\[n-upper\]) provides an estimate of$n\qff -\qff 1$from the above.
[**A lower estimate of$n\qff -\qff 1$.**]{} There is another way to estimate $p$ from below.By a miracle,this other estimate of $p$ from the same side leads to an estimate of$n\qff -\qff 1$from the other side.Let $z$ be a point in$L\dff \cap\dff K$if $L\dff \cap\dff K\qff \neq\qff \emptyset$,and an arbitrary point of $L$ otherwise.For every$x\qff \in\qff L\dff \smallsetminus\dff \{\dff z\dff\}$,$y\qff \in\qff K\dff \smallsetminus\dff \{\dff z\dff\}$there is a unique line $l$ containing$\{\dff x\fff,\pff y \dff\}$.All these lines are distinct,not equal to $L$, and do not contain $z$. Clearly,there are$(a\qff -\qff 1)(b\qff -\qff 1)$of such lines.A lower estimate of number $k_{\fff z}$ of lines containing $z$ is provided by(\[lemma\]).It follows that $$\quad
n
\off \geqslant\off
p
\off \geqslant\off
1
\qff +\qff
\frac{n\qff -\qff a}{b\qff -\qff 1}
\qff +\qff
(a\qff -\qff 1)(b\qff -\qff 1)$$
$$\quad
\hspace*{12em}
\phantom{n
\off \geqslant\off
p
\off }
=\off
1
\qff +\qff
\frac{(n\qff -\qff 1)\qff -\qff (a\qff -\qff 1)}{b\qff -\qff 1}
\qff +\qff
(a\qff -\qff 1)(b\qff -\qff 1)$$
and hence${\displaystyle}(n\qff -\qff 1)(b\qff -\qff 1)
\off \geqslant\off
(n\qff -\qff 1)
\qff -\qff
(a\qff -\qff 1)
\qff +\qff
(a\qff -\qff 1)(b\qff -\qff 1)^{\fff 2}$and $$\quad
(n\qff -\qff 1)(b\qff -\qff 2)
\off \geqslant\off
(a\qff -\qff 1)(b^{\fff 2}\qff -\qff 2\fff b)
\off =\off
(a\qff -\qff 1)\fff b\fff (b\qff -\qff 2).$$
Since$b\qff \geqslant\qff 2$,it follows that either$b\qff =\qff 2$,or${\displaystyle}n\qff -\qff 1
\off \geqslant\off
(a\qff -\qff 1)\fff b$.If$b\qff =\qff 2$, then all lines except $L$ consist of $2$ points and the inequality (\[n-upper\])implies that$a\qff \geqslant\qff n\qff -\qff 1$.But$L\qff \neq\qff E$and hence$a\qff \leqslant\qff n\qff -\qff 1$.It follows that$a\qff =\qff n\qff -\qff 1$and hence $L$ contains all points of $E$ except one and$(\dff E\fff,\pff \mathcal{L}\dff)$is a near-pencil,contrary to the assumption.Therefore $$\label{n-lower}
\quad
n\qff -\qff 1
\off \geqslant\off
(a\qff -\qff 1)\fff b\dff.$$
The inequality(\[n-lower\]) provides an estimate of$n\qff -\qff 1$from the below.
[**Combining the two estimates.**]{} After multiplying the inequality(\[n-lower\])by$(a\qff -\qff b\qff +\qff 1)$and combining the result with the inequality(\[n-upper\]), we see that $$\quad
a\fff (a\qff -\qff 1)
\off \geqslant\off
(a\qff -\qff 1)\fff b\fff (a\qff -\qff b\qff +\qff 1)$$
and hence${\displaystyle}a
\off \geqslant\off
b\fff(a\qff -\qff b\qff +\qff 1)
\off =\off
b\fff (a\qff -\qff b)
\qff +\qff
b$,or,what is the same $$\quad
0
\off \geqslant\off
(b\qff -\qff 1) (a\qff -\qff b)\dff.$$ Since$b\qff >\qff 1$,this implies that$b\qff \geqslant\qff a$.On the other hand,$b\qff \leqslant\qff a$by the definition of$a\fff,\pff b$.It follows that$a\qff =\qff b$.By combining$a\qff =\qff b$with the inequalities(\[n-upper\])and(\[n-lower\])we conclude,respectively,that $a\fff(a\qff -\qff 1)
\off \geqslant\off
n\qff -\qff 1$and$n\qff -\qff 1
\off \geqslant\off
a\fff(a\qff -\qff 1)$.It follows that$n\qff -\qff 1
\off =\off
a\fff(a\qff -\qff 1)$andhence$n
\off =\off
a\fff(a\qff -\qff 1)\qff +\qff 1$.By combining this with(\[p-lower-1\])we see that $$\quad
a\fff(a\qff -\qff 1)\qff +\qff 1
\off =\off
n
\off \geqslant\off
p
\off \geqslant\off
1\qff +\qff a\qff \frac{a\fff(a\qff -\qff 1)\qff +\qff 1\qff -\qff a}{a\qff -\qff 1}
\off =\off
1\qff +\qff a\fff(a\qff -\qff 1)\dff.$$
It follows that$n\qff =\qff p$,and therefore$p\qff \geqslant\qff n$if the inequality$p\qff \leqslant\qff n$is not assumed.
[**The case$p\qff =\qff n$.**]{} As we just saw,in this case$a\qff =\qff b$and$n\qff =\qff a\fff(a\qff -\qff 1)\qff +\qff 1$.
Let us prove first that every line belonging to $\mathcal{P}$ consists of exactly $a$ points.Consider all pairs$(\fff l\fff,\pff y\fff)$such that$l\qff \in\qff \mathcal{P}$and$y\qff \in\qff E\dff \smallsetminus\dff L$.The line $l$is uniquely determined by its point of intersection with $L$(which can be any point of$L$)and the point $y$.Therefore there are$a\fff(n\qff -\qff a)
\off =\off
a\fff n\qff -\qff a^{\fff 2}$such pairs.On the other hand,for every line $l\qff \in\qff \mathcal{P}\dff \smallsetminus\dff \{\qff L \qff\}$ there are$\leqslant\qff a\qff -\qff 1$choices of the point $y$ and hence the number of such pairs is$\leqslant\qff (p\qff -\qff 1)(a\qff -\qff 1)$.Moreover,if at least one line $l\qff \in\qff \mathcal{P}\dff \smallsetminus\dff \{\qff L \qff\}$has$<\qff a$points,then the number of such pairs is$<\qff (p\qff -\qff 1)(a\qff -\qff 1)$.But $(p\qff -\qff 1)(a\qff -\qff 1)
\off =\off
(n\qff -\qff 1)(a\qff -\qff 1)
\off =\off
n\fff a\qff -\qff a^{\fff 2}$.It follows that every line belonging to$\mathcal{P}\dff \smallsetminus\dff \{\qff L \qff\}$,and hence every line belonging to $\mathcal{P}$,consists of exactly $a$ points.
Now we are ready to prove that$\mathcal{L}\off =\off \mathcal{P}$.By the definition,every line containing a point of $L$ belongs to $\mathcal{P}$.Let$y\qff \in\qff E\dff \smallsetminus\dff L$.For every$x\qff \in\qff L$there is a unique line containing$\{\fff x\fff,\pff y\dff\}$.These lines are pairwise distinct,intersect only at $y$, and belong to $\mathcal{P}$.Moreover,every line containing $y$ and belonging to $\mathcal{P}$ is equal to one of these $a$ lines.Each of these lines contains $a\qff -\qff 1$points different form $y$.It follows that the total number of points on these lines is equal to$a\dff(a\qff -\qff 1)\qff +\qff 1$,i.e.to the number $n$ of points in $E$.Therefore for every point$z\qff \neq\qff y$there is a line belonging to $\mathcal{P}$ and containing$\{\fff z\fff,\pff y \dff\}$.Since there is only one line containing any two given points,it follows that all lines containing a point$y\qff \in\qff E\dff \smallsetminus\dff L$belong to $\mathcal{P}$.It follows that$\mathcal{L}\off =\off \mathcal{P}$and every point in$E\dff \smallsetminus\dff L$belongs to exactly $a$ lines.In view of the previous paragraph,$\mathcal{L}\off =\off \mathcal{P}$implies that every line consists of exactly $a$ points.
By the previous paragraph$k_{\fff y}\qff =\qff a$ if$y\qff \in\qff E\dff \smallsetminus\dff L$.If$y\qff \in\qff L$, thenby(\[lemma\]) $$\quad
k_{\fff y}
\off \geqslant\off
1
\qff +\qff
\frac{n\qff -\qff a}{b\qff -\qff 1}
\off =\off
1
\qff +\qff
\frac{a\fff(a\qff -\qff 1)\qff +\qff 1\qff -\qff a}{a\qff -\qff 1}
\off =\off
a\dff.$$
If$k_{\fff y}\off >\off a$, then the arguments of the previous paragraph show that $n\off >\off a\fff(a\qff -\qff 1)\qff +\qff 1$,contrary to$n\qff =\qff a\fff(a\qff -\qff 1)\qff +\qff 1$.The contradiction shows that$k_{\fff y}\off =\off a$also for$y\qff \in\qff L$.It follows that$(\dff E\fff,\pff \mathcal{L}\dff)$is a projective plane.This completes the proof of Hanani’s theorem.
[**Deducing the de Bruijn–Erdös theorem.**]{} Suppose that$m\qff \leqslant\qff n$.Obviously,$p\qff \leqslant\qff m$and hence$p\qff \leqslant\qff n$.By Hanani’s theorem this implies that$p\qff =\qff n$and$(\dff E\fff,\pff \mathcal{L}\dff)$is either a near-pencil,or a projective plane.Since$p\qff \leqslant\qff m\qff \leqslant\qff n$and$p\qff =\qff n$,it follows that$m\qff =\qff n$.
[**Remarks.**]{} In contrast with[@db-e]and many papers written much later,Hanani’s proof of his version of the de Bruijn–Erdös theorem in[@h2]is quite modern.The points and lines are not enumerated;in fact,there are no subscripts at all.But when he turns to the $3$-dimensional case,he returns to the tradition of enumerating almost everything in sight …
Also,in contrast with almost every other proof,Hanani’s proof does not use the [de Bruijn–Erdös ]{}inequalities,at least not directly.But the proofof the fact that$\mathcal{P}\qff =\qff \mathcal{L}$includes a proof of the [de Bruijn–Erdös ]{}inequalities$s_{\dff L}\qff \leqslant\qff k_{\fff y}$for$y\qff \not\in\qff L$.
[[.]{}]{.nodecor} {#another-h-proof .unnumbered}
==================
This proof follows the outline of the Hanani’s one,but brings into the play the smallest number $k_{\fff u}$ among all $k_{\fff z}$.Also,“the second largest”line is chosen not among all lines,but among the lines containing $u$.This allows to avoid Hanani’sLemma and to replace“miraculous”estimates by rather straightforward ones.The proof was partially inspired byV.Napolitano[@n].If one is interested only in the de Bruijn–Erdös theorem,it can be simplified even further.
Suppose that$n\qff \geqslant\qff p$.Following de Bruijn–Erdös[@db-e],let us consider a point $u$ such that $k_{\fff u}$ is the smallest number among all numbers $k_{\fff z}$.Let$a\qff =\qff s_{\dff L}$and$k\qff =\qff k_{\fff u}$.There are two cases to consider:the case when$k\qff \geqslant\qff a$and the case when$k\qff <\qff a$.The arguments in both cases are similar and can be unified,but the first case is simpler and we will deal with it first.
[**The case$k\qff \geqslant\qff a$.**]{} Every point is contained in one of the $k$ lines containing $u$,and each of these lines contains$\leqslant\qff a\qff -\qff 1$points in addition to $u$.Therefore the total number of points $$\label{n-upper-first}
\quad
n
\off \leqslant\off
1\qff +\qff k\fff(a\qff -\qff 1)\dff.$$ For every point$x\qff \in\qff L$there are$\geqslant\qss k\qff -\qff 1$lines containing $x$ and different from $L$.All these lines belong to $\mathcal{P}$ and are pairwise distinct.Therefore $$\label{p-lower-first}
\quad
p
\off \geqslant\off
1\qff +\qff a\fff(k\qff -\qff 1)\dff.$$ If$n\off \geqslant\off p$,then the inequalities(\[n-upper-first\]) and(\[p-lower-first\])imply that $$\quad
1\qff +\qff k\fff(a\qff -\qff 1)
\off \geqslant\off
1\qff +\qff a\fff(k\qff -\qff 1)$$ andhence$a\qff \geqslant\qff k$.Together with$k\qff \geqslant\qff a$this implies that$a\qff =\qff k$and the inequalities(\[n-upper-first\]) and(\[p-lower-first\])are actually equalities. It follows that$n\qff =\qff p\qff =\qff 1\qff +\qff a\fff(a\qff -\qff 1)$,every line containing $u$ consists of exactly $a$ points,and every point belonging to $L$ is contained in exactly $k$ lines.In other terms,$k_{\fff y}\qff =\qff k$if$y\qff \in\qff L$.In particular,every point of$L$can be taken as $u$ andhence every line intersecting $L$ consists of exactly $a$ points.In other terms,$s_l\qff =\qff a$if$l\qff \in\qff \mathcal{P}$.
Let$y\qff \in\qff E\dff \smallsetminus\dff L$.Then there are $a$ lines containing $y$ and belonging to $\mathcal{P}$,and together they contain $1\qff +\qff a\fff(a\qff -\qff 1)\qff =\qff n$points.It follows that for every point$y'\qff \neq\qff y$there is a line belonging to $\mathcal{P}$ and containing$\{\dff y\fff,\pff y' \dff\}$.Since there is only one line containing$\{\dff y\fff,\pff y' \dff\}$,this implies that$\mathcal{L}\qff =\qff \mathcal{P}$.This implies that$s_l\qff =\qff a$for all $l\qff \in\qff \mathcal{L}$and$k_{\fff y}\qff =\qff a$for all$y\qff \in\qff E\dff \smallsetminus\dff L$. Since we already proved that$k_{\fff y}\qff =\qff k\qff =\qff a$for all$y\qff \in\qff L$,we see that$k_{\fff y}\qff =\qff a$for all points $y$.It follows that$(\dff E\fff,\pff \mathcal{L}\dff)$is a projective plane.
[**The case$k\qff <\qff a$.**]{} By the [de Bruijn–Erdös ]{}inequalities in this case$u\qff \in\qff L$.Let $M$ be a line containing $u$ and such that $s_{\dff M}$ is the largest number among all numbers $s_l$ for lines $l$ containing $u$ and different from $L$.Let$a'\qff =\qff s_{\dff M}$.Then$a\qff \geqslant\qff a'$.The strategy is to use the fact that$u\qff \in\qff L$to refine the inequalities(\[n-upper-first\]) and(\[p-lower-first\])by using $a'$.
Every point is contained either in $L$ or in one of the other$k\qff -\qff 1$lines containing $u$.Each of these lines contains $\leqslant\qff a'\qff -\qff 1$ points in addition to $u$.Therefore the total number of points $$\label{n-above}
n
\off \leqslant\off
a\qff +\qff (k\qff -\qff 1)(a'\qff -\qff 1)\dff.$$ There are $k$ lines containing $u$,and for every point$x\qff \in\qff L$and different from $u$ there are$k_{\fff x}\qff -\qff 1$of lines containing $x$ and different from $L$.All these lines belong to $\mathcal{P}$ and are pairwise distinct.If$x\qff \in\qff L$and$x\qff \neq\qff u$,then$x\qff \not\in\qff M$andhence$k_{\fff x}\qff \geqslant\qff s_{\dff M}\qff =\qff a'$.It follows that $$\label{m-below}
p
\off \geqslant\off
k\qff +\qff (a\qff -\qff 1)(a'\qff -\qff 1)\dff.$$ The inequalities(\[n-above\])and(\[m-below\])together with the assumption$n\qff \geqslant\qff p$imply that $$\quad
a\qff +\qff (k\qff -\qff 1)(a'\qff -\qff 1)
\off \geqslant\off
k\qff +\qff (a\qff -\qff 1)(a'\qff -\qff 1)\dff.$$ By simplifying this inequality we see that$a\qff +\qff k\dff(a'\qff -\qff 1)
\off \geqslant\off
k\qff +\qff a\dff(a'\qff -\qff 1)$andhence $$\quad
k\dff(a'\qff -\qff 2)
\off \geqslant\off
a\dff(a'\qff -\qff 2)\dff.$$ Since $a'$ is the number of points in a line,$a'\qff \geqslant\qff 2$.It follows that either$k\qff \geqslant\qff a$,or$a'\qff =\qff 2$.But$k\qff \geqslant\qff a$ contradicts to the assumption $k\qff <\qff a$,and hence$a'\qff =\qff 2$.
The equality$a'\qff =\qff 2$means that $M$ consists of $2$ points.By the choice of $M$,this implies that every line containing $u$ and different from $L$ consists of $2$ points. Since $L$ and these other lines contain all points and pairwise intersect only in $u$,it follows that$n\qff =\qff a\qff +\qff k\qff -\qff 1$.
One of the points of $M$ is $u$.Let $z$ be the other point.Then$z\qff \not\in\qff L$because$M\qff \neq\qff L$,andhence there are $a$ lines containing $z$ and belonging to $\mathcal{P}$. Among these lines only $M$ contains $u$.There are also$k\qff -\qff 1$lines containing $u$ and not equal to $M$,and all of them belong to$\mathcal{P}$.It follows that$p\qff \geqslant\qff k\qff +\qff a\qff -\qff 1\qff =\qff n$.Since$n\qff \geqslant\qff p$,this implies that$p\qff =\qff n$and every line belonging to $\mathcal{P}$ contains either $u$ or $z$.
Suppose that there is a line $l$ containing $u$ and different from$L\fff,\pff M$.Let$y\qff \in\qff l$and$y\qff \neq\qff u$.Then$y\qff \not\in\qff L$and hence there are $a$ lines containing $y$ and belonging to $\mathcal{P}$.Among these lines only one contains $u$.By the previous paragraph,the other$a\qff -\qff 1$ lines contain $z$.Since there is only one line containing$\{\dff y\fff,\pff z \dff\}$,it follows that$a\qff -\qff 1\qff \leqslant\qff 1$and hence$a\qff =\qff 2$.Since$k\qff <\qff a$, this implies that$k\qff \leqslant\qff 1$contrary to the fact that$k_{\fff x}\qff \geqslant\qff 2$for all $x$.The contradiction shows that only the lines$L\fff,\pff M$contain $u$.
It follows that$E\qff =\qff L\dff \cup\dff M$and hence $z$ is the only point not belonging to $L$.In turn,this implies that the set of lines $\mathcal{L}$ consists of $L$ and the lines of the form$\{\dff x\fff,\pff z \dff\}$,where$x\qff \in\qff L$.Therefore $\mathcal{L}\qff =\qff \mathcal{P}$and$(\dff E\fff,\pff \mathcal{L}\dff)$is a near-pencil.
[[.]{}]{.nodecor} {#all .unnumbered}
==================
[**The Basterfield–Kelly–Conway argument.**]{} Suppose that$m\qff <\qff n$.Then $$\quad
(n\qff -\qff s_{\fff l})\bigl/(m\qff -\qff s_{\fff l})
\off >\off
{n}\bigl/{m}$$ for every$l\qff \in\qff \mathcal{L}$.If$z\qff \not\in\qff l$,then$s_{\fff l}\qff \leqslant\qff k_{\fff z}$andhence $m\qff -\qff k_{\fff z}\qff \leqslant\qff m\qff -\qff s_{\fff l}$.It follows that $$n
\off =\off
\sum_{z\qff \in\qff E}\off \frac{m\qff -\qff k_{\fff z}}{m\qff -\qff k_{\fff z}}
\off =\off
\sum_{l\qff \not\ni\qff z}\off \frac{1}{m\qff -\qff k_{\fff z}}
\off \geqslant\off
\sum_{l\qff \not\ni\qff z}\off \frac{1}{m\qff -\qff s_{\fff l}}
\off =\off
\sum_{l\qff \in\qff \mathcal{L}}\off \frac{n\qff -\qff s_{\fff l}}{m\qff -\qff s_{\fff l}}
\off >\off
\sum_{l\qff \in\qff \mathcal{L}}\off \frac{n}{m}
\off =\off
n\dff.$$ The contradiction leads to the conclusion that$m\qff \geqslant\qff n$.
This argument is the main part of the proof ofTheorem2.1(dealing with a more general situation)of the paper[@bk]byJ.G.BasterfieldandL.M.Kelly.Basterfield and Kelly[@bk]wrote that they are *“indebted toJ.Conway for the simplicity of the present formulation of the proofofTheorem2.1.”*By some reason this acknowledgmentled to attributing this argumenttoJ.Conway alone even by some authors referring directly to[@bk].By replacing the strict inequalities $<$ by the non-strict ones $\leqslant$,one can use this argument also to deal with the case$m\qff =\qff n$along the lines of Sections\[solution\]–\[reps\].This observation is apparently due toP.de Witte[@dw].
This is a sharp-witted,but also the most obscure and puzzling proof.It appears as a rabbit from a hat without any context or explanations and tells nothing about why the theorem is true.In the rest of this section I will explain a natural line of thinking which leads to such a proof.There is no evidence suggesting that it was discovered in this way,but it could have been.
[**Summing the [de Bruijn–Erdös ]{}inequalities.**]{} Summing [de Bruijn–Erdös ]{}inequalities and then comparing the result with(\[sums\]) is the key step of both the [de Bruijn–Erdös ]{}proof and the proof from Section\[solution\].A natural idea is to use all [de Bruijn–Erdös ]{}inequalities on an equal footing.One way to do this is to use systems of distinct representatives as in Section\[reps\].
One may hope for a proof using all [de Bruijn–Erdös ]{}inequalities in a way closer to the proof of inequalities(\[sum-all-pairs\])and(\[sum-divided\])in Section\[solution\]than to the cyclic order argument of de Bruijn–Erdös.Let us sum the inequalities$s_{\fff l}\qff \leqslant\qff k_{\fff z}$over allpairs$(\fff l\fff,\pff z\fff)\qff \in\qff \mathcal{L}\dff \times\dff E$ such that $z\qff \not\in\qff l$.Every $s_{\fff l}$ appears$n\qff -\qff s_{\fff l}$times in the left hand side of these inequalities,and every $k_{\fff z}$ appears$m\qff -\qff k_{\fff z}$times in the right hand side.Therefore,taking the sum results in the inequality $$\quad
\sum_{l\qff \in\qff \mathcal{L}}\qff s_{\fff l}\dff (n\qff -\qff s_{\fff l})
\off\off \leqslant\off\off
\sum_{z\qff \in\qff E}\qff k_{\fff z}\dff (m\qff -\qff k_{\fff z})\dff.$$ This inequality does not lead to a proof of the desired sort,but it admits a natural generalization.Let$F$be an increasing function.Instead of$s_{\fff l}\qff \leqslant\qff k_{\fff z}$,one can sum the inequalities$F\dff(\fff s_{\fff l}\fff)\pff \leqslant\off F\dff(\fff k_{\fff z}\fff)$.In fact,there is no need to apply the same function to$s_{\fff l}$and$k_{\fff z}$.
Let$F\fff,\pff G$be a pair of functions such that$s\qff \leqslant\qff k$implies $F\dff(\fff s\fff)\pff \leqslant\off G\dff(\fff k\fff)$.Taking the sum of the inequalities$F\dff(\fff s_{\fff l}\fff)\pff \leqslant\off G\dff(\fff k_{\fff z}\fff)$over allpairs$l\fff,\pff z$ such that $z\qff \not\in\qff l$results in the inequality $$\quad
\sum_{l\qff \in\qff \mathcal{L}}\qff F\dff(\fff s_{\fff l}\fff)\dff (n\qff -\qff s_{\fff l})
\off\off \leqslant\off\off
\sum_{z\qff \in\qff E}\qff G\dff (\fff k_{\fff z}\fff)\dff (m\qff -\qff k_{\fff z})\dff.$$
It remains to realize that the functions$F\fff,\pff G$may depend on the numbers$m\fff,\pff n$and that one can get rid of the factors$n\qff -\qff s_{\fff l}$and$m\qff -\qff k_{\fff z}$by dividing by these factors.
[**A proof of the de Bruijn–Erdös-Hanani theorem.**]{} As usual,we may assume that$m\qff \leqslant\qff n$.Let $$\quad
F\dff(\fff s\fff)\off =\off \frac{s}{n\qff -\qff s}
\hspace*{1.5em}\mbox{ and }\hspace*{1.5em}
G\dff(\fff k\fff)\off =\off \frac{k}{m\qff -\qff k}\qff.$$
Then$s\qff \leqslant\qff k$implies $F\dff(\fff s\fff)\pff \leqslant\off G\dff(\fff k\fff)$.Indeed,the latter inequality is equivalent to the inequality$s\fff(m\qff -\qff k)\pff \leqslant\off k\fff(n\qff -\qff s)$,and hence to the inequality$s\halfff m\pff \leqslant\off k\halfff n$,which is obviously true if$s\qff \leqslant\qff k$and$m\qff \leqslant\qff n$. By summing the inequalities$F\dff(\fff s_{\fff l}\fff)\pff \leqslant\off G\dff(\fff k_{\fff z}\fff)$over all pairs$(\fff l\fff,\pff z\fff)\qff \in\qff \mathcal{L}\dff \times\dff E$ such that $z\qff \not\in\qff l$we get the inequality $$\label{sum-frac}
\quad
\sum_{l\qff \in\qff \mathcal{L}}\off \frac{s_{\fff l}}{n\qff -\qff s_{\fff l}}\qff (n\qff -\qff s_{\fff l})
\off\off \leqslant\off\off
\sum_{z\qff \in\qff E}\off \frac{k_{\fff z}}{m\qff -\qff k_{\fff z}}\qff (m\qff -\qff k_{\fff z})\dff,$$
which is obviously equivalent to $$\quad
\sum_{l\qff \in\qff \mathcal{L}}\qff s_{\fff l}
\off\off \leqslant\off\off
\sum_{z\qff \in\qff E}\qff k_{\fff z}\qff.$$ In view of(\[sums\])the sides of the latter inequality are actually equal,andhence the sides of the inequality(\[sum-frac\])are also equal.Since the inequality(\[sum-frac\])is obtained by summing inequalities$F\dff(\fff s_{\fff l}\fff)\pff \leqslant\off G\dff(\fff k_{\fff z}\fff)$,itfollows thatif$(\fff l\fff,\pff z\fff)\qff \in\qff \mathcal{L}\dff \times\dff E$ and $z\qff \not\in\qff l$,then $$\quad
\frac{s_{\fff l}}{n\qff -\qff s_{\fff l}}
\off =\off
\frac{k_{\fff z}}{m\qff -\qff k_{\fff z}}\qff.$$
andhence$s_{\fff l}\fff m\off =\off k_{\fff z}\fff n$.Since$m\qff \leqslant\qff n$and$s_{\fff l}\qff \leqslant\qff k_{\fff z}$,the last equality implies that$m\qff =\qff n$and$s_{\fff l}\qff =\qff k_{\fff z}$.In particular,$z\qff \not\in\qff l$implies that$s_{\fff l}\qff =\qff k_{\fff z}$and hence every line containing $z$ intersects $l$.It remains to repeat the last paragraph ofSection\[reps\].
This proof has the advantage of explicitly using the equality(\[sums\]).The Basterfield–Kelley–Conway argument implicitly uses double sums and a change of the order of summation.This change of the order of summation is a stronger tool than the double counting proving(\[sums\]).
[**A version ofthis proof.**]{} Suppose that$m\qff \leqslant\qff n$.One can take as$F\fff,\pff G$the functions $$\quad
F\dff(\fff s\fff)\off =\off \frac{n}{n\qff -\qff s}
\hspace*{1.5em}\mbox{ and }\hspace*{1.5em}
G\dff(\fff k\fff)\off =\off \frac{m}{m\qff -\qff k}\qff.$$
They can be obtained by adding $1$ to the previous choice of the functions$F\fff,\pff G$.Therefore$s\qff \leqslant\qff k$againimplies $F\dff(\fff s\fff)\pff \leqslant\off G\dff(\fff k\fff)$.This can be also verified in the same way as before.By summing the inequalities$F\dff(\fff s_{\fff l}\fff)\pff \leqslant\off G\dff(\fff k_{\fff z}\fff)$over all pairs$l\fff,\pff z$ such that $z\qff \not\in\qff l$we get $$\label{sum-frac-alt}
\quad
\sum_{l\qff \in\qff \mathcal{L}}\off \frac{n}{n\qff -\qff s_{\fff l}}\qff (n\qff -\qff s_{\fff l})
\off\off \leqslant\off\off
\sum_{z\qff \in\qff E}\off \frac{m}{m\qff -\qff k_{\fff z}}\qff (m\qff -\qff k_{\fff z})\dff,$$
which is obviously equivalent to$m\halfff n\pff \leqslant\pff n\halfff m$.But the sides of the latter inequality are equal.Itfollows thatif$(\fff l\fff,\pff z\fff)\qff \in\qff \mathcal{L}\dff \times\dff E$ and $z\qff \not\in\qff l$,then $$\quad
\frac{n}{n\qff -\qff s_{\fff l}}
\off =\off
\frac{m}{m\qff -\qff k_{\fff z}}$$
andhence$s_{\fff l}\fff m\off =\off k_{\fff z}\fff n$.The rest of the proof is exactly the same as above.
Dividing everything in this version of the proofby $m$,which amounts to taking $$\quad
F\dff(\fff s\fff)\off =\off \frac{n}{m\fff(n\qff -\qff s)}
\hspace*{1.5em}\mbox{ and }\hspace*{1.5em}
G\dff(\fff k\fff)\off =\off \frac{1}{m\qff -\qff k}\qff,$$
and omitting the explanations turns this version into the Basterfield–Kelly–Conway argument.
[ABCD]{}
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May,
https://nikolaivivanov.com
E-mail:nikolai.v.ivanov@icloud.com
|
---
abstract: 'In this paper we explore speaker identification using electroencephalography (EEG) signals. The performance of speaker identification systems degrades in presence of background noise, this paper demonstrates that EEG features can be used to enhance the performance of speaker identification systems operating in presence and absence of background noise. The paper further demonstrates that in presence of high background noise, speaker identification system using only EEG features as input demonstrates better performance than the system using only acoustic features as input.'
address: 'Brain Machine Interface Lab, The University of Texas at Austin\'
bibliography:
- 'strings.bib'
- 'refs.bib'
title: Speaker Identification using EEG
---
Speaker Identification, EEG, Deep Learning
Introduction {#sec:intro}
============
Speaker identification is the task of determining an unknown speaker’s identity using speaker’s speech. The problem of speaker identification differs from the related problem of speaker verification or authentication. In speaker verification or authentication, if the speaker claims to be of a certain identity, then his or her voice is used to verify the claim. In this paper we study only speaker identification problem and not the verification problem.
Recently researchers have started using deep learning models to implement speaker identification systems. The references [@dhakal2019near; @zhao2015deep; @chung2018voxceleb2; @lukic2016speaker; @ravanelli2018speaker; @snyder2018x; @ghahabi2014deep; @richardson2015deep; @lei2014novel; @li2017deep] explains some of the related works on deep learning based speaker identification systems. Even though deep learning models have helped to establish a new state-of-the art performance for speaker identification, their performance degrades in presence of background noise like in the case of automatic speech recognition (ASR) systems. Recently researchers have started exploring the use of physiological signals to improve the performance of automatic speech recognition systems in presence of background noise as demonstrated by the work explained in reference [@krishna2019speech] whether authors demonstrated that electroencephalography (EEG) signals can help isolated word based ASR systems to overcome performance loss in presence of background noise. EEG is a non invasive way of measuring electrical activity of human brain. The EEG sensors are placed on the scalp of subject to obtain EEG electrical signal readings. The EEG signals demonstrate excellent temporal resolution which makes them ideal signals to be used in brain computer interface (BCI) systems. Recently researchers have also demonstrated continuous speech recognition using EEG signals on limited English vocabulary where EEG signals are translated to English text as demonstrated by the work explained in references [@krishna20; @krishna2019state]. Similarly in [@krishna2020synthesis] authors demonstrated preliminary results for synthesizing speech from EEG features.
A recent work described in [@han2019robust] demonstrates that in fact EEG features can improve the performance of speaker verification systems operating in presence of background noise. However in [@han2019robust] authors did not study the problem of speaker identification. In this paper we investigate whether it is possible to improve the performance of speaker identification systems using EEG features. In [@krishna2019voice] authors demonstrated that EEG features are also helpful in enhancing the performance of voice activity detection (VAD) systems operating in presence of background noise. If we are able to perform speaker identification using EEG features it will also help people with speaking disabilities or people who are not able to speak to use identification systems. Face identification systems performance degrades significantly in presence of darkness. Thus if we are able to design speaker identification systems that are robust to high background noise, that will give them significant advantage over face identification systems.
In this paper we demonstrate that EEG features can be used to enhance the performance of speaker identification systems operating in presence and absence of background noise. The paper further demonstrates that in presence of high background noise, speaker identification system using only EEG features as input demonstrates better performance than the system using only acoustic features as input.
Speaker Identification Model {#sec:format}
============================
Our speaker identification model takes EEG or acoustic or concatenation of EEG and acoustic features as input and predicts the identity of the speaker. The model architecture is described in Figure 1. The model consists of a single layer of temporal convolutional network (TCN) [@bai2018empirical] consisting of 128 filters followed by a single layer of gated recurrent unit (GRU) [@chung2014empirical] consisting of 128 hidden units followed by a dense or fully connected layer. The last time step output of the GRU layer is fed into the dense layer. The dense layer consists of linear activation function and number of hidden units same as total number of subjects. In this work we demonstrated our results for two data sets, the first data set had 4 subjects and second one had 8 subjects, thus in our case the dense layer can have 4 or 8 hidden units. Finally the dense layer output is passed to a softmax activation function which outputs the prediction probabilities. The labels were one hot vector encoded.
The model was trained for 300 epochs when experimented with the first data set and for 500 epochs when experimented using the second data set. We used categorical cross entropy as the loss function with adam [@kingma2014adam] as the optimizer. The batch size was set to 100. The validation split hyper parameter was set to a value of 0.1. All the scripts were written using Keras deep learning framework.
\[fig:asrmodel\]
Data Sets used for performing experiments {#sec:pagestyle}
=========================================
We used two data sets for performing experiments. Both the data sets consists of simultaneous speech and EEG recordings. The first data set we used was the data set used by authors in [@krishna2020synthesis]. It consists of speech-EEG data recorded in absence of background noise for four subjects. We use only the spoken speech and EEG, ie: The EEG recorded in parallel with spoken speech, not the listening utterances or listen EEG. We will refer to this data set as Data set A in this work.
The second data set we used was the data set B used by authors in [@krishna20]. It consists of speech-EEG data recorded in presence of a background noise of 65dB for eight subjects. We will refer to this data set as Data set B in this work.
More details of the data sets, description of the EEG recording hardware used are explained in [@krishna20; @krishna2020synthesis]. For each data set we used 80% of the data as training set, 10% as validation set and remaining 10% as test set. A figure describing the EEG sensor locations used in the EEG cap is shown in Figure 2.
EEG and Speech feature extraction details {#sec:typestyle}
=========================================
We followed the same preprocessing methods used by authors in [@krishna20] to preprocess EEG and speech signal. The EEG signals were sampled at 1000Hz and a fourth order infinite impulse response (IIR) band pass filter with cut off frequencies 0.1Hz and 70Hz was applied. A notch filter with cut off frequency 60 Hz was used to remove the power line noise. The EEGlab’s [@delorme2004eeglab] Independent component analysis (ICA) toolbox was used to remove other biological signal artifacts like electrocardiography (ECG), electromyography (EMG), electrooculography (EOG) etc from the EEG signals. We then extracted the five EEG features explained by authors in [@krishna20]. The details of EEG features set are covered in [@krishna20]. The EEG features were extracted at a sampling frequency of 100 Hz for each EEG channel. The dimension of EEG feature space was 155.
The recorded speech signal was sampled at 16KHz frequency. We extracted mel-frequency cepstral coefficients (MFCC) as features for speech signal. We extracted MFCC features of dimension 13. The MFCC features were also sampled at 100Hz same as the sampling frequency of EEG features.
EEG Feature Dimension Reduction Algorithm Details {#sec:majhead}
=================================================
We used kernel principal component analysis (KPCA) [@mika1999kernel] to de-noise the EEG feature space by performing dimension reduction as demonstrated by authors in [@krishna20]. We reduced the 155 EEG feature space to a dimension of 30 after identifying the right number of components by plotting the cumulative explained variance plot [@krishna2019state].
Results {#sec:print}
=======
We used classification test accuracy as the performance metric to evaluate the speaker identification model during test time. Classification test accuracy is defined as the ratio of number of correct predictions given by the model on test set to total number of predictions given by the model on test set data. Table 1 demonstrates speaker identification results obtained during test time for Data set A. As seen from Table 1 for Data set A, the highest performance was observed when the model was trained and tested using concatenation of EEG and acoustic features.
Table 2 demonstrates speaker identification results obtained during test time for Data set B. As seen from Table 2 for Data set B, the highest performance was observed when the model was trained and tested using only EEG features but for Data set B we also observed that when the model was trained and tested using concatenation of acoustic and EEG features, it resulted in better performance than training and testing the model with only acoustic features.
The Figure 3 shows the training and validation accuracy plot for the identification model when trained with concatenation of acoustic and EEG features for Data set A. The Figure 4 shows the training and validation accuracy plot for the identification model when trained using only EEG features for Data set B.
The overall results from Tables 1, 2 demonstrates that EEG features are helpful in enhancing the performance of speaker identification systems. We noted an interesting observation for speaker identification experiment done in presence of background noise, as seen from Table 2 the performance of the system using only EEG was better than the performance of the system using concatenation of acoustic and EEG features. One possible explanation for this observation might be that the acoustic features were extremely noisy and the model might have needed more training examples of MFCC+EEG features to achieve better generalization. Another reason might be the nature of EEG data set, in Data set A the subjects speak out loud the utterances that they listened to and their EEG signals were recorded whereas in Data set B the subjects read out loud English sentences shown to them on computer screen, in both cases the EEG signals corresponding to speech production might have different properties, this needs further understanding which will be considered for our future work. However we observed that in presence of background noise the speaker identification system trained with EEG features demonstrated better performance than the system trained with acoustic features. In the both the experiments ( from Tables 1, 2) **MFCC+EEG** always demonstrated better performance than only **MFCC** for speaker identification during test time.
------- ------- --
45.56 43.33
------- ------- --
: Test time results for speaker identification for **Data set A (Absence of background noise)**
------- ----------- --
25.69 **59.72**
------- ----------- --
: Test time results for speaker identification for **Data set B (Presence of background noise)**
\[fig:asrmodel\]
\[fig:asrmodel\]
conclusion {#sec:refs}
==========
In this paper we demonstrated that EEG features can be used to enhance the performance of speaker identification systems operating in presence and absence of background noise. We further demonstrated that in presence of high background noise, speaker identification system using only EEG features as input demonstrates better performance than the system using only acoustic features as input.
Future work will focus on improving current results by training the model with more number of examples and also validating the results on data sets consisting of more number of subjects or speakers.
Acknowledgement
===============
We would like to thank Kerry Loader and Rezwanul Kabir from Dell, Austin, TX for donating us the GPU to train the models used in this work.
|
---
abstract: 'We consider a curved Sitnikov problem, in which an infinitesimal particle moves on a circle under the gravitational influence of two equal masses in Keplerian motion within a plane perpendicular to that circle. There are two equilibrium points, whose stability we are studying. We show that one of the equilibrium points undergoes stability interchanges as the semi-major axis of the Keplerian ellipses approaches the diameter of that circle. To derive this result, we first formulate and prove a general theorem on stability interchanges, and then we apply it to our model. The motivation for our model resides with the $n$-body problem in spaces of constant curvature.'
address:
- 'Departamento de Matemáticas Aplicadas y Sistemas, UAM-Cuajimalpa, Av. Vasco de Quiroga 4871, México, D.F. 05348, México.'
- 'Department of Mathematical Sciences, Yeshiva University, 245 Lexington Ave, New York, NY 10016, USA.'
- 'Mathematics Department, Penn State University, University Park, PA 16802, USA.'
- 'Departamento de Matemáticas, ITAM México, Río Hondo 1, Col. Progreso Tizapán, México D.F. 01080 .'
author:
- 'Luis Franco-Pérez'
- Marian Gidea
- Mark Levi
- 'Ernesto Pérez-Chavela'
title: Stability interchanges in a curved Sitnikov problem
---
Introduction {#sec:introduction}
============
A curved Sitnikov problem {#intro_confined}
-------------------------
We consider the following curved Sitnikov problem: Two bodies of equal masses (primaries) move, under mutual gravity, on Keplerian ellipses about their center of mass. A third, massless particle is confined to a circle passing through the center of mass of the primaries, denoted by $P_0$, and perpendicular to the plane of motion of the primaries; the second intersection point of the circle with that plane is denoted by $P_1$. We assume that the massless particle moves under the gravitational influence of the primaries without affecting them. The dynamics of the massless particle has two equilibrium points, at $P_0$ and $P_1$. We focus on the local dynamics near these two points, more precisely, on the dependence of the linear stability of these points on the parameters of the problem.
When the Keplerian ellipses are not too large or too small, $P_0$ is a local center and $P_1$ is a hyperbolic fixed point. When we increase the size of the Keplerian ellipses, as the distance between $P_1$ and the closest ellipse approaches zero, then $P_1$ undergoes stability interchanges. That is, there exists a sequence of open, mutually disjoint intervals of values of the semi-major axis of the Keplerian ellipses, such that, on each of these intervals the linearized stability of $P_1$ is strongly stable, and each complementary interval contains values where the linearized stability is not strongly stable, i.e., it is either hyperbolic or parabolic. The length of these intervals approaches zero when the semi-major axis of the Keplerian ellipses approaches the diameter of the circle on which the massless particle moves. This phenomenon is the main focus in the paper.
[It is stated in [@Alfaro] and suggested by numerical evidence [@Hagel_Lothka; @Kalas] that the linearized stability of the point $P_0$ also undergoes stability interchanges when the size of the binary is kept fixed and the eccentricity of the Keplerian ellipses approaches $1$. ]{}
Stability interchanges of the type described above are ubiquitous in systems of varying parameters; they appear, for example, in the classical Hill’s equation and in the Mathieu equation [@Magnus]. To prove the occurrence of this phenomenon in our curved Sitnikov problem, we first formulate a general result on stability interchanges for a general class of simple mechanical systems. More precisely, we consider the motion of two bodies — one massive and one massless — which are confined to a pair of curves and move under Newtonian gravity. We let the distance between the two curves be controlled by some parameter $\lambda$. We assume that the position of the infinitesimal particle that achieves the minimum distance between the curves is an equilibrium point. We show that, in the case when the minimum distance between the two curves approaches zero, corresponding to $\lambda\rightarrow 0$, there existence a sequence of mutually disjoint open intervals $(\lambda_{2n-1},\lambda_{2n})$, whose lengths approach zero as $\lambda\to 0$, such that whenever $\lambda \in (\lambda_{2n-1},\lambda_{2n})$ the linearized stability of the equilibrium point is strongly stable, and each complementary interval contains values of $\lambda$ where the linearized stability is not strongly stable. From this result we derive the above mentioned stability interchange result for the curved Sitnikov problem.
The curved Sitnikov problem considered in this paper is an extension of the classical Sitnikov problem described in Section \[classical\_sitnikov\] (also, see e.g., [@Mos]). When the radius of the circle approaches infinity, in the limit we obtain the classical Sitnikov problem — the infinitesimal mass moves along the line perpendicular to the plane of the primaries and passing through the center of mass. The equilibrium point $P_1$ becomes the point at infinity and is of a degenerate hyperbolic type. Thus, stability interchanges of $P_1$ represent a new phenomenon that we encounter in the curved Sitnikov problem but not in the classical one. [Also in the last case, it is well known that for ${\varepsilon}=0$ the classical Sitnikov problem is integrable. In the case of the curved one, numerical evidence suggests that it is not (see Figure \[fig:poincare\]).]{}
The motivation for considering the curved Sitnikov problem resides in the $n$-body problem in spaces with constant curvature, and with models of planetary motions in binary star systems, as discussed in Section \[n\_body\_curvature\].
Classical Sitnikov problem {#classical_sitnikov}
--------------------------
We recall here the classical Sitnikov problem. Two bodies (primaries) of equal masses $m_1=m_2=1$ move in a plane on Keplerian ellipses of eccentricity ${\varepsilon}$ about their center of mass, and a third, massless particle moves on a line perpendicular to the plane of the primaries and passing through their center of mass. By choosing the plane of the primaries the $xy$-plane and the line on which the massless particle moves the $z$-axis, the equations of motion of the massless particle can be written, in appropriate units, as $$\label{eqn:sitnikov1} \ddot z=-\frac{2z}{(z^2+r^2(t))^{3/2}},$$ where $r(t)$ is the distance from the primaries to their center of mass given by $$\label{eqn:sitnikov2} r(t)=1-{\varepsilon}\cos u(t),$$ where $u(t)$ is the eccentric anomaly in the Kepler problem. By normalizing the time we can assume that the period of the primaries is $2\pi$, and $$\label{eqn:sitnikov3} r(t)=(1-{\varepsilon}\cos t)+O({\varepsilon}^2),$$ for small ${\varepsilon}$.
When ${\varepsilon}=0$, i.e., the primaries move on a circular orbit and the dynamics of the massless particle is described by a 1-degree of freedom Hamiltonian and so is integrable. Depending on the energy level, one has the following types of solutions: an equilibrium solution, when the particle rests at the center of mass of the primaries; periodic solutions around the center of mass; escape orbits, either parabolic, that reach infinity with zero velocity, or hyperbolic, that reach infinity with positive velocity.
When ${\varepsilon}\in(0,1)$, the differential equation is non-autonomous and the system is non-integrable. Consider the case ${\varepsilon}\ll 1$. The system also has bounded and unbounded orbits, as well as unbounded oscillatory orbits and capture orbits (oscillatory orbits are those for which $\limsup_{t\to \pm\infty}|z(t)|=+\infty$ and $\liminf_{t\to \pm\infty}|z(t)|<+\infty$, and capture orbits are those for which $\limsup_{t\to -\infty}|z(t)|=+\infty$ and $\limsup_{t\to +\infty}|z(t)|<+\infty$). In his famous paper about the final evolutions in the three body problem, Chazy introduced the term [*oscillatory motions*]{} [@Chazy], although he did not find examples of these, leaving the question of their existence open. Sitnikov’s model yielded the first example of oscillatory motions [@Sitnikov]. There are many relevant works on this problem, including [@Alekseev1968a; @Alekseev1968b; @Alekseev1969; @McGehee; @Mos; @Robinson; @Dankowicz; @GarciaPerezChavela; @GorodetskiKaloshin].
The curved Sitnikov problem introduced in Section \[intro\_confined\] is a modification of the classical problem when the massless particle moves on a circle rather than a line. Here we regard the circle as a very simple restricted model of a space with constant curvature. In Subsection \[n\_body\_curvature\] we introduce and summarize some aspects of this problem.
The $n$-body problem in spaces with [constant]{} curvature {#n_body_curvature}
----------------------------------------------------------
The $n$-body problem on spaces with [constant]{} curvature is a natural extension of the $n$-body problem in the Euclidean space; in either case the gravitational law considered is Newtonian. The extension was first proposed independently by the founders of hyperbolic geometry, Nikolai Lobachevsky and János Bolyai. It was subsequently studied in the late 19th, early 20th century, by Serret, Killing, Lipschitz, Liebmann, Schering, etc. Schrödinger developed a quantum mechanical analogue of the Kepler problem on the two-sphere in 1940. The interest in the problem was revived by Kozlov, Harin, Borisov, Mamaev, Kilin, Shchepetilov, Vozmischeva, and others, in the 1990’s. A more recent surge of interest was stimulated by the works on relative equilibria in spaces with constant curvature (both positive and negative) by Diacu, Pérez-Chavela, Santoprete, and others, starting in the 2010’s. See [@Diacu] for a history of the problem and a comprehensive list of references.
A distinctive aspect of the $n$-body problem on curved spaces is that the lack of (Galilean) translational invariance results in the lack of center-of-mass and linear-momentum integrals. Hence, the study of the motion cannot be reduced to a barycentric coordinate system.
As a consequence, the two-body problem on a sphere can no longer be reduced to the corresponding problem of motion in a central potential field, as is the case for the Kepler problem in the Euclidean space. As it turns out, the two-body problem on the sphere is not integrable [@Ziglin].
Studying the three-body problem on spaces with curvature is also challenging. Perhaps the simplest model is the restricted three-body problem on a circle. This was studied in [@Fr]. First, they consider the motion of the two primaries on the circle, which is integrable, collisions can be regularized, and all orbits can be classified into three different classes (elliptic, hyperbolic, parabolic). Then they consider the motion of the massless particle under the gravity of the primaries, when one or both primaries are at a fixed position. They obtain once again a complete classification of all orbits of the massless particle.
In this paper we take the ideas from above one step further, by considering the curved Sitnikov problem, with the massless particle moving on a circle under the gravitational influence of two primaries that move on Keplerian ellipses in a plane perpendicular to that circle. In the limit case, when the primaries are identified with one point, that is when the primaries coalesce into a single body, the Keplerian ellipses degenerate to a point, and the limit problem coincides with the two-body problem on a circle described above.
While the motivation of this work is theoretical, there are possible connections with the dynamics of planets in binary star systems. About 20 planets outside of the Solar System have been confirmed to orbit about binary stars systems; since more than half of the main sequence stars have at least one stellar companion, it is expected that a substantial fraction of planets form in binary stars systems. The orbital dynamics of such planets can vary widely, with some planets orbiting one star and some others orbiting both stars. Some chaotic-like planetary orbits have also been observed, e.g. planet Kepler-413b orbiting Kepler-413 A and Kepler-413 B in the constellation Cygnus, which displays erratic precession. This planet’s orbit is tilted away from the plane of binaries and deviating from Kepler’s laws. It is hypothesized that this tilt may be due to the gravitational influence of a third star nearby [@Kostov2014]. Of a related interest is the relativistic version of the Sitnikov problem [@Kovacs]. Thus, mathematical models like the one considered in this paper could be helpful to understand possible types of planetary orbits in binary stars systems.
To complete this introduction, the paper is organized as follows: In Section \[curved\] we go deeper in the description of the curved Sitnikov problem, studying the limit cases and its general properties. In Section \[general\] we present a general result on stability interchanges. In Section \[section\_stability\] we show that the equilibrium points in the curved Sitnikov problem present stability interchanges. Finally, in order to have a self contained paper, we add an Appendix with general results (without proofs) from Floquet theory.
The curved Sitnikov problem {#curved}
===========================
Description of the model
------------------------
We consider two bodies with equal masses (primaries) moving under mutual Newtonian gravity on identical elliptical orbits of eccentricity $\varepsilon$, about their center of mass. For small values of $\varepsilon$, the distance $r(t)$ from either primary to the center of mass of the binary is given by $$\begin{split}\label{primarias}
r_{\varepsilon}(t;r)=& r\rho(t;\varepsilon), \qquad r>0,\\\rho(t;\varepsilon)=&\left(1-\varepsilon\cos (u(t))\right)= \left(1-\varepsilon\cos (t)\right)+\mathcal{O}(\varepsilon^{2}),\end{split}$$ [where $u(t)$ is the eccentric anomaly, which satisfies Kepler’s equation $u-{\varepsilon}\sin u=(2\pi/\tau)t$, where $\tau/2\pi$ is the mean motion[^1] of the primaries. The expansion in is convergent for ${\varepsilon}< {\varepsilon}_c = 0.6627...$; see [@Plummer].]{}
A massless particle is confined on a circle of radius $R$ passing through the center of mass of the binary and perpendicular to the plane of its motion. We assume that the only force acting on the infinitesimal particle is the component along the circle of the resultant of the gravitational forces exerted by the primaries. The motion of the primaries take place in the $xy$-coordinate plane and the circle with radius $R$ is in the $yz$-coordinate plane. See Figure \[problema\].
![*The curved Sitnikov problem.*[]{data-label="problema"}](figure1){width="8cm"}
We place the center of mass at the point $(0,R,0)$ in the $xyz$-coordinate system. The position of the primaries are determined by the functions $$\begin{aligned}
\mathbf{x}_{1}(t)&=&(r_{\varepsilon}(t;r)\sin t ,R+r_{\varepsilon}(t;r)\cos t ,0),\\
\mathbf{x}_{2}(t)&=&(-r_{\varepsilon}(t;r)\sin t ,R-r_{\varepsilon}(t;r)\cos t ,0).
\end{aligned}$$ [Note that $t=0$ corresponds to the passage of the primaries through the pericenter at $y=R\pm r(1-\varepsilon)$ and $t=\pi$ to the passage of the primaries through the apocenter at $y=R\pm r(1+\varepsilon)$; both peri- and apo-centers lie on the plane of the circle of radius $R$ in the $y$-axis.]{}
The position of the infinitesimal particle is $\mathbf{x}(t)=(0,y(t),z(t))$ (taking into account the restriction of motion for the infinitesimal particle to the circle $y^{2}+z^{2}=R^{2}$). We will derive the equations of motion by computing the gravitational forces exerted by the primaries: $$\begin{aligned}
\label{fuerza}
\mathbf{F}_{\varepsilon}(y,t;R,r)&=&-\frac{\mathbf{x}-\mathbf{x}_{1}}{||\mathbf{x}-\mathbf{x}_{1}||^{3}}-
\frac{\mathbf{x}-\mathbf{x}_{2}}{||\mathbf{x}-\mathbf{x}_{2}||^{3}}\\\nonumber
&=&-\frac{(-r_{\varepsilon}(t;r)\sin t ,y-R-r_{\varepsilon}(t;r)\cos t ,z)}
{||\mathbf{x}-\mathbf{x}_{1}||^{3}}\\\nonumber
&&-\frac{(r_{\varepsilon}(t;r)\sin t ,y-R+r_{\varepsilon}(t;r)\cos t ,z)}
{||\mathbf{x}-\mathbf{x}_{2}||^{3}}\,,
\end{aligned}$$ where the distance from the particle to each primary is $$\begin{aligned}
||\mathbf{x}-\mathbf{x}_{1}||=
\left[r^{2}_{\varepsilon}(t;r)+2R^{2}+2Rr_{\varepsilon}(t;r)\cos t -2y(R+r_{\varepsilon}(t;r)\cos t )\right]^{1/2},\\
||\mathbf{x}-\mathbf{x}_{2}||=\left[r^{2}_{\varepsilon}(t;r) +2R^{2}-2Rr_{\varepsilon}(t;r)\cos t -2y(R-r_{\varepsilon}(t;r)\cos t )\right]^{1/2}.
\end{aligned}$$
[We note that when $r(1+\varepsilon)=2R$ the elliptical orbit of the primary with the apo-center at $y<R$ crosses the circle of radius $R$, hence collisions between the primary and the infinitesimal mass are possible. Therefore we will restrict to $r<\frac{2R}{1+\varepsilon}$; when $\varepsilon=0$, this means $r<2R$. ]{}
We write (\[fuerza\]) in polar coordinates, that is $y=R\cos q$, $z=R\sin q$, and we obtain $$\begin{aligned}
\label{fuerzaangulo}
\mathbf{F}_{\varepsilon}(q,t;R,r)&=&-\frac{(-r_{\varepsilon}(t;r)\sin t ,R\cos q-R-r_{\varepsilon}(t;r)\cos t ,R\sin q)}{||\mathbf{x}-\mathbf{x}_{1}||^{3}}\\\nonumber
&&-\frac{(r_{\varepsilon}(t;r)\sin t ,R\cos q-R+r_{\varepsilon}(t;r)\cos t ,R\sin q)}{||\mathbf{x}-\mathbf{x}_{2}||^{3}}.
\end{aligned}$$
The origin $q=0$ corresponds to the point $(0,R,0)$ in the $xyz$-coordinate system. Thus, the primaries move on elliptical orbits around this point.
Next we will retain the component along the circle of the resulting force (\[fuerzaangulo\]). That is, we will ignore the constraint force that confines the motion of the particle to the circle, as this force acts perpendicularly to the tangential component of the gravitational attraction force. The unit tangent vector to the circle of radius $R$ at $(0,R\cos( q),R\sin( q))$ pointing in the positive direction is given by $\mathbf{u}( q)=(0,-\sin( q),\cos( q))$. The component of the force $\mathbf{F}_{\varepsilon}( q,t;R,r)$ along the circle is computed as $$\label{fuerzaangulorestringida}
\mathbf{F}_{\varepsilon}( q,t;R,r)\cdot\mathbf{u}( q)=-\frac{(R+r_{\varepsilon}(t;r)\cos(t))\sin( q)}{||\mathbf{x}-\mathbf{x}_{1}||^{3}}-\frac{(R-r_{\varepsilon}(t;r)\cos(t))\sin( q)}{||\mathbf{x}-\mathbf{x}_{2}||^{3}}.$$
The motion of the particle, as a Hamiltonian system of one-and-a-half degrees of freedom, corresponds to $$\begin{aligned}
\label{sistemaestudio}
\dot{ q}&=&p\,,\\\nonumber
\dot{p}&=&f_{\varepsilon}(q,t;R,r),
\end{aligned}$$ where $$\begin{aligned}
f_{\varepsilon}( q,t;R,r)&:=&\mathbf{F}_{\varepsilon}( q,t;R,r)\cdot\mathbf{u}( q)\,,\\
||\mathbf{x}-\mathbf{x}_{1}||&=&\left[ r^{2}_{\varepsilon}(t;r) +2R(1-\cos q)(R+r_{\varepsilon}(t;r)\cos t) \right]^{1/2}\,,\\
||\mathbf{x}-\mathbf{x}_{2}||&=&\left[ r^{2}_{\varepsilon}(t;r)
+2R(1-\cos q)(R-r_{\varepsilon}(t;r)\cos t)\right]^{1/2}\,,
\end{aligned}$$
Hence $$H_{\varepsilon}(q,p,t;R,r)=\frac{p^2}{2}+V_{\varepsilon}(q,t;R,r),$$ where the potential is given by $$\begin{aligned}
V_{\varepsilon}(q,t;R,r)&= -\frac{1}{R}\left(\frac{1}{||\mathbf{x}-\mathbf{x}_{1}||}+\frac{1}{||\mathbf{x}-\mathbf{x}_{2}||}\right)
.\end{aligned}$$
Limit cases
-----------
The curved Sitnikov problem can be viewed as a link between the classical Sitnikov problem and the Kepler problem on the circle, mentioned in Section \[sec:introduction\].
### The limit $R\rightarrow\infty$.
We express in terms of the arc length $w=R q$, obtaining
$$\begin{gathered}
\label{fuerzalongituddearco}
{f}_{\varepsilon}(w,t;R,r)=-\frac{(R+r_{\varepsilon}(t;r)\cos t )\sin\left(w/R\right)}
{\left[r^{2}_{\varepsilon}(t;r) +2R(1-\cos \left(w/R\right))(R+r_{\varepsilon}(t;r)\cos t) \right]^{\frac{3}{2}}}\\
-\frac{(R-r_{\varepsilon}(t;r)\cos t )\sin\left(w/R\right)}
{\left[r^{2}_{\varepsilon}(t;r) +2R(1-\cos \left(w/R\right))(R-r_{\varepsilon}(t;r)\cos t) \right]^{\frac{3}{2}}}
\end{gathered}$$
which we can write in a suitable form as $$\begin{gathered}
{f}_{\varepsilon}(w,t;R,r)=-\frac{w\frac{\sin\left(w/R\right)}{\left(w/R\right)}
\left(1+\frac{r_{\varepsilon}(t;r)}{R}\cos t \right)}{
\left[r^{2}_{\varepsilon}(t;r) +2w^2\frac{(1-\cos \left(w/R\right))}{\left ( w/R\right)^2}\left(1+\frac{r_{\varepsilon}(t;r)}{R}\cos t\right) \right]^{\frac{3}{2}}}\\
-\frac{w\frac{\sin\left(w/R\right)}{\left(w/R\right)}
\left(1-\frac{r_{\varepsilon}(t;r)}{R}\cos t \right)}{
\left[r^{2}_{\varepsilon}(t;r) +2w^2\frac{(1-\cos \left(w/R\right))}{\left ( w/R\right)^2}\left(1-\frac{r_{\varepsilon}(t;r)}{R}\cos t\right) \right]^{\frac{3}{2}}}.
\end{gathered}$$ Letting $R$ tend to infinity we obtain $$\lim_{R\rightarrow\infty} {f}_{\varepsilon}(w,t;R,r)=-\frac{2w}{\left(r^{2}_{\varepsilon}(t;r)+w^{2}\right)^{3/2}}\,,$$ which is the classical Sitnikov Problem.
### The limit $r\rightarrow0$. {#section_limit_0}
When we take the limit $r\rightarrow0$ in (\[fuerzaangulorestringida\]) we are fusing the primaries into a large mass at the center of mass and we obtain a two-body problem on the circle. The component force along the circle corresponds to $$\label{eq:A1}
\lim_{r\rightarrow0}f_{\varepsilon}( q,t;R,r)=-\frac{\sin( q)}{\sqrt{2}R^{2}(1-\cos( q))^{3/2}}\,.$$ This problem was studied in [@Fr] with a different force given by $$\label{eq:A2}
-\frac{1}{R q^{2}}+\frac{1}{R(2\pi- q)^{2}},$$ the distance between the large mass and the particle is measured by the arc length (in that paper the authors assume that $R=1$). The potential of the force (\[eq:A1\]) is $$V_{1}(q_{1})=-\frac{1}{R^2\sqrt{2}(1-\cos (q_{1}))^{1/2}},$$ where $q_1$ denotes the angular coordinate, and the potential for (\[eq:A2\]) is $$V_{2}(q_{2})=-\frac{1}{Rq_{2}}-\frac{1}{R(2\pi-q_{2})},$$ where $q_2$ denotes the angular coordinate. Each problem defines an autonomous system with Hamiltonian $$\label{energia}
H_{i}(p_{i},q_{i})=\frac{1}{2}p_{i}^{2}+V_{i}(q_{i})\,,$$ taking $p_{1}=dq_{1}/dt$, $p_{2}=dq_{2}/dt$ and $i=1,2$. Let $\phi^i_{t}$ be the flow of the Hamiltonian $H_{i}$, and let $ A_i $ denote the phase space, $ i=1, \ 2 $
[Using that all orbits are determined by the energy relations given by (\[energia\]), it is not difficult to define a homeomorphism $g:A_1 \to A_2$ which maps orbits of system (\[eq:A1\]) into orbits of system (\[eq:A2\]). In the same way we can define a homeomorphism $h:A_2 \to A_1$ which in fact is $g^{-1}$. This shows the $C^0$ equivalence of the respective flows.]{} One can show that the two corresponding flows are $C^0$–equivalent.
We recall from [@Fr] that the solutions of the two-body problem on the circle (apart from the equilibrium antipodal to the fixed body) are classified in three families (elliptic, parabolic and hyperbolic solutions) according to their energy level. The elliptic solutions come out of a collision, stop instantaneously, and reverse their path back to the collision with the fixed body. The parabolic solutions come out of a collision and approach the equilibrium as $ t \rightarrow \infty $. Hyperbolic motions comes out of a collision with the fixed body, traverse the whole circle and return to a collision.
We remark that the two limit cases $R\to\infty$ and $r\to 0$ are not equivalent. Indeed, in the case $r\to 0$ the resulting system is autonomous, the point $q=0$ is a singularity for the system, and the point $q=\pi$ is a hyperbolic fixed point, while in the case $R\to\infty$ the resulting system is non-autonomous (for ${\varepsilon}\neq 0$), the point $q=0$ is a fixed point of elliptic type, and the point $q=\infty$ is a degenerate hyperbolic periodic orbit.
General properties {#generalproperties}
------------------
### Extended phase space, symmetries, and equilibrium points
It is clear that, besides the limit cases $R\to\infty$ and $r\to 0$, the dynamics of the system depends only on the ratio $r/R$, so we can fix $R=1$ and study the dependence of the global dynamics on $r$ where $0<r<2$. In this case using (\[sistemaestudio\]) and (\[primarias\]) we get $$\begin{gathered}
\label{fuerzaparametrogamma}
{f}_{\varepsilon}(q,t;r)=-\frac{(1+r\rho(t;\varepsilon)\cos(t))\sin( q)}
{\left[r^{2}\rho^2(t;\varepsilon) +2(1-\cos q)(1+r\rho(t;\varepsilon)\cos t)
\right]^{3/2}}\\
-\frac{(1-r\rho(t;\varepsilon)\cos(t))\sin( q)}{\left[r^{2}\rho^2(t;\varepsilon) +2(1-\cos q)(1-r\rho(t;\varepsilon)\cos t)
\right]^{3/2}}.
\end{gathered}$$
To study the non-autonomous system we will make the system autonomous by introducing the time as an extra dependent variable $$\label{eqn:autonomous}
\mathcal{X}_{\varepsilon}(q,p,s;r)=\left\{
\begin{array}{rcl}
\dot{q}&=&p\\
\dot{p}&=& {f}_{\varepsilon}(q,s;r)\\
\dot{s}&=&1
\end{array}\right..$$
This vector field is defined on $[0,2\pi]\times\mathbb{R}\times[0,2\pi]$, where we identify the boundary points of the closed intervals. The flow of $\mathcal{X}_ \varepsilon$ possesses symmetries defined by the functions $$\begin{aligned}
\mathbb{S}_{1}(q,p,s)&=&(-q,-p,s)\,,\\
\mathbb{S}_{2}(q,p,s)&=&(q,p,s+2\pi)\,,\\
\mathbb{S}_{3}(q,p,s)&=&(q+2\pi,p,s)\,,\\
\mathbb{S}_{4}(q,p,s)&=&(q,-p,-s)\,
\end{aligned}$$ in the sense that
- $\mathbb{S}_{1}(\mathcal{X}_{\varepsilon}(q,p,s))=\mathcal{X}_{\varepsilon}(\mathbb{S}_{1}(q,p,s))$,
- $\mathcal{X}_{\varepsilon}(q,p,s)=\mathcal{X}_{\varepsilon}(\mathbb{S}_{2}(q,p,s))$,
- $\mathcal{X}_{\varepsilon}(q,p,s)=\mathcal{X}_{\varepsilon}(\mathbb{S}_{3}(q,p,s))$
- $\mathbb{S}_{4}(\mathcal{X}_{\varepsilon}(q,p,s))=-\mathcal{X}_{\varepsilon}(\mathbb{S}_{4}(q,p,s))$,
as can be verified by a direct computation.
[The function $\mathbb{S}_{1}$ describes the symmetry respect to the trajectory $(0,0,s)$, $\mathbb{S}_{2}$ and $\mathbb{S}_{3}$ describe the bi-periodicity of ${f}_{\varepsilon}(q,s;r)$ and $\mathbb{S}_{4}$ describes the reversibility of the system.]{}
System has two equilibria $(0,0)$ and $(\pi,0)$, which correspond to periodic orbits for $\mathcal{X}_{\varepsilon}$.
While the classical Sitnikov equation is autonomous for $ \varepsilon = 0$, our equation is not, and thus we expect it to be non–integrable, as is borne out by numerical simulation. Figure \[fig:poincare\] shows a Poincaré section corresponding to $s=0$ (mod $2\pi$), with $\varepsilon=0$ and $r=1$. This simulation suggests that the invariant KAM circles coexist with chaotic regions.
![*Poincaré section for the curved Sitnikov problem, for $\varepsilon=0$ and $r=1$.*[]{data-label="fig:poincare"}](figure2){width="9.5cm"}
[In the sequel, we will analyze the dynamics around the equilibrium points $(\pi,0)$ and $(0,0)$. One important phenomenon that we will observe is that both equilibrium points undergo stability interchanges as parameters are varied. More precisely, when [$\varepsilon$ sufficiently small is kept fixed]{} and $r\rightarrow \frac{2R}{1+\varepsilon}$, the point $(\pi,0)$ undergoes infinitely many changes in stability, and when $r$ is kept fixed and $\varepsilon\to 1$, the point $(0,0)$ undergoes infinitely many changes in stability.]{}
In the next section we will first prove a general result.
A general result on stability interchanges {#general}
==========================================
[In this section we switch to a more general mechanical model which exhibits stability interchanges. We consider the motion under mutual gravity of an infinitesimal particle and a heavy mass each constrained to its own curve and moving under gravitational attraction, and study the linear stability of the equilibrium point corresponding to the closest position between the particles along the curves they are moving on. In Section \[section\_stability\], we will apply this general result to the equilibrium point $P_1$ of the curved Sitnikov problem described in Section \[intro\_confined\]. The fact that in the curved Sitnikov problem there are two heavy masses, rather than a single one as considered in this section, does not change the validity of the stability interchanges result, since, as we shall see, what it ultimately matters is the time-periodic gravitational potential acting on the infinitesimal particle.]{} To describe the setting of this section, consider a particle constrained to a curve $ {\bf x} = {\bf x} (s, \lambda ) $ in $ {\mathbb R} ^3 $, where $s$ is the arc length along the curve and $\lambda$ is a parameter with values in some interval $ [0, \lambda_0] $, $\lambda_0> 0 $. Another (much larger) gravitational mass undergoes a [*prescribed* ]{} periodic motion according to $ {\bf y} = {\bf y} (t, \lambda) = {\bf y} (t+1, \lambda )$; see Figure \[fig:generalcase\_A\]. We assume that the mass of the particle at $ {\bf x} (s) $ is negligible compared to the mass at ${\bf y}(t) $, treating the particle at ${\bf x} (s) $ as massless.
![ An infinitesimal particle with coordinates ${\bf x}(s)$ is constrained to a curve and moves under the influence of the larger mass with coordinates ${\bf y}(t)$. []{data-label="fig:generalcase_A"}](figure3){width="9.5cm"}
To write the equation of motion for the unknown coordinate $s$ of the massless particle, let: $${\bf z} (s,t, \lambda ) = {\bf x} (s, \lambda) - {\bf y} (t, \lambda)
\label{eq:q}.$$ Assume that $ s=0, \ t=0 $ minimize the distance between the two curves: $$| {\bf z} (0,0, \lambda) | = \min_{s,t} |{\bf z} (s,t, \lambda)|\buildrel{def}\over{=} \delta( \lambda ),
\label{eq:min}$$ for all $ \lambda \in [0, \lambda_0]$, and that this minimum point is non-degenerate with respect to $t$, in the sense that $$\frac{\partial^2 }{\partial t^2}|z(0,t,\lambda)|_{\mid t=0} \neq 0 .\label{eq:min02}$$
Moreover, we make the following orthogonality assumption: $$\label{eqn:orthogonality}
\dot{\bf x}(0,\lambda)\cdot \dot{\bf y}(0,\lambda)=0,\, \textrm{ for all }\lambda\in[0,\lambda_0].$$
[In the sequel we will study the case when the minimum distance $\min_{s,t} |{\bf z} (s,t, \lambda)| \rightarrow 0 $, that is, the shortest distance from the orbit ${\bf y}(t,\lambda)$ of the massive body to the curve ${\bf x}(s,\lambda)$ drawn by the infinitesimal mass approaches $0$ as $\lambda\to\lambda_0$. See Figure \[fig:generalcase\_A\]. In the curved Sitnikov problem, this corresponds to the case when $r\to \frac{2R}{1+\varepsilon}$ ($r\to 2R$ when [ $\varepsilon = 0$).]{} ]{}
To write the equation of motion for $s$, we note that the Newtonian gravitational potential of the particle at $ {\bf x} (s) $ is a function of $s$ and $t$ given by $$U( s, t, \lambda ) =- | {\bf z} (s,t, \lambda ) | ^{-1},
\label{eq:U}$$ and the evolution of $s$ is governed by the Euler–Lagrange equation $ \frac{d}{dt} L_{\dot s}-L_s=0 $ with the Lagrangian $$L = \frac{1}{2} \dot s ^2 - U( s, t, \lambda ),$$ leading to[^2] $$\ddot s + U ^\prime ( s , t,\lambda) = 0, \ \textrm{ where } ^\prime = \frac{\partial}{\partial s} .
\label{eq:governing.eq}$$
Note that $s=0 $ is an equilibrium for any $\lambda$, since $ U ^\prime (0,t, \lambda ) = 0 $ for all $ t $ and for all $\lambda$, according to (\[eq:min\]). Linearizing (\[eq:governing.eq\]) around the equilibrium $ s=0 $ we obtain $$\ddot S+ a(t, \lambda ) S = 0, \ \ a(t+1, \lambda ) = a (t, \lambda ),
\label{eq:lineq}$$ where $a(t, \lambda ) = U ^{\prime\prime} (0,t,\lambda)$.
We have the following general result:
\[thm:thm1\] Assume that , , hold, that $ {\bf x} $ and $ {\bf y} $ are both bounded in the $ C^2 $–norm uniformly in $\lambda$, and $$\label{eqn:A}\min_{s,t}\|x(s,\lambda)-y(t,\lambda)\|\to 0 \textrm{ as }\lambda\to 0.$$ Then there exists an infinite sequence $$\lambda_1 > \lambda_2 \geq \lambda _3> \lambda _4\geq \cdots >
\lambda_ {2n-1}>\lambda_ {2n} \geq \cdots \rightarrow 0
\label{eq:sequence}$$ such that the equilibrium solution $ s=0 $ of (\[eq:governing.eq\]) is linearly strongly stable[^3] for all $ \lambda \in (\lambda_{2n}, \lambda _{2n-1}) $. Furthermore, each complementary $\lambda $–interval contains points where the linearized equilibrium is not strongly stable, i.e. is either hyperbolic or parabolic.
The proof of Theorem \[thm:thm1\] relies on Lemmas \[lem:arginf\], \[lem:arg\] and \[lem:uniformlimit\] stated below.
\[lem:arginf\] Consider the linear system $$\ddot x + a(t, \lambda ) x=0,
\label{eq:hill1}$$ where $a(t,\lambda)$ is a continuous function of $ t\in [0,1] $, $ \lambda \in (0, \bar \lambda] $, where $ \bar \lambda > 0 $. Let $ z(t, \lambda) = x+ i \dot x $, where $x=x(t, \lambda ) $ is a nontrivial solution of (\[eq:hill1\]). Assume that there exists an interval $ [t_0( \lambda ), t_1( \lambda ) ]\subset [0,1]$, possibly depending on $\lambda$, such that $$\lim_{ \lambda \downarrow 0} \arg z(t, \lambda)\biggl|_{t_0}^{t_1} \rightarrow -\infty.
\label{eq:arginf}$$ Here $\arg z(t, \lambda )$ is defined as a continuous function of $t$. Although this choice of $\arg$ is unique only modulo $ 2 \pi $, its increment as stated in equation (\[eq:arginf\]) is uniquely defined.
Then there exists a sequence $\{ \lambda_k\}_{k=0}^ \infty$ satisfying (\[eq:sequence\]) such that the Floquet matrix of (\[eq:lineq\]) is strongly stable for all $\lambda \in (\lambda_{2n}, \lambda _{2n-1}) $, for any $ n > 0 $. Furthermore, every complementary $\lambda$–interval contains values of $\lambda$ for which the Floquet matrix is not strongly stable.
Proof of this lemma can be found in [@Levi].
\[lem:arg\] Consider the linear system $$\ddot x + a(t) x=0,
\label{eq:hill}$$ and assume that $ a(t) > 0 $ on some interval $ [t_0, t_1] $. For any (nontrivial) solution $ x(t) $ the corresponding phase vector $ z(t ) = x + i \dot x $ rotates by $$\theta [z]\buildrel{def}\over{=} \arg z(t) \biggl|_{t_0}^{t_1}\leq -\min_{[t_0,t_1] } \sqrt{ a(t) } (t_1-t_0) + \pi.
\label{eq:argestimate}$$
Writing the differential equation $ \ddot x + a (t) x = 0 $ as a system $ \dot x = y, \ \dot y = - a(t)x $, we obtain (using complex notation): $$\dot \theta = \frac{d}{dt} \arg (x+iy)= - \frac{d}{dt} \hbox{Im} ( \ln z) = - \hbox{Im} \frac{\dot z }{z} =
-\hbox{Im} \frac{(y-iax)(x-iy)}{x ^2 + y ^2 } = -(a \cos^2 \theta+ \sin ^2 \theta ).$$ We conclude that for any solution of (\[eq:hill\]), the angle $\theta=\arg (x+iy)$ satisfies $$\dot \theta \leq -(a_m\cos ^2 \theta + \sin^2 \theta ), \ \ \hbox{for} \ \ t\in [t_0,t_1] , \ \ \hbox{where} \ \ a_m=\min_{[t_0,t_1] }a(t).$$ To invoke comparison estimates, consider $ \bar\theta (t) $ which satisfies $$\frac{d}{dt} \bar \theta = -(a_m \cos^2 \bar \theta + \sin^2 \bar\theta ), \ \ \bar \theta (t_0) = \theta (t_0).
\label{eq:thetabar}$$ By the comparison estimate, we conclude: $$\theta \biggl|_{t_0}^{t_1}\leq \bar\theta \biggl|_{t_0}^{t_1},
\label{eq:comparison}$$ and the proof of the lemma will be complete once we show that $ \bar \theta $ satisfies the estimate (\[eq:argestimate\]). To that end we consider a solution of $$\ddot {\bar x } + a_m \bar x = 0$$ with the initial condition satisfying $$\arg(\bar x(t_0)+i\dot {\bar x}(t_0)) = \bar \theta (t_0).
\label{eq:ic}$$ This solution is of the form $${\bar x}(t)= A\cos (\sqrt{ a_m} t- \varphi ), \ \ A ={\rm const.} ,$$ where $\varphi$ is chosen so as to satisfy (\[eq:ic\]).
Since $ \arg( \bar x+ i\dot {\bar x}) $ satisfies the same differential equation as $ \bar \theta $, and since the initial conditions match, we conclude that $ \bar \theta(t) = \arg( \bar x+ i\dot {\bar x})$, so that$$\bar \theta(t) =
\arg( \cos (\sqrt{ a_m } t- \varphi) - i\sqrt{ a_m }\sin (\sqrt{ a_m } t- \varphi ) )=
- (\sqrt{a_m} t- \varphi )+ \widehat{ \pi /2},
\label{eq:barest}$$ where $ \widehat X $ denotes a quantity whose absolute value does not exceed $X$. In other words, $ \bar \theta(t) $ is given by a linear function with coefficient $ - \sqrt{ a_m} $, up to an error $ < \pi /2$. The last inequality is due to the fact that the complex numbers $\cos(\sqrt{a_m}t-\phi)-i\sin(\sqrt{a_m}t-\phi)$ and $\cos(\sqrt{a_m}t-\phi)-i\sqrt{a_m}t\sin(\sqrt{a_m}t-\phi)$ lie in the same quadrant, so the difference between their arguments is no more than $\pi/2$.
Therefore, over the interval $ [t_0,t_1] $ the function $ \bar \theta $ changes by the amount $ \sqrt{ a_m}(t_1-t_0) $ with the error of at most $ \frac{\pi}{2} + \frac{\pi}{2} = \pi$: $$\bar \theta \biggl|_{t_0}^{t_1}\leq -\sqrt{ a_m } (t_1-t_0) + \pi;
\label{eq:barest1}$$ restating this more formally, (\[eq:barest\]) implies $$\bar \theta(t_1)< - (\sqrt{ a_m} t_1- \varphi)+ \frac{\pi}{2} , \ \ \ \bar \theta(t_0)> - (\sqrt{ a_m}t_0- \varphi)- \frac{\pi}{2}.$$ Subtracting the second inequality from the first gives (\[eq:barest\]).
Substituting (\[eq:barest1\]) into (\[eq:comparison\]) yields (\[eq:argestimate\]) and completes the proof of Lemma \[lem:arg\]. $\diamondsuit$
\[lem:uniformlimit\] Consider the potential $U$ defined by equation (\[eq:U\]). Assume that the functions ${\bf x}={\bf x}(s,\lambda)$, ${\bf y}={\bf y}(t,\lambda)$ satisfy
- the minimum $\min_{s,t} |{\bf z}(\cdot,\cdot,\lambda) |=\delta(\lambda)$ is non-degenerate and is achieved at $s=t=0$, where ${\bf z}(s,t,\lambda)={\bf x}(s,\lambda)-{\bf y}(t,\lambda)$,
- $||{\bf z}(\cdot,\cdot,\lambda) ||_{C^2} \leq M $ uniformly in [$0<\lambda < \lambda_0$]{}, and
- $\delta ( \lambda ) \rightarrow 0$ as $ \lambda \rightarrow 0 $.
Then there exists time $\tau= \tau (\lambda)$ (approaching zero as $\lambda \rightarrow 0$) such that
$$\lim_{ \lambda \rightarrow 0} \left(\tau(\lambda) \cdot \min_{|t|\leq\tau (\lambda)} \sqrt{U ^{\prime\prime} (0,t,\lambda)} \right) \rightarrow \infty.
\label{eq:lemma4}$$
Differentiating (\[eq:U\]) with respect to $s$ twice, we get $$U^{\prime \prime} (0,t, \lambda ) = \left[\frac{({\bf z}^\prime \cdot {\bf z}^\prime + {\bf z} \cdot {\bf z} ^{\prime\prime})
({\bf z} \cdot {\bf z} ) -3({\bf z} \cdot {\bf z} ^\prime )^2 }
{({\bf z} \cdot {\bf z})^{5/2}}\right]_{\mid s=0}.
\label{eq:Upp}$$ We now estimate all the dot products in the above expression to obtain a lower bound.
First, $${\bf z} ^\prime \cdot {\bf z} ^\prime = 1,
\label{eq:1}$$ since $s$ is the arc length, and from here ${\bf z} ^\prime \cdot {\bf z} ^{\prime \prime} = 0$. Now, to estimate $ {\bf z}\cdot {\bf z} $ and $ {\bf z}\cdot {\bf z}^\prime$ we observe that ${\bf z}\cdot {\bf z}^\prime= \frac{1}{2} ({\bf z}\cdot {\bf z})^\prime $ and we note that the first expression, as a function of $t$ with $ s=0 $ fixed has a minimum at $t=0$ that we call $\delta ^2 $, and that the second function vanishes at $ t=0 $. Applying Taylor’s formula with respect to $t$ we then have $${\bf z} \cdot {\bf z} =\delta^2 + \widehat{k t ^2}, \hspace{.7cm} {\bf z} \cdot {\bf z}^\prime = \frac{1}{2}
({\bf z} \cdot {\bf z})^\prime = \widehat {\ kt\ },
\label{eq:zz}$$ where the constant $k$ is determined by the $ C^2$–norm $M$ of $ {\bf z} $. For the remaining dot product we have (still keeping $ s=0 $ and $t$ arbitrary): $$|{\bf z} \cdot {\bf z} ^{\prime\prime}| \leq | {\bf z} | |{\bf z} ^{\prime\prime} | \buildrel{ (\ref{eq:zz}) }\over{\leq} M
\sqrt{ \delta ^2 + k t ^2 }.
\label{eq:zz''}$$
Using the above estimates in (\[eq:Upp\]), we obtain $$U ^{\prime \prime}(0,t, \lambda ) \geq
\frac{(1- M \sqrt{ \delta^2 + k t ^2 })\delta^2- 3k ^2 t^2 }
{(\delta^2 + k t ^2)^{5/2}} .
\label{eq:inequality1}$$
Now we restrict $t$ to have $\delta ^2 + kt ^2\leq 2 \delta^2 $; this guarantees that the denominator in (\[eq:inequality1\]) does not exceed $ (2 \delta )^{5/2}$; to bound the numerator, we further restrict $t$ so that the dominant part $ \delta ^2 - 3 k ^2 t^2 \geq \frac{1}{2} \delta ^2 $, thus bounding the numerator from below by $$( \delta ^2 - 3 k ^2 t ^2 ) - M \sqrt{2 \delta ^2} \delta^2 \geq \frac{1}{2} \delta ^2 - M \sqrt{2} \delta ^3 > \frac{1}{4} \delta ^2$$ if $ \delta $ is sufficiently small. Summarizing, we restricted $t$ to $$| t | \leq c \delta \buildrel{def}\over{=} \tau ( \lambda ), \ \ \hbox{where} \ \ c = \min(k^{-1/2} ,
(k \sqrt{ 6} ) ^{-1} ),
\label{eq:tau}$$ and showed that for all such $t$ and for $ \delta$ small enough $$U^{\prime \prime}(0,t, \lambda ) \geq
\frac{ \frac{1}{4} \delta ^2 }
{(2\delta^2 )^{5/2}} = \frac{c_1}{\delta ^3},
\label{eq:inequality2}$$ where $ c_1=2 ^{-9/2} $. With $ \tau $ defined in (\[eq:tau\]) we obtain $\lim_{\tau\to 0} \left( \tau\cdot\min_{|t| \leq \tau} \sqrt{a(t,\lambda)}\right) = \infty$, thus completing the proof of the lemma. $\diamondsuit$
[**Proof of Theorem \[thm:thm1\]**]{} Consider the linearized equation $$\ddot x + U^{\prime \prime} (0,t, \lambda)x = 0,
\label{eq:lineq1}$$ [and consider the phase point $ z=x+i \dot x $ of a nontrivial solution. Lemma \[lem:arg\] gives us the rotation estimate (\[eq:argestimate\]) for any time interval $ [t_0, t_1] $; let this interval be $ [- \tau ( \lambda ) ,\tau ( \lambda )] $ where $ \tau ( \lambda ) $ is taken from the statement of Lemma \[lem:uniformlimit\]. We then have from (\[eq:argestimate\]): (\[eq:argestimate\]) $$\theta [z]\buildrel{def}\over{=} \arg z(t) \biggl|_{t_0}^{t_1}\leq
- 2\tau(\lambda )\cdot\min_{|t|\leq \tau(\lambda )}\sqrt{U^{\prime \prime}(0,t, \lambda)} + \pi.
\label{eq:argestimate1}$$ According to the conclusion (\[eq:lemma4\]) of Lemma \[lem:uniformlimit\], $ \theta \rightarrow - \infty $ as $ \lambda \rightarrow 0 $. This satisfies the condition (\[eq:arginf\]) Lemma \[lem:arginf\], which now applies and its conclusion comples the proof of Theorem \[thm:thm1\]. $\diamondsuit$ ]{}
Stability of the equilibrium points in the curved Sitnikov problem {#section_stability}
==================================================================
The system has two equilibria $(0,0)$ and $(\pi,0)$, which correspond to periodic orbits for $\mathcal{X}_{\varepsilon}$. The associated linear system around the fixed point $(q_{*},p_{*})$ can be written as $$\label{partelineal}
\dot{\mathbf{v}}=A(t)\mathbf{v}\,,\qquad \mathbf{v}=\left(\begin{array}{c}x \\y \\\end{array} \right)\,,$$ with $$A(t)=\left.\left(
\begin{array}{cc}
0 & 1 \\
\frac{\partial {f}_{\varepsilon}}{\partial q} & 0 \\
\end{array}
\right)\right|_{q=q_{*}}.$$
Let $X(t)$ be a fundamental matrix solution of system (\[partelineal\]) given by $$X(t)=\left(
\begin{array}{cc}
x_{1}(t) & x_{2}(t) \\
y_{1}(t) & y_{2}(t) \\
\end{array}
\right),$$ with the initial condition $X(0)=I$, the identity matrix; $x_{1}$ is an even function and $x_{2}$ and odd one since $\partial {f}_{\varepsilon}/\partial q$ is an even function with respect to $t$. The monodromy matrix is given by $X(2\pi)$ and we denote $\lambda_{1},\lambda_{2}$ its eigenvalues, the Floquet multipliers associated to (\[partelineal\]). These are given by $$\label{valorespropios}
\lambda_{1},\lambda_{2}=\frac{x_{1}(2\pi)+y_{2}(2\pi)\pm\sqrt{\left(x_{1}(2\pi)+y_{2}(2\pi)\right)^{2}-4}}{2}\,,$$ the trace $\textrm{Tr}(X(2\pi))=x_{1}(2\pi)+y_{2}(2\pi)$ determines the linearized dynamics around the fixed point. Moreover, since the function $(\partial {f}_{\varepsilon}/\partial q)_{\mid q=q^*}$ is an even function, we know $x_{1}(2\pi)=y_{2}(2\pi)$ (see [@Magnus]) and then $$\label{valorespropiosenpi}
\lambda_{1},\lambda_{2}=y_{2}(2\pi)\pm\sqrt{\left(y_{2}(2\pi)\right)^{2}-1}\,.$$ To emphasize the dependence on the parameters ${\varepsilon}, r$ we write $$\label{funciondelta}
y_{2}(2\pi;\varepsilon,r)=y_2(2\pi).$$ Thus the linear stability of the equilibrium is
1. Elliptic type: $| y_{2}(2\pi;\varepsilon,r) |<1$.
2. Parabolic type: $|y_{2}(2\pi;\varepsilon,r)|=1$.
3. Hyperbolic type: $|y_{2}(2\pi;\varepsilon,r)|>1$.
Since the Wronskian is equal to $1$ for all $t$, then $$W(\varepsilon, r)=(y_{2}(2\pi;\varepsilon,r))^{2}-x_{2}(2\pi;\varepsilon,r)y_{1}(2\pi;\varepsilon,r)=1,$$ and we have: $$\label{wronskiano}
\left(y_{2}(2\pi;\varepsilon,r)\right)^{2}=1+x_{2}(2\pi;\varepsilon,r)y_{1}(2\pi;\varepsilon,r)\,.$$ From this last expression it follows that, in the parabolic case, the periodic orbit corresponding to the equilibrium point $(\pi,0)$ is associated to $x_{1}(t)$ or to $x_{2}(t)$ (or to both).
Stability of the equilibrium point $(\pi,0)$. Case ${\varepsilon}=0$.
---------------------------------------------------------------------
In this subsection we consider the system - for the case $\varepsilon=0$, namely when the primaries are following circular trajectories. We also fix $R=1$. The extended vector field under study is $$\label{funcionestudio_0}
\mathcal{X}_{0}(q,p,t;\gamma)=\left\{
\begin{array}{rcl}
\dot{q}&=&p\\
\dot{p}&=&{f}_{0}(q,t;r)\\
\dot{t}&=&1
\end{array}\right.$$ where $$\label{eq:f0}
{f}_{0}(q,t;r):= -\frac{(1+r\cos(t))\sin(q)}{||\mathbf{x}-\mathbf{x}_{1}||^{3}}-\frac{(1-r\cos(t))\sin(q)}
{ ||\mathbf{x}-\mathbf{x}_{2}||^{3}}\,,\\$$ $$\begin{aligned}
||\mathbf{x}-\mathbf{x}_{1}||&=&\left[r^{2}+2+2 r\cos(t)-2\cos (q)-2r\cos(q)\cos(t))\right]^{1/2}\,,\nonumber\\
||\mathbf{x}-\mathbf{x}_{2}||&=&\left[r^{2}+2-2 r\cos(t)-2\cos (q)+2r\cos(q)\cos(t))\right]^{1/2}\,,\nonumber
\end{aligned}$$ with $0<r<2$. We observe that in this case, function (\[eq:f0\]) is $\pi$–periodic (remember that ${f}_{\varepsilon}(q,t;r)$ is $2\pi$–periodic if $\varepsilon >0$).
The linear system associated to (\[funcionestudio\_0\]) around the fixed point $(\pi,0)$ is defined by the function $$\label{parcialenpi}
\frac{\partial {f}_{0}}{\partial q}(\pi,t;r)=\frac{(1+r\cos(t))}{\left[r^{2}+4+4r\cos(t)\right]^{3/2}}
+\frac{(1-r\cos(t))}{\left[r^{2}+4-4r\cos(t)\right]^{3/2}}\,,$$ which is $C^{1}$ respect to $t$ and $r$.
\[monotone\] The function $\frac{\partial {f}_{0}}{\partial q}(\pi,t;r)$ is monotone decreasing with respect to $r$, that is, for all $t\in[0,\pi)$ $$\label{equality2}
\frac{\partial {f}_{0}}{\partial q}(\pi,t;r_{1})>\frac{\partial {f}_{0}}{\partial q}(\pi,t;r_{2})$$ if $r_{1}<r_{2}$ (see Figure \[partelinealenpifig\]).
Let be $F(t,r)=\frac{\partial {f}_{0}}{\partial q}(\pi,t;r)$, then by straightforward computation we get $$\begin{aligned}
\frac{\partial F}{\partial r}(t,r) &=& \frac{(\cos(t))[r^2+4r\cos(t)+4] - 3(1+r\cos(t))[r+2\cos(t)]}{\left[r^{2}+4+4r\cos(t)\right]^{5/2}} \\
&+& \frac{(- \cos(t))[r^2-4r\cos(t)+4] - 3(1-r\cos(t))[r-2\cos(t)]}{\left[r^{2}+4-4r\cos(t)\right]^{5/2}} \\
&\leq & \frac{-4r\cos(t)^2 - 6r}{\min \{ \left[r^{2}+4+4r\cos(t)\right]^{5/2},\left[r^{2}+4-4r\cos(t)\right]^{5/2} \}} < 0.
\end{aligned}$$ $\diamondsuit$
![*Plots for $\partial \bar{f}_{0}/\partial q$ varying $r$ from $0$ to $2$.*[]{data-label="partelinealenpifig"}](figure4){width="7cm"}
\[proposicion2\] The equilibrium point $(\pi,0)$ is of hyperbolic type if $r\leq\left(\sqrt{17}-3\right)^{1/2}=1.059\dots$.
We first show that $F(t,r)\geq 0$ for all $t$ if and only if $r\leq\left(\sqrt{17}-3\right)^{1/2}$. We compute $$\begin{split}\label{eqnderf}\frac{\partial F}{\partial t}(t,r)=r\sin t\left [-\frac{1}{(r^2+4r\cos t+4)^{1/2}} +\frac{1}{(r^2-4r\cos t+4)^{1/2}} \right.\\ \left.
+\frac{6(1+r\cos t)}{(r^2+4r\cos t+4)^{5/2}}-\frac{6(1-r\cos t)}{(r^2-4r\cos t+4)^{5/2}}\right ].\end{split}$$
We have $F(t,r)=F\left( \pi-t, r \right)$ so it is enough to restrict $t\in[0,\frac{\pi}{2}]$. Note that $\frac{\partial F}{\partial t}(t,r)=0$ for $t=0,\pi/2$.
For $0<r<2$, we have $$F \left (\frac{\pi}{2},r \right)=\frac{2}{(r^2+4)^{3/2}}>0,$$ and $$F(0,r)=\frac{1+r}{(2+r)^{3}}+\frac{1-r}{(2-r)^{3}}=\frac{-2(r^4+6r^2-8)}{(4-r^2)^{3}}.$$
It follows immediately that $F(0,r)<0$ if $r>(\sqrt{17}-3)^{1/2}$. If $r\cos t\leq 1$ then implies $F(t,r)\geq 0$. Hence $F(t,r)\geq 0$ for all $t$ provided $r\leq 1$. Let $1<r\leq (\sqrt{17}-3)^{1/2}$. If $t\in[0,\cos^{-1}(1/r)]$, which is equivalent to $r\cos t>1$, then $$\begin{split} \frac{1}{(r^2-4r\cos t+4)^{1/2}}
-\frac{1} {(r^2+4r\cos t+4)^{1/2}}\geq 0,\\
\frac{6(1+r\cos t)}{(r^2+4r\cos t+4)^{5/2}}-\frac{6(1-r\cos t)}{(r^2-4r\cos t+4)^{5/2}}\geq 0.\end{split}$$ Therefore implies that $F(t,r)$ is increasing in $t$ for $t\in [0,\cos^{-1}(1/r)]$, and, since $F(0,r)\geq 0$, it follows that $F(t,r)\geq 0$ for all $t$.
Now let $x(t)=x_1(t) + x_2(t)$ be a particular solution of system (\[partelineal\]); by hypothesis it satisfies $x(0)=1, \dot{x}(0)=1$. From with $\varepsilon = 0$ we obtain $$\ddot{x} = \frac{\partial {f}_{0}}{\partial q}(\pi,t;r) x \quad {\rm or} \quad \ddot{x} - F(t,r)x = 0.$$
Then, since $F(t,r)\leq 0$ for all $t$, we can apply directly Lyapunov’s instability criterion (see for instance page 60 in [@Ces]) to show that $(\pi,0)$ is of hyperbolic type. $\diamondsuit$
We can in fact estimate the first value $r_1$ of $r$ at which the equilibrium point $(\pi,0)$ becomes of parabolic type for the first time. [The idea is to extend slightly the result beyond $r=\left(\sqrt{17}-3\right)^{1/2}$,]{} and then find the maximum of the $r$ for which Proposition \[proposicion2\] holds. In the way we have to do straightforward analytic but tedious computations that we decide to avoid in this paper. Finally we get that the value of $r$ to have the first parabolic solution is $r_1 \approx 1.2472\cdots$.
In Figure \[estabilidad\] we show a couple of numerical simulations which illustrate the stability interchanges of the equilibrium point $(\pi,0)$ for $r \in (0,2)$ and $\varepsilon = 0$. The figure on the right hand side is a plot for $r$ close to 2.
Stability of the equilibrium point $(\pi,0)$. Case ${\varepsilon}\neq 0$.
-------------------------------------------------------------------------
In this case, for every $0<r<(\sqrt{17}-3)^{1/2}$, $\partial f_0/\partial q (\pi,t;r)>0$, hence $\partial f_{\varepsilon}/\partial q (\pi,t;r)>0$ for all ${\varepsilon}>0$ sufficiently small (depending on $r$). Therefore $(\pi,0)$ remains an equilibrium point of hyperbolic type in the case when the primaries move on Keplerian ellipses of sufficiently small eccentricity $\varepsilon>0$.
We remark that Theorem \[thm:thm1\] does not depend of the shape of the curves where ${\bf x}(x)$ or ${\bf y}(t)$ are moving, in other words we can apply Theorem \[thm:thm1\] independently if the primaries are moving on a circle or on ellipses of eccentricity $\varepsilon>0$. Thus we obtain the following result on stability interchanges:
\[thm:Sit1\] In the curved Sitnikov problem, let us fix any $\varepsilon\in [0,1) $, let $R=1$, and consider $r$ (the semi-major axes of the Keplerian ellipses traced out by the primaries) as the parameter. As $r$ approaches $\frac{2}{1+\varepsilon}$, the distance between a Keplerian ellipse and the circle of the massless particle approaches zero. There exists a sequence $r_n\uparrow \frac{2}{1+\varepsilon}$ satisfying $$\label{secesion}
r_0 \leq r_1 < r _2 \leq r _3 \cdots <
r_{2n}\leq r_{2n+1} < \cdots,$$ such that the equilibrium point $(\pi, 0)$ of equation (\[partelineal\]) is strongly stable for $ r \in (r_{2n-1}, r _{2n}) $ and not strongly stable for some $r$ in the complementary intervals $(r_{2n }, r _{2n+1}) $. In other words, the equilibrium point $(\pi, 0)$ loses and then regains its strong stability infinitely many times as $ r $ increases towards $\frac{2}{1+\varepsilon}$.
We verify that Theorem \[thm:thm1\] applies. The curve ${\bf x}(s,\lambda)$ of Theorem \[thm:thm1\] is represented by the circle of radius $R=1$, the curve $ {\bf y} (t,\lambda)$ is represented by the orbit of the primary that gets closer to $(\pi,0)$ (when $\varepsilon =0$ the primaries are co-orbital), and the parameter $\lambda$ corresponds to $r$. The planes of the two curves are perpendicular, as in , and the minimum distance between the curves, given by , is non-degenerate as in , and it corresponds to the infinitesimal mass being at $y=-1$ and the closest primary to this point being at $y=1-r(1+\varepsilon)$. Thus, the minimum distance is $\delta(r)=2-r(1+\varepsilon)$, and it approaches $0$ when $r\to \frac{2}{1+\varepsilon}$. To apply Theorem \[thm:thm1\] we only need to verify that ${\bf x}$ and ${\bf y}$ are bounded in the $C^2$-norm uniformly in $r$. This is obviously true for ${\bf y}(t,r)$ since the motion of the primary is not affected by the motion of the infinitesimal mass; it is also true for ${\bf x}(s,r)$ since $s=$arclength and the motion lies on the circle of radius $R=1$, hence $\|{\bf x}(s,r)\|=1$ (the radius of the circle), $\|{\bf x}^\prime(s,r)\|=1$ (the unit speed of a curve parametrized by arc-length), and $\|{\bf x}^{\prime\prime}(s,r)\|=1$ (the curvature of the circle). Hence the conclusion of Theorem \[thm:thm1\] follows immediately. $\diamondsuit$
Stability of the equilibrium point $q=(0,0)$.
---------------------------------------------
[The linear stability of $(0,0)$ is the same as of the barycenter in the classical Sitnikov problem, since $$\dfrac{\partial{f_\varepsilon}}{\partial q}(0)=-\frac{2R}{r^3\rho(t;\varepsilon)^3}.$$]{} The stability of this point can be treated very similarly to that of the origin for Hill’s equation, so in this analysis we use results from that theory.
As in the study of the other equilibrium point we start with the case $\varepsilon = 0.$ Here the function $f_0(q,t;r)$ defined in equation (\[sistemaestudio\]) around $q=(0,0)$ is given by $$\label{funcionencero}
\frac{\partial f_{0}}{\partial q}(q)=-\frac{2}{r^{3}}q+\frac{9+r^{2}+9r^{2}(\cos(t))^{2}}{3r^{5}}q^{3}+\mathcal{O}(q^{5})\,.$$ The local dynamics is determined by the linear part. The eigenvalues are on the unit circle, given by $\sigma_{1,2}=\pm i\sqrt{2/r^{3}}$, and the Floquet multipliers, which come from the monodromy matrix $$X(\pi)=
\left(
\begin{array}{cc}
\cos \left(\sqrt{\frac{2}{r^{3}}}\pi\right)& \left(\sqrt{\frac{r^{3}}{2}}\right)\sin \left(\sqrt{\frac{2}{r^{3}}}\pi\right) \\
-\left(\sqrt{\frac{2}{r^{3}}}\pi\right)\sin \left(\sqrt{\frac{2}{r^{3}}}\pi\right) & \cos \left(\sqrt{\frac{2}{^{3}}}\pi\right) \\
\end{array}
\right), \qquad X(0)=I\,,$$ are $\lambda_{1,2}=e^{\pm i\sqrt{2/r^{3}}\pi}$.
Hence $q=(0,0)$ is of elliptic type if $\sqrt{2/r^{3}}\neq k$ for any $k\in\mathbb{Z}$, and it is of parabolic type if $\sqrt{2/r^{3}}=k$ for some $k\in\mathbb{Z}$. In fact, in the parabolic case, if $\sqrt{2/r^{3}}=2m$, $m\in\mathbb{Z}$, then there exists a $\pi$-periodic solution, and if $\sqrt{2/r^{3}}=2m+1$, $m\in\mathbb{Z}$, then there exists a $2\pi$-periodic solution.
\[proposicion8\] The equilibium point $q=(0,0)$ of the system defined by is stable for all $r\in(0,2)$.
The linear part possess Floquet multipliers which place the system in either the elliptic or the parabolic case. In the last case, when $\sqrt{2/r^{3}}=k$ for some $k\in\mathbb{Z}$, there exists two independent eigenvectors associated to the eigenvalues $\lambda_{1,2}$. Then the conjugacy of $A$ and $\pm I$ implies that the monodromy matrix is $A = \pm I$ and $q=(0,0)$ is stable (see [@Ortega2] for more details).
When the origin is of elliptic type for the linear system, we observe that the coefficient of the term of order $3$ in equation (\[funcionencero\]) is greater than zero for all $r\in(0,2)$, then we can apply directly Ortega’s theorem (see Appendix and [@Ortega]), and therefore we obtain that the equilibrium point $(0,0)$ is stable for the whole system, that is, including the nonlinear part. $\diamondsuit$
We note that, in the case when $\varepsilon=0$, stability interchanges in a weak sense appear as $r\to 0$, since $(0,0)$ switches between elliptic type when $r\neq (2/k^2)^{1/3}$ and parabolic type when $r= (2/k^2)^{1/3}$, $k\in\mathbb{Z}^+$, as noted before.
[In the case when $\varepsilon\neq 0$, as we mentioned earlier, the linear stability of $(0,0)$ is the same as in the classical Sitnikov problem, so it only depends on the eccentricity parameter $\varepsilon$. The papers [@Alfaro; @Hagel_Lothka; @Kalas] state that there are stability interchanges when the size $r$ of the binary is kept fixed and the eccentricity $\varepsilon$ of the Keplerian ellipses approaches $1$. We should point out that Theorem \[thm:thm1\] does not apply to this case since the function $r_\varepsilon(t,r)$ describing the motion of the primaries — corresponding to $y(t,\lambda)$ in Theorem \[thm:thm1\] — does not remain bounded in the $C^2$ norm uniformly in $\varepsilon$.]{}
Floquet Theory
==============
In order to have a self contained paper we add this appendix with the main results on Floquet theory, must of them are very well known for people in the field.
Consider the linear system $$\label{linearsystem}
\dot{\mathbf{x}}=A(t)\mathbf{x}\,,\qquad \mathbf{x}\in\mathbb{R}^{2}\,,$$ where $A(t)$ is a $T$-periodic matrix-valued function. Let $X(t)$ be the fundamental matrix solution $$\label{fundmatrix} X(t)=\left(
\begin{array}{cc}
x_{1}(t) & x_{2}(t) \\
y_{1}(t) & y_{2}(t) \\
\end{array}
\right)$$ with the initial condition $X(0)=I$, the identity matrix. Let $\lambda_{1}$ and $\lambda_{2}$ be the eigenvalues (Floquet multipliers) of the monodromy matrix matrix $X(T)$ and let $\mu_{1},\mu_{2}$ (Floquet exponents) be such that $\lambda_{1}=e^{\mu_{1}T}, \lambda_{2}=e^{\mu_{2}T}$.
Suppose $X(t)$ is a fundamental matrix solution for (\[linearsystem\]), then $$X(t+T)=X(t)X(T)$$ for all $t\in\mathbb{R}$. Also there exists a constant matrix $B$ such that $e^{TB}=X(T)$ and a $T$-periodic matrix $P(t)$, so that, for all $t$, $$X(t)=P(t)e^{Bt}.$$
Let be $\lambda$ a Floquet multiplier for (\[linearsystem\]) and $\lambda=e^{\mu T}$, then there exists a nontrivial solution $x(t)=e^{\mu t}p(t)$, with $p(t)$ a $T$-periodic function. Moreover, $x(t+T)=\lambda x(t)$.
Thus, the Floquet multipliers lead to the following characterization:
- If $|\lambda|<1\Leftrightarrow\textrm{Re}(\mu)<0$ then $x(t)\rightarrow 0$ as $t\rightarrow \infty$.
- If $|\lambda|=1\Leftrightarrow\textrm{Re}(\mu)=0$ then $x(t)$ is a pseudo-periodic, bounded solution. In particular when $\lambda=1$ then $x(t)$ is $T$-periodic and when $\lambda=-1$ then $x(t)$ is $2T$-periodic.
- If $|\lambda|>1$ then $\textrm{Re}(\mu)>0$ and therefore $x(t)\rightarrow \infty$ as $t
\rightarrow \infty$, an unbounded solution.
If the Floquet multipliers satisfy $\lambda_{1}\neq\lambda_{2}$, then the equation (\[linearsystem\]) has two linearly independent solutions $$x_{1}(t)=p_{1}(t)e^{\mu_{1}t}\,,\qquad x_{2}(t)=p_{2}(t)e^{\mu_{2}t}\,,$$ where $p_{1}(t)$ and $p_{2}(t)$ are $T$-periodic functions and $\mu_{1}$ and $\mu_{2}$ are the respective Floquet exponents.
In this way, the stability of the solution to (\[linearsystem\]) is
- Asymptotically stable if $|\lambda_{i}|<1$ for $i=1,2$.
- Lyapunov stable if $|\lambda_{i}|\leq1$ for $i=1,2$, or if $|\lambda_{i}|=1$ and the algebraic multiplicity equals the geometric multiplicity.
- Unstable if $|\lambda_{i}|>1$ for at least one $i$, or if $|\lambda_{i}|=1$ and the algebraic multiplicity is greater than the geometric multiplicity.
A particular case for the equation (\[linearsystem\]) is the so-called Hill’s equation, namely the periodic linear second order differential equation: $$\label{ecapen}
\ddot{z}+f(t)z=0\,,$$ where $f(t)$ is a $\pi$-periodic function. Equation (\[ecapen\]), as a first order system, is $$\label{linealecapen}
\dot{\mathbf{v}}=A(t)\mathbf{v}\,,\qquad \mathbf{v}=\left(\begin{array}{c}x \\y\\ \end{array}\right)\,,\qquad
A(t)=\left(
\begin{array}{cc}
0 & 1 \\
-f(t) & 0 \\
\end{array}
\right)\,.$$ Let (\[fundmatrix\]) be a fundamental matrix solution of (\[linealecapen\]). The monodromy matrix corresponds to $X(\pi)$ and, as before, let $\lambda_{1}$ and $\lambda_{2}$ be the Floquet multipliers associated to (\[linealecapen\]) and $\mu_{1}$ and $\mu_{2}$ be the corresponding Floquet exponents.
In this paper we assume that $f(t)$ is an even function; from Floquet theory we know that $x_{1}$ is an even and $x_{2}$ is an odd function. Also, the trace of $A(t)$ vanishes and $$\lambda_{1}\lambda_{2}=e^{\int_{0}^{\pi}\textrm{tr}(A(t))dt}=1\,.$$ Assuming $\mu_1=a+ib$ is the Floquet exponent corresponding to the Floquet multiplier $\lambda_{1}$, the general solution to (\[linealecapen\]) is characterized as follows:
- Elliptic type: $\lambda_{1}\in\mathbb{C}\setminus\mathbb{R}$, with $|\lambda_{1}|=1$ (and $\lambda_{2}=\bar{\lambda}_{1}$). The general solution is pseudo-periodic and can be written as $$\mathbf{v}(t)=c_{1}\textrm{Re}\left(\mathbf{p}(t)e^{ibt/\pi}\right)+c_{2}\textrm{Im}\left(\mathbf{p}(t)e^{ibt/\pi}\right)\,.$$ The origin is Lyapunov stable.
- Parabolic type: $\lambda_{1}=\lambda_{2}=\pm 1$.
- If $\lambda_{1}=1$ and there are two linearly independent eigenvectors of the monodromy matrix, the general solution is $$\mathbf{v}(t)=c_{1}\mathbf{p}_{1}(t)+c_{2}\mathbf{p}_{2}(t)\,,$$ and is $\pi$-periodic and Lyapunov stable.
- If $\lambda_{1}=1$ and there is just one eigenvector associated to this eigenvalue, thus the general solution is $$\mathbf{v}(t)=(c_{1}+c_{2}t)\mathbf{p}_{1}(t)+c_{2}\mathbf{p}_{2}(t)\,,$$ and is unstable.
- If $\lambda_{2}=-1$ and there are two linearly independent eigenvectors of the monodromy matrix, the general solution is $$\mathbf{v}(t)=c_{1}\mathbf{p}_{1}(t)e^{it}+c_{2}\mathbf{p}_{2}(t)e^{it}\,,$$ and is $2\pi$-periodic and Lyapunov stable.
- If $\lambda_{1}=-1$ and there is just one eigenvector associated to this eigenvalue, thus the general solution is $$\mathbf{v}(t)=(c_{1}+c_{2}t)\mathbf{p}_{1}(t)e^{it}+c_{2}\mathbf{p}_{2}(t)e^{it}\,,$$ and is unstable.
- Hyperbolic type: $\lambda_{1}\in\mathbb{R}$, but $|\lambda_{1}|\neq 1$ (and $\lambda_{2}=1/\lambda_{1}$).
- If $\lambda_{1}>1$, then the solution is $$\mathbf{v}(t)=c_{1}\mathbf{p}_{1}(t)e^{\mu_{1} t}+c_{2}\mathbf{p}_{2}(t)e^{-\mu_{1} t}\,.$$
- If $\lambda_{1}<-1$, then the solution is $$\mathbf{v}(t)=c_{1}\mathbf{p}_{1}(t)e^{\mu_{1} t}e^{it}+c_{2}\mathbf{p}_{2}(t)e^{-\mu_{1} t}e^{it}\,.$$
Thus, the origin is unstable.
A useful result from Rafael Ortega [@Ortega] considers the nonlinear Hill equation $$\label{ecortega}
y''+a(t)y+c(t)y^{2n-1}+d(t,y)=0,$$ with $n\geq2$, where the functions $a,c:\mathbb{R}\rightarrow\mathbb{R}$ are continuous, $T$-periodic, and $\int_{0}^{T}|c(t)|\neq 0$, and the function $d:\mathbb{R}\times(-\epsilon,\epsilon)\rightarrow\mathbb{R}$, for $\epsilon>0$, is continuous, has continuous derivatives of all orders respect to $y$, is $T$-periodic function respect to $t$, and $d(t,y)=\mathcal{O}\left(|y|^{2n}\right)$ as $y\rightarrow0$ uniformly with respect to $t\in\mathbb{R}$.
The linear part around the solution $y=0$ of (\[ecortega\]) is $$\label{ortegalineal}
y''+a(t)y=0\,.$$
Assume the following:
1. The equation (\[ortegalineal\]) is stable.
2. $c\geq0$ or $c\leq0$.
Then $y=0$ is a stable solution of (\[ecortega\]).
Acknowledgements {#acknowledgements .unnumbered}
----------------
We thank to the anonymous referees, their remarks and suggestions help us to improve this paper. Research of M.G. was partially supported by NSF grant DMS-1515851. M. L. gratefully acknowledges support by the NSF grant DMS-1412542. The fourth author (EPC) has received partial support by the Asociación Mexicana de Cultura A.C. Parts of this work have been done while the authors visited CIMAT, Guanajuato, LFP and EPC visit Yeshiva University and M.G. visited UAM-I in Mexico City. All authors are grateful for the hospitality of these institutions.
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[^1]: The mean motion is the time-average angular velocity over an orbit.
[^2]: to explain this form of the Lagrangian, we note that the equations for our massless particle are obtained by taking the limit of the particle of small mass $m$; for such a particle, in the ambient potential $U$, the Lagrangian is $ \frac{1}{2} m \dot s ^2 -m U( s, t, \lambda ) $ – the factor $m$ in front of $U$ is due to the fact that $U$ is the potential energy of the unit mass. Dividing the Euler–Lagrange equation by $m$ gives (\[eq:governing.eq\]).
[^3]: The equilibrium solution is linearly strongly stable if the Floquet multipliers of (\[eq:lineq\]) lie on the unit circle and are not real, or equivalently, if the linearized system lies in the interior of the set of stable systems.
|
---
abstract: |
We extend some results of Bonahon, Bullock, Turaev and Wong concerning the skein algebras of closed surfaces to Lê’s stated skein algebra associated to open surfaces. We prove that the stated skein algebra with deforming parameter $+1$ embeds canonically into the centers of the stated skein algebras whose deforming parameter is an odd root unity. We also construct an isomorphism between the stated skein algebra at $+1$ and the algebra of regular function of a generalization of the $\SL_2$-character variety of the surface. As a result, we associate to each isomorphism class of irreducible or local representations of the stated skein algebra, an invariant which is a point in the character variety.\
**Keywords**: Stated skein algebras, Character varieties.\
**Mathematics Subject Classification**: $57$R$56$, $57$N$10$, $57$M$25$.
author:
- Julien Korinman and Alexandre Quesney
bibliography:
- 'biblio.bib'
title: Classical shadows of stated skein representations at roots of unity
---
\[section\] \[theorem\][Main Theorem]{} \[theorem\][Proposition]{} \[theorem\][Corollary]{} \[theorem\][Corollaire]{} \[theorem\][Lemma]{} \[theorem\][Notations]{} \[theorem\][Definition]{} \[theorem\][Theorem-Definition]{} \[theorem\][Remark]{} \[theorem\][Example]{}
Introduction
============
In this paper, we will consider two related objects associated to a punctured surface, namely the Kauffman-bracket skein algebra and the $\SL_2(\mathbb{C})$-character variety. These objects have been well studied in the case where the surface is closed. They were recently generalized to open surfaces [in such a way that they have]{} a nice behavior relatively to gluings of punctured surfaces. The goal of this paper is to extend some classical results concerning skein algebras and character varieties to the case of open surfaces. Before we state the main results, let us give a brief historical background.
### Historical background {#historical-background .unnumbered}
*Closed surfaces.* In [@CullerShalenCharVar], Culler and Shalen defined the $\SL_2(\mathbb{C})$ character variety $\mathcal{X}_{\SL_2}(M)$ of a manifold $M$ whose fundamental group is finitely generated. This affine variety is closely related to the moduli space of flat connexions on a trivial $\SL_2(\mathbb{C})$ bundle over $M$ and, therefore, it is related to Chern-Simons topological quantum field theory, gauge theory and low-dimensional topology; see [@LabourieCharVar; @MarcheCours09; @MarcheCharVarSkein] for surveys. If $\Sigma$ is a closed oriented surface, the smooth part of $\mathcal{X}_{\SL_2}(\Sigma)$ carries a symplectic form, first defined in [@AB] in the context of gauge theory. This symplectic structure was used by Goldman [@Goldman86] to equip the algebra of regular functions $\mathbb{C}[\mathcal{X}_{\SL_2}(\mathbf{\Sigma})]$ with a Poisson bracket. A similar Poisson structure for character varieties of punctured closed surfaces was introduced by Fock and Rosly in [@FockRosly] (see also [@AlekseevKosmannMeinrenken] for an alternative construction) in the differential geometric context.
Turaev [@Tu88], Hoste and Przytycki [@HP92] independently defined the *Kauffman-bracket skein algebra* $\mathcal{S}_A(\mathbf{\Sigma})$ as a tool to study the Jones polynomial and the $SU(2)$ Witten-Reshetikhin-Turaev TQFTs. Skein algebras are defined for any commutative unital ring $\mathcal{R}$ together with an invertible element $A\in \mathcal{R}^{\times}$ and a closed punctured surface $\mathbf{\Sigma}$.
Skein algebras are deformations of the algebra of regular functions of character varieties of closed punctured surfaces. In particular, this means that there is an isomorphism of Poisson algebras between $\mathcal{S}_{+1}(\mathbf{\Sigma})$ and $\mathbb{C}[\mathcal{X}_{\SL_2}(\mathbf{\Sigma})]$. In more details, this relies on a (non canonical) isomorphism from $\mathcal{S}_{+1}(\mathbf{\Sigma})$ to $\mathcal{S}_{-1}(\mathbf{\Sigma})$ ([@Barett]). The latter algebra carries a natural Poisson bracket (see Section 2.5). An isomorphism of algebras between $\mathcal{S}_{-1}(\mathbf{\Sigma})$ and $\mathbb{C}[\mathcal{X}_{\SL_2}(\mathbf{\Sigma})]$ was defined by Bullock [@Bullock], assuming that the skein algebra is reduced. This latter fact was latter proved independently in [@PS00] and [@ChaMa]. Turaev showed in [@Turaev91] that Bullock’s isomorphism is Poisson.
In TQFT, skein algebras appear through their non-trivial finite dimensional representations. Skein algebras admit such representations if and only if the parameter $A$ is a root of unity. A recent result of Bonahon and Wong in [@BonahonWong1] states, in particular, that when $A$ has odd order, there exists an embedding of the skein algebra with parameter $+1$ into the center of $\mathcal{S}_A(\mathbf{\Sigma})$. Since each simple representation induces a character on the center of the skein algebra, using Bullock’s isomorphism, one can associate to each isomorphism class of simple representation a point in the character variety. This invariant is called *the classical shadow* of the representation.\
*Open surfaces.* In [@LeStatedSkein], Lê generalized the Kauffman-bracket skein algebras to open punctured surfaces. He called it the *stated skein algebra* and we denote it by $\mathcal{S}_{\omega}(\mathbf{\Sigma})$. It depends on an invertible element $\omega \in \mathcal{R}^{\times}$. When the surface is closed, it coincides with the classical skein algebra with parameter $A=\omega^{-2}$. An important feature of the stated skein algebra is its behavior under gluing of surfaces. More precisely, let $a$ and $b$ be two boundary arcs of an open punctured surface $\mathbf{\Sigma}$, and let us denote by $\mathbf{\Sigma}_{|a\#b}$ the surface obtained from $\mathbf{\Sigma}$ by gluing $a$ and $b$. Lê showed that there is an injective algebra morphism $$\label{eq: le morph}
i_{|a\#b} : \mathcal{S}_{\omega}(\mathbf{\Sigma}_{| a\#b}) \hookrightarrow \mathcal{S}_{\omega}(\mathbf{\Sigma})$$ which is coassociative in that it does not depend on the order we glue the arcs *i.e.* for four distinct boundary arcs $a,b,c,d$, one has $i_{|a\#b}\circ i_{|c\#d} = i_{|c\#d}\circ i_{|a\#b}$. In particular, for each topological triangulation $\Delta$ of $\mathbf{\Sigma}$, one has an injective morphism of algebras $$\label{eq: le morph2}
i^{\Delta} : \mathcal{S}_{\omega}(\mathbf{\Sigma}) \hookrightarrow \otimes_{\mathbb{T} \in F(\Delta)} \mathcal{S}_{\omega}(\mathbb{T}).$$ Here $\mathbb{T}$ denotes the triangle (*i.e.* a disc with three punctures on its boundary) and the tensor product runs over the faces of the triangulation; see Section 2 for precise definitions.
As applications, Lê provided a simple proof that the algebra $\mathcal{S}_{\omega}(\mathbf{\Sigma})$ is reduced when the surface is closed without punctures (which was proved earlier in [@BonahonWongqTrace]) and he obtained a simpler formulation of Bonahon and Wong’s quantum trace map of [@BonahonWongqTrace].
Motivated by Lê’s construction, the first author defined in [@KojuTriangularCharVar] a generalization of character varieties to open surfaces. We denote it by $\mathcal{X}_{\SL_2}(\mathbf{\Sigma})$. This character variety is a Poisson affine variety which coincides with the classical one when the surface is closed. It shares a similar gluing property than the stated skein algebra, namely, there exist injective Poisson morphisms $i_{|a\#b}: \mathbb{C}[\mathcal{X}_{\SL_2}(\mathbf{\Sigma}_{|a\#b})] \hookrightarrow \mathbb{C}[\mathcal{X}_{\SL_2}(\mathbf{\Sigma})] $ and $i^{\Delta} : \mathbb{C}[\mathcal{X}_{\SL_2}(\mathbf{\Sigma})] \hookrightarrow \otimes_{\mathbb{T}\in F(\Delta)} \mathbb{C}[\mathcal{X}_{\SL_2}(\mathbb{T})]$ between the Poisson algebras of regular functions. However, the Poisson structure on $\mathbb{C}[\mathcal{X}_{\SL_2}(\mathbf{\Sigma})]$ depends on a choice of an orientation $\mathfrak{o}$ of the boundary arcs of the punctured surface. We denote by $\{\cdot, \cdot\}^{\mathfrak{o}}$ its Poisson bracket.
### Main results {#main-results .unnumbered}
Let $\mathbf{\Sigma}$ be a punctured surface. Lê’s morphism embeds the skein algebra of a triangulated surface into a tensor product of the skein algebras of the triangle. However, it does not provide a full description of the stated skein algebra in terms of these smaller pieces. In a first result we provide such description; it goes as follows. Remark that endows the skein algebra of the bigon $\mathbb{B}$ (*i.e.* a disc with two punctures on its boundary) with a bialgebra structure. It is in fact a Hopf algebra and one can show that it is canonically isomorphic to the classical quantum $\SL_2$ algebra $\mathbb{C}_q[\SL_2]$ described in [@Kassel; @ChariPressley] (with $q= \omega^{-4}$). Note also that induces Hopf comodule maps: $\Delta_a^L : \mathcal{S}_{\omega}(\mathbf{\Sigma}) \rightarrow \mathcal{S}_{\omega}(\mathbb{B}) \otimes \mathcal{S}_{\omega}(\mathbf{\Sigma})$ and $\Delta_b^R : \mathcal{S}_{\omega}(\mathbf{\Sigma}) \rightarrow \mathcal{S}_{\omega}(\mathbf{\Sigma}) \otimes \mathcal{S}_{\omega}(\mathbb{B})$ obtained by gluing a bigon on a boundary arc, $a$ or $b$, of $\mathbf{\Sigma}$; see Section $2.2$ for details.
\[theorem1\] The following sequence is exact: $$0 \rightarrow \mathcal{S}_{\omega}(\mathbf{\Sigma}_{|a\#b}) \xrightarrow{i_{|a\#b}} \mathcal{S}_{\omega}(\mathbf{\Sigma}) \xrightarrow{\Delta_a^L - \sigma \circ \Delta_b^R} \mathcal{S}_{\omega}(\mathbb{B}) \otimes \mathcal{S}_{\omega}(\mathbf{\Sigma}),$$ where $\sigma(x\otimes y)= y\otimes x$.
Theorem \[theorem2\] can be reformulated using coHochschild cohomology, whose zeroth group (see Definition \[def\_coHochschild\]) computes the skein algebra: $$\mathcal{S}_{\omega}(\mathbf{\Sigma}_{|a\#b}) \cong \mathrm{coHH}^0 (\mathbb{C}_q[\SL_2] , _a{\mathcal{S}_{\omega}(\mathbf{\Sigma})}_b),$$ where $_a{\mathcal{S}_{\omega}(\mathbf{\Sigma})}_b$ is seen as a bicomodule over $\mathbb{C}_q[\SL_2]$ via the comodule maps $\Delta_a^L$ and $\Delta_b^R$.
Theorem \[theorem1\] provides, for any topological triangulation $\Delta$ of $\mathbf{\Sigma}$, an isomorphism of algebras $$\mathcal{S}_{\omega}(\mathbf{\Sigma})\cong \mathrm{coHH}^0\left( \otimes_{e\in \mathring{\mathcal{E}}(\Delta)} \mathbb{C}_q[\SL_2], \otimes_{\mathbb{T}\in F(\Delta)} \mathcal{S}_{\omega}(\mathbb{T})\right),$$ where the first tensor product runs over the inner edges of the triangulation and the second over the faces of the triangulation. Hence $\mathcal{S}_{\omega}(\mathbf{\Sigma})$ is completely determined by the combinatoric of the triangulation together with $\mathcal{S}_{\omega}(\mathbb{T})$ and its appropriated structures of comodule over $\mathbb{C}_q[\SL_2]$. This is a key feature in the proofs of the next two theorems.
Our second result is a generalization to open surfaces of Bonahon and Wong’s main theorem in [@BonahonWong1]. Given $N\geq 1$, denote by $T_N(X)$ the $N$-th Chebyshev polynomial of first kind.
\[theorem2\] Suppose that $\mathbf{\Sigma}$ has at least one puncture per connected component and suppose that $\omega$ is a root of unity of odd order $N>1$. There exists an embedding $$j_{\mathbf{\Sigma}} : \mathcal{S}_{+1}(\mathbf{\Sigma}) \hookrightarrow \mathcal{Z}\left( \mathcal{S}_{\omega}(\mathbf{\Sigma}) \right)$$ of the (commutative) stated skein algebra with parameter $+1$ into the center of the stated skein algebra with parameter $\omega$. Moreover, the morphism $j_{\mathbf{\Sigma}}$ is characterized by the property that it sends a closed curve $\gamma$ to $T_N(\gamma)$ and a stated arc $\alpha_{\varepsilon \varepsilon'}$ to $(\alpha_{\varepsilon \varepsilon'})^N$.
In the last result we generalize to open surfaces Bullock’s isomorphism of [@Bullock] and Turaev’s theorem of [@Turaev91]; we prove that the stated skein algebra is a deformation of the character variety. The fundamental result in this direction is as follows.
The $\mathbb{C}[[\hbar]]$-module $\mathcal{S}_{+1}(\mathbf{\Sigma})[[\hbar]]:= \mathcal{S}_{+1}(\mathbf{\Sigma})\otimes_{\mathbb{C}}\mathbb{C}[[\hbar]]$ is endowed with a star product $\star_{\hbar}$. The latter is obtained by pulling-back the product of $\mathcal{S}_{+1}(\mathbf{\Sigma})$ along an isomorphism $\mathcal{S}_{+1}(\mathbf{\Sigma})[[\hbar]] \xrightarrow{\cong} \mathcal{S}_{\omega_{\hbar}}(\mathbf{\Sigma})$ of vector spaces, where $\omega_{\hbar}:= \exp\left( -\hbar/4 \right)$ (see Section $2.5$ for details). This equips $\mathcal{S}_{+1}(\mathbf{\Sigma})$ with a Poisson algebra structure; its Poisson bracket $\left\{ \cdot, \cdot \right\}^{s}$ is defined by $$f \star_{\hbar} g - g\star_{\hbar} f = \hbar \{ f, g\}^s \pmod{\hbar^2} \mbox{, for all }f,g \in \mathcal{S}_{+1}(\mathbf{\Sigma}).$$ See Section \[sec: explicit formula of bracket\] for an explicit description.
\[theorem3\] Suppose that $\mathbf{\Sigma}$ has a topological triangulation $\Delta$. Let $\mathfrak{o}_{\Delta}$ be an orientation of the edges of $\Delta$ and $\mathfrak{o}$ be the induced orientation of the boundary arcs of $\mathbf{\Sigma}$. There exists an isomorphism of Poisson algebras $$\Psi^{(\Delta, \mathfrak{o}_{\Delta})} : \left( \mathcal{S}_{+1}(\mathbf{\Sigma}), \{ \cdot, \cdot \}^s \right) \xrightarrow{\cong} \left( \mathbb{C}[\mathcal{X}_{\SL_2}(\mathbf{\Sigma})], \{\cdot, \cdot \}^{\mathfrak{o}} \right).$$ Moreover, the above isomorphism exists for small punctured surfaces (see Definition \[def small\]), for which it only depends on $\mathfrak{o}$.
The isomorphism $\Psi^{(\Delta, \mathfrak{o}_{\Delta})}$ induces, by tensoring with $\mathbb{C}[[\hbar]]$, an isomorphism of vector spaces $\mathbb{C}[\mathcal{X}_{\SL_2}(\mathbf{\Sigma})] [[\hbar]] \xrightarrow{\cong}\mathcal{S}_{+1}(\mathbf{\Sigma})[[\hbar]] $. Denote by $\star_{(\Delta, \mathfrak{o}_{\Delta})}$ the product on $\mathbb{C}[\mathcal{X}_{\SL_2}(\mathbf{\Sigma})] [[\hbar]]$ obtained by pulling back the product $\star_{\hbar}$ by this isomorphism.
\[corollary1\] The algebra $\left( \mathbb{C}[\mathcal{X}_{\SL_2}(\mathbf{\Sigma})] [[\hbar]] , \star_{(\Delta, \mathfrak{o}_{\Delta})} \right)$ is a deformation quantization of the character variety with Poisson structure given by $\mathfrak{o}$.
Theorems \[theorem2\] and \[theorem3\] allow us to extend Bonahon and Wong’s *classical shadow* [@BonahonWong1] to open surfaces. Indeed, suppose that $\omega$ is a root of unity of odd order. A finite dimensional representation $\mathcal{S}_{\omega}(\mathbf{\Sigma}) \rightarrow \End(V)$ that sends each element of the image of $j_{\mathbf{\Sigma}} : \mathcal{S}_{+1}(\mathbf{\Sigma}) \hookrightarrow \mathcal{S}_{\omega}(\mathbf{\Sigma})$ to scalar operators, induces a character on the algebra $\mathcal{S}_{+1}(\mathbf{\Sigma}) \cong \mathbb{C}[\mathcal{X}_{\SL_2}(\mathbf{\Sigma})]$, hence defines a point in $\mathcal{X}_{\SL_2}(\mathbf{\Sigma})$. To sum up, and calling *central* these representations, one has the following.
\[corollary2\] When $\omega$ is a root of unity of odd order, to each isomorphism class of central representations of the stated skein algebra $\mathcal{S}_{\omega}(\mathbf{\Sigma})$, one can associate an invariant which is a point in the character variety $\mathcal{X}_{\SL_2}(\mathbf{\Sigma})$.
Central representations include the families of irreducible representations, local representations and representations induced by simple modules of the balanced Chekhov-Fock algebras using the quantum trace map (see Section $3.3$ for details).
The authors were recently informed that Costantino and Lê have found results very similar to Theorem \[theorem1\] and Theorem \[theorem3\] ([@CostantinoLe19]).
### Plan of the paper {#plan-of-the-paper .unnumbered}
In the second section we briefly recall from [@LeStatedSkein] the definition and general properties of the stated skein algebra and prove Theorem \[theorem1\]. We then use the triangular decomposition to reduce the proof of Theorem \[theorem2\] to the cases of the bigon and the triangle for which the proof is a simple computation. We eventually characterize the Poisson bracket arising in skein theory. In the third section, we briefly recall from [@KojuTriangularCharVar] the definition of character varieties for open surfaces. Again, using triangular decompositions, we reduce the proof of Theorem \[theorem3\] to the cases of the bigon and the triangles for which the proof is elementary. In the appendix, we prove a technical result needed in the proof of Theorem \[theorem2\] and derive a generalization of the main theorem of [@BonahonMiraculous].
#### Acknowledgments.
The first author is thankful to F.Bonahon, F.Costantino, L.Funar, and J.Toulisse for useful discussions and to the University of South California and the Federal University of São Carlos for their kind hospitality during the beginning of this work. He acknowledges support from the grant ANR ModGroup, the GDR Tresses, the GDR Platon, CAPES and the GEAR Network. The second author was supported by PNPD/CAPES-2013 during the first period of this project, and by “grant \#2018/19603-0, São Paulo Research Foundation (FAPESP)” during the second period.
All along the paper we reserve the following notations: $A:=\omega^{-2}$ and $q:=\omega^{-4}$.
Stated skein algebras
=====================
Definitions and general properties of the stated skein algebras
---------------------------------------------------------------
We briefly review from [@LeStatedSkein] the definition and main properties of the stated skein algebras.
A *punctured surface* is a pair $\mathbf{\Sigma}=(\Sigma,\mathcal{P})$ where $\Sigma$ is a compact oriented surface and $\mathcal{P}$ is a finite subset of $\Sigma$ which intersects non-trivially each boundary component. A *boundary arc* is a connected component of $\partial \Sigma \setminus \mathcal{P}$.
Let $\mathbf{\Sigma}=(\Sigma, \mathcal{P})$ be a punctured surface and write $\Sigma_{\mathcal{P}}:= \Sigma \setminus \mathcal{P}$. A *tangle* in $ \Sigma_{\mathcal{P}}\times (0,1)$ is a compact framed, properly embedded $1$-dimensional manifold $T\subset \Sigma_{\mathcal{P}}\times (0,1)$ such that for every point of $\partial T \subset \partial \Sigma_{\mathcal{P}}\times (0,1)$ the framing is parallel to the $(0,1)$ factor and points to the direction of $1$. Here, by framing, we refer to a thickening of $T$ to an oriented surface. Define the *height* of a point $(v,h)\in \Sigma_{\mathcal{P}}\times (0,1)$ to be $h$. If $b$ is a boundary arc and $T$ a tangle, the points of $\partial_b T := \partial T \cap b\times(0,1)$ are totally ordered by their height and we impose that no two points in $\partial_bT$ have the same height. A tangle has *vertical framing* if for each of its points, the framing is parallel to the $(0,1)$ factor and points in the direction of $1$. Two tangles are isotopic if they are isotopic through the class of tangles that preserves the partial boundary height orders. By convention, the empty set is a tangle only isotopic to itself.
Every tangle is isotopic to a tangle with vertical framing. We can further isotope a tangle such that it is in general position with the standard projection $\pi : \Sigma_{\mathcal{P}}\times (0,1)\rightarrow \Sigma_{\mathcal{P}}$ with $\pi(v,h)=v$, that is such that $\pi_{\big| T} : T\rightarrow \Sigma_{\mathcal{P}}$ is an immersion with at most transversal double points in the interior of $\Sigma_{\mathcal{P}}$. We call *diagram* of $T$ the image $D=\pi(T)$ together with the over/undercrossing information at each double point. An isotopy class of diagram $D$ together with a total order of $\partial_b D=\partial D\cap b$ for each boundary arc $b$, define uniquely an isotopy class of tangle. When choosing an orientation $\mathfrak{o}(b)$ of a boundary arc $b$ and a diagram $D$, the set $\partial_bD$ receives a natural order by setting that the points are increasing when going in the direction of $\mathfrak{o}(b)$. We will represent tangles by drawing a diagram and an orientation (an arrow) for each boundary arc. When a boundary arc $b$ is oriented we assume that $\partial_b D$ is ordered according to the orientation. The data of an isotopy class of diagram and a choice $\mathfrak{o}$ of orientations of the boundary arcs define uniquely an isotopy class of tangle. A *state* of a tangle is a map $s:\partial T \rightarrow \{-, +\}$. A couple $(T,s)$ is called a *stated tangle*. We define a *stated diagram* $(D,s)$ in a similar manner.
Let $\mathcal{R}$ be a commutative unital ring and $\omega\in \mathcal{R}^{\times}$ an invertible element.
\[def\_stated\_skein\] The *stated skein algebra* $\mathcal{S}_{\omega}(\mathbf{\Sigma})$ is the free $\mathcal{R}$-module generated by isotopy classes of stated tangles in $\Sigma_{\mathcal{P}}\times (0, 1)$ modulo the relations and , which are,\
the Kauffman bracket relations: $$\label{eq: skein 1}
\begin{tikzpicture}[baseline=-0.4ex,scale=0.5,>=stealth]
\draw [fill=gray!45,gray!45] (-.6,-.6) rectangle (.6,.6) ;
\draw[line width=1.2,-] (-0.4,-0.52) -- (.4,.53);
\draw[line width=1.2,-] (0.4,-0.52) -- (0.1,-0.12);
\draw[line width=1.2,-] (-0.1,0.12) -- (-.4,.53);
\end{tikzpicture}
=\omega^{-2}
\begin{tikzpicture}[baseline=-0.4ex,scale=0.5,>=stealth]
\draw [fill=gray!45,gray!45] (-.6,-.6) rectangle (.6,.6) ;
\draw[line width=1.2] (-0.4,-0.52) ..controls +(.3,.5).. (-.4,.53);
\draw[line width=1.2] (0.4,-0.52) ..controls +(-.3,.5).. (.4,.53);
\end{tikzpicture}
+\omega^{2}
\begin{tikzpicture}[baseline=-0.4ex,scale=0.5,rotate=90]
\draw [fill=gray!45,gray!45] (-.6,-.6) rectangle (.6,.6) ;
\draw[line width=1.2] (-0.4,-0.52) ..controls +(.3,.5).. (-.4,.53);
\draw[line width=1.2] (0.4,-0.52) ..controls +(-.3,.5).. (.4,.53);
\end{tikzpicture}
\hspace{.5cm}
\text{ and }\hspace{.5cm}
\begin{tikzpicture}[baseline=-0.4ex,scale=0.5,rotate=90]
\draw [fill=gray!45,gray!45] (-.6,-.6) rectangle (.6,.6) ;
\draw[line width=1.2,black] (0,0) circle (.4) ;
\end{tikzpicture}
= -(\omega^{-4}+\omega^{4})
\begin{tikzpicture}[baseline=-0.4ex,scale=0.5,rotate=90]
\draw [fill=gray!45,gray!45] (-.6,-.6) rectangle (.6,.6) ;
\end{tikzpicture}
;$$ the boundary relations: $$\label{eq: skein 2}
\begin{tikzpicture}[baseline=-0.4ex,scale=0.5,>=stealth]
\draw [fill=gray!45,gray!45] (-.7,-.75) rectangle (.4,.75) ;
\draw[->] (0.4,-0.75) to (.4,.75);
\draw[line width=1.2] (0.4,-0.3) to (0,-.3);
\draw[line width=1.2] (0.4,0.3) to (0,.3);
\draw[line width=1.1] (0,0) ++(90:.3) arc (90:270:.3);
\draw (0.65,0.3) node {\scriptsize{$+$}};
\draw (0.65,-0.3) node {\scriptsize{$+$}};
\end{tikzpicture}
=
\begin{tikzpicture}[baseline=-0.4ex,scale=0.5,>=stealth]
\draw [fill=gray!45,gray!45] (-.7,-.75) rectangle (.4,.75) ;
\draw[->] (0.4,-0.75) to (.4,.75);
\draw[line width=1.2] (0.4,-0.3) to (0,-.3);
\draw[line width=1.2] (0.4,0.3) to (0,.3);
\draw[line width=1.1] (0,0) ++(90:.3) arc (90:270:.3);
\draw (0.65,0.3) node {\scriptsize{$-$}};
\draw (0.65,-0.3) node {\scriptsize{$-$}};
\end{tikzpicture}
=0,
\hspace{.2cm}
\begin{tikzpicture}[baseline=-0.4ex,scale=0.5,>=stealth]
\draw [fill=gray!45,gray!45] (-.7,-.75) rectangle (.4,.75) ;
\draw[->] (0.4,-0.75) to (.4,.75);
\draw[line width=1.2] (0.4,-0.3) to (0,-.3);
\draw[line width=1.2] (0.4,0.3) to (0,.3);
\draw[line width=1.1] (0,0) ++(90:.3) arc (90:270:.3);
\draw (0.65,0.3) node {\scriptsize{$+$}};
\draw (0.65,-0.3) node {\scriptsize{$-$}};
\end{tikzpicture}
=\omega
\begin{tikzpicture}[baseline=-0.4ex,scale=0.5,>=stealth]
\draw [fill=gray!45,gray!45] (-.7,-.75) rectangle (.4,.75) ;
\draw[-] (0.4,-0.75) to (.4,.75);
\end{tikzpicture}
\hspace{.1cm} \text{ and }
\hspace{.1cm}
\omega^{-1}
{
\begin{tikzpicture}[baseline=-0.4ex,scale=0.5, >=stealth]
\draw [fill=gray!60,gray!45] (-.7,-.75) rectangle (.4,.75) ;
\draw[->] (0.4,-0.75) to (.4,.75);
\draw[line width=1.2] (0.4,-0.3) to (-.7,-.3);
\draw[line width=1.2] (0.4,0.3) to (-.7,.3);
\draw (0.65,0.3) node {\scriptsize{$-$}};
\draw (0.65,-0.3) node {\scriptsize{$+$}};
\end{tikzpicture}
}
- \omega^{-5}
{
\begin{tikzpicture}[baseline=-0.4ex,scale=0.5, >=stealth]
\draw [fill=gray!60,gray!45] (-.7,-.75) rectangle (.4,.75) ;
\draw[->] (0.4,-0.75) to (.4,.75);
\draw[line width=1.2] (0.4,-0.3) to (-.7,-.3);
\draw[line width=1.2] (0.4,0.3) to (-.7,.3);
\draw (0.65,0.3) node {\scriptsize{$+$}};
\draw (0.65,-0.3) node {\scriptsize{$-$}};
\end{tikzpicture}
}
=
{
\begin{tikzpicture}[baseline=-0.4ex,scale=0.5]
\draw [fill=gray!20,gray!45] (-.7,-.75) rectangle (.4,.75) ;
\draw[-] (0.4,-0.75) to (.4,.75);
\draw[line width=1.2] (-.7,-0.3) to (-.4,-.3);
\draw[line width=1.2] (-.7,0.3) to (-.4,.3);
\draw[line width=1.15] (-.4,0) ++(-90:.3) arc (-90:90:.3);
\end{tikzpicture}
}.$$ The product of two classes of stated tangles $[T_1,s_1]$ and $[T_2,s_2]$ is defined by isotoping $T_1$ and $T_2$ in $\Sigma_{\mathcal{P}}\times (1/2, 1) $ and $\Sigma_{\mathcal{P}}\times (0, 1/2)$ respectively and then setting $[T_1,s_1]\cdot [T_2,s_2]=[T_1\cup T_2, s_1\cup s_2]$.
A closed component of a diagram $D$ is trivial if it bounds an embedded disc in $\Sigma_{\mathcal{P}}$. An open component of $D$ is trivial if it can be isotoped, relatively to its boundary, inside some boundary arc. A diagram is *simple* if it has neither double points nor trivial component. The empty set is considered as a simple diagram. Let $\mathfrak{o}$ be an orientation of the boundary arcs of $\mathbf{\Sigma}$ and denote by $\leq_{\mathfrak{o}}$ the total orders induced on each boundary arc. A state $s: \partial D \rightarrow \{ - , + \}$ is $\mathfrak{o}-$*increasing* if for any boundary arc $b$ and any points $x,y \in \partial_bD$, then $x<_{\mathfrak{o}} y$ implies $s(x)< s(y)$. Here we choose the convention $- < +$.
\[def\_basis\] We denote by $\mathcal{B}^{\mathfrak{o}}\subset \mathcal{S}_{\omega}(\mathbf{\Sigma})$ the set of classes of stated diagrams $[D,s]$ such that $D$ is simple and $s$ is $\mathfrak{o}$-increasing.
By [@LeStatedSkein Theorem $2.11$], the set $\mathcal{B}^{\mathfrak{o}}$ is an $\mathcal{R}$-module basis of $\mathcal{S}_{\omega}(\mathbf{\Sigma})$.
Important properties that we will use all along the paper are the following *height exchange moves* and proved in [@LeStatedSkein Lemma $2.4$]. Note that the formula (20) of Lemma 2.4 of *loc. cit.* contains a misprint. It is corrected here in . $$\label{eq: height exch 1}
{
\begin{tikzpicture}[baseline=-0.4ex,scale=0.5, >=stealth]
\draw [fill=gray!60,gray!45] (-.7,-.75) rectangle (.4,.75) ;
\draw[<-] (0.4,-0.75) to (.4,.75);
\draw[line width=1.2] (0.4,-0.3) to (-.7,-.3);
\draw[line width=1.2] (0.4,0.3) to (-.7,.3);
\draw (0.65,0.3) node {\scriptsize{$+$}};
\draw (0.65,-0.3) node {\scriptsize{$+$}};
\end{tikzpicture}
}=\omega^{2} {
\begin{tikzpicture}[baseline=-0.4ex,scale=0.5, >=stealth]
\draw [fill=gray!60,gray!45] (-.7,-.75) rectangle (.4,.75) ;
\draw[->] (0.4,-0.75) to (.4,.75);
\draw[line width=1.2] (0.4,-0.3) to (-.7,-.3);
\draw[line width=1.2] (0.4,0.3) to (-.7,.3);
\draw (0.65,0.3) node {\scriptsize{$+$}};
\draw (0.65,-0.3) node {\scriptsize{$+$}};
\end{tikzpicture}
}, ~~~
{
\begin{tikzpicture}[baseline=-0.4ex,scale=0.5, >=stealth]
\draw [fill=gray!60,gray!45] (-.7,-.75) rectangle (.4,.75) ;
\draw[<-] (0.4,-0.75) to (.4,.75);
\draw[line width=1.2] (0.4,-0.3) to (-.7,-.3);
\draw[line width=1.2] (0.4,0.3) to (-.7,.3);
\draw (0.65,0.3) node {\scriptsize{$+$}};
\draw (0.65,-0.3) node {\scriptsize{$-$}};
\end{tikzpicture}
}=\omega^{-2} {
\begin{tikzpicture}[baseline=-0.4ex,scale=0.5, >=stealth]
\draw [fill=gray!60,gray!45] (-.7,-.75) rectangle (.4,.75) ;
\draw[->] (0.4,-0.75) to (.4,.75);
\draw[line width=1.2] (0.4,-0.3) to (-.7,-.3);
\draw[line width=1.2] (0.4,0.3) to (-.7,.3);
\draw (0.65,0.3) node {\scriptsize{$+$}};
\draw (0.65,-0.3) node {\scriptsize{$-$}};
\end{tikzpicture}
}, ~~~
{
\begin{tikzpicture}[baseline=-0.4ex,scale=0.5, >=stealth]
\draw [fill=gray!60,gray!45] (-.7,-.75) rectangle (.4,.75) ;
\draw[<-] (0.4,-0.75) to (.4,.75);
\draw[line width=1.2] (0.4,-0.3) to (-.7,-.3);
\draw[line width=1.2] (0.4,0.3) to (-.7,.3);
\draw (0.65,0.3) node {\scriptsize{$-$}};
\draw (0.65,-0.3) node {\scriptsize{$-$}};
\end{tikzpicture}
}=\omega^{2} {
\begin{tikzpicture}[baseline=-0.4ex,scale=0.5, >=stealth]
\draw [fill=gray!60,gray!45] (-.7,-.75) rectangle (.4,.75) ;
\draw[->] (0.4,-0.75) to (.4,.75);
\draw[line width=1.2] (0.4,-0.3) to (-.7,-.3);
\draw[line width=1.2] (0.4,0.3) to (-.7,.3);
\draw (0.65,0.3) node {\scriptsize{$-$}};
\draw (0.65,-0.3) node {\scriptsize{$-$}};
\end{tikzpicture}
}$$ $$\label{eq: height exch corr}
\omega^{-3}{
\begin{tikzpicture}[baseline=-0.4ex,scale=0.5, >=stealth]
\draw [fill=gray!60,gray!45] (-.7,-.75) rectangle (.4,.75) ;
\draw[<-] (0.4,-0.75) to (.4,.75);
\draw[line width=1.2] (0.4,-0.3) to (-.7,-.3);
\draw[line width=1.2] (0.4,0.3) to (-.7,.3);
\draw (0.65,0.3) node {\scriptsize{$-$}};
\draw (0.65,-0.3) node {\scriptsize{$+$}};
\end{tikzpicture}
}-\omega^{3}{
\begin{tikzpicture}[baseline=-0.4ex,scale=0.5, >=stealth]
\draw [fill=gray!60,gray!45] (-.7,-.75) rectangle (.4,.75) ;
\draw[->] (0.4,-0.75) to (.4,.75);
\draw[line width=1.2] (0.4,-0.3) to (-.7,-.3);
\draw[line width=1.2] (0.4,0.3) to (-.7,.3);
\draw (0.65,0.3) node {\scriptsize{$-$}};
\draw (0.65,-0.3) node {\scriptsize{$+$}};
\end{tikzpicture}
}=(\omega^{-4} -\omega^{4}) {
\begin{tikzpicture}[baseline=-0.4ex,scale=0.5]
\draw [fill=gray!20,gray!45] (-.7,-.75) rectangle (.4,.75) ;
\draw[-] (0.4,-0.75) to (.4,.75);
\draw[line width=1.2] (-.7,-0.3) to (-.4,-.3);
\draw[line width=1.2] (-.7,0.3) to (-.4,.3);
\draw[line width=1.15] (-.4,0) ++(-90:.3) arc (-90:90:.3);
\end{tikzpicture}
}.$$
An important case that we will be led to consider is the stated skein algebra at parameter $\omega=+1$. As shown in [@LeStatedSkein Corollary 2.5] it is commutative; this is a direct consequence of and the height exchange formulas and .
Suppose that $\mathbf{\Sigma}$ has two boundary arcs, say $a$ and $b$. Let $\mathbf{\Sigma}_{|a\#b}$ be the punctured surface obtained from $\mathbf{\Sigma}$ by gluing $a$ and $b$. Denote by $\pi : \Sigma_{\mathcal{P}} \rightarrow (\Sigma_{|a\#b})_{\mathcal{P}_{|a\#b}}$ the projection and $c:=\pi(a)=\pi(b)$. Let $(T_0, s_0)$ be a stated framed tangle of ${\Sigma_{|a\#b}}_{\mathcal{P}_{|a\#b}} \times (0,1)$ transversed to $c\times (0,1)$ and such that the heights of the points of $T_0 \cap c\times (0,1)$ are pairwise distinct. Let $T\subset \Sigma_{\mathcal{P}}\times (0,1)$ be the framed tangle obtained by cutting $T_0$ along $c$. Any two states $s_a : \partial_a T \rightarrow \{-,+\}$ and $s_b : \partial_b T \rightarrow \{-,+\}$ give rise to a state $(s_a, s, s_b)$ on $T$. Both the sets $\partial_a T$ and $\partial_b T$ are in canonical bijection with the set $T_0\cap c$ by the map $\pi$. Hence the two sets of states $s_a$ and $s_b$ are both in canonical bijection with the set $\mathrm{St}(c):=\{ s: c\cap T_0 \rightarrow \{-,+\} \}$. Let $i_{|a\#b}: \mathcal{S}_{\omega}(\mathbf{\Sigma}_{|a\#b}) \rightarrow \mathcal{S}_{\omega}(\mathbf{\Sigma})$ be the linear map given, for any $(T_0, s_0)$ as above, by: $$i_{|a\#b} \left( [T_0,s_0] \right) := \sum_{s \in \mathrm{St}(c)} [T, (s, s_0 , s) ].$$
[@LeStatedSkein Theorem $3.1$] The linear map $i_{|a\#b}$ is an injective morphism of algebras. Moreover the gluing operation is coassociative in the sense that if $a,b,c,d$ are four distinct boundary arcs, then we have $i_{|a\#b} \circ i_{|c\#d} = i_{|c\#d} \circ i_{|a\#b}$.
\[def small\] A *small* punctured surface is one of the following four connected punctured surfaces: the sphere with one or two punctures; the disc with only one puncture (on its boundary); the bigon (disc with two punctures on its boundary).
A punctured surface is said to *admit a triangulation* if each of its connected components has at least one puncture and is not small.
Suppose $\mathbf{\Sigma}=(\Sigma, \mathcal{P})$ admits a triangulation. A *topological triangulation* $\Delta$ of $\mathbf{\Sigma}$ is a collection $\mathcal{E}(\Delta)$ of arcs in $\Sigma$ (named edges) which satisfy the following conditions: the endpoints of the edges belong to $\mathcal{P}$; the interior of the edges are pairwise disjoint; the edges are not contractible and are pairwise non isotopic, if fixed their endpoints; the boundary arcs of $\mathbf{\Sigma}$ belong to $\mathcal{E}(\Delta)$. Moreover, the collection $\mathcal{E}(\Delta)$ is required to be maximal for these properties.
The closure of each connected component of $\Sigma \setminus \mathcal{E}(\Delta)$ is called a *face* and the set of faces is denoted by $F(\Delta)$. Given a topological triangulation $\Delta$, the punctured surface is obtained from the disjoint union $\bigsqcup_{\mathbb{T}\in F(\Delta)} \mathbb{T}$ of triangles by gluing the triangles along the boundary arcs corresponding to the edges of the triangulation. Note that, here (and all along the paper), the indexing $\mathbb{T}\in F(\Delta)$ denotes a face while the indexed $\mathbb{T}$ denotes the triangle. We hope that this abuse of notation is harmless. By composing the associated gluing maps, one obtains an injective morphism of algebras: $$i^{\Delta} : \mathcal{S}_{\omega}(\mathbf{\Sigma}) \hookrightarrow \otimes_{\mathbb{T} \in F(\Delta)} \mathcal{S}_{\omega}(\mathbb{T}).$$
The stated skein algebra has natural filtrations defined as follows. Let $S=\{a_1, \ldots, a_n\}$ be a set of boundary arcs of $\mathbf{\Sigma}$ and fix an orientation $\mathfrak{o}$ of the boundary arcs of $\mathbf{\Sigma}$. For a basis element $[D,s]$ of $\mathcal{B}^{\mathfrak{o}}$, write $d([D,s]):= \sum_{a\in S} \big| \partial_a D \big|$. The map $d$ extends to a linear map $d: \mathcal{S}_{\omega}(\mathbf{\Sigma}) \rightarrow \mathbb{Z}^{\geq 0}$. It follows from the relations and that for each $x,y\in \mathcal{S}_{\omega}(\mathbf{\Sigma})$, we have $d(xy)\leq d(x)+d(y)$. Given $m\geq 0$, denote by $\mathcal{F}_m\subset \mathcal{S}_{\omega}(\mathbf{\Sigma})$ the sub-vector space of those vectors $x$ satisfying $d(x)\leq m$. These sub-spaces satisfy $\mathcal{F}_{m}\subset \mathcal{F}_{m+1}$, $\mathcal{S}_{\omega}(\mathbf{\Sigma}) = \bigcup_{m\geq 0} \mathcal{F}_m$ and $\mathcal{F}_{m_1} \cdot \mathcal{F}_{m_2} \subset \mathcal{F}_{m_1+m_2}$, hence they form an algebra filtration of the stated skein algebra.
\[def\_filtration\] The sequence $(\mathcal{F}_m)_{m\geq 0}$ is called the *filtration* of $\mathcal{S}_{\omega}(\mathbf{\Sigma})$ associated to the orientation $\mathfrak{o}$ and the set $S$ of boundary arcs. For an element $X= \sum_{i\in I} x_i [D_i, s_i] \in \mathcal{S}_{\omega}(\mathbf{\Sigma})$, developed in the basis $\mathcal{B}^{\mathfrak{o}}$, we call *leading term* of $X$ the element: $$\lt (X) := \sum_{j\in I | d([D_j, s_j])= d(X)} x_j [D_j, s_j].$$
Hopf comodule maps
------------------
Recall that the bigon $\mathbb{B}$ is a disc with two punctures on its boundary. It has two boundary arcs, say $b_L$ and $b_R$. Consider the simple diagram $\alpha$ made of a single arc joining $b_L$ and $b_R$. For $n\geq 0$, denote by $\alpha^{(n)}$ the diagram made of $n$ parallel copies of $\alpha$. Denote by $\alpha_{\varepsilon \varepsilon'}$ the class in $\mathcal{S}_{\omega}(\mathbb{B})$ of the stated diagram $(\alpha, s)$ where $s(\alpha\cap b_L)=\varepsilon$ and $s(\alpha\cap b_R)=\varepsilon'$. Fix an arbitrary orientation $\mathfrak{o}_{\mathbb{B}}$ of $b_L$ and $b_R$. It is proved in [@LeStatedSkein Theorem $4.1$] that the stated skein algebra $\mathcal{S}_{\omega}(\mathbb{B})$ is presented by the four generators $\alpha_{\varepsilon \varepsilon'}$, with $\varepsilon, \varepsilon' = \pm$, and the following relations, where we put $q:= \omega^{-4}$: $$\begin{aligned}
\label{relbigone}
\alpha_{++}\alpha_{+-} &= q^{-1}\alpha_{+-}\alpha_{++} & \alpha_{++}\alpha_{-+}&=q^{-1}\alpha_{-+}\alpha_{++}
\\ \alpha_{--}\alpha_{+-} &= q\alpha_{+-}\alpha_{--} & \alpha_{--}\alpha_{-+}&=q\alpha_{-+}\alpha_{--}
\\ \alpha_{++}\alpha_{--}&=1+q^{-1}\alpha_{+-}\alpha_{-+} & \alpha_{--}\alpha_{++}&=1 + q\alpha_{+-}\alpha_{-+}
\\ \alpha_{-+}\alpha_{+-}&=\alpha_{+-}\alpha_{-+} & &\end{aligned}$$
Consider a disjoint union $\mathbb{B}\bigsqcup \mathbb{B}'$ of two bigons. When gluing the boundary arcs $b_R$ with $b'_L$, we obtain another bigon. Denote by $\Delta : \mathcal{S}_{\omega}(\mathbb{B})\rightarrow \mathcal{S}_{\omega}(\mathbb{B}) \otimes \mathcal{S}_{\omega}(\mathbb{B})$ the composition: $$\Delta: \mathcal{S}_{\omega}(\mathbb{B}) \xrightarrow{i_{|b_R\#b_L'}} \mathcal{S}_{\omega}(\mathbb{B}\bigsqcup \mathbb{B}') \xrightarrow{\cong} \mathcal{S}_{\omega}(\mathbb{B}) \otimes \mathcal{S}_{\omega}(\mathbb{B}).$$
The map $\Delta$ is characterized by the formula $\Delta( \alpha_{\varepsilon \varepsilon'})= \alpha_{\varepsilon +}\otimes \alpha_{+ \varepsilon'} + \alpha_{\varepsilon -} \otimes \alpha_{- \varepsilon'}$. Define linear maps $\epsilon: \mathcal{S}_{\omega}(\mathbb{B}) \rightarrow \mathcal{R}$ and $S : \mathcal{S}_{\omega}(\mathbb{B}) \rightarrow \mathcal{S}_{\omega}(\mathbb{B})$ by the formulas $ \epsilon(\alpha_{\varepsilon \varepsilon'})= \delta_{\varepsilon \varepsilon'}$, $S(\alpha_{++}) = \alpha_{--}, S(\alpha_{--})=\alpha_{++}, S(\alpha_{+-})=-q \alpha_{+-}$ and $S(\alpha_{-+})=-q^{-1}\alpha_{-+}$. The coproduct $\Delta$, the counit $\epsilon$ and the antipode $S$ endow $\mathcal{S}_{\omega}(\mathbb{B})$ with a structure of Hopf algebra. This Hopf algebra is canonically isomorphic to the so-called *quantum* $\SL_2$ Hopf algebra $\mathbb{C}_q[\SL_2]$ as defined in ([@Manin_QGroups], [@Kassel] Chapter $IV$ Section $6$, [@ChariPressley] Definition $7.1.1$).\
For later use, let us write the coproduct, counit and antipode by the following more compact form:
$$\begin{pmatrix} \Delta (\alpha_{++}) & \Delta (\alpha_{+-}) \\ \Delta(\alpha_{-+}) & \Delta(\alpha_{--}) \end{pmatrix}
=
\begin{pmatrix} \alpha_{++} & \alpha_{+-} \\ \alpha_{-+} & \alpha_{--} \end{pmatrix}
\otimes
\begin{pmatrix} \alpha_{++} & \alpha_{+-} \\ \alpha_{-+} & \alpha_{--} \end{pmatrix}$$
$$\begin{pmatrix} \epsilon(\alpha_{++}) & \epsilon(\alpha_{+-}) \\ \epsilon(\alpha_{-+}) & \epsilon(\alpha_{--}) \end{pmatrix} =
\begin{pmatrix} 1 &0 \\ 0& 1 \end{pmatrix}
\text{ and }
\begin{pmatrix} S(\alpha_{++}) & S(\alpha_{+-}) \\ S(\alpha_{-+}) & S(\alpha_{--}) \end{pmatrix}
=
\begin{pmatrix} \alpha_{--} & -q\alpha_{+-} \\ -q^{-1}\alpha_{-+} & \alpha_{++} \end{pmatrix} .$$
Remark that when $q=+1$, we recover the Hopf algebra of regular functions of $\SL_2(\mathbb{C})$.
Consider a punctured surface $\mathbf{\Sigma}$ with boundary arc $a$. When gluing the boundary $a$ of $\mathbf{\Sigma}$ with the boundary arc $b_L$ of $\mathbb{B}$ we obtain the same punctured surface $\mathbf{\Sigma}$. Define a left Hopf comodule map (see *e.g.* [@Kassel Definition $III.7.1$]) $\Delta_a^L : \mathcal{S}_{\omega}(\mathbf{\Sigma})\rightarrow \mathcal{S}_{\omega}(\mathbb{B}) \otimes \mathcal{S}_{\omega}(\mathbf{\Sigma})$ as the composition: $$\Delta_a^L : \mathcal{S}_{\omega}(\mathbf{\Sigma}) \xrightarrow{i_{|a\# b_L }} \mathcal{S}_{\omega}(\mathbb{B} \bigsqcup \mathbf{\Sigma}) \xrightarrow{\cong} \mathcal{S}_{\omega}(\mathbb{B}) \otimes \mathcal{S}_{\omega}(\mathbf{\Sigma}).$$
Similarly, define a right Hopf comodule map $\Delta_a^R : \mathcal{S}_{\omega}(\mathbf{\Sigma})\rightarrow \mathcal{S}_{\omega}(\mathbf{\Sigma})\otimes \mathcal{S}_{\omega}(\mathbb{B}) $ as the composition: $$\Delta_a^R : \mathcal{S}_{\omega}(\mathbf{\Sigma}) \xrightarrow{i_{|b_R\# a }} \mathcal{S}_{\omega}(\mathbf{\Sigma} \bigsqcup \mathbb{B}) \xrightarrow{\cong} \mathcal{S}_{\omega}(\mathbf{\Sigma}) \otimes \mathcal{S}_{\omega}(\mathbb{B}).$$
The coassociativity of $\Delta_a^L$ and $\Delta_a^R$ follows from the coassociativity of the gluing maps.
The image of the gluing map
---------------------------
All along this subsection, we fix a punctured surface $\mathbf{\Sigma}$ and an orientation $\mathfrak{o}$ of its boundary arcs.
For a boundary arc $a$ and a diagram $D$, we write $n_a(D):= |\partial_a D|$. Given $n\geq 1$, define the set $\St(n):= \{ -, + \}^n$ and the subset $\St^{\uparrow}(n)\subset \St(n)$ which consists of $n$-tuples $(\varepsilon_1, \ldots, \varepsilon_n)$ such that $i<j$ implies $\varepsilon_i \leq \varepsilon_j$. If $s=(\varepsilon_1, \ldots, \varepsilon_n)\in \St(n)$, denote by $s^{\uparrow}=(\varepsilon'_1, \ldots, \varepsilon'_n)\in \St^{\uparrow}(n)$ the unique element such that the number of indices $i$ such that $\varepsilon_i=+$ is equal to the number of indices $j$ such that $\varepsilon'_j=+$. Given $s=(\varepsilon_1, \ldots, \varepsilon_n)\in \St(n)$, denote by $k(s)$ the number of pairs $(i,j)$ such that $i<j$ and $\varepsilon_i > \varepsilon_j$. For $s\in \St^{\uparrow}(n)$, let $$H_s(q) := \sum_{s'\in \St(n)| s'^{\uparrow}=s} q^{2 k(s')}.$$
Let $a$ and $b$ be two boundary arcs of $\mathbf{\Sigma}$ and consider the filtration associated to $S:=\{a, b\}$ and $\mathfrak{o}$.
\[lemma\_leading\_term\] Let $(D,s)$ be a stated diagram and consider $v_1, v_2$ two points which both belong either to $\partial_a D$ or to $\partial_b D$. Suppose that $v_1<_{\mathfrak{o}} v_2$ and that there is no $v\in \partial D$ such that $v_1 <_{\mathfrak{o}} v <_{\mathfrak{o}} v_2$. Further assume that $s(v_1)=+$ and $s(v_2)=-$. Let $s'$ be the state of $D$ such that $s'(v_1)=-$, $s'(v_2)=+$ and $s'(v)=s(v)$ if $v\in \partial D \setminus \{v_1, v_2\}$. One has $\lt ([D,s])= q \lt ([D, s'])$.
This is a straightforward consequence of the boundary relations and the height exchange formulas and .
Let $(D,s)$ be a stated diagram of $\mathbf{\Sigma}$ and write $s=(s_a, s_0, s_b)$, where $s_a$ and $s_b$ are the restrictions of $s$ to $\partial_a D$ and $\partial_b D$ respectively. It results from Lemma \[lemma\_leading\_term\] that we have the equality: $$\lt ([D,(s_a,s_0,s_b)]) = q^{k(s_a) +k(s_b)} \lt([D, (s_a^{\uparrow}, s_0, s_b^{\uparrow})]).$$
Fix an orientation $\mathfrak{o}_{\mathbb{B}}$ of the boundary arcs of the bigone. Consider the filtration of $\mathcal{S}_{\omega}(\mathbb{B})\otimes \mathcal{S}_{\omega}(\mathbf{\Sigma})\cong \mathcal{S}_{\omega}(\mathbf{\Sigma} \bigsqcup \mathbb{B})$ associated to the set of boundary arcs $S':=\{b_L, b_R, a,b\}$ and the orientations $\mathfrak{o}$ and $\mathfrak{o}_{\mathbb{B}}$. Given $X' \in \mathcal{S}_{\omega}(\mathbb{B})\otimes \mathcal{S}_{\omega}(\mathbf{\Sigma})$, we denote by $\lt'(X')$ the associated leading term. By definition of the left comodule map, we have the formula: $$\Delta_a^L ([D, (s_a,s_0,s_b)]) = \sum_{s\in \St(n_a(D))} [\alpha^{(n_a(D))}, (s_a,s)] \otimes [D, (s, s_0,s_b)]$$
\[lemmA\] Let $[D, (s_a,s_0,s_b)]$ be an element of the basis $\mathcal{B}^{\mathfrak{o}}$. One has $$\lt'\left(\Delta_a^L ([D, (s_a, s_0,s_b)])\right)=\sum_{s\in \St^{\uparrow}(|\partial_a(D)|)} H_s(q) [\alpha^{(|\partial_a(D)|)}, (s_a,s)] \otimes [D, (s,s_0,s_b)]$$ and $$\lt'\left(\sigma \circ \Delta_b^R([D, (s_a,s_0,s_b)]) \right)= \sum_{s\in \St^{\uparrow}(|\partial_b(D)|)} H_{s}(q) [\alpha^{(|\partial_b(D)|)}, (s,s_b)] \otimes [D, (s_a, s_0,s)],$$ where the summands are written in the canonical basis of $\mathcal{S}_{\omega}(\mathbb{B})\otimes \mathcal{S}_{\omega}(\mathbf{\Sigma})$.
This is a straightforward consequence of Lemma \[lemma\_leading\_term\].
The inclusion $\mathrm{Im}(i_{|a\#b}) \subset \ker (\Delta_a^L - \sigma \circ \Delta_b^R)$ follows from the coassociativity of the comodule maps. To prove the reverse inclusion, consider an element $X:= \sum_{i\in I} x_i [D_i,s_i] \in \ker (\Delta_a^L - \sigma \circ \Delta_b^R)$ developed in the basis $\mathcal{B}^{\mathfrak{o}}$. If $\lt(X)=0$, then $X$ is a linear combination of diagrams which do not intersect $a$ and $b$, hence $X$ belongs to the image of $i_{|a\#b}$. Suppose that $\lt(X)>0$. We will find an element $Y\in \mathcal{S}_{\omega}(\mathbf{\Sigma}_{|a\#b})$ such that $\lt(i_{a\#b}(Y)) = \lt(X)$. Hence $X$ will belong to the image of $i_{|a\#b}$ if and only if $Z:= X- i_{|a\#b}(Y)$ belongs to this image. Since $\lt(Z)<\lt(X)$, the proof will follow by induction on $\lt(X)$.
Consider the set $\widetilde{\mathcal{D}}$ of pairs $(D,s_0)$ for which there exists some states $s_a$ and $s_b$ such that the basis element $[D,(s_a,s_0,s_b)]$ appears in the expression of $X$. Given $\widetilde{D}=(D,s_0)\in \widetilde{\mathcal{D}}$, denote by $\St_X(\widetilde{D})$ the set of couples $(s_a,s_b)$ such that $[D, (s_a,s_0,s_b)]$ appears in the expression of $X$. We re-write the development of $X$ in the canonical basis as: $$X= \sum_{\widetilde{D}=(D,s_0)\in \widetilde{\mathcal{D}}} \sum_{(s_a,s_b)\in \St_X(\widetilde{D})} x_{[D,(s_a,s_0,s_b)]} [D, (s_a,s_0,s_b)].$$ Consider the subset $\widetilde{\mathcal{D}}_{\mathrm{max}} \subset \widetilde{\mathcal{D}}$ of pairs $(D,s_0)$ such that $d(X)=n_a(D)+n_b(D)$. By Lemma \[lemmA\], one has: $$\begin{gathered}
\lt'(\Delta_a^L(X)) = \sum_{(D,s_0)\in \widetilde{\mathcal{D}}_{\mathrm{max}}} \sum_{(s_a,s_b)\in \St_X((D,s_0))} x_{[D,(s_a,s_0,s_b)]}
\\ \sum_{s\in \St^{\uparrow}(n_a(D))} H_s(q) [\alpha^{(n_a(D))}, (s_a,s)] \otimes [D, (s,s_0,s_b)].
\end{gathered}$$ Similarly, one has: $$\begin{gathered}
\lt'(\sigma\circ\Delta_b^R(X)) = \sum_{(D,s_0)\in \widetilde{\mathcal{D}}_{\mathrm{max}}} \sum_{(s_a,s_b)\in \St_X((D,s_0))} x_{[D,(s_a,s_0,s_b)]}
\\ \sum_{s'\in \St^{\uparrow}(n_b(D))} H_{s'}(q) [\alpha^{(n_b(D))}, (s',s_b)] \otimes [D, (s_a,s_0,s')].
\end{gathered}$$
From the equality $\lt'(\Delta_a^L(X)) = \lt'(\sigma\circ \Delta_b^R(X))$, we find that for any pair $(D,s_0)\in \widetilde{\mathfrak{D}}_{\mathrm{max}}$, for any pair $(s_a,s_b) \in \St_X((D,s_0))$ and for any state $s\in \St^{\uparrow}(n_a(D))$, there exists a pair $(s_a',s_b')\in \St_X((D,s_0))$ and a state $s'\in \St^{\uparrow}(n_b(D))$ such that: $$\begin{gathered}
x_{[D,(s_a,s_0,s_b)]} H_s(q) [\alpha^{(n_a(D))}, (s_a,s)] \otimes [D, (s,s_0,s_b)] \\
= x_{[D, (s_a',s_0,s'_b)]} H_{s'}(q) [\alpha^{(n_b(D))}, (s',s_b)] \otimes [D, (s_a,s_0,s')].
\end{gathered}$$
We deduce the followings:
- For any $(D,s_0)\in \widetilde{\mathcal{D}}_{\mathrm{max}}$, we have $n_a(D)=n_b(D)= \frac{1}{2}d(X)$. We will denote by $n$ this integer.
- We have the equalities $s'=s_a=s_b$ and $s=s'_a=s'_b$. Hence for any $(D,s_0)\in \widetilde{\mathcal{D}}_{\mathrm{max}}$, we have $\St_X((D,s_0))= \{ (s,s), s\in \St^{\uparrow}(n) \}$.
- For any $(D,s_0)\in \widetilde{\mathcal{D}}_{\mathrm{max}}$ and $s \in \St^{\uparrow}(n)$, the coefficient $x_{[D,(s,s_0,s)]}$ is independent of $s$. We will denote this coefficient by $x_{(D,s_0)}$.
With the above notations, we re-write the leading term of $X$ as: $$\lt(X) = \sum_{(D,s_0)\in \widetilde{\mathcal{D}}_{\mathrm{max}}} x_{(D,s_0)} \sum_{s\in \St^{\uparrow}(n)} H_s(q)[D,(s,s_0,s)].$$
Given $(D,s_0)\in \widetilde{\mathcal{D}}_{\mathrm{max}}$, since $n_a(D)=n_b(D)=n$, there exists a diagram $D_0$ of $\mathbf{\Sigma}_{|a\#b}$ such that $D$ is obtained from $D_0$ by cutting along the common image in $\Sigma_{|a\#b}$ of $a$ and $b$ by the projection. Define the following element: $$Y:= \sum _{(D,s_0)\in \widetilde{\mathcal{D}}_{\mathrm{max}}} x_{(D,s_0)} [D_0, s_0] \in \mathcal{S}_{\omega}(\mathbf{\Sigma}).$$
By the above expression, we have the equality $\lt(X) = \lt(i_{|a\#b}(Y))$. This concludes the proof.
Consider a topological triangulation $\Delta$ of $\mathbf{\Sigma}$. The punctured surface $\mathbf{\Sigma}$ is obtained from the disjoint union $\mathbf{\Sigma}_{\Delta}:=\bigsqcup_{\mathbb{T} \in F(\Delta)} \mathbb{T}$ by gluing the triangles along their common edges. Denote by $\mathring{\mathcal{E}}(\Delta) \subset \mathcal{E}(\Delta)$ the subset of edges which are not boundary arcs. Each edge $e\in \mathring{\mathcal{E}}(\Delta)$ lifts in $\mathbf{\Sigma}_{\Delta}$ to two boundary arcs $e_L$ and $e_R$. By composing all the left comodule maps $\Delta_{e_L}^L$ together (the order does not matter) one gets a Hopf comodule map $$\Delta^L : \otimes_{\mathbb{T}\in F(\Delta)} \mathcal{S}_{\omega}(\mathbb{T}) \rightarrow \left( \otimes_{e\in \mathring{\mathcal{E}}(\Delta)} \mathcal{S}_{\omega}(\mathbb{B})\right) \otimes \left( \otimes_{\mathbb{T}\in F(\Delta)} \mathcal{S}_{\omega}(\mathbb{T}) \right).$$ Similarly, composing all the right comodule maps $\Delta_{e_R}^R$ together gives $$\Delta^R : \otimes_{\mathbb{T}\in F(\Delta)} \mathcal{S}_{\omega}(\mathbb{T}) \rightarrow \left( \otimes_{\mathbb{T}\in F(\Delta)} \mathcal{S}_{\omega}(\mathbb{T}) \right)\otimes \left( \otimes_{e\in \mathring{\mathcal{E}}(\Delta)} \mathcal{S}_{\omega}(\mathbb{B})\right).$$
The following sequence is exact. $$0 \rightarrow \mathcal{S}_{\omega}(\mathbf{\Sigma}) \xrightarrow{i^{\Delta}}
\otimes_{\mathbb{T}\in F(\Delta)} \mathcal{S}_{\omega}(\mathbb{T}) \xrightarrow{\Delta^L -\sigma\circ\Delta^R} \left( \otimes_{e\in \mathring{\mathcal{E}}(\Delta)} \mathcal{S}_{\omega}(\mathbb{B})\right) \otimes \left( \otimes_{\mathbb{T}\in F(\Delta)} \mathcal{S}_{\omega}(\mathbb{T}) \right).$$
Theorem \[theorem1\] applied to each inner edge provides an isomorphism between $\mathcal{S}_{\omega}(\mathbf{\Sigma})$ and the intersection, over the inner edges $e$, of $\text{Ker} ( \Delta_{e_L}^L-\sigma\circ \Delta_{e_R}^R)$. We conclude by observing that the latter intersection is $\text{Ker} ( \Delta^L-\sigma\circ \Delta^R)$.
We can reformulate the above exact sequence in terms of coHochschild cohomology.
\[def\_coHochschild\] Given a coalgebra $C$ with a bi-comodule $M$, with comodules maps $\Delta^L : M \rightarrow A\otimes M$ and $\Delta^R: M \rightarrow M \otimes A$, the $0$-th coHochschild cohomology group is $\mathrm{coHH}^0(A, M):= \ker \left(\Delta^L - \sigma \circ \Delta^R\right)$.
The above triangular decomposition of skein algebra can be re-written as: $$\mathcal{S}_{\omega}(\mathbf{\Sigma})\cong \mathrm{coHH}^0\left( \otimes_{e\in \mathring{\mathcal{E}}(\Delta)} \mathbb{C}_q[\SL_2], \otimes_{\mathbb{T}\in F(\Delta)} \mathcal{S}_{\omega}(\mathbb{T})\right).$$
The center of stated skein algebras at odd roots of unity
---------------------------------------------------------
Here we prove Theorem \[theorem2\]. We prove it for the bigon, then for the triangle, and we conclude with the general case. Let us start by the following classical result.
\[lemma\_qbinomial\] Let $\mathcal{R}$ be a ring, $N> 1$ an integer and $q\in \mathcal{R}^{\times}$ an element such that $q^N=1$. Suppose that $\mathcal{A}$ is an $\mathcal{R}$-algebra and $x,y\in \mathcal{A}$ are such that $yx=qxy$. One has $(x+y)^N= x^N +y^N$.
By [@Kassel Proposition $IV.2.2$], one has: $$(x+y)^N = \sum_{k=0}^N \begin{pmatrix} N \\ k \end{pmatrix}_q x^k y^{N-k},$$ where $\begin{pmatrix} N \\ k \end{pmatrix}_q := \prod_{i=0}^{k-1} \left( \frac{1-q^{N-i}}{1-q^{i+1}} \right)$. Since $q^N=1$, the coefficients $\begin{pmatrix} N \\ k \end{pmatrix}_q$ vanish for $1\leq k \leq N-1$, and we get the desired formula.
### The case of the bigon
Let $\mathcal{R}_q[\GL_2]$ be the following Hopf algebra. As an algebra, it is generated by elements $\alpha_{\varepsilon \varepsilon'}$ with relations:
$$\begin{aligned}
\alpha_{++}\alpha_{+-} &= q^{-1}\alpha_{+-}\alpha_{++} & \alpha_{++}\alpha_{-+}&=q^{-1}\alpha_{-+}\alpha_{++}
\\ \alpha_{--}\alpha_{+-} &= q\alpha_{+-}\alpha_{--} & \alpha_{--}\alpha_{-+}&=q\alpha_{-+}\alpha_{--}
\\ \alpha_{++}\alpha_{--}-\alpha_{--}\alpha_{++} &= (q^{-1}-q) \alpha_{+-}\alpha_{-+} & \alpha_{-+}\alpha_{+-}&=\alpha_{+-}\alpha_{-+}\end{aligned}$$
The counit, coproduct and antipode are given successively by $\epsilon(\alpha_{\varepsilon \varepsilon'})=\delta_{\varepsilon \varepsilon'}$, $\Delta(\alpha_{\varepsilon \varepsilon'})=\alpha_{\varepsilon +} \otimes \alpha_{+\varepsilon'} + \alpha_{\varepsilon - } \otimes \alpha_{-\varepsilon'}$ and $S(\alpha_{++}) = \alpha_{--}, S(\alpha_{--})=\alpha_{++}, S(\alpha_{+-})=-q \alpha_{+-}$, $S(\alpha_{-+})=-q^{-1}\alpha_{-+}$.
Note that the element $\det_q:= \alpha_{++}\alpha_{--} - q^{-1} \alpha_{+-}\alpha_{-+}$ is central and group-like. The Hopf algebra $\mathcal{S}_{\omega}(\mathbb{B})$ is isomorphic the quotient of $\mathcal{R}_q[\GL_2]$ by the Hopf ideal generated by $(\det_q -1)$.
\[lemma\_GL2\] In $\mathcal{R}_q[\GL_2]$ the only group-like elements are of the form $\det_q^n$ for some $n\geq 0$.
Since $\mathcal{R}_q[\GL_2]$ does not depend on $q$ as a bi-algebra, it is sufficient to prove the lemma in the case where $q=+1$. In this case, $\mathcal{R}[\GL_2]$ is the Hopf algebra of the affine algebraic group $\GL_2(\mathcal{R})$ and group-like elements correspond to regular one-dimensional representations of $\GL_2(\mathcal{R})$. The one dimensional regular representation associated to $\det^n$ is the representation $\GL_2(\mathcal{R})\rightarrow \mathcal{R}^{\times}$ sending a matrix $A$ to $\det(A)^n$ and it is well-known that these are the only ones. This concludes the proof.
We first prove Theorem \[theorem2\] for the bigone.
\[lemma\_center\_bigone\] Suppose that $q:=\omega^{-4}$ is a root of unity of odd order $N>1$. There exists a morphism of Hopf algebras $j_{\mathbb{B}}:\mathcal{S}_{+1}(\mathbb{B}) \rightarrow \mathcal{S}_{\omega}(\mathbb{B})$ characterized by $j_{\mathbb{B}}(\alpha_{\varepsilon \varepsilon'}):= (\alpha_{\varepsilon \varepsilon'})^N$ whose image lies in the center of $\mathcal{S}_{\omega}(\mathbb{B})$.
First, let us construct a morphism of Hopf algebras $\widetilde{j} : \mathcal{R}_{+1}[\GL_2] \rightarrow \mathcal{R}_q[\GL_2]$ as follows. For each generator $\alpha_{\varepsilon,\varepsilon'}$, let $\widetilde{j}(\alpha_{\varepsilon \varepsilon'}):= \alpha_{\varepsilon \varepsilon'}^N$. Let us show that it extends to $\mathcal{R}_{+1}[\GL_2]$ as an algebra morphism. Since $\mathcal{R}_{+1}[\GL_2]$ is a free commutative algebra, it is enough to show that each $\widetilde{j}(\alpha_{\varepsilon \varepsilon'})$ lies in the center of $\mathcal{R}_q[\GL_2]$. We prove this for $\alpha_{++}$ and $\alpha_{-+}$, the two other cases follow by “symmetry” in the relations.
$\alpha_{++}^N$ is central: One has $\alpha_{++}^N\alpha_{+-}= q^{-N}\alpha_{+-}\alpha_{++}^N = \alpha_{+-}\alpha_{++}^N$ and $\alpha_{++}^N\alpha_{-+}= q^{-N}\alpha_{-+}\alpha_{++}^N = \alpha_{-+}\alpha_{++}^N$. One also has: $$\begin{aligned}
\alpha_{++}^N \alpha_{--} - \alpha_{--}\alpha_{++}^N &=& \alpha_{++}^{N-1}( {\det}_q +q^{-1}\alpha_{+-}\alpha_{-+}) - ( {\det}_q +q\alpha_{+-}\alpha_{-+})\alpha_{++}^{N-1}
\\ &=& q^{-1}\alpha_{++}^{N-1} \alpha_{+-}\alpha_{-+} - q \alpha_{+-}\alpha_{-+}\alpha_{++}^{N-1}
\\ &=& (q^{-N}-q^N) \alpha_{+-}\alpha_{++}^{N-1}\alpha_{-+} = 0.
\end{aligned}$$
$\alpha_{-+}^N$ is central: it commutes with $\alpha_{+-}$ by definition. Furthermore, one has $\alpha_{+-}^N \alpha_{++}= q^N \alpha_{++}\alpha_{+-} = \alpha_{++}\alpha_{+-}$ and $\alpha_{+-}^N \alpha_{--}= q^{-N} \alpha_{--}\alpha_{+-} = \alpha_{--}\alpha_{+-}$.
It remains to prove that $\widetilde{j}$ is a morphism of bialgebras, hence a morphism of Hopf algebras. This is done by a direct inspection on each generator; for $\alpha_{++}$ it is as follows. Recall that $\Delta(\alpha_{++})=\alpha_{++}\otimes \alpha_{++} + \alpha_{+-}\otimes \alpha_{-+}$ and write $x:= \alpha_{++}\otimes \alpha_{++}$ and $y:=\alpha_{+-}\otimes \alpha_{-+}$. Since $xy=q^{-2} yx$ in $\mathcal{R}_q[\GL_2]^{\otimes 2}$ and $q^{2N}=1$, by Lemma \[lemma\_qbinomial\] one has $(x+y)^N=x^N+y^N$. Therefore one has $\widetilde{j}\otimes \widetilde{j}(\Delta(\alpha_{++})) = \Delta(\widetilde{j}(\alpha_{++}))$. The other cases are similar.
Since the element $\det_{+1} \in \mathcal{R}_{+1}[\GL_2]$ is group-like and $\widetilde{j}$ is a bi-algebra morphism, the element $\widetilde{j}(\det_{+1})$ is group-like. By Lemma \[lemma\_GL2\] there exists some $n\geq 0$ such that $\alpha_{++}^N\alpha_{--}^N - \alpha_{+-}^N\alpha_{-+}^N = \det_q^n$. Hence $\widetilde{j}$ induces, by passing to the quotient, a morphism of Hopf algebras $j_{\mathbb{B}}:\mathcal{S}_{+1}(\mathbb{B}) \rightarrow \mathcal{S}_{\omega}(\mathbb{B})$ whose image lies in the center of $\mathcal{S}_{\omega}(\mathbb{B})$.
It remains to prove the injectivity of $j_{\mathbb{B}}$. Consider an orientation $\mathfrak{o}$ of the boundary arcs of $\mathbb{B}$ such that both $b_L$ and $b_R$ point towards the same puncture. The basis $\mathcal{B}^{\mathfrak{o}}$ of $\mathcal{S}_{\omega}(\mathbb{B})$ given by Definition \[def\_basis\] consists in the set of vectors $\alpha_{++}^a \alpha_{+-}^b \alpha_{--}^c$ and $\alpha_{++}^{a'} \alpha_{-+}^{b'} \alpha_{--}^{c'}$ for which $a,b,c,a',b',c' \geq 0$ and $b'\neq 0$. Since $\mathcal{B}^{\mathfrak{o}}$ is also a basis of $\mathcal{S}_{+1}(\mathbb{B})$, it follows from the definition of $j_{\mathbb{B}}$ that it is injective.
### The case of the triangle
Denote by $\alpha, \beta, \gamma$ the three arcs of Figure \[figtriangle\] and $\tau$ the automorphism of $\mathcal{S}_{\omega}(\mathbb{T})$ induced by the rotation sending $\alpha, \beta, \gamma$ to $\beta, \gamma, \alpha$ respectively . In [@LeStatedSkein Theorem $4.6$], it was proved that the stated skein algebra $\mathcal{S}_{\omega}(\mathbb{T})$ is presented by the generators $\alpha_{\varepsilon \varepsilon'}, \beta_{\varepsilon \varepsilon'}, \gamma_{\varepsilon \varepsilon'}$ and the following relations together with their images through $\tau$ and $\tau^2$: $$\begin{aligned}
\alpha_{-\varepsilon}\alpha_{+\varepsilon'}&=& A^2 \alpha_{+\varepsilon}\alpha_{-\varepsilon'}-\omega^{-5}C_{\varepsilon'}^{\varepsilon} \label{eq1} \\
\alpha_{\varepsilon -}\alpha_{\varepsilon' +}&=& A^2 \alpha_{\varepsilon +}\alpha_{\varepsilon'-}-\omega^{-5}C_{\varepsilon'}^{\varepsilon} \label{eq2} \\
\beta_{\mu \varepsilon} \alpha_{\mu' \varepsilon'} &=& A\alpha_{\varepsilon \varepsilon'}\beta_{\mu\mu'} - A^2 C_{\mu'}^{\varepsilon} \gamma_{\varepsilon' \mu} \label{eq3}
\\ \alpha_{- \varepsilon}\beta_{\varepsilon' +} &=& A^2 \alpha_{+ \varepsilon} \beta_{\varepsilon' -} - \omega^{-5}\gamma_{\varepsilon \varepsilon'} \label{eq4}
\\ \alpha_{\varepsilon -} \gamma_{+ \varepsilon'} &=& A^2 \alpha_{\varepsilon +} \gamma_{- \varepsilon'} + \omega \beta_{\varepsilon' \varepsilon} \label{eq5}\end{aligned}$$
Here we use the notation $A:= \omega^{-2}$, $C_-^- = C_+^+:=0$, $C_+^-:= -\omega^5$ and $C_-^+:=\omega$.
![The three diagrams $\alpha, \beta, \gamma$, the stated diagram representing $\alpha_{\varepsilon \varepsilon'}$ and the diagram $\theta^{(2,1,1)}$.[]{data-label="figtriangle"}](triangle_generateurs.eps){width="7cm"}
When $\omega=+1$, the algebra $\mathcal{S}_{+1}(\mathbb{T})$ has the following simpler presentation. Consider the commutative unital polynomial algebra $\mathcal{A}:= \mathcal{R}[\alpha_{\varepsilon \varepsilon'}, \beta_{\varepsilon \varepsilon'}, \gamma_{\varepsilon \varepsilon'} | \varepsilon, \varepsilon'=\pm ]$. Given $\delta \in \{\alpha, \beta, \gamma \}$, denote by $M_{\delta}$ the $2\times 2$ matrix with coefficients in $\mathcal{A}$ defined by $M_{\delta}:= \begin{pmatrix} \delta_{++} & \delta_{+-} \\ \delta_{-+}& \delta_{--} \end{pmatrix}$ and write $C:= \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}$ and $\mathds{1}:=\begin{pmatrix} 1&0 \\ 0&1 \end{pmatrix}$.
\[lemma\_triangle+1\] The algebra $\mathcal{S}_{+1}(\mathbb{T})$ is isomorphic to $$\quotient{\mathcal{R}[\alpha_{\varepsilon \varepsilon'}, \beta_{\varepsilon \varepsilon'}, \gamma_{\varepsilon \varepsilon'} | \varepsilon, \varepsilon'=\pm ]}{\left( \begin{array}{l} \det(M_{\alpha})=\det(M_{\beta})=\det(M_{\gamma})=1, \\ M_{\gamma}CM_{\beta} C M_{\alpha} C = \mathds{1} \end{array} \right)}.$$
For $\omega=1$, part of equation , and and their image by rotations make $\mathcal{S}_{+1}(\mathbb{T})$ commutative; equations and coincide; is the image of by rotation, and the latter is a particular case of . Moreover, a direct inspection shows that the other part of and of correspond to $\det(M_{\alpha})=1$ and $(M_{\gamma}C)^{-1}=M_{\beta} C M_{\alpha} C$, respectively.
\[lemma\_center\_triangle\] Suppose that $\omega$ is a root of unity of odd order $N>1$. There exists an injective morphism of algebras $j_{\mathbb{T}}:\mathcal{S}_{+1}(\mathbb{T}) \rightarrow \mathcal{S}_{\omega}(\mathbb{T})$, whose image lies in the center of $\mathcal{S}_{\omega}(\mathbb{T})$, characterized by $j_{\mathbb{T}}(\delta_{\varepsilon \varepsilon'}):= (\delta_{\varepsilon \varepsilon'})^N$ for $\delta\in \{\alpha, \beta, \gamma\} $ and $\varepsilon, \varepsilon' = \pm$. Moreover, if $a$ is a boundary arc of $\mathbb{T}$, the following diagrams commute: $$\begin{array}{ll}
\begin{tikzcd}
\mathcal{S}_{+1}(\mathbb{T}) \arrow[r, "\Delta_a^L"] \arrow[d, hook, "j_{\mathbb{T}}"] & \mathcal{S}_{+1}(\mathbb{B})\otimes \mathcal{S}_{+1}(\mathbb{T}) \arrow[d, hook, "j_{\mathbb{B}}\otimes j_{\mathbb{T}}"] \\
\mathcal{S}_{\omega}(\mathbb{T}) \arrow[r, "\Delta_a^L"] & \mathcal{S}_{\omega}(\mathbb{B})\otimes \mathcal{S}_{\omega}(\mathbb{T})
\end{tikzcd}
&
\begin{tikzcd}
\mathcal{S}_{+1}(\mathbb{T}) \arrow[r, "\Delta_a^R"] \arrow[d, hook, "j_{\mathbb{T}}"] & \mathcal{S}_{+1}(\mathbb{T})\otimes \mathcal{S}_{+1}(\mathbb{B}) \arrow[d, hook, "j_{\mathbb{B}}\otimes j_{\mathbb{T}}"] \\
\mathcal{S}_{\omega}(\mathbb{T}) \arrow[r, "\Delta_a^R"] & \mathcal{S}_{\omega}(\mathbb{\mathbb{T}})\otimes \mathcal{S}_{\omega}(\mathbb{B})
\end{tikzcd}
\end{array}$$
We proceed in a similar way than Lemma \[lemma\_center\_bigone\], by showing first that the extension of the assignment $j_{\mathbb{T}}(\delta_{\varepsilon \varepsilon'}):= \delta_{\varepsilon \varepsilon'}^N$ to a morphism of algebras is well-defined. In virtue of Lemma \[lemma\_triangle+1\] and by the rotation automorphism, it is enough to show that $\alpha_{\varepsilon \varepsilon'}^N$ lies in the center of $\mathcal{S}_{\omega}(\mathbb{T})$ and that $j_{\mathbb{T}}$ sends $\det( M_{\alpha})-1$ and $M_{\gamma}CM_{\beta} C M_{\alpha} C - \mathds{1}$ to zero.
First remark that the relations and put together coincide with the defining relations of $\mathcal{S}_{\omega}(\mathbb{B})$, hence one has an inclusion of algebras $\phi: \mathcal{S}_{\omega}(\mathbb{B}) \hookrightarrow \mathcal{S}_\omega(\mathbb{T})$ defined by $\phi(\alpha_{\varepsilon\varepsilon'})=\alpha_{\varepsilon\varepsilon'}$. By applying Lemma \[lemma\_center\_bigone\], one obtains an inclusion $\phi\circ j_{\mathbb{B}}: \mathcal{S}_{+1}(\mathbb{B}) \hookrightarrow \mathcal{S}_\omega(\mathbb{T})$ which coincides with $j_{\mathbb{T}}$ on the $\alpha_{\varepsilon \varepsilon'}$’s. It remains to show that the $\alpha_{\varepsilon \varepsilon'}^N$’s commute with the $\beta_{\mu \mu'}$’s and the $\gamma_{\mu \mu'}$’s, and that $j_{\mathbb{T}}$ vanishes on $M_{\gamma}CM_{\beta} C M_{\alpha} C - \mathds{1}$.
We have $\alpha_{\varepsilon \varepsilon'}^N \beta_{\mu \varepsilon} = A^{-N}\beta_{\mu \varepsilon}\alpha_{\varepsilon\varepsilon'}^N = \beta_{\mu \varepsilon}\alpha_{\varepsilon\varepsilon'}^N$. From $$\begin{aligned}
\alpha_{+ \varepsilon}^N \beta_{\varepsilon' -} &= \alpha_{+ \varepsilon}^{N-1}(A^{-2}\alpha_{- \varepsilon}\beta_{\varepsilon' +} +\omega^{-1} \gamma_{\varepsilon \varepsilon'})
\\ &= (A^{-3N+1}\alpha_{- \varepsilon} \beta_{\varepsilon' +} + \omega^{-1}A^{N-1} \gamma_{\varepsilon \varepsilon'})\alpha_{+ \varepsilon}^{N-1}\end{aligned}$$ and $$\beta_{\varepsilon' -} \alpha_{+ \varepsilon}^N = (A\alpha_{- \varepsilon}\beta_{\varepsilon' +} + \omega \gamma_{\varepsilon \varepsilon'})\alpha_{+ \varepsilon}^{N-1},$$ one obtains $$\alpha_{+ \varepsilon}^N\beta_{\varepsilon' -} - \beta_{\varepsilon' -}\alpha_{+ \varepsilon}^N = (A(A^{-3N}-1)\alpha_{- \varepsilon}\beta_{\varepsilon' +} + \omega(A^N-1)\gamma_{\varepsilon \varepsilon'})\alpha_{+ \varepsilon}^{N-1} = 0.$$
Similarly, we compute: $$\begin{aligned}
\alpha_{- \varepsilon}^N \beta_{\varepsilon' +} &=& \alpha_{- \varepsilon}^{N-1}(A^{2}\alpha_{+ \varepsilon}\beta_{\varepsilon' -} -\omega^{-5} \gamma_{\varepsilon \varepsilon'})
\\ &=& (A^{N+1}\alpha_{+ \varepsilon} \beta_{\varepsilon' -} - \omega^{-3}A^{N} \gamma_{\varepsilon \varepsilon'})\alpha_{- \varepsilon}^{N-1};
\\ \beta_{\varepsilon' +} \alpha_{- \varepsilon}^N &=& (A\alpha_{+ \varepsilon}\beta_{\varepsilon' -} - \omega^{-3} \gamma_{\varepsilon \varepsilon'})\alpha_{- \varepsilon}^{N-1}.\end{aligned}$$ Thus we find: $$\alpha_{- \varepsilon}^N\beta_{\varepsilon' +} - \beta_{\varepsilon' +}\alpha_{- \varepsilon}^N = (A(A^{N}-1)\alpha_{+ \varepsilon}\beta_{\varepsilon' -} - \omega^{-3}(A^N-1)\gamma_{\varepsilon \varepsilon'})\alpha_{- \varepsilon}^{N-1} = 0.$$ So we have proven that $\alpha_{\varepsilon \varepsilon'}^N$ commutes with every elements $\beta_{\mu \mu'}$. The commutativity of $\alpha_{\varepsilon \varepsilon'}^N$ with each element $\gamma_{\mu \mu'}$ is shown in a very similar way.
Next, showing that $j_{\mathbb{T}}$ vanishes on $M_{\gamma}CM_{\beta} C M_{\alpha} C - \mathds{1}$ amounts to showing that $$\beta_{\mu \varepsilon}^N \alpha_{\mu' \varepsilon'}^N - \alpha_{\varepsilon \varepsilon'}^N \beta_{\mu \mu'}^N = \gamma_{\varepsilon', \mu}^N \text{ for } \varepsilon\neq \mu'.$$ Suppose that $\varepsilon=+$ and $\mu'=-$. Let $x:= \beta_{\mu -}\alpha_{+\varepsilon'}$ and $y:=\alpha_{-\varepsilon'}\beta_{\mu+}$. In $\mathcal{S}_\omega(\mathbb{T})$ one has $xy=q^2yx$; by Lemma \[lemma\_qbinomial\], one has $(x+y)^N=x^N + y^N$. On the other hand, equations and give $\alpha_{-\varepsilon'}\beta_{\mu +}= A \beta_{\mu +} \alpha_{-\varepsilon'}$, hence $y^N= \alpha_{-\varepsilon'}^N \beta_{\mu +}^N$. Similarly, one has $x^N=\beta_{\mu - }^N\alpha_{+\varepsilon'}^N$. Finally, from $$\beta_{\mu +}\alpha_{-\varepsilon'}-A\alpha_{+\varepsilon'}\beta_{\mu - } = \omega^{-3}\gamma_{\varepsilon' \mu},$$ we find that $$\gamma_{\varepsilon' \mu}^N = (x-Ay)^N = x^N-y^N = \beta_{\mu - }^N\alpha_{+\varepsilon'}^N - \alpha_{-\varepsilon'}^N \beta_{\mu +}^N.$$ The case $\varepsilon = -$ and $\mu'=+$ is handled similarly.
Now let us prove that $j_{\mathbb{T}}$ is injective. To this end, let us consider the following basis of $\mathcal{S}_{\omega}(\mathbb{T})$.
Consider the counter-clockwise orientation $\mathfrak{o}$ of the boundary arcs of $\mathbb{T}$ as in Figure \[figtriangle\]. Given $\mathbf{k}=(k_{\alpha}, k_{\beta}, k_{\gamma}) \in (\mathbb{Z}^{\geq 0})^3$, denote by $\theta^{\mathbf{k}}$ the (not simple) diagram $\alpha^{k_{\alpha}}\beta^{k_{\beta}}\gamma^{k_{\gamma}}$; see Figure \[figtriangle\] for an example. It is proven in [@LeStatedSkein Proof of Theorem 4.6] that the set of classes $[\theta^{\mathbf{k}}, s]$, where $s$ is $\mathfrak{o}$-increasing, forms a basis of $\mathcal{S}_{\omega}(\mathbb{T})$.
By construction, $j_{\mathbb{T}}$ sends the elements $[\theta^{\mathbf{k}}, s]$ of $\mathcal{S}_{+1}(\mathbb{T})$, where $s$ is $\mathfrak{o}$-increasing, to some basis elements $[\theta^{N \mathbf{k}}, s']$, where $s'$ is also $\mathfrak{o}$ increasing, therefore $j_{\mathbb{T}}$ is injective.
The proof that $j_{\mathbb{T}}$ is a morphism of Hopf comodules is similar to the proof in Lemma \[lemma\_center\_bigone\] of the fact that $j_{\mathbb{B}}$ is a morphism of Hopf algebras and is left to the reader. This concludes the proof.
### The general case: proof of Theorem \[theorem2\]
In this section we prove Theorem \[theorem2\]; we do this in five steps.\
In Step $1$, we show that the decomposition Theorem \[theorem1\] together with the two previous sections provide an injective morphism of algebras $$\label{eq: morph j surf triang}
j_{(\mathbf{\Sigma},\Delta)} : \mathcal{S}_{+1}(\mathbf{\Sigma}) \hookrightarrow \mathcal{S}_{\omega}(\mathbf{\Sigma}),$$ which is central. We study further properties of $j_{(\mathbf{\Sigma},\Delta)}$ and we show that it is does *not* depend on a topological triangulation $\Delta$. The other steps are devoted to making explicit the morphism $j_{(\mathbf{\Sigma},\Delta)}$ on arcs and loops. In Step $2$ to $4$, we suppose that the punctured surface has a non-degenerated triangulation (see below); in Step $5$ we treat the other punctured surfaces.\
In Step $2$, we prove that $j_{(\mathbf{\Sigma},\Delta)}$ sends the stated arcs that have their endpoints on *two different* boundary arcs of $\Sigma$, to their $N$-th power.\
In Step $3$, we prove that $j_{(\mathbf{\Sigma},\Delta)}$ sends some particular closed curves of $\Sigma_{\mathcal{P}}$ to their $N$-th Chebyshev polynomial of first kind.\
Step $4$ is more involved. We first prove a structural result. Adding a puncture on a surface $\mathbf{\Sigma}$ gives rise to a surjective map from the skein algebra of the new punctured surface to that of the initial one. We show that $j_{(\mathbf{\Sigma},\Delta)}$ commutes with these surjections (see Proposition \[prop\_add\_puncture\]). This uses results of the previous steps. From this, we prove that $j_{(\mathbf{\Sigma},\Delta)}$ sends the stated arcs that have their endpoints on *one* boundary arc of $\Sigma$ to their $N$-th power, and we prove that it sends *any* closed curve of $\Sigma_{\mathcal{P}}$ to its $N$-th Chebyshev polynomial of first kind.\
In Step $5$, we treat the remaining cases of connected punctured surfaces that do not admit a non-degenerated topological triangulation.\
These five steps prove Theorem \[theorem2\].\
All along this section, $\mathbf{\Sigma}$ is a punctured surface, $\Delta$ a topological triangulation $\mathbf{\Sigma}$ and $\omega$ a root of unity of odd order $N>1$. Except for Step $1$ and $5$, the triangulation $\Delta$ is required to be *non-degenerated*, that is, such that each of its inner edges separates two distinct faces.
#### Step 1: formal definition.
Consider the following diagram, where both lines are exact by Theorem \[theorem1\] and the vertical maps are given by Lemmas \[lemma\_center\_bigone\] and \[lemma\_center\_triangle\]. $$\label{diag j}
\begin{tikzcd}
0 \arrow[r,""] &
\mathcal{S}_{+1}(\mathbf{\Sigma}) \arrow[r, "i^{\Delta}"] \arrow[d, hook, dotted, "\exists!", "j_{(\mathbf{\Sigma}, \Delta)}"'] &
\otimes_{\mathbb{T}\in F(\Delta)} \mathcal{S}_{+1}(\mathbb{T}) \arrow[r,"\Delta^L - \sigma \circ \Delta^R"] \arrow[d, hook,"\otimes_{\mathbb{T}} j_{\mathbb{T}}"] &
\left( \otimes_{e\in \mathring{\mathcal{E}}(\Delta)} \mathcal{S}_{+1}(\mathbb{B}) \right) \otimes \left( \otimes_{\mathbb{T}\in F(\Delta)} \mathcal{S}_{+1}(\mathbb{T}) \right)
\arrow[d, hook,"(\otimes_e j_{\mathbb{B}})\otimes (\otimes_{\mathbb{T}} j_{\mathbb{T}})"] \\
0 \arrow[r,""] &
\mathcal{S}_{\omega}(\mathbf{\Sigma}) \arrow[r, "i^{\Delta}"] &
\otimes_{\mathbb{T}\in F(\Delta)} \mathcal{S}_{\omega}(\mathbb{T}) \arrow[r,"\Delta^L - \sigma \circ \Delta^R"] &
\left( \otimes_{e\in \mathring{\mathcal{E}}(\Delta)} \mathcal{S}_{+1}(\mathbb{B}) \right) \otimes \left( \otimes_{\mathbb{T}\in F(\Delta)} \mathcal{S}_{+1}(\mathbb{T}) \right)
\end{tikzcd}$$ The existence of an injective morphism $j_{(\mathbf{\Sigma},\Delta)} : \mathcal{S}_{+1}(\mathbf{\Sigma}) \hookrightarrow \mathcal{S}_{\omega}(\mathbf{\Sigma})$ follows from the exactness of the lines and the injectivity of $\otimes_{\mathbb{T}\in F(\Delta)}j_{\mathbb{T}}$. Moreover, since $j_{\mathbb{T}}$ is central, so is $j_{(\mathbf{\Sigma},\Delta)}$.
Let us show that $j_{(\mathbf{\Sigma},\Delta)}$ is compatible with the gluing maps.
\[lem: j commute with i\] If $a$ and $b$ are two boundary arcs of $\mathbf{\Sigma}$, the following diagram commutes: $$\begin{tikzcd}
\mathcal{S}_{+1}(\mathbf{\Sigma}_{|a\#b})
\arrow[r, hook, "j_{\mathbf{\Sigma}_{a\#b}}"]
\arrow[d, hook, "i_{a\#b}"]
& \mathcal{S}_{\omega} (\mathbf{\Sigma}_{|a\#b}) \arrow[d, hook, "i_{a\#b}"] \\
\mathcal{S}_{+1}(\mathbf{\Sigma})
\arrow[r, hook, "j_{\mathbf{\Sigma}}"]
& \mathcal{S}_{\omega}(\mathbf{\Sigma})
\end{tikzcd}$$
Let $\Delta_{a\#b}$ the topological triangulation of $\mathbf{\Sigma}_{|a\#b}$ that is induced by $\Delta$. Let us consider the following diagram. $$\begin{tikzcd}
\mathcal{S}_{+1}(\mathbf{\Sigma}_{|a\#b}) \arrow[r, hook, "i_{a\#b}"] \arrow[d, hook, "j_{(\mathbf{\Sigma}_{|a\#b}, \Delta_{a\#b})}"]
\arrow[rr, bend left=20, "i^{\Delta_{a\#b}}"]
&
\mathcal{S}_{+1}(\mathbf{\Sigma}) \arrow[r, hook, "i^{\Delta}"] \arrow[d, hook, "j_{(\mathbf{\Sigma}, \Delta)}"] &
\otimes_{\mathbb{T}}\mathcal{S}_{+1}(\mathbb{T}) \arrow[d, hook, "\otimes_{\mathbb{T}}j_{\mathbb{T}}"] \\
\mathcal{S}_{\omega}(\mathbf{\Sigma}_{|a\#b}) \arrow[r, hook, "i_{a\#b}"]
\arrow[rr, bend right=20, "i^{\Delta_{a\#b}}"]
&
\mathcal{S}_{\omega}(\mathbf{\Sigma}) \arrow[r, hook, "i^{\Delta}"] &
\otimes_{\mathbb{T}}\mathcal{S}_{\omega}(\mathbb{T})
\end{tikzcd}$$ The outer triangles commute by coassociativity of the gluing maps. Two of the three squares commute by Diagram . Since $i^{\Delta}$ is injective, the remaining (left-hand side) square commutes.
We now prove that the morphism $j_{(\mathbf{\Sigma}, \Delta)}$ does not depend on $\Delta$. We first need a preliminary result.
\[lemma\_square\] Let $Q$ be a square (*i.e.* a disc with four punctures on its boundary) and $\Delta_Q$ a topological triangulation of $Q$. If $\alpha_{\varepsilon \varepsilon'} \in \mathcal{S}_{\omega}(Q)$ is the class of a stated arc, then $j_{(Q, \Delta_Q)}(\alpha_{\varepsilon \varepsilon'})= \alpha_{\varepsilon \varepsilon'}^N$. In particular, $j_{(Q,\Delta_Q)}$ does not depend on $\Delta_Q$.
Let $e$ be the inner edge of $\Delta_Q$ which is a common boundary arc of two triangles $\mathbb{T}_1$ and $\mathbb{T}_2$. Make the intersection $\alpha\cap e$ transversal and minimal via an isotopy on $\alpha$. If the intersection is empty, then $\alpha$ is included in one of the triangles and the lemma follows from Lemma \[lemma\_center\_triangle\]. If $\alpha \cap e$ is not empty, then it has only one element. Therefore, by letting $\alpha^{\mathbb{T}_i}:= \alpha \cap \mathbb{T}_i$ for $i=1,2$, one has $i^{\Delta_Q}(\alpha_{\varepsilon \varepsilon'}) = \alpha^{\mathbb{T}_1}_{\varepsilon +} \otimes \alpha^{\mathbb{T}_2}_{+ \varepsilon'} + \alpha^{\mathbb{T}_1}_{\varepsilon -} \otimes \alpha^{\mathbb{T}_2}_{- \varepsilon'}$. Write $x:= \alpha^{\mathbb{T}_1}_{\varepsilon +} \otimes \alpha^{\mathbb{T}_2}_{+ \varepsilon'}$ and $y:= \alpha^{\mathbb{T}_1}_{\varepsilon -} \otimes \alpha^{\mathbb{T}_2}_{- \varepsilon'}$ and remark that $xy=q^{-2}yx$. By Lemma \[lemma\_qbinomial\] one has $$i^{\Delta_Q}(\alpha_{\varepsilon \varepsilon'}^N)= i^{\Delta_Q}(\alpha_{\varepsilon \varepsilon'})^N = (x+y)^N = x^N +y^N = (j_{\mathbb{T}_1} \otimes j_{\mathbb{T}_2}) \circ i^{\Delta_Q} (\alpha_{\varepsilon \varepsilon'}).$$ Hence one has the equality $j_{(Q, \Delta_Q)}(\alpha_{\varepsilon \varepsilon'})= \alpha_{\varepsilon \varepsilon'}^N$.
\[lemma\_independance\] The morphism $j_{(\mathbf{\Sigma}, \Delta)}$ does not depend on $\Delta$.
Every two triangulations can be related by a finite sequence of flips on the edges. Therefore, it is enough to prove that if $\Delta'$ differs from $\Delta$ by a flip of one edge, then $j_{(\mathbf{\Sigma}, \Delta)}=j_{(\mathbf{\Sigma}, \Delta')}$.
Let $e$ be an inner edge of $\Delta$ that bounds two distinct faces $\mathbb{T}_1$ and $\mathbb{T}_2$. Consider the topological triangulation $\Delta'$ obtained from $\Delta$ by flipping the edge $e$ inside the square $Q=\mathbb{T}_1\cup \mathbb{T}_2$. Let $i: \mathcal{S}_{\omega}(\mathbf{\Sigma}) \hookrightarrow \mathcal{S}_{\omega}(\mathbf{\Sigma}\setminus Q) \otimes \mathcal{S}_{\omega}(Q)$ be the gluing morphism. By Lemma \[lemma\_square\], the morphism $j_Q : \mathcal{S}_{+1}(Q) \hookrightarrow \mathcal{S}_{\omega}(Q)$ does not depend on the triangulation of $Q$. Therefore, by Lemma \[lem: j commute with i\], both the morphisms $j_{(\mathbf{\Sigma}, \Delta)}$ and $j_{(\mathbf{\Sigma}, \Delta')}$ make the following diagram commutative: $$\begin{tikzcd}
\mathcal{S}_{+1}(\mathbf{\Sigma}) \arrow[r, hook, "i"] \arrow[d, hook, shift right=1.5ex, "j_{(\mathbf{\Sigma}, \Delta')}"'] \arrow[d, hook, "j_{(\mathbf{\Sigma}, \Delta)}"] &
\mathcal{S}_{+1}(\mathbf{\Sigma}\setminus Q)\otimes \mathcal{S}_{+1}(Q) \arrow[d, hook, "j_{(\mathbf{\Sigma}\setminus Q, \Delta_{\mathbf{\Sigma}\setminus Q})}\otimes j_Q"] \\
\mathcal{S}_{\omega}(\mathbf{\Sigma}) \arrow[r, hook, "i"] &
\mathcal{S}_{\omega}(\mathbf{\Sigma}\setminus Q)\otimes \mathcal{S}_{+1}(Q).
\end{tikzcd}$$ This proves that $j_{(\mathbf{\Sigma}, \Delta)}=j_{(\mathbf{\Sigma}, \Delta')}$ and concludes the proof.
#### Step 2: arcs with endpoints in distinct boundary arcs.
\[lemma\_baleze\] If $\alpha_{\varepsilon \varepsilon'}\in \mathcal{S}_{\omega}(\mathbf{\Sigma})$ is the class of a stated arc such that its endpoints lie on two different boundary arcs, then $j_{\mathbf{\Sigma}} (\alpha_{\varepsilon \varepsilon'}) = \alpha_{\varepsilon \varepsilon'}^N$.
In virtue of Diagram , it is enough to prove that $$\label{eq: coprod j is j coprod}
i^{\Delta}(\alpha_{\varepsilon\varepsilon'}^N)=(\otimes_{\mathbb{T}\in F(\Delta)}j_{\mathbb{T}}) i^{\Delta}(\alpha_{\varepsilon\varepsilon'}).$$
Without lost of generality, we suppose that the arc $\alpha$ is in minimal and transverse position with the edges of $\Delta$. Let $T$ be a (vertical framed) tangle of $\Sigma_{\mathcal{P}}\times (0,1)$ that projects on $\alpha$ and such that its height projection is an injective map. Note that for each $\mathbb{T} \in F(\Delta)$, the tangle $T_{\mathbb{T}}:=T \cap \mathbb{T}$ may have various connected components; since the height projection is injective, these components are ordered by height. Let $T^{(N)}$ be a tangle of $N$ parallel copies of $T$ obtained by stacking $N$ copies of $T$, but close enough to have the following property. For each $\mathbb{T} \in F(\Delta)$, if $T_1$ and $T_2$ are two connected components of $T_{\mathbb{T}}$ such that $T_1$ is below $T_2$, then, in $T^{(N)}_{\mathbb{T}}:=T^{(N)} \cap \mathbb{T}$, each copy of $T_1$ is below all the copies of $T_2$. See Figure \[figtriangle3d\] for an illustration. Note that since $\alpha$ is an arc with boundary points at two distinct boundary arcs, the tangle $T^{(N)}$ is a representative of the $N$-th product of $\alpha_{\varepsilon\varepsilon'}$ in $\mathcal{S}_{\omega}(\mathbf{\Sigma})$; otherwise it may not be true.
![Instance of tangles $T_{\mathbb{T}}$ and $T^{(N)}_{\mathbb{T}}$. []{data-label="figtriangle3d"}](triangle3d.eps){width="8cm"}
The left-hand term of can be described as the cutting of $T^{(N)}$ along each edge of the triangulation, and summing the result over all possible states at each edge. More formally, it is described as follows.
Let $K$ be a subset of edges of $\Delta$ that intersect $\alpha$. We let $\St_K(\alpha)$ be the set of maps $s: T \cap (K\times (0,1)) \rightarrow \{-,+\}$. Note that $\St_K(\alpha)= \sqcup_{e\in K} \St_{\{e\}}(\alpha)$, which allows us to write $s\in \St_K(\alpha)$ as $\sqcup s_e$. We will only consider the two sets $K$: the set $E$ of all the *internal* edges of $\Delta$ that intersect $\alpha$, and the set $K=\{e\}$ for an edge $e$.
For $s\in \St_E(\alpha)$, write $s^{(N)}:=(s, \ldots, s)\in \St_E(\alpha)^{\times N}$. We denote by $s_0$ the state of $\alpha_{\varepsilon\varepsilon'}$ (so, one has $\alpha_{\varepsilon\varepsilon'}=[T,s_0]$).
For $\mathbf{s}= (s_{1}, \ldots, s_{N})\in \St_E(\alpha)^{\times N}$, we let $$\alpha(\mathbf{s}) := \otimes_{\mathbb{T}\in F(\Delta)} [T^{(N)}_{\mathbb{T}}, ( \mathbf{s}\sqcup s_0^{(N)})_{|\partial\mathbb{T}}] \in \otimes_{\mathbb{T}\in F(\Delta)}\mathcal{S}_{\omega}(\mathbb{T}),$$ where we associate, to the $k$-th copy of $T^{(N)}_{\mathbb{T}}$, the restriction of the state $s_{k}$. With this notation, the left-hand term of can be written as $$\label{eq:j1}
i^{\Delta}(\alpha_{\varepsilon \varepsilon'}^N) = \sum_{\mathbf{s}\in \St_E(\alpha)^{\times N}} \alpha(\mathbf{s}).$$
Now, let us describe the right-hand term of . Note that the construction of $T^{(N)}$ ensures that, for each triangle $\mathbb{T}$ and each state $s$ of $T_{\mathbb{T}}$, one has $j_{\mathbb{T}}([T_{\mathbb{T}},s])=[T^{(N)}_{\mathbb{T}},s^{(N)}]$. Therefore, one has $$\label{eq:j2}
\left( \otimes_{\mathbb{T}\in F(\Delta)} j_{\mathbb{T}} \right) i^{\Delta} (\alpha_{\varepsilon \varepsilon'}) = \sum_{s\in \St_E(\alpha)} \alpha(s^{(N)}).$$
Let $Y$ be the set of non-diagonal states $\St_E(\alpha)^{\times N}\setminus \{(s,...,s)| s\in \St_E(\alpha)\}$. The sum in and in differ by the sum of $\alpha(\mathbf{s})$ for $\mathbf{s}\in Y$.
Let us fix an edge $e$ of $E$ and let us split $Y$ into $J\sqcup Y_e$ where $Y_e$ is the set of $N$-tuples of states at $e$, that is, $Y_e=\{\mathbf{s}\in Y~|~ \mathbf{s}: T^{(N)} \cap (e\times (0,1))\to \{-,+\}\}$. Therefore, showing amounts to showing that $$\sum_{\mathbf{s'}\in J} \sum_{\mathbf{s}\in Y_e} \alpha(\mathbf{s}'\sqcup \mathbf{s})=0.$$ In fact, let us show that, for each $\mathbf{s}'\in J$, one has $\sum_{\mathbf{s}\in Y_e} \alpha(\mathbf{s}'\sqcup \mathbf{s})=0$.
Let $\mathbb{T}_1$ and $\mathbb{T}_2$ be the two triangles adjoining $e$ and let $Q\subset \Sigma_{\mathcal{P}}$ be the resulting square. Denote by $i_Q: \mathcal{S}_{\omega}(Q)\hookrightarrow \otimes_{\mathbb{T}\in F(\Delta)} \mathcal{S}_{\omega}(\mathbb{T})$ the corresponding embedding and write $T_Q:= T\cap (Q\times (0,1))$ . For each $\mathbf{s}'\in J$, one has $$\begin{gathered}
\sum_{\mathbf{s}\in Y_{e}} \alpha(\mathbf{s}'\sqcup \mathbf{s}) = \left( \otimes_{\mathbb{T}\neq \mathbb{T}_1, \mathbb{T}_2} [T^{(N)}_{\mathbb{T}}, \mathbf{s}'_{|\partial \mathbb{T}} ]\right) \\
\otimes \left( i_Q([T_Q, \mathbf{s}'_{|\partial Q} ]) - (j_{\mathbb{T}_1} \otimes j_{\mathbb{T}_2}) \circ i_Q ([T_Q, \mathbf{s}'_{|Q}])\right).\end{gathered}$$ The last term is zero by Lemma \[lemma\_square\]. This concludes the proof.
#### Step 3: closed curves that intersect $\Delta$ nicely.
The $N$-th Chebyshev polynomial of first kind is the polynomial $T_N(X) \in \mathbb{Z}[X]$ defined by the recursive formulas $T_0(X)=2$, $T_1(X)=X$ and $T_{n+2}(X)=XT_{n+1}(X) -T_n(X)$ for $n\geq 0$.
The following proposition is at the heart of (our proof of) the so-called “miraculous cancelations” from [@BonahonWong1]. We postpone its proof to the Appendix A.
\[proptchebychev\] If $\omega$ is a root of unity of odd order $N\geq 1$, then in $\mathcal{S}_{\omega}(\mathbb{B})$, the following equality holds: $$T_N(\alpha_{++}+\alpha_{--}) = \alpha_{++}^N + \alpha_{--}^N.$$
\[lemma\_curve\] Let $\gamma \in \mathcal{S}_{\omega}(\mathbf{\Sigma})$ be the class of a closed curve. If the closed curve can be chosen such that it intersects an edge of $\Delta$ once and only once, then $j_{\mathbf{\Sigma}} (\gamma) = T_N(\gamma)$.
Consider the punctured surface $\mathbf{\Sigma}(e)$ obtained from $\mathbf{\Sigma}$ by replacing $e$ by two arcs $e'$ and $e''$ parallel to $e$ with the same endpoints and removing the bigone between $e'$ and $e''$. Consider the injective morphism $i_{|e'\#e''} : \mathcal{S}_{\omega}(\mathbf{\Sigma}) \hookrightarrow \mathcal{S}_{\omega}(\mathbf{\Sigma}(e))$. By Lemma \[lem: j commute with i\], the following diagram commutes: $$\begin{tikzcd}
\mathcal{S}_{+1}(\mathbf{\Sigma}) \arrow[r, hook, "j_{\mathbf{\Sigma}}"] \arrow[d, hook, "i_{|e'\#e''}"] & \mathcal{S}_{\omega}(\mathbf{\Sigma}) \arrow[d, hook, "i_{|e'\#e''}"] \\
\mathcal{S}_{+1}(\mathbf{\Sigma}(e)) \arrow[r, hook, "j_{\mathbf{\Sigma}(e)}"] & \mathcal{S}_{\omega}(\mathbf{\Sigma}(e))
\end{tikzcd}$$ By cutting $\gamma$ along $e$, we get an arc $\beta\subset \Sigma(e)$ such that $i_{e'\#e''}(\gamma)=\beta_{++}+\beta_{--}$. Consider the algebra morphism $\varphi : \mathcal{S}_{\omega}(\mathbb{B}) \rightarrow \mathcal{S}_{\omega}(\mathbf{\Sigma}(e))$ sending $\alpha_{\varepsilon \varepsilon'}$ to $\beta_{\varepsilon \varepsilon'}$. One has: $$\begin{aligned}
j_{\mathbf{\Sigma}(e)} \circ i_{|e'\#e''} (\gamma) &= j_{\mathbf{\Sigma}(e)} (\beta_{++} + \beta_{--})& \\
&= \varphi(\alpha_{++}^N + \alpha_{--}^N) &\mbox{ by Lemma \ref{lemma_baleze}} \\
&= \varphi(T_N(\alpha_{++} + \alpha_{--})) &\mbox{ by Proposition \ref{proptchebychev}} \\
&= i_{|e'\#e''} (T_N(\gamma)). &
\end{aligned}$$ Hence one has $j_{\mathbf{\Sigma}}(\gamma) = T_N(\gamma)$.
#### Step 4: adding a puncture.
Let $\mathbf{\Sigma}'=(\Sigma, \mathcal{P}\cup \{p\})$ be a punctured surface obtained from $\mathbf{\Sigma}=(\Sigma, \mathcal{P})$ by adding one puncture $p\in \Sigma_{\mathcal{P}}$. The inclusion map $ \Sigma_{\mathcal{P}\cup \{p\}} \times (0,1) \hookrightarrow \Sigma_{\mathcal{P}}\times (0,1)$ induces a morphism of algebras $\pi : \mathcal{S}_{\omega}(\mathbf{\Sigma}') \rightarrow \mathcal{S}_{\omega}(\mathbf{\Sigma})$ which is surjective.
\[prop\_add\_puncture\] The following diagram is commutative. $$\begin{tikzcd}
\mathcal{S}_{+1}(\mathbf{\Sigma}') \arrow[r, hook, "j_{\mathbf{\Sigma}'}"] \arrow[d, two heads, "\pi"] &
\mathcal{S}_{\omega}(\mathbf{\Sigma}') \arrow[d, two heads, "\pi"] \\
\mathcal{S}_{\omega}(\mathbf{\Sigma}) \arrow[r, hook, "j_{\mathbf{\Sigma}}"] &
\mathcal{S}_{\omega}(\mathbf{\Sigma})
\end{tikzcd}$$
**Case 1:** Suppose $\mathbf{\Sigma}= \mathbb{T}$ is the triangle and $p$ is an inner puncture. The skein algebra $\mathcal{S}_{\omega}(\mathbb{T}')$ is generated by the classes of the stated arcs $\alpha'_{\varepsilon \varepsilon'}, \beta'_{\varepsilon \varepsilon'}, \gamma'_{\varepsilon \varepsilon'}$ and by the peripheral closed curve $\gamma_p$ encircling $p$ as shown in the left part of Figure \[fig\_add\_puncture\]. One has $\pi(\delta'_{\varepsilon \varepsilon'})= \delta_{\varepsilon \varepsilon'}$ for $\delta\in \{\alpha, \beta, \gamma\}, \varepsilon, \varepsilon'\in \{-,+\}$ and $\pi(\gamma_p)= -q-q^{-1}$. By Lemma \[lemma\_baleze\], for $\delta\in \{\alpha, \beta, \gamma\}$ one has $j_{\mathbb{T}'}(\delta'_{\varepsilon \varepsilon'})= (\delta'_{\varepsilon \varepsilon'})^N$, hence $\pi \circ j_{\mathbb{T}'} (\delta'_{\varepsilon \varepsilon'}) = (\delta_{\varepsilon \varepsilon'})^N= j_{\mathbb{T}}\circ \pi (\delta'_{\varepsilon \varepsilon})$. By Lemma \[lemma\_curve\], one has $j_{\mathbb{T}'}(\gamma_p)= T_N(\gamma_p)$, hence $\pi\circ j_{\mathbb{T}'}(\gamma_p)= T_N(-q-q^{-1})=-2= j_{\mathbb{T}}\circ \pi(\gamma_p)$. Therefore, one has $\pi\circ j_{\mathbb{T}'} = j_{\mathbb{T}}\circ \pi$.
![A triangle to which we add a puncture on its interior (on the left) and on its boundary (on the right).[]{data-label="fig_add_puncture"}](triangle_add_puncture.eps){width="4cm"}
**Case 2:** Suppose $\mathbf{\Sigma}= \mathbb{T}$ is the triangle and $p$ is a puncture on the boundary. The skein algebra $\mathcal{S}_{\omega}(\mathbb{T}')$ is generated by the classes of the stated arcs $\alpha'_{\varepsilon \varepsilon'}, \beta'_{\varepsilon \varepsilon'}, \gamma'_{\varepsilon \varepsilon'}$ and $\eta_{\varepsilon, \varepsilon'}$ as shown in the right part of Figure \[fig\_add\_puncture\]. One has $\pi(\delta'_{\varepsilon \varepsilon'})= \delta_{\varepsilon \varepsilon'}$ for $\delta\in \{\alpha, \beta, \gamma\}, \varepsilon, \varepsilon'\in \{-,+\}$ and $\pi(\eta_{\varepsilon \varepsilon'})= -A^{3}C_{\varepsilon}^{\varepsilon'}$ by [@LeStatedSkein Lemma $2.3$]. By Lemma \[lemma\_baleze\], for $\delta\in \{\alpha, \beta, \gamma\}$ one has $\pi \circ j_{\mathbb{T}'} (\delta'_{\varepsilon \varepsilon'}) = (\delta_{\varepsilon \varepsilon'})^N= j_{\mathbb{T}}\circ \pi (\delta'_{\varepsilon \varepsilon})$. Remark that $( -A^{3}C_{\varepsilon}^{\varepsilon'}(\omega) )^N = -C_{\varepsilon}^{\varepsilon'}(+1)$. By lemma \[lemma\_baleze\], one has $\pi \circ j_{\mathbb{T}'} (\eta_{\varepsilon \varepsilon'}) =( -A^{3}C_{\varepsilon}^{\varepsilon'}(\omega) )^N = j_{\mathbb{T}}\circ \pi (\eta_{\varepsilon \varepsilon})$. Therefore, one has $\pi\circ j_{\mathbb{T}'} = j_{\mathbb{T}}\circ \pi$.
**Case 3:** Let $\mathbb{T}_0\in F(\Delta)$ be a triangle such that, either $p$ is in the interior of $\mathbb{T}_0$ if $p$ is an inner puncture, or $p$ lies in the boundary of $\mathbb{T}_0$ if $p$ is in the boundary of $\Sigma_{\mathcal{P}}$. Consider the gluing maps $i: \mathcal{S}_{\omega}(\mathbf{\Sigma}) \hookrightarrow \mathcal{S}_{\omega}(\mathbb{T}_0)\otimes \mathcal{S}_{\omega}(\mathbf{\Sigma}\setminus \mathbb{T}_0)$ and $i': \mathcal{S}_{\omega}(\mathbf{\Sigma}') \hookrightarrow \mathcal{S}_{\omega}(\mathbb{T}'_0)\otimes \mathcal{S}_{\omega}(\mathbf{\Sigma}\setminus \mathbb{T}_0)$. One has the following diagram: $$\begin{tikzpicture}[>=stealth,thick,draw=black!70, arrow/.style={->,shorten >=1pt}, point/.style={coordinate}, pointille/.style={draw=red, top color=white, bottom color=red},scale=1,baseline={([yshift=-.5ex]current bounding box.center)}]
\matrix[row sep=6mm,column sep=6mm,ampersand replacement=\&]
{
\node (00) {$\mathcal{S}_{+1}(\mathbb{T}'_0)\otimes \mathcal{S}_{+1}(\mathbf{\Sigma}\setminus \mathbb{T}_0)$} ; \& \node (10){} ; \& \node (20) {} ; \& \node (30){$\mathcal{S}_{\omega}(\mathbb{T}'_0) \otimes \mathcal{S}_{\omega}(\mathbf{\Sigma}\setminus \mathbb{T}_0)$} ; \\
\node (01) {} ; \& \node (11){$\mathcal{S}_{+1}(\mathbf{\Sigma}')$} ; \& \node (21) {$\mathcal{S}_{\omega}(\mathbf{\Sigma}')$} ; \& \node (31){} ; \\
\node (02) {} ; \& \node (12){$\mathcal{S}_{+1}(\mathbf{\Sigma})$} ; \& \node (22) {$\mathcal{S}_{\omega}(\mathbf{\Sigma})$} ; \& \node (32){} ; \\
\node (03) {$\mathcal{S}_{+1}(\mathbb{T}_0)\otimes \mathcal{S}_{+1}(\mathbf{\Sigma}\setminus \mathbb{T}_0)$} ; \& \node (13){} ; \& \node (23) {} ; \& \node (33){$\mathcal{S}_{\omega}(\mathbb{T}_0) \otimes \mathcal{S}_{\omega}(\mathbf{\Sigma}\setminus \mathbb{T}_0)$.} ; \\
};
\path
(11) edge[above right,right hook-latex] node {\scriptsize{$i'$}} (00)
(21) edge[above left,right hook-latex] node {\scriptsize{$i'$}} (30)
(12) edge[below right,right hook-latex] node {\scriptsize{$i$}} (03)
(22) edge[below left,right hook-latex] node {\scriptsize{$i$}} (33)
(11) edge[above,right hook-latex] node {\scriptsize{$j_{\mathbf{\Sigma}'}$}} (21)
(11) edge[left,,->>] node {\scriptsize{$\pi$}} (12)
(21) edge[right,,->>] node {\scriptsize{$\pi$}} (22)
(12) edge[below,right hook-latex] node {\scriptsize{$j_{\mathbf{\Sigma}}$}} (22)
(00) edge[above,right hook-latex] node {\scriptsize{$j_{\mathbb{T}'_0}\otimes j_{\mathbf{\Sigma}\setminus \mathbb{T}_0}$}} (30)
(00) edge[right,->>] node {\scriptsize{$\pi_{\mathbb{T}_0}\otimes \id$}} (03)
(30) edge[left,,->>] node {\scriptsize{$\pi_{\mathbb{T}_0}\otimes \id$}} (33)
(03) edge[below,right hook-latex] node {\scriptsize{$j_{\mathbb{T}_0}\otimes j_{\mathbf{\Sigma}\setminus \mathbb{T}_0}$}} (33)
;
\end{tikzpicture}$$ Let us show that the inner middle square commutes. By Case 1 and 2, the big outer square commutes. By definition of $\pi, i$ and $i'$, both the left-hand side and the right-hand side squares commute. By Lemma \[lem: j commute with i\], both the top and the bottom squares commute. Because $i$ and $i'$ are injective, these facts make the inner middle square commutative.
\[lemma\_T1\] If $\alpha_{\varepsilon \varepsilon'}\in \mathcal{S}_{\omega}(\mathbf{\Sigma})$ is the class of a stated arc such that its endpoints lie on the same boundary arcs, then $j_{\mathbf{\Sigma}} (\alpha_{\varepsilon \varepsilon'}) = \alpha_{\varepsilon \varepsilon'}^N$.
Since the two endpoints of $\alpha$ lie on the same boundary component, we can pick a puncture $p$ on this boundary that lies between these two endpoints. Denote by $\mathbf{\Sigma}'=(\Sigma, \mathcal{P}\cup \{p\})$ the punctured surface obtained by adding this puncture. Denote by $\alpha' \subset \Sigma_{\mathcal{P}\cup \{p\}}$ the arc induced by $\alpha$ (so one has $\pi(\alpha'_{\varepsilon \varepsilon'})=\alpha_{\varepsilon \varepsilon'}$). By Lemma \[lemma\_baleze\], one has $j_{\mathbf{\Sigma}'}(\alpha'_{\varepsilon \varepsilon'})= (\alpha'_{\varepsilon \varepsilon'})^N$ hence $\pi \circ j_{\mathbf{\Sigma}'} (\alpha'_{\varepsilon \varepsilon'})= \alpha_{\varepsilon \varepsilon'}^N$. Proposition \[prop\_add\_puncture\] implies that $j_{\mathbf{\Sigma}}(\alpha_{\varepsilon \varepsilon'})= \alpha_{\varepsilon \varepsilon'}^N$.
\[lemma\_T2\] If $\gamma \in \mathcal{S}_{\omega}(\mathbf{\Sigma})$ is the class of a closed curve, then $j_{\mathbf{\Sigma}} (\gamma) = T_N(\gamma)$.
If the closed curve can be chosen such that it intersects an edge of $\Delta$ once and only once, then this is Lemma \[lemma\_curve\]. Otherwise, we can refine the triangulation by adding an inner puncture in order to have this property. Denote by $\mathbf{\Sigma}'$ the resulting punctured surface and let $\gamma'\in \mathcal{S}_{+1}(\mathbf{\Sigma}')$ be such that $\pi(\gamma')=\gamma$. Lemma \[lemma\_curve\] implies that $j_{\mathbf{\Sigma}'}(\gamma')=T_N(\gamma')$ and Proposition \[prop\_add\_puncture\] implies that $j_{\mathbf{\Sigma}}(\gamma)=T_N(\gamma)$.
#### Step 5: exceptional cases.
We treat the four connected punctured surfaces that do not have a non-degenerated topological triangulation.
The disc with only one puncture (on its boundary) and the sphere with only one puncture have both trivial skein algebra, while the sphere with two punctures has a commutative skein algebra. Therefore, Theorem \[theorem2\] holds trivially for them.\
Let $\mathbb{D}$ be a disc with one puncture on its boundary and one inner puncture and let $\Delta$ be its unique topological triangulation drawn in Figure \[fig\_degenerate\]. Let $\delta$ be the unique non trivial arc of $\mathbb{D}$ and $\eta$ be the peripheral curve encircling the inner puncture.
\[lemma\_degenerate\] For any $\varepsilon, \varepsilon' \in \{-,+\}$, one has $j_{\mathbb{D}}(\delta_{\varepsilon \varepsilon'})= \delta_{\varepsilon \varepsilon'}^N$ and $j_{\mathbb{D}}(\eta)=T_N(\eta)$.
![The punctured surface $\mathbb{D}$ obtained from the triangle by gluing two boundary arcs. []{data-label="fig_degenerate"}](triangle_degenerate.eps){width="6cm"}
The punctured surface $\mathbb{D}$ is obtained from the triangle $\mathbb{T}$ by gluing two boundary arcs $a$ and $b$. Using the notations of Figure \[fig\_degenerate\], one has $i_{a\#b}(\delta_{\varepsilon \varepsilon'})= \alpha_{+\varepsilon'}\beta_{\varepsilon +} + \alpha_{- \varepsilon'}\beta_{\varepsilon -}$ and $i_{a\#b}(\eta) = \gamma_{++} + \gamma_{--}$. First, using Proposition \[proptchebychev\], one has $$j_{\mathbb{T}}(i_{a\#b}(\eta)) = \gamma_{++}^N +\gamma_{--}^N = T_N(\gamma_{++} + \gamma_{--}) = i_{a\#b}(T_N(\eta)).$$ This implies $j_{\mathbb{D}}(\eta)=T_N(\eta)$. Next write $x:= \alpha_{+\varepsilon'}\beta_{\varepsilon +} $ and $y:=\alpha_{- \varepsilon'}\beta_{\varepsilon -}$. Using the height exchange moves one has $ \alpha_{+\varepsilon'}\beta_{\varepsilon +} = \omega^{-2} \beta_{\varepsilon +} \alpha_{+\varepsilon'}$, hence $x^N = \alpha_{+\varepsilon'}^N\beta_{\varepsilon +}^N$ and similarly $y^N=\alpha_{- \varepsilon'}^N\beta_{\varepsilon -}^N$. Using four times, one obtains $xy=\omega^{8}yx$ hence $(x+y)^N=x^N+y^N$ by Lemma \[lemma\_qbinomial\]. Therefore, one has $$j_{\mathbb{T}}(i_{a\#b}(\delta_{\varepsilon \varepsilon'})) = x^N +y^N = (x+y)^N = i_{a\#b}( \delta_{\varepsilon \varepsilon'}^N).$$ This implies $j_{\mathbb{D}}(\delta_{\varepsilon \varepsilon'})= \delta_{\varepsilon \varepsilon'}^N$.
A Poisson bracket on $\mathcal{S}_{+1}(\mathbf{\Sigma})$
--------------------------------------------------------
In this section, we define and make explicit a Poisson structure on $\mathcal{S}_{+1}(\mathbf{\Sigma})$.
### Preliminaries
We briefly recall some general facts concerning deformation quantization.
Let $\mathcal{A}$ be a complex commutative unital algebra, $\mathbb{C}[[\hbar]]$ be the ring of formal series in a parameter $\hbar$ and $\mathcal{A}[[\hbar]]:=\mathcal{A}\otimes_{\mathbb{C}}\mathbb{C}[[\hbar]]$. A *star product* $\star$ on $\mathcal{A}$ is an associative product on $\mathcal{A}[[\hbar]]$ such that if $f=\sum_i f_i \hbar^i $ and $g=\sum_i g_i \hbar^i$ are elements of $\mathcal{A}[[h]]$, then: $$f\star g = f_0g_0 \mod{\hbar},$$ where $f_0g_0$ denotes the product of $f_0$ and $g_0$ in $\mathcal{A}$. A star product induces a Poisson structure on $\mathcal{A}$ by the formula: $$\label{eq_poisson} f\star g -g\star f = \hbar \{ f, g \} \mod{\hbar^2},$$ for all $f,g\in \mathcal{A}$. The algebra $(\mathcal{A}[[\hbar]], \star)$ is called a *deformation quantization* of the commutative Poisson algebra $(\mathcal{A}, \{\cdot, \cdot \})$. We refer to ([@KontsevichQuantizationPoisson], [@GRS_QuantizationDeformation] $II.2$) for detailed discussions. A *morphism of star products* between $(\mathcal{A}, \star_{\mathcal{A}})$ and $(\mathcal{B}, \star_{\mathcal{B}})$ is an algebra morphism $\psi : \mathcal{A}[[\hbar]] \rightarrow \mathcal{B}[[\hbar]]$ whose restriction to $\mathcal{A}\subset \mathcal{A}[[\hbar]]$ induces a morphism $\phi : \mathcal{A} \rightarrow \mathcal{B}$. Note that such a $\phi$ is, in fact, a morphism of Poisson algebras for the induced Poisson algebra structures. An isomorphism $\psi : (\mathcal{A}[[\hbar]], \star_1) \xrightarrow{\cong} (\mathcal{A}[[\hbar]], \star_2)$ of star products is called a *gauge equivalence* if $\psi(f)= f \pmod{\hbar}$. If two star products are gauge equivalent, they induce the same Poisson bracket on $\mathcal{A}$.
To end this preamble, let us mention that deformation quantization is well-behaved relatively to the tensor product. Indeed, if $\mathcal{A}[[h]]$ and $\mathcal{B}[[h]]$ are deformation quantizations of $\mathcal{A}$ and $\mathcal{B}$ respectively, then $\mathcal{A}[[h]]\otimes \mathcal{B}[[h]]\cong (\mathcal{A}\otimes \mathcal{B})[[h]]$ is a deformation quantization of $\mathcal{A}\otimes \mathcal{B}$. Note also that the Poisson structure on $\mathcal{A}\otimes \mathcal{B}$ given by is $$\label{eq: tens poiss quant}
\ \{ f\otimes g, f'\otimes g'\} = ff' \otimes \{g,g'\} + \{f,f'\}\otimes gg',$$ for $f,f'\in \mathcal{A}$ and $g,g' \in \mathcal{B}$.
### Formal definition
Let $\mathbf{\Sigma}$ be a punctured surface and $\mathfrak{o}$ an orientation of its boundary arc. Denote by $\mathcal{S}_{+1}(\mathbf{\Sigma})$ the stated skein algebra associated to the ring $\mathbb{C}$ with $\omega=+1$ and denote by $\mathcal{S}_{\omega_{\hbar}}(\mathbf{\Sigma})$ the stated skein algebra associated to the ring $\mathbb{C}[[\hbar]]$ with $\omega_{\hbar}:= \exp\left( -\hbar/4 \right)$. The convention is chosen so that $q=\exp(\hbar)$. Recall the basis $\mathcal{B}^{\mathfrak{o}}$ from Definition \[def\_basis\]. Since $\mathcal{B}^{\mathfrak{o}}$ is independent of $\omega$, one has a canonical isomorphism of $\mathbb{C}[[\hbar]]$-modules $$\label{eq: iso psi Bo}
\psi^{\mathfrak{o}} : \mathcal{S}_{+1}(\mathbf{\Sigma})[[\hbar]] \xrightarrow{\cong} \mathcal{S}_{\omega_{\hbar}}(\mathbf{\Sigma}).$$
\[de:poiss brack\] Pulling-back the product of $\mathcal{S}_{\omega_{\hbar}}(\mathbf{\Sigma})$ along $\psi^{\mathfrak{o}}$ gives a star product $\star_{\hbar}$ on $\mathcal{S}_{+1}(\mathbf{\Sigma})$. We denote by $\{\cdot, \cdot\}^s$ the resulting Poisson bracket on $\mathcal{S}_{+1}(\mathbf{\Sigma})$ given by Equation .
Note that for any two orientations $\mathfrak{o}_1$ and $\mathfrak{o}_2$ of the boundary arcs of $\mathbf{\Sigma}$, the automorphism $(\psi^{\mathfrak{o}_2})^{-1}\circ \psi^{\mathfrak{o}_1} : \mathcal{S}_{+1}(\mathbf{\Sigma})[[\hbar]] \xrightarrow{\cong} \mathcal{S}_{+1}(\mathbf{\Sigma})[[\hbar]]$ is a gauge equivalence, hence the Poisson bracket $\{\cdot, \cdot \}^s$ does not depend on $\mathfrak{o}$.
By definition, $(\mathcal{S}_{+1}(\mathbf{\Sigma})[[\hbar]], \star_{\hbar})$ is a quantization deformation of the Poisson algebra $(\mathcal{S}_{+1}(\mathbf{\Sigma}), \{\cdot, \cdot\}^s)$. Moreover, this structure of Poisson algebra is compatible with decompositions of surfaces. More precisely, one has the following.
\[lemma: poisson morph at +1 skein\] The gluing maps $i_{|a\#b} : \mathcal{S}_{+1}(\mathbf{\Sigma}_{|a\#b})\hookrightarrow \mathcal{S}_{+1}(\mathbf{\Sigma})$, the maps $i^{\Delta}: \mathcal{S}_{+1}(\mathbf{\Sigma}) \hookrightarrow \otimes_{\mathbb{T}\in F(\Delta)}\mathcal{S}_{+1}(\mathbb{T})$ and the coproduct maps $\Delta^L, \Delta^R$ are Poisson morphisms.
This results from the fact that each of these morphisms arises from a morphism of star products.
### Explicit formula {#sec: explicit formula of bracket}
This section is devoted to making explicit the Poisson bracket $\{\cdot, \cdot\}^s$ on stated diagrams. It will be expressed in terms of *resolutions* of stated diagrams, which are defined at crossings and at points on the boundary arcs.
All along this section, $\mathbf{\Sigma}$ is a punctured surface.
#### Resolution at a crossing.
Let $(D,s)$ be a stated diagram and $c$ a crossing of $D$. Denote by $D_+$ and $D_-$ the diagrams obtained from $D$ by replacing the crossing $c$ by its positive and negative resolution respectively: $$\text{ the crossing $c$ }
\begin{tikzpicture}[baseline=-0.4ex,scale=0.5,>=stealth]
\draw [fill=gray!45,gray!45] (-.6,-.6) rectangle (.6,.6) ;
\draw[line width=1.2,-] (-0.4,-0.52) -- (.4,.53);
\draw[line width=1.2,-] (0.4,-0.52) -- (0.1,-0.12);
\draw[line width=1.2,-] (-0.1,0.12) -- (-.4,.53);
\end{tikzpicture}
\text{ and its positive }
\begin{tikzpicture}[baseline=-0.4ex,scale=0.5,>=stealth]
\draw [fill=gray!45,gray!45] (-.6,-.6) rectangle (.6,.6) ;
\draw[line width=1.2] (-0.4,-0.52) ..controls +(.3,.5).. (-.4,.53);
\draw[line width=1.2] (0.4,-0.52) ..controls +(-.3,.5).. (.4,.53);
\end{tikzpicture}
\text{ and negative }
\begin{tikzpicture}[baseline=-0.4ex,scale=0.5,rotate=90]
\draw [fill=gray!45,gray!45] (-.6,-.6) rectangle (.6,.6) ;
\draw[line width=1.2] (-0.4,-0.52) ..controls +(.3,.5).. (-.4,.53);
\draw[line width=1.2] (0.4,-0.52) ..controls +(-.3,.5).. (.4,.53);
\end{tikzpicture}
\text{ resolution.}$$ The resolution of $(D,s)$ at the crossing $c$ is defined by $$\Res_c(D,s) := \left[D_+ , s\right] - \left[D_-, s\right] \in \mathcal{S}_{+1}(\mathbf{\Sigma}).$$
#### Resolution at boundary points.
Let $b_1,...,b_k$ be the boundary arcs of $\Sigma_{\mathcal{P}}$.
A *height order* on a stated diagram $(D,s)$ of $\Sigma_{\mathcal{P}}$ is a $k$–tuple $\mathfrak{o}=(\mathfrak{o}_1,...,\mathfrak{o}_k)$ of bijections of sets $\mathfrak{o}_i: \partial_{b_i} D \to \{1<...<n_{i}\}$.
Note that the product of symmetric groups $\mathbb{S}_{n_1} \times \cdots \times \mathbb{S}_{n_k}$ acts freely and transitively on the set of height orders by left composition.
To a height order $\mathfrak{o}$ on $(D,s)$ corresponds a stated tangle with same height order and which projects to $(D,s)$. Therefore, one can consider the class of $(D,s,\mathfrak{o})$ in $\mathcal{S}_{\omega}(\mathbf{\Sigma})$. If $\omega=+1$, the class $[D,s, \mathfrak{o}]\in \mathcal{S}_{+1}(\mathbf{\Sigma})$ is independent of $\mathfrak{o}$, and we denote it simply by $[D,s]$.\
Let us choose a boundary arc $b_i$ and suppose there are two points $p_1$ and $p_2$ of $\partial_{b_i}D$ such that $\mathfrak{o}_i(p_1)=\mathfrak{o}_i(p_2)+1$ (*i.e.* $p_1$ is the $\mathfrak{o}_i$–successor of $p_2$). Let $\widetilde{\mathfrak{o}}$ be the order on $b_i$ that is induced by the orientation of $\Sigma$. To alleviate notation, we write $p<_{\widetilde{\mathfrak{o}}}q$ for $\widetilde{\mathfrak{o}}(p)<\widetilde{\mathfrak{o}}(q)$.
Let $\tau \in \mathbb{S}_{n_i}$ be the transposition that exchanges the $\mathfrak{o}_i$ order of $p_1$ and $p_2$. The resolution of $(D,s)$ along $\tau$, denoted by $\Res_{\tau} (D,s, \mathfrak{o})\in \mathcal{S}_{+1}(\mathbf{\Sigma})$, is given by $$\Res_{\tau} (D,s, \mathfrak{o}) =
\begin{cases}
\frac{1}{2} [ D, s], & \text{if either } s(p_1)=s(p_2)\mbox{ and } p_1<_{\widetilde{\mathfrak{o}}} p_2, \\
& \mbox{or } (s(p_1), s(p_2))=(-,+)\mbox{ and } p_2<_{\widetilde{\mathfrak{o}}} p_1; \\
-\frac{1}{2} [D,s], & \mbox{if either } s(p_1)=s(p_2)\mbox{ and }p_2<_{\widetilde{\mathfrak{o}}} p_1, \\
& \mbox{or } (s(p_1), s(p_2))=(+,-)\mbox{ and } p_1<_{\widetilde{\mathfrak{o}}}p_2; \\
\frac{1}{2}[D,s] +[{D}, {\tau s}], & \mbox{if } (s(p_1), s(p_2))=(+,-)\mbox{ and }p_2<_{\widetilde{\mathfrak{o}}}p_1;
\\
-\frac{1}{2}[D,s] -[{D}, {\tau s}], & \mbox{if } (s(p_1), s(p_2))=(-,+)\mbox{ and }p_1<_{\widetilde{\mathfrak{o}}} p_2,
\end{cases}$$ where $\tau s$ is the state that differs from $s$ only by exchanging the states of $p_1$ and $p_2$.\
Let us extend the resolution to several points. For two transpositions $\sigma_1$ and $\sigma_2$ of $\mathfrak{o}$–consecutive points, let $$\label{eq: extension res}
\Res_{\sigma_1\circ\sigma_2}(D,s,\mathfrak{o}) = \Res_{\sigma_1}(D,s, \sigma_2\circ \mathfrak{o}) + \Res_{\sigma_2}(D,s,\mathfrak{o}).$$
For a permutation $\sigma \in \mathbb{S}_{n_{1}} \times \cdots \times \mathbb{S}_{n_{k}}$, the resolution $\Res_{\sigma}(D,s,\mathfrak{o})$ is defined via , by considering the decomposition of $\sigma$ into transpositions of $\mathfrak{o}$–consecutive points.
The resolution $\Res_{\sigma}(D,s,\mathfrak{o})$ is not invariant under isotopy of $(D,s)$. Also, one has $\Res_{id}(D,s,\mathfrak{o}) =0$.
\[lemmabracket\] In the skein algebra $\mathcal{S}_{\omega_{\hbar}}(\mathbf{\Sigma})$, the following two statements holds.
1. Let $D\crosspos$ and $D\crossneg$ be two diagrams that differ from each other only by a change of a crossing $c$. One has $$[D\crosspos, s, \mathfrak{o}]-[D\crossneg, s, \mathfrak{o}] = \hbar \Res_c(D\crosspos, s) \mod{\hbar^2}.$$
2. Let $(D,s,\mathfrak{o})$ be an $\mathfrak{o}$-ordered stated diagram. For $\pi\in \mathbb{S}_{n_1}\times \ldots \times \mathbb{S}_{n_k}$ one has $$[D, s , \mathfrak{o}] - [D,s,\pi\circ\mathfrak{o}] = \hbar\Res_{\pi}(D,s,\mathfrak{o}) \mod{\hbar^2}.$$
In the two statements, the resolutions $\Res$ are seen in $\mathcal{S}_{\omega_{\hbar}}(\mathbf{\Sigma})$ via the isomorphism $\psi^{\widetilde{\mathfrak{o}}}$ of .
The first equality is a consequence of the Kauffman bracket crossing relation whereas the second is a consequence of the height exchange formulas. Let us give a few details on the less straightforward case. Remark that is equivalent to $$\label{eq: height exch equivalent}
{
\begin{tikzpicture}[baseline=-0.4ex,scale=0.5, >=stealth]
\draw [fill=gray!60,gray!45] (-.7,-.75) rectangle (.4,.75) ;
\draw[<-] (0.4,-0.75) to (.4,.75);
\draw[line width=1.2] (0.4,-0.3) to (-.7,-.3);
\draw[line width=1.2] (0.4,0.3) to (-.7,.3);
\draw (0.65,0.3) node {\scriptsize{$-$}};
\draw (0.65,-0.3) node {\scriptsize{$+$}};
\end{tikzpicture}
}
= \omega^{-2}
{
\begin{tikzpicture}[baseline=-0.4ex,scale=0.5, >=stealth]
\draw [fill=gray!60,gray!45] (-.7,-.75) rectangle (.4,.75) ;
\draw[->] (0.4,-0.75) to (.4,.75);
\draw[line width=1.2] (0.4,-0.3) to (-.7,-.3);
\draw[line width=1.2] (0.4,0.3) to (-.7,.3);
\draw (0.65,0.3) node {\scriptsize{$-$}};
\draw (0.65,-0.3) node {\scriptsize{$+$}};
\end{tikzpicture}
}
+(\omega^{2}-\omega^{-6})
{
\begin{tikzpicture}[baseline=-0.4ex,scale=0.5, >=stealth]
\draw [fill=gray!60,gray!45] (-.7,-.75) rectangle (.4,.75) ;
\draw[->] (0.4,-0.75) to (.4,.75);
\draw[line width=1.2] (0.4,-0.3) to (-.7,-.3);
\draw[line width=1.2] (0.4,0.3) to (-.7,.3);
\draw (0.65,0.3) node {\scriptsize{$+$}};
\draw (0.65,-0.3) node {\scriptsize{$-$}};
\end{tikzpicture}
};$$ by applying it to $\omega_{\hbar}=\exp(-\hbar/4)$, one obtains $${
\begin{tikzpicture}[baseline=-0.4ex,scale=0.5, >=stealth]
\draw [fill=gray!60,gray!45] (-.7,-.75) rectangle (.4,.75) ;
\draw[<-] (0.4,-0.75) to (.4,.75);
\draw[line width=1.2] (0.4,-0.3) to (-.7,-.3);
\draw[line width=1.2] (0.4,0.3) to (-.7,.3);
\draw (0.65,0.3) node {\scriptsize{$-$}};
\draw (0.65,-0.3) node {\scriptsize{$+$}};
\end{tikzpicture}
}
-
{
\begin{tikzpicture}[baseline=-0.4ex,scale=0.5, >=stealth]
\draw [fill=gray!60,gray!45] (-.7,-.75) rectangle (.4,.75) ;
\draw[->] (0.4,-0.75) to (.4,.75);
\draw[line width=1.2] (0.4,-0.3) to (-.7,-.3);
\draw[line width=1.2] (0.4,0.3) to (-.7,.3);
\draw (0.65,0.3) node {\scriptsize{$-$}};
\draw (0.65,-0.3) node {\scriptsize{$+$}};
\end{tikzpicture}
}
=\left( \frac{1}{2}
{
\begin{tikzpicture}[baseline=-0.4ex,scale=0.5, >=stealth]
\draw [fill=gray!60,gray!45] (-.7,-.75) rectangle (.4,.75) ;
\draw[->] (0.4,-0.75) to (.4,.75);
\draw[line width=1.2] (0.4,-0.3) to (-.7,-.3);
\draw[line width=1.2] (0.4,0.3) to (-.7,.3);
\draw (0.65,0.3) node {\scriptsize{$-$}};
\draw (0.65,-0.3) node {\scriptsize{$+$}};
\end{tikzpicture}
}
+
{
\begin{tikzpicture}[baseline=-0.4ex,scale=0.5, >=stealth]
\draw [fill=gray!60,gray!45] (-.7,-.75) rectangle (.4,.75) ;
\draw[->] (0.4,-0.75) to (.4,.75);
\draw[line width=1.2] (0.4,-0.3) to (-.7,-.3);
\draw[line width=1.2] (0.4,0.3) to (-.7,.3);
\draw (0.65,0.3) node {\scriptsize{$+$}};
\draw (0.65,-0.3) node {\scriptsize{$-$}};
\end{tikzpicture}
}
\right) \hbar \text{ mod } \hbar^2.$$ This provides the second formula for $\pi$ a transposition of two consecutive points $p_2<_{\widetilde{\mathfrak{o}}}p_1$ such that $ (s(p_1), s(p_2))=(+,-)$.
\[propbracket\] Let $\mathfrak{o}$ be an orientation of the boundary arcs of $\mathbf{\Sigma}$ and $(D_1, s_2, \mathfrak{o}_1)$ and $(D_2,s_2, \mathfrak{o}_2)$ be two $\mathfrak{o}$-ordered stated diagrams such that $D_1$ and $D_2$ intersect transversally in the interior of $\Sigma_{\mathcal{P}}$. Let $(D_1D_2, s_1s_2)$ be the stated diagram obtained by staking $D_1$ on top of $D_2$, $\mathfrak{o}_1\mathfrak{o}_2$ the resulting height order and $\pi$ the permutation sending $\mathfrak{o}_2\mathfrak{o}_1$ to $\mathfrak{o}_1\mathfrak{o}_2$. In $\mathcal{S}_{+1}(\mathbf{\Sigma})$, the Poisson bracket from Definition \[de:poiss brack\] satisfies $$\left\{ [D_1, s_1], [D_2, s_2]\right\}^s = \sum_{c\in D_1\cap D_2} \Res_c(D_1D_2, s_1s_2) + \Res_{\pi}(D_1D_2, s_1s_2, \mathfrak{o_1}\mathfrak{o_2}).$$
In the algebra $\mathcal{S}_{\omega_{\hbar}}(\mathbf{\Sigma})$, the products writes $[D_1, s_1, \mathfrak{o}_1] \cdot [D_2, s_2, \mathfrak{o}_2] = [D_1D_2, s_1s_2, \mathfrak{o}_1\mathfrak{o}_2]$ and $[D_2, s_2, \mathfrak{o}_2] \cdot [D_1, s_1, \mathfrak{o}_1] = [D_2D_1, s_2s_1, \mathfrak{o}_2\mathfrak{o}_1]$. We pass from the diagram $D_1D_2$ to $D_2D_1$ by changing each crossing in the intersection of the diagrams and changing the height order using $\pi$, so the formula is a consequence of Lemma \[lemmabracket\].
When $\Sigma$ is a closed surface, we recover Goldman’s formula from [@Goldman86]. When $\Sigma$ has non trivial boundary and no inner punctures, the sub-algebra of the stated skein algebra generated by tangles with states having only value $+$ is isomorphic to the Müller algebra defined in [@Muller] (see [@LeStatedSkein Section $6$] ). The Poisson bracket restricts to the corresponding sub-algebra of $\mathcal{S}_{+1}(\mathbf{\Sigma})$ and the resulting Poisson algebra is isomorphic to Yuasa’ s Poisson algebra in [@Yuasa].
\[exampleBigone\] The Poisson bracket $\{-,-\}^s$ on $\mathcal{S}_{+1}(\mathbb{B})$ is given by
$$\begin{aligned}
\{ \alpha_{++}, \alpha_{+-}\}^s &= -\alpha_{+-}\alpha_{++} & \{\alpha_{++}, \alpha_{-+}\}^s &= -\alpha_{-+}\alpha_{++}
\\ \{ \alpha_{--}, \alpha_{+-} \}^s &= \alpha_{+-}\alpha_{--} & \{\alpha_{--}, \alpha_{-+} \}^s &= \alpha_{-+}\alpha_{--}
\\ \{ \alpha_{+-}, \alpha_{-+} \}^s &= 0 & \{ \alpha_{++}, \alpha_{--}\}^s &= -2\alpha_{+-}\alpha_{-+}\end{aligned}$$
\[exampleTriangle\] For the triangle $\mathbb{T}$, the Poisson structure is described by the previous formulas in Example \[exampleBigone\] by replacing $\alpha$ by each of the three arcs $\alpha, \beta$ and $\gamma$, together with the following relations and their images through the automorphisms $\tau$ and $\tau^2$: $$\begin{aligned}
\{ \gamma_{\varepsilon \mu}, \alpha_{ \mu' \varepsilon}\}^s &= -\frac{1}{2} \gamma_{\varepsilon \mu} \alpha_{\mu' \varepsilon}
\\ \{ \gamma_{- \mu}, \alpha_{\mu' +} \}^s &= \frac{1}{2} \gamma_{- \mu}\alpha_{\mu' +}
\\ \{ \gamma_{+ \mu}, \alpha_{\mu' -}\}^s &= -\frac{3}{2} \gamma_{+ \mu}\alpha_{\mu' -} + 2 \beta_{\mu \mu'}.\end{aligned}$$
Character varieties {#sec 3}
===================
Character varieties for open surfaces
-------------------------------------
In this subsection we briefly recall from [@KojuTriangularCharVar] the definition and main properties of character varieties for open surfaces.
Recall that the character variety of a closed punctured connected surface $\mathbf{\Sigma}$ is the GIT quotient $$\mathcal{X}_{\SL_2}(\mathbf{\Sigma}):= \Hom\left( \pi_1(\Sigma_{\mathcal{P}}) , \SL_2(\mathbb{C}) \right) \sslash \SL_2(\mathbb{C})$$ under the action by conjugation of $\SL_2(\mathbb{C}) $. In [@Goldman86], Goldman defined a Poisson structure on its algebra of regular functions. It follows from [@Bullock; @PS00; @Barett; @Turaev91] that, given a spin structure $S$ on $\Sigma$ with quadratic form $w_S$, there is a Poisson isomorphism $$\phi^S : (\mathcal{S}_{+1}(\mathbf{\Sigma}), \{\cdot, \cdot \}^s) \xrightarrow{\cong} (\mathbb{C}[\mathcal{X}_{\SL_2}(\mathbf{\Sigma})], \{\cdot, \cdot \}).$$ For each non-contractible closed curve $\gamma$, it is given by $\phi^S(\gamma)= (-1)^{w_S([\gamma])} \tau_{\gamma} $, where $\tau_{\gamma}$ is the regular function $\tau_{\gamma}([\rho]):= \tr (\rho(\gamma))$.
In [@KojuTriangularCharVar], the first author introduced a generalization of the character varieties to punctured surfaces which are not necessarily closed. We will also denote it by $ \mathcal{X}_{\SL_2}(\mathbf{\Sigma})$. Loosely speaking, it can be described as an algebraic quotient of the following form $$\mathcal{X}_{\SL_2}(\mathbf{\Sigma}):= \Hom\left( \Pi_1(\Sigma_{\mathcal{P}},\partial\Sigma_{\mathcal{P}}) , \underline{\SL_2} \right) \sslash \mathcal{G},$$ where $\Hom\left( \Pi_1(\Sigma_{\mathcal{P}},\partial\Sigma_{\mathcal{P}}) , \underline{\SL_2} \right)$ denotes the set of functors from the relative fundamental groupoid to the one object category $\underline{\SL_2}$, and $\mathcal{G}$ is the group of their natural transformations; see [@KojuTriangularCharVar] for details.
The character variety turns out to be an affine Poisson variety (whose Poisson structure depends on a choice of orientation of the boundary arcs) and it is proved in [@KojuTriangularCharVar Theorem $1.1$] that its algebra of regular functions $ \mathbb{C}[\mathcal{X}_{\SL_2}(\mathbf{\Sigma})]$ is well-behaved under triangular decomposition: for a topological triangulation $\Delta$, there are an injective morphism $i^{\Delta} : \mathbb{C}[\mathcal{X}_{\SL_2}(\mathbf{\Sigma})] \hookrightarrow \otimes_{\mathbb{T}\in F(\Delta)} \mathbb{C}[\mathcal{X}_{\SL_2}(\mathbb{T})]$ and Hopf comodule maps $\Delta^L$ and $\Delta^R$ such that the following sequence is exact: $$\begin{gathered}
0 \to \mathbb{C}[\mathcal{X}_{\SL_2}(\mathbf{\Sigma})] \xrightarrow{i^{\Delta}} \otimes_{\mathbb{T}\in F(\Delta)} \mathbb{C}[\mathcal{X}_{\SL_2}(\mathbb{T})]
\\ \xrightarrow{\Delta^L - \sigma \circ \Delta^R} \left( \otimes_{e \in \mathring{\mathcal{E}}(\Delta)} \mathbb{C}[\SL_2] \right) \otimes \left( \otimes_{\mathbb{T}\in F(\Delta)} \mathbb{C}[\mathcal{X}_{\SL_2}(\mathbb{T})]\right).
\end{gathered}$$
In the present paper, we proceed by describing the character variety for the bigone and the triangle, together with the Hopf comodule maps $\Delta^L$ and $\Delta^R$. Then, in virtue of the above property, we define the character variety for any punctured surface as the kernel of $\Delta^L - \sigma \circ \Delta^R$.\
First, recall that $\mathfrak{sl}_2$ denotes the Lie algebra of the $2\times 2$ traceless matrices. It has a basis formed by $$H:=\begin{pmatrix} 1 & 0 \\ 0& -1 \end{pmatrix},
E:= \begin{pmatrix} 0 & 1 \\ 0&0 \end{pmatrix} \text{ and }
F:= \begin{pmatrix} 0&0 \\ 1 & 0 \end{pmatrix}.$$ In order to define the Poisson structure, we will need the following.
The *classical* r*-matrices* $r^{\pm}\in \mathfrak{sl}_2^{\otimes 2}$ are the bi-vectors $r^+ := \frac{1}{2} H\otimes H + 2 E\otimes F$ and $r^-:= \frac{1}{2} H\otimes H +2 F\otimes E$. Their symmetric part $\tau = \frac{1}{2} H\otimes H + E\otimes F +F \otimes E$ is the invariant bi-vector associated to the (suitably normalised) Killing form and we denote by $\overline{r}^+:= E\otimes F -F\otimes E=:-\overline{r}^-$ their skew-symmetric part.
The classical $r$-matrices satisfy the classical Yang-Baxter equation (see [@DrinfeldrMatrix], [@ChariPressley Section $2.1$] for details).
Given $a$ a boundary arc of $\mathbf{\Sigma}$, we write $\mathfrak{o}(a)=+$ if the $\mathfrak{o}$-orientation of $a$ coincides with the orientation induced by the orientation of $\Sigma_{\mathcal{P}}$ and write $\mathfrak{o}(a)=-$ if the orientation are opposite.
### The bigon
Consider the bigon $\mathbb{B}$ and write $\mathfrak{o}(b_L)=\varepsilon_1$ and $\mathfrak{o}(b_R)=\varepsilon_2$.
\[def\_bigone\_poisson\] The character variety of the bigon is $\mathcal{X}_{\SL_2}(\mathbb{B}):= \SL_2(\mathbb{C})$. Denote by $N= \begin{pmatrix} x_{++} & x_{+-} \\ x_{-+} & x_{--} \end{pmatrix}$ the $2\times 2$ matrix with coefficients in $\mathbb{C}[\mathcal{X}_{\SL_2}(\mathbb{B})]$. The Poisson bracket associated to $\mathfrak{o}$ is defined by: $$\left\{ N \otimes N \right\}^{\varepsilon_1, \varepsilon_2} := \overline{r}^{\varepsilon_1} (N \otimes N) + (N\otimes N)\overline{r}^{\varepsilon_2}.$$
Here we used the classical notation $\{N \otimes N\}$ to denote the matrix defined by $\{N\otimes N \}_{\varepsilon \varepsilon' \mu \mu'}= \{x_{\varepsilon \varepsilon'}, x_{\mu \mu'} \}$.
Denote the Poisson variety $(\mathbb{C}[\SL_2], \{\cdot, \cdot \}^{\varepsilon_1, \varepsilon_2})$ by $\mathbb{C}[\SL_2]^{\varepsilon_1, \varepsilon_2}$. Remark that $\{\cdot, \cdot\}^{\varepsilon_1, \varepsilon_2} = - \{\cdot, \cdot\}^{-\varepsilon_1, -\varepsilon_2}$. By [@KojuTriangularCharVar Lemma $4.1$], the coproduct $\Delta : \mathbb{C}[\SL_2]^{\varepsilon_1, \varepsilon_2} \rightarrow \mathbb{C}[\SL_2]^{\varepsilon_1, \varepsilon} \otimes \mathbb{C}[\SL_2]^{-\varepsilon, \varepsilon_2}$ and the antipode $S: \mathbb{C}[\SL_2]^{\varepsilon_1, \varepsilon_2} \rightarrow \mathbb{C}[\SL_2]^{-\varepsilon_1, -\varepsilon_2}$ are Poisson morphisms. In particular, the Poisson brackets $\{\cdot, \cdot \}^{-, +}$ and $\{\cdot, \cdot\}^{+,-}$ are the only ones which endow $\SL_2(\mathbb{C})$ with a Poisson-Lie structure.
### The triangle
Consider the triangle $\mathbb{T}$ and fix an orientation $\mathfrak{o}$ of each of its three boundary arcs $a,b$ and $c$. We will use the notation $s(\alpha)=t(\beta):=c$, $s(\gamma)=t(\alpha):=b$ and $s(\beta)=t(\gamma):=a$. Here, for instance, we think of $\alpha$ as an oriented path joining a point in $c=s(\alpha)$ (source) to a point in $b=t(\alpha)$ (target).
\[def\_triangle\_poisson\] The character variety of the triangle is the affine variety: $$\mathcal{X}_{\SL_2}(\mathbb{T}):= \left\{ (M_{\alpha}, M_{\beta}, M_{\gamma}) \in \SL_2(\mathbb{C})^3 | M_{\gamma}M_{\beta}M_{\alpha}=\mathds{1} \right\}$$
Given $\delta \in \{ \alpha, \beta, \gamma \}$, denote by $N_{\delta}:=\begin{pmatrix} \delta(+,+) & \delta(+,-)\\ \delta(-,+) & \delta(-,-) \end{pmatrix}$ and the $2\times 2$ matrix with coefficients in $\mathbb{C}[\mathcal{X}_{\SL_2}(\mathbb{T})]$. The Poisson bracket $\{\cdot, \cdot\}^{\mathfrak{o}}$ is defined by the formulas: $$\begin{aligned}
\left\{ N_{\delta} \otimes N_{\delta} \right\}^{\mathfrak{o}} &:=& \overline{r}^{\mathfrak{o}(s(\delta))} (N_{\delta} \otimes N_{\delta}) + (N_{\delta}\otimes N_{\delta})\overline{r}^{\mathfrak{o}(t(\delta))}, \delta \in \{\alpha, \beta, \gamma\}; \\
\{ N_{\alpha} \otimes N_{\gamma} \}^{\mathfrak{o}} &:=& - (N_{\alpha} \otimes \mathds{1}) r^{\mathfrak{o}(b)} (\mathds{1} \otimes N_{\gamma}); \\
\{ N_{\gamma}\otimes N_{\beta} \}^{\mathfrak{o}} &:=& - (N_{\gamma} \otimes \mathds{1}) r^{\mathfrak{o}(a)} (\mathds{1} \otimes N_{\beta} ); \\
\{ N_{\beta}\otimes N_{\alpha} \}^{\mathfrak{o}} &:=& - (N_{\beta} \otimes \mathds{1}) r^{\mathfrak{o}(c)} (\mathds{1} \otimes N_{\alpha} ).
\end{aligned}$$
Remark that, writing $S(N_{\delta}):= \begin{pmatrix} \delta(-,-) & -\delta(+,-) \\ -\delta(-,+) & \delta(+,+) \end{pmatrix}$, the last expressions can be re-written in the form: $$\begin{aligned}
\{ N_{\alpha} \otimes S(N_{\gamma}) \}^{\mathfrak{o}} = (N_{\alpha}\otimes S(N_{\gamma}))r^{\mathfrak{o}(b)} \\
\{ N_{\gamma} \otimes S(N_{\beta}) \}^{\mathfrak{o}} = (N_{\gamma}\otimes S(N_{\beta}))r^{\mathfrak{o}(a)} \\
\{ N_{\beta} \otimes S(N_{\alpha}) \}^{\mathfrak{o}} = (N_{\beta}\otimes S(N_{\alpha}))r^{\mathfrak{o}(c)}
\end{aligned}$$
Given a boundary arc $d\in \{a,b,c\}$, we define a left Hopf-comodule $\Delta_d^L: \mathbb{C}[\mathcal{X}_{\SL_2}(\mathbb{T})] \rightarrow \mathbb{C}[\SL_2]^{(+\mathfrak{o}(d), - \mathfrak{o}(d))} \otimes \mathbb{C}[\mathcal{X}_{\SL_2}(\mathbb{T})]$ by: $$\begin{pmatrix} \Delta_d^L (\delta(+,+)) & \Delta_d^L(\delta(+,-)) \\ \Delta_d^L(\delta(-,+)) & \Delta_d^L(\delta(-,-)) \end{pmatrix} :=
\left\{ \begin{array}{ll}
\begin{pmatrix} x_{++} & x_{+-} \\ x_{-+} & x_{--} \end{pmatrix} \otimes N_{\delta} &\mbox{, if }s(\delta)=d; \\
\mathds{1}\otimes N_{\delta} &\mbox{, else.}
\end{array} \right.$$
Similarly, define a right Hopf-comodule $\Delta_d^R: \mathbb{C}[\mathcal{X}_{\SL_2}(\mathbb{T})] \rightarrow \mathbb{C}[\mathcal{X}_{\SL_2}(\mathbb{T})] \otimes \mathbb{C}[\SL_2]^{(-\mathfrak{o}(d), + \mathfrak{o}(d))} $ by: $$\begin{pmatrix} \Delta_d^R (\delta(+,+)) & \Delta_d^R(\delta(+,-)) \\ \Delta_d^R(\delta(-,+)) & \Delta_d^R(\delta(-,-)) \end{pmatrix} :=
\left\{ \begin{array}{ll}
N_{\delta} \otimes \begin{pmatrix} x_{++} & x_{+-} \\ x_{-+} & x_{--} \end{pmatrix} &\mbox{, if }t(\delta)=d; \\
N_{\delta} \otimes \mathds{1} &\mbox{, else.}
\end{array} \right.$$
By [@KojuTriangularCharVar Lemma $4.6$ ], both $\Delta_d^L$ and $\Delta_d^R$ are Poisson morphisms.
### The general case
Let $\mathbf{\Sigma}$ be a punctured surface, $\Delta$ a topological triangulation of $\mathbf{\Sigma}$, and $\mathfrak{o}_{\Delta}$ an orientation of each edge of $\Delta$. For a face $\mathbb{T}\in F(\Delta)$, let $\mathfrak{o}_{\mathbb{T}}$ be the orientation of its boundary arcs given by $\mathfrak{o}_{\Delta}$. Equip the algebra $\otimes_{\mathbb{T}\in F(\Delta)}\mathbb{C}[\mathcal{X}_{\SL_2}(\mathbb{T})]^{\mathfrak{o}_{\mathbb{T}}}$ with the Poisson bracket defined in Definition \[def\_triangle\_poisson\] . Each inner edge $e\in \mathring{\mathcal{E}}(\Delta)$ lifts to two oriented boundary arcs in $\mathbf{\Sigma}_{\Delta}:= \bigsqcup_{\mathbb{T}\in F(\Delta)} \mathbb{T}$. We denote by $e_L$ the lift of $e$ whose orientation coincides with the induced orientation of $\mathbf{\Sigma}_{\Delta}$ and by $e_R$ the other lift. The comodule maps $\Delta_{e_L}^L$ and $\Delta_{e_R}^R$ induce the comodule maps: $$\Delta^L : \otimes_{\mathbb{T}\in F(\Delta)} \mathbb{C}[\mathcal{X}_{\SL_2}(\mathbb{T})]^{\mathfrak{o}_{\mathbb{T}}} \rightarrow \left( \otimes_{e \in \mathring{\mathcal{E}}(\Delta)} \mathbb{C}[\SL_2]^{-,+} \right) \otimes \left( \otimes_{\mathbb{T}\in F(\Delta)} \mathbb{C}[\mathcal{X}_{\SL_2}(\mathbb{T})]^{\mathfrak{o}_{\mathbb{T}}}\right);$$ $$\Delta^R : \otimes_{\mathbb{T}\in F(\Delta)} \mathbb{C}[\mathcal{X}_{\SL_2}(\mathbb{T})]^{\mathfrak{o}_{\mathbb{T}}} \rightarrow \left( \otimes_{\mathbb{T}\in F(\Delta)} \mathbb{C}[\mathcal{X}_{\SL_2}(\mathbb{T})]^{\mathfrak{o}_{\mathbb{T}}} \right) \otimes \left( \otimes_{e \in \mathring{\mathcal{E}}(\Delta)} \mathbb{C}[\SL_2]^{-,+} \right).$$
The character variety $\mathcal{X}_{\SL_2}(\mathbf{\Sigma})$ is the affine variety whose algebra of regular functions is the kernel of $\Delta^L - \sigma \circ \Delta^R$.
Since the morphisms $\Delta^L$ and $\Delta^R$ are Poisson morphisms, the algebra $\mathbb{C}[\mathcal{X}_{\SL_2}(\mathbf{\Sigma})]$ is a sub-Poisson algebra of $\otimes_{\mathbb{T}\in F(\Delta)}\mathbb{C}[\mathcal{X}_{\SL_2}(\mathbb{T})]^{\mathfrak{o}_{\mathbb{T}}}$. Moreover, this algebra is finitely generated and reduced (see [@KojuTriangularCharVar Theorem $1.1$]). As an affine variety, the character variety does not depend on $\mathfrak{o}_{\Delta}$. Moreover it follows from [@KojuTriangularCharVar Theorem $1.4$] that the Poisson structure does not depend on the orientation of the inner edges, hence only depends on the orientation $\mathfrak{o}$ of the boundary arcs. We denote by $\{.,.\}^{\mathfrak{o}}$ the Poisson bracket on $\mathbb{C}[\mathcal{X}_{\SL_2}(\mathbf{\Sigma})]$. By [@KojuTriangularCharVar Proposition $2.11$], the character variety does not depend, up to canonical isomorphism, on the triangulation. Moreover when $\Sigma$ is closed, the Poisson variety $\mathcal{X}_{\SL_2}(\mathbf{\Sigma})$ is canonically isomorphic to the “classical” (Culler-Shalen) character variety with its Goldman Poisson structure ([@KojuTriangularCharVar Theorem $1.1$]).
Relation between character varieties and stated skein algebras
--------------------------------------------------------------
We first prove Theorem \[theorem3\] for the bigon and the triangle, then we prove the general case using a topological triangulation.
### The case of the bigon
Let $$M:=\begin{pmatrix} \alpha_{++} & \alpha_{+-} \\ \alpha_{-+} & \alpha_{--} \end{pmatrix},
N:=\begin{pmatrix} x_{++} & x_{+-} \\ x_{-+} & x_{--}\end{pmatrix} \text{ and }
C:=\begin{pmatrix} 0 & 1 \\ -1 & 0\end{pmatrix}$$ be three matrices with coefficients in $\mathcal{S}_{+1}(\mathbb{B})$, $\mathbb{C}[\SL_2]$ and $\mathbb{C}$ respectively.
\[lemma\_poisson\_bigone\] For $\varepsilon_1, \varepsilon_2 \in \{ -,+\}$, there is a Poisson isomorphism $\Psi^{\varepsilon_1,\varepsilon_2} : (\mathcal{S}_{+1}(\mathbb{B}), \{\cdot, \cdot \}^s) \xrightarrow{\cong} \mathbb{C}[\SL_2]^{\varepsilon_1, \varepsilon_2}$ defined by the formula: $$\Psi^{\varepsilon_1, \varepsilon_2} (M) := \left\{
\begin{array}{ll}
N &\mbox{, if }(\varepsilon_1,\varepsilon_2) = (-,+); \\
CNC &\mbox{, if }(\varepsilon_1,\varepsilon_2) = (+,-);\\
-CN &\mbox{, if }(\varepsilon_1,\varepsilon_2) = (+,+); \\
-NC &\mbox{, if }(\varepsilon_1,\varepsilon_2) = (-,-). \\
\end{array}\right.$$
That $\Psi^{\varepsilon_1, \varepsilon_2}$ is an isomorphism of algebras follows from the fact that $\det(C)=1$. Let us see the compatibility of $\Psi^{\varepsilon_1, \varepsilon_2}$ with the Poisson structures. For $(\varepsilon_1, \varepsilon_2)=(-,+)$, this follows from a direct comparison of Definition \[def\_bigone\_poisson\] and Example \[exampleBigone\]. Indeed, one has: $$\begin{aligned}
\left\{ N\otimes N\right\}^{-,+} &=& \overline{r}^-(N\otimes N) + (N\otimes N)\overline{r}^+
\\ &=& (F\otimes E - E\otimes F)(N\otimes N) +(N\otimes N)(E\otimes F - F\otimes E) \\
&=& \begin{pmatrix} 0 & x_{++} \\ 0 & x_{-+} \end{pmatrix} \otimes \begin{pmatrix} x_{+-} & 0 \\ x_{--} & 0\end{pmatrix} -
\begin{pmatrix} x_{+-} & 0 \\ x_{--} & 0\end{pmatrix} \otimes \begin{pmatrix} 0 & x_{++} \\ 0 & x_{-+} \end{pmatrix} + \\
&&
\begin{pmatrix} 0 & 0 \\ x_{++} & x_{+-} \end{pmatrix} \otimes \begin{pmatrix} x_{-+} & x_{--} \\ 0 & 0 \end{pmatrix} -
\begin{pmatrix} x_{-+} & x_{--} \\ 0 & 0 \end{pmatrix} \otimes \begin{pmatrix} 0 & 0 \\ x_{++} & x_{+-} \end{pmatrix}.
\end{aligned}$$
We recover the formulas computed in Example \[exampleBigone\]. For $(\varepsilon_1, \varepsilon_2)=(+,+)$, we prove that the isomorphism $\varphi : \mathbb{C}[\SL_2]^{-,+} \xrightarrow{\cong} \mathbb{C}[\SL_2]^{+,+}$ given by $\varphi := \Psi^{+,+} \circ(\Psi^{-,+})^{-1}$, is a Poisson morphism. Note that $\varphi(N)= -CN$ and that $ (C\otimes C) \overline{r}^{\varepsilon} = \overline{r}^{- \varepsilon} (C\otimes C)$. It follows that $$\begin{aligned}
\{ \varphi(N) \otimes \varphi(N) \}^{+, +} &=& \overline{r}^+ (CN \otimes CN) + (CN\otimes CN) \overline{r}^+ \\
&=& (C\otimes C) \left( \overline{r}^- (N\otimes N) + (N\otimes N) \overline{r}^+ \right) = \varphi^{\otimes 2} \left( \{ N\otimes N\}^{-, +} \right)
\end{aligned}$$ which proves the claim. The two remaining cases for $(\varepsilon_1, \varepsilon_2)$ are proved similarly.
### The case of the triangle
For $\delta \in \{ \alpha, \beta, \gamma \}$, let $$M_{\delta} := \begin{pmatrix} \delta_{++} & \delta_{+-} \\ \delta_{-+} & \delta_{--} \end{pmatrix} \text{ and }
N_{\delta}:= \begin{pmatrix} \delta(+,+) & \delta(+,-) \\ \delta(-,+) & \delta(-,-) \end{pmatrix}$$ be two matrices with coefficients in $\mathcal{S}_{+1}(\mathbb{T})$ and $\mathbb{C}[\mathcal{X}_{\SL_2}(\mathbb{T})]$ respectively.
\[lemma\_poisson\_triangle\] There is a Poisson isomorphism $\Psi^{\mathfrak{o}} : (\mathcal{S}_{+1}(\mathbb{T}), \{\cdot, \cdot \}^s ) \xrightarrow{\cong} (\mathbb{C}[\mathcal{X}_{\SL_2}(\mathbb{T})], \{\cdot, \cdot \}^{\mathfrak{o}} )$ defined by the formula: $$\Psi^{\mathfrak{o}} (M_{\delta}) := \left\{
\begin{array}{ll}
N_{\delta} &\mbox{, if }(\mathfrak{o}(s(\alpha)),\mathfrak{o}(t(\alpha))) = (-,+); \\
CN_{\delta}C &\mbox{, if }(\mathfrak{o}(s(\alpha)),\mathfrak{o}(t(\alpha))) = (+,-);\\
-CN_{\delta} &\mbox{, if }(\mathfrak{o}(s(\alpha)),\mathfrak{o}(t(\alpha))) = (+,+); \\
-N_{\delta}C &\mbox{, if }(\mathfrak{o}(s(\alpha)),\mathfrak{o}(t(\alpha))) = (-,-). \\
\end{array}\right.$$ for each $\delta \in \{ \alpha, \beta, \gamma \}$. Moreover, if $d\in \{a,b,c\}$ is a boundary arc of $\mathbb{T}$, the following diagrams commute:
$$\begin{aligned}
\begin{tikzcd}
\mathcal{S}_{+1}(\mathbb{T}) \arrow[r, "\Delta_d^L"] \arrow[d, "\Psi^{\mathfrak{o}}", "\cong"'] &
\mathcal{S}_{+1}(\mathbb{B}) \otimes \mathcal{S}_{+1}(\mathbb{T}) \arrow[d, "\Psi^{\mathfrak{o}(d), - \mathfrak{o}(d)} \otimes \Psi^{\mathfrak{o}}", "\cong"'] \\
\mathbb{C}[\mathcal{X}_{\SL_2}(\mathbb{T})] \arrow[r, "\Delta_d^L"] & \mathbb{C}[\SL_2]\otimes \mathbb{C}[\mathcal{X}_{\SL_2}(\mathbb{T})]
\end{tikzcd} &
\begin{tikzcd}
\mathcal{S}_{+1}(\mathbb{T}) \arrow[r, "\Delta_d^R"] \arrow[d, "\Psi^{\mathfrak{o}}", "\cong"'] &
\mathcal{S}_{+1}(\mathbb{T}) \otimes \mathcal{S}_{+1}(\mathbb{B}) \arrow[d, "\Psi^{\mathfrak{o}} \otimes \Psi^{-\mathfrak{o}(d), \mathfrak{o}(d)} ", "\cong"'] \\
\mathbb{C}[\mathcal{X}_{\SL_2}(\mathbb{T})] \arrow[r, "\Delta_d^R"] & \mathbb{C}[\mathcal{X}_{\SL_2}(\mathbb{T})] \otimes \mathbb{C}[\SL_2]
\end{tikzcd}
\end{aligned}$$
That $\Psi^{\mathfrak{o}}$ is an algebra morphism follows from Lemma \[lemma\_triangle+1\]. For $\delta\in \{\alpha, \beta, \gamma\}$, the equality $(\Psi^{\mathfrak{o}})^{\otimes 2} (\{ \delta_{\varepsilon \varepsilon'} , \delta_{\mu \mu'} \}^{\mathfrak{o}}) = \{ \Psi^{\mathfrak{o}}(\delta_{\varepsilon \varepsilon'}) , \Psi^{\mathfrak{o}}(\delta_{\mu \mu'}) \}^s$ follows from the same computation that the proof of Lemma \[lemma\_poisson\_bigone\]. For $\mathfrak{o}(a)=\mathfrak{o}(b)=\mathfrak{o}(c)=+$, one has: $$\begin{aligned}
\{ N_{\alpha} \otimes N_{\gamma} \}^{\mathfrak{o}} &=& - (N_{\alpha} \otimes \mathds{1})(\frac{1}{2} H\otimes H +2E \otimes F)(\mathds{1}\otimes N_{\gamma}) \\
&=& - \frac{1}{2} \begin{pmatrix} \alpha(+,+) & - \alpha(+,-) \\ \alpha(-,+) & -\alpha(-,-) \end{pmatrix} \otimes \begin{pmatrix} \gamma(+,+) & \gamma(+,-) \\ - \gamma(-,+) & -\gamma(-,-) \end{pmatrix} \\
&& -2 \begin{pmatrix} 0 & \alpha(+,+) \\ 0 & \alpha(-,+) \end{pmatrix} \otimes \begin{pmatrix} 0 & 0 \\ \gamma(+,+) & \gamma(+,-) \end{pmatrix}.
\end{aligned}$$
We recover the formulas of Example \[exampleTriangle\], hence one has $(\Psi^{\mathfrak{o}})^{\otimes 2} (\{ \alpha_{\varepsilon \varepsilon'} , \gamma_{\mu \mu'} \}^{\mathfrak{o}}) = \{ \Psi^{\mathfrak{o}}(\alpha_{\varepsilon \varepsilon'}) , \Psi^{\mathfrak{o}}(\gamma_{\mu \mu'}) \}^s$. We get similar formulas by permuting cyclically the arcs $\gamma, \beta$ and $\alpha$. This proves that $\Psi^{\mathfrak{o}}$ is a Poisson morphism when $\mathfrak{o}(a)=\mathfrak{o}(b)=\mathfrak{o}(c)=+$. For another choice $\mathfrak{o}'$ of orientation of the boundary arcs, we prove that $\Psi^{\mathfrak{o}'}$ is Poisson by showing that the isomorphism $\Psi^{\mathfrak{o}'} \circ (\Psi^{\mathfrak{o}})^{-1}$ is Poisson. This follows from a similar computation than the one in the proof of Lemma \[lemma\_poisson\_bigone\] by using the fact that $(C\otimes C)r^{\varepsilon} = r^{- \varepsilon}(C\otimes C)$. The fact that the two diagrams in the lemma commute follows from a straightforward computation.
### The general case: proof of Theorem \[theorem3\]
Consider a topological triangulation $\Delta$ of a punctured surface $\mathbf{\Sigma}$, together with a choice $\mathfrak{o}_{\Delta}$ of orientation of its edges. Consider the following commutative diagram:
$$\begin{tikzcd}
0 \arrow[r,""] &
\mathcal{S}_{+1}(\mathbf{\Sigma}) \arrow[r, "i^{\Delta}"] \arrow[d, dotted, "\cong", "\exists! \Psi^{(\Delta, \mathfrak{o}_{\Delta})}"'] &
\otimes_{\mathbb{T}} \mathcal{S}_{+1}(\mathbb{T}) \arrow[r,"\Delta^L - \sigma \circ \Delta^R"] \arrow[d, "\cong","\otimes_{\mathbb{T}} \Psi^{\mathfrak{o}_{\mathbb{T}}}"'] &
\left( \otimes_{e} \mathcal{S}_{+1}(\mathbb{B}) \right) \otimes \left( \otimes_{\mathbb{T}} \mathcal{S}_{+1}(\mathbb{T}) \right)
\arrow[d, "\cong", "(\otimes_e \Psi^{-,+})\otimes (\otimes_{\mathbb{T}} \Psi^{\mathfrak{o}(\mathbb{T})})"'] \\
0 \arrow[r,""] &
\mathbb{C}[\mathcal{X}_{\SL_2}(\mathbf{\Sigma})] \arrow[r, "i^{\Delta}"] &
\otimes_{\mathbb{T}} \mathbb{C}[\mathcal{X}_{\SL_2}(\mathbb{T})] \arrow[r,"\Delta^L - \sigma \circ \Delta^R"] &
\left(\otimes_{e}\mathbb{C}[\SL_2]^{-,+}\right) \otimes \left( \otimes_{\mathbb{T}} \mathbb{C}[\mathcal{X}_{\SL_2}(\mathbb{T})] \right)
\end{tikzcd}$$ In this diagram, both lines are exact and all morphisms are Poisson by Lemma \[lemma: poisson morph at +1 skein\] and [@KojuTriangularCharVar], hence there exists a unique Poisson isomorphism $\Psi^{(\Delta, \mathfrak{o}_{\Delta})} : \left( \mathcal{S}_{+1}(\mathbf{\Sigma}), \{ \cdot, \cdot \}^s \right) \xrightarrow{\cong} \left( \mathbb{C}[\mathcal{X}_{\SL_2}(\mathbf{\Sigma})], \{\cdot, \cdot \}^{\mathfrak{o}} \right)$ induced by restriction of $\otimes_{\mathbb{T}} \Psi^{\mathfrak{o}(\mathbb{T})}$. This concludes the proof.
Classical Shadows
-----------------
Suppose that $\omega\in \mathbb{C}$ is a root of unity of odd order $N>1$. A *central representation* of the stated skein algebra is a finite dimensional representation $r: \mathcal{S}_{\omega}(\mathbf{\Sigma}) \rightarrow \End(V)$ which sends each element of the image of the morphism $j$ of Theorem \[theorem2\] to scalar operators. Fix a topological triangulation $\Delta$ of $\mathbf{\Sigma}$ and an orientation $\mathfrak{o}_{\Delta}$ of its edges. Then $r$ induces a character on $\mathcal{S}_{+1}(\mathbf{\Sigma})\xrightarrow[\cong]{\Psi^{(\Delta, \mathfrak{o}_{\Delta})}} \mathbb{C}[\mathcal{X}_{\SL_2}(\mathbf{\Sigma})]$ and this character induces a point in the character variety $\mathcal{X}_{\SL_2}(\mathbf{\Sigma})$ that we call the *classical shadow* of $r$, as in [@BonahonWong1] in the closed case. By definition, the classical shadow only depends on the isomorphism class of $r$.
To motivate the results of this paper, we list three families of central representations. First irreducible representations are obviously central. Then choose for each triangle $\mathbb{T}\in F(\Delta)$ an irreducible representation $r^{\mathbb{T}} : \mathcal{S}_{\omega}(\mathbb{T}) \rightarrow \End(V_{\mathbb{T}})$ and consider the composition: $$r: \mathcal{S}_{\omega}(\mathbf{\Sigma}) \xrightarrow{i^{\Delta}} \otimes_{\mathbb{T}\in F(\Delta)}\mathcal{S}_{\omega}(\mathbb{T}) \xrightarrow{\otimes_{\mathbb{T}} r^{\mathbb{T}}} \End( \otimes_{\mathbb{T}} V_{\mathbb{T}}).$$
Such a representation is central and were called *local representations* in [@BonahonBaiLiuLocalRep]. Eventually, consider the balanced Chekhov-Fock algebra $\mathcal{Z}_{\omega}(\mathbf{\Sigma}, \Delta)$ defined in [@BonahonWongqTrace] after the original construction of [@ChekhovFock]. Given a closed punctured surface whose set of punctures in non-empty, the authors of [@BonahonWongqTrace] defined an algebra morphism (the quantum trace) $\tr : \mathcal{S}_{\omega}(\mathbf{\Sigma}) \rightarrow \mathcal{Z}_{\omega}(\mathbf{\Sigma}, \Delta)$ that was extended to open punctured surface in [@LeStatedSkein]. One motivation is the fact that the representation theory of the balanced Chekhov-Fock algebra is easier to study than the one of the skein algebras (see [@BonahonLiu; @BonahonWong2]). For an irreducible representation $\pi : \mathcal{Z}_{\omega}(\mathbf{\Sigma}, \Delta) \rightarrow \End{V}$ of the balanced Chekhov-Fock algebra, we call *quantum Teichmüller representation*, the composition: $$r: \mathcal{S}_{\omega}(\mathbf{\Sigma}) \xrightarrow{\tr} \mathcal{Z}_{\omega}(\mathbf{\Sigma}, \Delta) \xrightarrow{\pi} \End(V).$$ Quantum Teichmüller representations are central.
Proof of Proposition \[proptchebychev\] and application
=======================================================
Proof of Proposition \[proptchebychev\]
---------------------------------------
We divide the proof of Proposition \[proptchebychev\] in five lemmas.
[Along this section, we write $A:=\omega^{-2}$.]{} Denote by $\mathbb{A}=([0,1]\times S^1, \{p, p'\})$ the annulus with punctures $p=\{0\} \times \{1\}$ and $p'=\{1\}\times \{1\}$ in each of its boundary components and denote by $b=\{0\}\times S^1\setminus \{p\}$ and $b'=\{1\}\times S^1\setminus \{p'\}$ its boundary arcs. Let $\gamma\subset [0,1]\times S^1$ be the curve $\{\frac{1}{2}\} \times S^1$. Let $\delta^{(n)}, \eta^{(n)}\subset [0,1]\times S^1$ be the arcs with endpoints $b$ and $b'$ such that $\delta^{(n)}$ spirals $n$ times in the counterclockwise direction and $\eta^{(n)}$ spirals $n$ times in the clockwise direction. The arcs are drawn in Figure \[figannulus\]. By convention, $\delta^{(0)}$ and $\eta^{(0)}$ represent the empty diagram. Denote by $\beta$ the arc $[0, 1]\times\{-1\}$. By convention, if $\alpha$ is one of the arcs $\beta, \delta^{(n)}, \eta^{(n)}$, we denote by $\alpha_{\varepsilon \varepsilon'}\in \mathcal{S}_{\omega}(\mathbb{A})$ the class of the corresponding stated tangle with sign $\varepsilon$ in $b$ and $\varepsilon'$ in $b'$. The following lemma and its proof are quite similar, though stated in a different skein algebra, to [@LeKauffmanBracket Proposition $2.2$].
![The annulus $\mathbb{A}$, the square $Q$ and some arcs and curves.[]{data-label="figannulus"}](annulus.eps){width="8cm"}
\[lemmaannulus\] In $\mathcal{S}_{\omega}(\mathbb{A})$ the elements $T_N(\gamma)$ and $\beta_{\varepsilon \varepsilon'}$ commute.
First note that a direct application of the Kauffman bracket skein relations implies that $\gamma \cdot \delta^{(n)}_{\varepsilon \varepsilon'}= A \delta^{(n+1)}_{\varepsilon \varepsilon'}+A^{-1} \delta^{(n)}_{\varepsilon \varepsilon'}$ and $\gamma\cdot \eta^{(n)}_{\varepsilon \varepsilon'}= A \eta^{(n)}_{\varepsilon \varepsilon'}+A^{-1}\eta^{(n+1)}_{\varepsilon \varepsilon'}$ when $n\geq 1$. Next we show by induction on $n\geq 0$ that $T_n(\gamma)\cdot \beta_{\varepsilon \varepsilon'} = A^n \delta^{(n)}_{\varepsilon \varepsilon'} + A^{-n}\eta^{(n)}_{\varepsilon \varepsilon'}$. The statements is an immediate consequence of the definitions when $n=0$ and a direct application of the Kauffman bracket relations when $n=1$. Suppose that the results holds for $n$ and $n+1$. Then: $$\begin{aligned}
T_{n+2}(\gamma)\beta_{\varepsilon \varepsilon'} &=& \gamma\cdot T_{n+1}(\gamma)\cdot \beta_{\varepsilon \varepsilon'} - T_{n}(\gamma)\cdot \beta_{\varepsilon \varepsilon'}
\\ &=& \gamma\cdot (A^{n+1}\delta^{(n+1)}_{\varepsilon \varepsilon'} + A^{-(n+1)}\eta^{(n+1)}_{\varepsilon \varepsilon'}) - (A^n\delta^{(n)}_{\varepsilon \varepsilon'} + A^{-n}\eta^{(n)})
\\ &=& A^{n+2} \delta^{(n+2)}_{\varepsilon \varepsilon'} + A^{-(n+2)}\eta^{n+2}_{\varepsilon \varepsilon'},
\end{aligned}$$ and the statement follows by induction. Similarly we show that $\beta_{\varepsilon \varepsilon'} \cdot T_n(\gamma)= A^{-n}\delta^{(n)}_{\varepsilon \varepsilon'} + A^n\eta^{(n)}$. Hence we have: $$T_N(\gamma)\cdot \beta_{\varepsilon \varepsilon'} - \beta_{\varepsilon \varepsilon'}\cdot T_N(\gamma) = (A^N-A^{-N})(\delta^{(N)}_{\varepsilon \varepsilon'} - \eta^{(N)}_{\varepsilon \varepsilon'}) =0.$$
Denote by $Q$ the square, *i.e.* a disc with four punctures on its boundary. Let $b_1, \ldots, b_4$ be its four boundary arcs labelled in the counter-clockwise order. When gluing $b_1$ along $b_3$, we obtain the annulus with $b_2$ sent to $b$ and $b_4$ sent to $b'$. We denote by $i_{|b_1\#b_3} : \mathcal{S}_{\omega}(\mathbb{A})\hookrightarrow \mathcal{S}_{\omega}(Q)$ the gluing morphism. Let $\alpha, \beta, \delta^{(n)}, \eta^{(n)}\subset Q$ be the arcs which are sent to $\gamma, \beta, \delta^{(n)}$ and $\eta^{(n)}$ respectively as in Figure \[figannulus\]. Fix an arbitrary orientation $\mathfrak{o}$ of the boundary arcs of $Q$ and consider the filtration $(\mathcal{F}_m)_{m\geq 0}$ associated to $S=\{ b_2, b_4\}$. Write $d:\mathcal{S}_{\omega}(Q)\rightarrow \mathbb{Z}^{\geq 0}$ the corresponding map and $\mathcal{G}_m:=\quotient{\mathcal{F}_m}{\mathcal{F}_{m-1}}$ the corresponding graduation.
\[lemma1\] The following holds: $$\lt\left( (\alpha_{++}+\alpha_{--})^N\right) = \lt \left(T_N(\alpha_{++}+\alpha_{--})\right) = \alpha_{++}^N + \alpha_{--}^N.$$
First note that in $\mathcal{G}_{4}$, we have $\alpha_{--}\alpha_{++}= q^2\alpha_{++}\alpha_{--}$. So it follows from Lemma \[lemma\_qbinomial\] that in $\mathcal{G}_{2N}$, we have $\lt\left( (\alpha_{++}+\alpha_{--})^N\right) = \alpha_{++}^N + \alpha_{--}^N $. Since $T_N(X) - X^N$ is a polynomial of degree strictly smaller that $N$, the degree of $T_N((\alpha_{++}+\alpha_{--}) - (\alpha_{++}+\alpha_{--}))$ is strictly smaller than $2N$, thus $ \lt \left(T_N(\alpha_{++}+\alpha_{--})\right) = \lt\left( (\alpha_{++}+\alpha_{--})^N\right) $.
Let $\alpha^{(n)}$ be the diagram made of $n$ parallel copies of $\alpha$. Using the identifications $\partial \delta^{(n)}=\partial \eta^{(n)} = \partial \alpha^{(n)} \cup \partial \beta$, we denote by $\delta^{(n)}_{(s,\varepsilon, \varepsilon')}, \eta^{(n)}_{(s, \varepsilon, \varepsilon')} \in \mathcal{S}_{\omega}(Q)$ the classes of the tangles $\delta^{(n)}$ and $\eta^{(n)}$ with states given by a state $s$ of $\alpha^{(n)}$ and a state $(\varepsilon, \varepsilon')$ of $\beta$.
\[lemma2\] For $n\geq 0$ and $s$ a state of $\alpha^{(n)}$, we have: $$\lt \left( \left[ [\alpha^{(n)}, s], \beta_{\varepsilon \varepsilon'} \right] \right) = (A^n-A^{-n})\left(\delta^{(n)}_{(s, \varepsilon, \varepsilon')} - \eta^{(n)}_{(s, \varepsilon, \varepsilon')} \right),$$ where we used the notation $[x,y]= xy-yx$.
The diagram obtained by stacking $\alpha^{(n)}$ on top of $\beta$ has $n$ crossings and thus $2^n$ resolutions using the Kauffman bracket relation. We remark that the resolution obtained by replacing each crossing by $\resneg$ is $A^n \delta^{(n)}_{(s,\varepsilon, \varepsilon')}$ while the resolution obtained by replacing each crossing by $\respos$ is $A^{-n} \eta^{(n)}_{(s, \varepsilon, \varepsilon')}$. These two resolutions have degree $2n$ and all the others resolutions have degrees strictly smaller, thus $\lt \left( [\alpha^{(n)}, s] \cdot \beta_{\varepsilon \varepsilon'} \right)= A^n \delta^{(n)}_{(s,\varepsilon, \varepsilon')} + A^{-n} \eta^{(n)}_{(s, \varepsilon, \varepsilon')}$. We prove similarly that $\lt \left( \beta_{\varepsilon \varepsilon'}\cdot [\alpha^{(n)}, s] \right)= A^{-n} \delta^{(n)}_{(s,\varepsilon, \varepsilon')} + A^{n} \eta^{(n)}_{(s, \varepsilon, \varepsilon')}$ and conclude by taking the difference.
\[lemma3\] If $x\in \mathcal{S}_{\omega}(Q)$ is a polynomial in $\mathcal{S}_{\omega}(Q)$ in the elements $\alpha_{\varepsilon \varepsilon'}$ such that $d(x)<2N$ and such that $x$ commutes with all elements $\beta_{\mu,\mu'}$, then $x$ is a constant.
Let $x=\sum_{i\in I} x_i [\alpha^{n_i}, s_i]$ be the decomposition in the basis of stated tangles with increasing states $s_i$ and denote by $2n<2N$ its degree. Suppose by contradiction that $n\neq 0$. Let $J=\{ j\in I \mbox{, such that } n_i=n \}\subset I$ so we have $\lt(x) = \sum_{j\in J} x_j [\alpha^{n}, s_j]$. The hypothesis on $x$ and Lemma \[lemma2\] imply that: $$0 = \lt \left( [x, \beta_{\varepsilon \varepsilon'} ] \right) = \sum_{j\in J} x_j (A^n -A^{-n})(\delta^{(n)}_{(s_j, \varepsilon, \varepsilon')} - \eta^{(n)}_{(s, \varepsilon, \varepsilon')} ).$$
Since the elements $\delta^{(n)}_{(s_j, \varepsilon, \varepsilon')}$ and $\eta^{(n)}_{(s_j, \varepsilon, \varepsilon')}$ are linearly independent for $n\geq 1$, we conclude that $x_j(A^n-A^{-n})=0$ for all $j\in J$. Since $0<n<N$, we obtain that $x_j=0$ for all $j\in J$ thus $\lt(x)=0$. This gives the contradiction.
The set $\mathcal{B}':=\{\alpha_{-+}^a \alpha_{++}^b \alpha_{+-}^c, a,b,c\geq 0\} \cup \{\alpha_{-+}^a \alpha_{--}^b \alpha_{+-}^c, a,b,c\geq 0\}$ forms a basis of the algebra $\mathcal{S}_{\omega}(\mathbb{B})$. This fact is Exercise $7$ in Chapter $IV$ Section $6$ of [@Kassel], and is proved as follows. Choose an orientation $\mathfrak{o}$ of the boundary arcs of $\mathbb{B}$ such that $b_L$ and $b_R$ points towards different punctures and consider the filtration associated to $S=\{b_L, b_R\}$. For each element of the basis $\mathcal{B}^{\mathfrak{o}}$, there exists exactly one element of $\mathcal{B}'$ which has the same leading term. For $x\in \mathcal{S}_{\omega}(\mathbb{B})$, denote by $c(x)\in \mathcal{R}$ the coefficient of $1$ in the decomposition of the basis $\mathcal{B}'$.
\[lemma4\] One has the equality: $ c(T_N(\alpha_{++} + \alpha_{--}))= 0$.
Let $n\geq 1$ be an odd integer and let us show that $c\left( (\alpha_{++} + \alpha_{--})^n \right) =0$. The proof will then follow from the fact that $T_N(X)$ is an odd polynomial, thus is a linear combination of such elements, and the fact that $c$ is linear. The product $\left( (\alpha_{++} + \alpha_{--})^n \right)$ develops as a sum of terms of the form $x=x_1\ldots x_n$ where $x_i$ is either $\alpha_{++}$ or $\alpha_{--}$. Using the defining relations of $\mathcal{S}_{\omega}(\mathbb{B})$, we can further develop each term $x$ as a linear combination of terms of the form $\alpha_{-+}^a \alpha_{++}^b \alpha_{+-}^a$ and $ \alpha_{-+}^a \alpha_{--}^b\alpha_{+-}^a$ where $2a+b$ has the same parity than $n$. Since $n$ is odd, each of these summands satisfies $b\neq 0$ so $c(x)=0$.
Consider the element $x:= T_N(\alpha_{++}+\alpha_{--}) - \alpha_{++}^N - \alpha_{--}^N \in \mathcal{S}_{\omega}(Q)$. By Lemma \[lemma1\], its degree is strictly smaller that $2N$. By Lemma \[lemmaannulus\], in $\mathcal{S}_{\omega}(\mathbb{A})$ the elements $T_N(\gamma)$ and $\beta_{\varepsilon \varepsilon'}$ commute. The image through the injective gluing morphism $i_{|b_1\#b_3} : \mathcal{S}_{\omega}(\mathbb{A})\hookrightarrow \mathcal{S}_{\omega}(Q)$ of $T_N(\gamma)$ and $\beta_{\varepsilon \varepsilon'}$ are respectively $T_N(\alpha_{++} + \alpha_{--})$ and $\beta_{\varepsilon \varepsilon'}$, thus they commute. By Lemma \[lemma2\], the elements $\alpha_{++}^N$ and $\alpha_{--}^N$ also commute with $\beta_{\varepsilon \varepsilon'}$ so $x$ commutes with each element $\beta_{\varepsilon \varepsilon'}$. Lemma \[lemma3\] implies that $x$ is a constant and Lemma \[lemma4\] implies that this constant is null. This concludes the proof.
A generalization of a theorem of Bonahon
----------------------------------------
Proposition \[proptchebychev\] provides the following generalization of the main theorem of [@BonahonMiraculous]. Let $\mathcal{A}$ be an $\mathcal{R}$-algebra and $\rho : \mathbb{C}_q[\SL_2]^{\otimes k} \rightarrow \mathcal{A}$ be a morphism of algebras. Let $\rho_i $ be the $i$-th component of $\rho$. For $1\leq i \leq k$, consider the following two matrices with coefficients in $\mathcal{A}$: $$\begin{aligned}
A_i := \begin{pmatrix} \rho_i(\alpha_{++}) & \rho_i(\alpha_{+-}) \\ \rho_i(\alpha_{-+}) & \rho_i(\alpha_{--}) \end{pmatrix} &
A_i^{(N)} := \begin{pmatrix} \rho_i(\alpha_{++})^N & \rho_i(\alpha_{+-})^N \\ \rho_i(\alpha_{-+})^N & \rho_i(\alpha_{--})^N \end{pmatrix}.
\end{aligned}$$
The following proposition was proved in [@BonahonMiraculous Theorem $1$] in the particular case where $\rho_i(\alpha_{+-})\rho_i(\alpha_{-+})=0$ for each $i\in \{1, \ldots, k\}$.
If $q$ is a root of unity of odd order $N>1$, then one has: $$T_N\left( \tr (A_1\ldots A_k)\right) = \tr\left( A_1^{(N)} \ldots A_k^{(N)} \right).$$
By Proposition \[proptchebychev\] and using that both $\rho$ and the $(k-1)$-th coproduct $\Delta^{(k-1)} : \mathbb{C}_q[\SL_2] \rightarrow \mathbb{C}_q[\SL_2]^{\otimes k}$ are morphisms of algebras, one has: $$T_N \circ \rho \circ \Delta^{(k-1)} (\alpha_{++} + \alpha_{--}) = \rho \circ \Delta^{(k-1)} (\alpha_{++}^N +\alpha_{--}^N).$$ We conclude by remarking that $ \rho \circ \Delta^{(k-1)} (\alpha_{++} + \alpha_{--}) = \tr (A_1\ldots A_k)$ and $ \rho \circ \Delta^{(k-1)} (\alpha_{++}^N +\alpha_{--}^N)= \tr\left( A_1^{(N)} \ldots A_k^{(N)} \right)$, where the second equality follows from the fact that $j_{\mathbb{B}}$ is a morphism of Hopf algebras (Lemma \[lemma\_center\_bigone\]) hence commutes with $\Delta^{(k-1)}$.
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Dynamic scaling behavior of extended nonlinear systems out of equilibrium have attracted much attention in different areas of physics [@scaling]. In continuum models dynamic scaling is intimately related to self-similar asymptotics of nonlinear partial differential equations [@Barenblatt]. Sometimes logarithmic corrections enter dynamic scaling laws [@scaling]. No general scenario for their appearance is known. One can say from experience that they appear in marginal cases, dividing (in an appropriate functional space) regimes with qualitatively different behavior. The aim of this work is to investigate one particular setting, of general interest, where logarithmic corrections to scaling appear somewhat unexpectedly: cooling dynamics of hot bubbles in gases.
Heat transfer in gases, strongly heated locally, looks quite differently from the simple picture provided by the linear heat diffusion equation. The difference is mainly due to the small-Mach-number gas flow that develops (even at zero gravity) owing to intrinsic pressure gradients. This conductive cooling flow (CCF) brings in cold gas from the periphery and strongly modifies the cooling dynamics. Some aspects of CCFs have been studied experimentally and theoretically, mainly in the context of the late stage of strong explosions [@M; @K; @GLM]. In this Letter we report a new, striking feature of a CCF. We find that, if the initial temperature profile decays rapidly enough at large distances \[like $\exp (-k|x|)\,, k>0$\], the hot bubble, while cooling down significantly, expands [*logarithmically*]{} slowly.
Starting from the continuity, momentum and energy equations for an inviscous ideal gas at zero gravity, and employing the small-Mach-number expansion, one arrives at the following nonlinear equation for the scaled gas temperature [@M]: $$\partial_t T = T^2 \partial_x(T^{\nu-1} \partial_x T)\,,
\label{1}$$ where indices $t$ and $x$ stand for partial derivatives (a slab geometry is assumed), and $\nu$ is the exponent in the power-law temperature dependence of the heat conductivity of the gas [@density].
The scaled gas pressure stays constant (and equal to unity) in this approximation, so the scaled gas density is simply $\rho(x,t) = T^{-1}(x,t)$, while the gas velocity is $v(x,t) = T^{\nu} \partial_x T$ [@M]. Therefore, once solving Eq. (\[1\]) for the temperature, one can easily find all other variables.
Eq. (\[1\]) has a multitude of similarity solutions: $$T_{\beta}(x,t) = t^{\frac{2\beta-1}{\nu+1}} \theta(x/t^{\beta}) \,,
\label{2}$$ where $\beta$ is an arbitrary parameter. Therefore, an interesting selection problem appears, like in many other situations in nonlinear dynamics of extended systems [@Barenblatt; @CH].
Eq. (\[1\]) has appeared in the context of cooling of the “fireball" produced by a strong local explosion in a gas [@M; @K; @GLM]. An explosion involves energy release on a time scale short compared to the characteristic acoustic time. In this case the preceding rapid stage of the dynamics produces an inverse power-law dependence of the gas temperature on the distance from the explosion site [@ZR]. It has been shown [@M; @K] that the exponent of this power law uniquely selects the scaling exponent $\beta$. As the result, the fireball expansion exhibits a power law in time.
A different type of local gas heating occurs when the time scale of the energy release is long compared to the acoustic time, but still short compared to the cooling time. In this case the initial temperature profile is more localized, as it reflects the spatial structure of the heating agent (for example, the radial intensity of laser beam). This regime will be in the focus of this Letter. We will see that there is [*no*]{} self-similar asymptotics to this problem. Instead, the solution approaches, at long times, a “quasi-similarity" asymptotics with logarithmic corrections to scaling.
Consider first a model problem when the initial temperature profile has compact support: $T(x,0)=T_0(x)>0$ at $x \in [-L,L]$, and zero elsewhere. We will limit ourselves to a temperature-independent heat conductivity, $\nu=0$. Despite this choice, the nonlinearity of Eq. (\[1\]) persists. Assume symmetry with respect to $x=0$ and impose the Neumann boundary conditions: $\partial_x T(0,t)= \partial_x T(L,t)=0$. A local analysis of Eq. (\[1\]) near the edge of support of its solution $T(x,t)$ shows that the support remains compact and [*unchanged*]{} for $t>0$. Therefore, this model problem exhibits a [*complete*]{} localization. What is the late-time behavior of the temperature? The constancy of support immediately selects $\beta=0$, so the similarity ansatz becomes $T_0(x,t)= t^{-1} \theta (x)$. Then Eq. (\[1\]) yields $\theta (x) = (a^2/2) \cos^2 (x/a)$ for $x \in [-L,L],
\theta (x) = 0$ elsewhere, and $a=\pi L/2$. This simple similarity solution describes cooling of the hot bubble (and filling it with the dense gas) without [*any*]{} change in the bubble size.
Remarkably, $T_0(x,t)$ represents a long-time asymptotics for [*any*]{} initial condition that has compact support $[-L,L]$ and obeys the Neumann boundary conditions. We will show here only that this solution is linearly stable with respect to small perturbations, and find the spectrum of the linearized problem. Introduce new variables: $u=t\,T(x,t)$ and $\tau=\ln t$. Eq. (\[1\]) assumes the form $$\partial_{\tau} u = u - (\partial_x u)^2 + u \partial_{xx} u\,,
\label{3}$$ while the similarity solution $T_0 (x,t)$ becomes steady-state solution $\theta (x)$. Introducing a small correction $v(x,\tau)$ to this solution and linearizing Eq. (\[3\]), we obtain $\partial_{\tau} v = \hat{L} (\xi) v$, where $$\hat{L}(\xi) = (1/2) \cos^2 \xi\, \partial_{\xi\xi} +
\sin 2\xi \,\partial_{\xi} +
2 \sin^2 \xi
\label{4}$$ and $\xi=x/a$. We look for the eigenfunctions in the form of $v(\xi,\tau) = e^{ \gamma \tau} \psi_\gamma (\xi)$. The general solution of the resulting ordinary differential equation is $$\begin{aligned}
\psi_\gamma (\xi) = C_1 \cos^2 \xi\, _2F_1 (a_-, a_+, 1/2, -\tan^2 \xi)
\nonumber\\
+\, C_2 \cos \xi \sin \xi\, _2F_1 (b_-, b_+, 3/2, -\tan^2 \xi)\,,
\label{6}\end{aligned}$$ where $_2F_1$ is the hypergeometric function, $C_1$ and $C_2$ are arbitrary constants, $a_{\pm} = \left[1 \pm (8\gamma+9)^{1/2}\right]/4$ and $b_{\pm} = \left[3 \pm (8\gamma+9)^{1/2}\right]/4$. Requiring that the perturbation remains small compared to the unperturbed solution \[and hence vanishes like $(\pi/2-\xi)^2$ or faster at $\xi \to \pi/2$\], we find the (continuous) spectrum of the linearized problem: $-\infty <\gamma \le -1$. This result proves linear stability of the similarity solution $T_0 (x,t)$. Notice the presence of gap between the upper edge of the spectrum $\gamma=-1$ and stability border $\gamma=0$. Going back to physical variables, we find that small [*temperature*]{} perturbations around the similarity solution exhibit a power law decay $t^{\gamma-1}$.
At this stage we notice that $\beta=0$ is a marginal case dividing two qualitatively different types of dynamics as described by the family of similarity solutions (\[2\]). Indeed, solutions with $\beta>0$ describe power-law [*expansions*]{} [@M; @K; @GLM], while solutions with $\beta<0$ correspond to power-law [*shrinkings*]{} [@shrinking]. One can expect logarithmic corrections to appear “on the background" of the special case $\beta=0$, when the initial condition does [*not*]{} have compact support, but decays rapidly at large distances. Therefore, we assume that the initial temperature profile of the bubble is symmetric with respect to $x=0$ and decays exponentially at large distances [@intermediate]. We will continue using the new variables and require $u(x,0)\to c \exp (-k |x|)$ at $|x|\to\infty$, where $k$ and $c$ are positive constants. One can always put $k=1$ [@k=1]. We will be interested in a long-time asymptotics of the solution: $\tau \gg 1$. Our first important observation is that $u (x,\tau) = c \exp(\tau-x)$ is an [*exact*]{} solution of Eq. (\[3\]). This traveling wave solution with a unit speed corresponds to a steady-state solution $T(x)=c \exp(-x)$ in physical variables, and it represents the correct asymptotics of the solution to our problem at $x \to +\infty$. What about the bubble “core"? We will show that it can be described, at $\tau \gg 1$, by a “quasi-similarity" solution plus small corrections: $$u (x,\tau) =
u_0(x,\tau) + u_1(x,\tau) + \dots\,,
\label{expansion}$$ where $$u_0 (x,\tau) = \frac{a^2 (\tau)}{2}\, \cos^2\, \frac{x}{a(\tau)}
\label{QS2}$$ and $ \dots \ll u_1 \ll u_0$. One of our goals is to find an asymptotic expansion for $a(\tau)$.
The leading term of $a(\tau)$ can be guessed immediately. Indeed, expansion of $u_0 (x,\tau)$ in powers of $x-\pi a(\tau)/2$ near the point $x=\pi a(\tau)/2$ begins with the term $(1/2)\left(x-\pi a(\tau)/2\right)^2$. This is a wave traveling with speed $\pi\dot{a}/2$ along the $x$-axis! Therefore, it is natural to look for a [*general*]{} traveling wave solution $v(x,\tau)=V(x-\tau)$ of Eq. (\[3\]) with a unit speed and require that it behaves like $(z+\mbox{const})^2/2$ at $z\to -\infty$ and like $c\, \exp (-z)$ at $z\to +\infty$, where $z=x-\tau$. If such a solution exists, we can match it with the leading term of the quasi-similarity solution (\[QS2\]) in the region $1\ll - (x-\pi a/2) \ll a, 1 \ll -z$, once $$a(\tau) = 2 \tau/\pi\,.
\label{leading}$$ Eqs. (\[QS2\]) and (\[leading\]) have important implications. First, the temperature scaling with physical time $t$ at the bubble center acquires a logarithmic correction. Second, the bubble core expands logarithmically slowly. We will show in the following that these are indeed correct results, calculate the subleading and sub-subleading terms for $a(\tau)$, and find other attributes of asymptotic solution.
The general traveling wave solution of Eq. (\[3\]), $V(z)$, obeys the second-order equation $$-V_z=V-V_z^2+VV_{zz}\,
\label{TR2}$$ that is soluble analytically. One integration yields $$V^{-1} (d V/d z)=-1-W\left(-\exp\left(-1-V^{-1}\right)\right)\,,
\label{TR8}$$ where $W(\eta)$ is the product log function defined as the solution of equation $W e^W =\eta$ (see, [*e.g.,*]{} Ref. , p. 751). The arbitrary constant in Eq. (\[TR8\]) has been chosen to satisfy the required asymptotic behavior $V(z) \to (z+\mbox{const})^2/2$ at $z \to -\infty$. Notice that, as $V>0$ and $V_z<0$, we should work with the negative branch of the product log function: $\eta<0$ and $W(\eta) <0$.
Integrating Eq. (\[TR8\]), we obtain the traveling wave solution in an implicit form: $$J(V)+ z + C = 0\,,
\label{TW}$$ where $$J(V) = \int\limits_1^{U(V)}\frac{d\zeta}{1-\zeta-e^{-\zeta}}\ ,
\label{TR12}$$ $U(V)=-\ln\left[-W\left(-\exp (-1-V^{-1})\right)\right]$ and $C$ is an arbitrary constant.
To understand the asymptotic behavior of this solution at $z\to -\infty$ and $z \to +\infty$, we need to know the asymptotics of $J(V)$. After some algebra we obtain $$J(V)= \left\{\begin{array}{lcl}
\ln V +\Delta_1 +{\cal O}\left(V
\right)\,,
\\
(2V)^{1/2}+\frac{1}{3} \ln V + \Delta_2 +
{\cal O}\left(V^{-1/2}\right)\,,
\end{array}\right.
\label{TR28}$$ at $V\to+0$ and $V\to +\infty$, respectively. Here $$\Delta_1=\int\limits_1^\infty \frac{\left(1- e^{-\zeta}\right)
d\zeta}
{\zeta (1-\zeta-e^{-\zeta})} = -1.46074400\dots\,,$$ $\Delta_2 = -2 -(1/3) \ln \,(2 e)- \Delta_3$, and $$\Delta_3 = \int\limits_0^1
\left(\frac{1}{1-\zeta-e^{-\zeta}} +\frac{2}{\zeta^2}
+\frac{2}{3\zeta}\right) d\zeta = -0.05361892\dots$$ Using Eqs. (\[TW\]) and (\[TR28\]) we obtain: $$V(z)= e^{-z-C-\Delta_1} + {\cal O}\left(e^{-2z}\right)\quad\mbox{at}\quad
z\to+\infty
\label{TR30}$$ and $$\begin{aligned}
V(z)&=&(1/2) (z+C+\Delta_2)^2 \nonumber \\
& &+ \,(2/3) (z+C+\Delta_2)\,
\ln \left(|z+C+\Delta_2|)/\sqrt{2}\right) \nonumber\\
& & +\, {\cal O}\left(\ln^2|z+C+\Delta_2|\right)
\quad\mbox{at}\quad z\to-\infty\,.
\label{TR31}\end{aligned}$$ The required asymptotic behavior $V\to c\, \exp (-z)$ at $z\to +\infty$ selects $C= -\Delta_1 - \ln c$, so the traveling wave solution (\[TW\]) is now fully determined. After some rearrangement, we rewrite the asymptotics (\[TR30\]) and (\[TR31\]) as $$V (z)=c e^{-z}+{\cal O}(e^{-2 z})
\mbox{~~~~~at~~~~~}
z\to+\infty\,,
\label{TR36}$$ and $$\begin{aligned}
V(z)&=&(1/2)(z-\Delta)^2+(2/3)\,
(z-\Delta)\, \ln |z-\Delta| \nonumber \\
& &+\, {\cal O}(\ln^2|z|)
\mbox{~~~~~at~~~~~} z\to-\infty\,,
\label{TR37}\end{aligned}$$ where $\Delta=\Delta_1 - \Delta_2 +(1/3) \ln 2+\ln c$.
The leading term in Eq. (\[TR36\]) corresponds to a steady-state solution in the physical variables, while the subleading term is [*exponentially*]{} small with respect to the leading one. This essentially static (“glassy") behavior of the solution at large distances reflects [*effective*]{} diffusion choking at small temperatures.
Now let us return to the bubble core description, Eq. (\[QS2\]). Our basic assumption here (supported by the results) is that, in the asymptotic stage $\tau\gg 1$, the terms $u_0, u_1, \dots$ depend on time only through the time dependences of $a$, of $a$ and $\dot{a}$, of $a, \dot{a},
\dots$, respectively. The small parameter of this expansion is $\dot{a}/a$. In the zeroth approximation of this perturbation scheme, $u_0$ obeys Eq. (\[3\]) without the time derivative term. In the first approximation we obtain the following linear equation: $$\hat{L} \, u_1(\xi) = a \dot{a}\, \cos^2 \xi\, (1+ \xi \tan \xi)\,,
\label{QS6}$$ where we have again used $\xi = x/a$.
The zero modes of the operator $\hat{L}$ are $\Upsilon (\xi) = \cos^2\xi + \xi\, \cos\xi\, \sin \xi$ and $\Phi (\xi) = \sin\xi\, \cos\xi$. Looking for the general solution of Eq. (\[QS6\]) in the form of $u_1=C_1(\xi)\, \Upsilon(\xi)+C_2(\xi)\, \Phi(\xi)$ and defining $a(\tau)$ by the condition $u(0,\tau) =a^2 (\tau)/2$, we arrive at $$\begin{aligned}
u_1 (\xi) &=&-(2/3)\, a\dot{a}\, \cos^2 \xi\,
[(\xi\tan\xi-1)\, \ln\cos\xi \nonumber \\
& &+\, 2\ln (2/e)\, \xi\tan\xi +
\tan\xi\, \mbox{Im}\, \left[\mbox{Li$_2$}(-e^{2i \xi})\right]]\,,
\label{QS10}\end{aligned}$$ where $\mbox{Li$_2$}(x)=\sum_{k=1}^\infty k^{-2} x^k$ is the dilogarithm (see, [*e.g.,*]{} Ref. , p. 743). In the vicinity of $\xi=\pi/2$ $$u_1=\frac{\pi}{3}\, a\dot{a}\, \tilde{\xi}\,
\ln\frac{4\,|\tilde{\xi}|}{e^2} + a \dot{a}\,
{\cal O}\left(\,|\tilde{\xi}|^3
\ln|\tilde{\xi}|\right )\,,
\label{QS11}$$ where $\tilde{\xi}=\xi-\pi/2$. Too close to $\xi=\pi/2$ the correction $u_1$ and its derivatives become larger than the zero-order solution $u_0$ and its corresponding derivatives, so the perturbation procedure breaks down. Therefore, the bubble core solution \[Eqs. (\[expansion\]), (\[QS2\]) and (\[QS10\])\] should be matched with the traveling wave solution \[Eq. (\[TW\])\] in the region where $|x-\pi a/2|$ is small enough (so that the leading term of the asymptotics of $u_0$ is much larger than the subleading terms) but, on the other hand, large enough (so that $u_1$ is small compared to $u_0$). Working in this region and collecting the leading contributions from $u_0$ and $u_1$, we obtain after some rearrangement: $$\begin{aligned}
u&=&\frac{1}{2}\, \tilde{x}^2\,
+\frac{\pi}{3}\dot{a}\,\tilde{x} \,\ln|\tilde{x}| +
\frac{1}{a^2}\, {\cal O}\left(\tilde{x}^4\right)
+\, \frac{\dot{a}}{a^2}\,{\cal O}\left(|\tilde{x}|^3\ln |\tilde{x}|\right)\nonumber \\
& & + \,{\cal O}\left(\dot{a}^2\ln^2a\right)\, {\cal O}\left(\ln
|\tilde{x}|\right)
+\dots\,,
\label{QS14}\end{aligned}$$ where $\tilde{x} = x-(\pi/2) \, a - (\pi/3)\,\dot{a}\, \ln(e^2 a/4)$.
Now we can perform the matching procedure. We require that the $z\to -\infty$ asymptotics of the traveling wave solution, Eq. (\[TR37\]), coincide with the asymptotics (\[QS14\]) of the bubble core solution. This yields $$a(\tau)=\frac{2}{\pi k}\, \left[\, \tau-\frac{2}{3}\, \ln\frac{\tau}{4\pi}
+ B + \ln ck^2+o(1)\right]\,,
\label{MT3}$$ where $B=1+\Delta_1+\Delta_3=-0.514362926\dots$, $o(1)$ denotes terms that vanish as $\tau \to \infty$ and we have restored the $k$-dependence [@k=1]. We see that the leading term in $a(\tau)$ is logarithmic in physical time $t$, and it coincides with Eq. (\[leading\]). The subleading term behaves like $\ln \tau \sim \ln \ln t$, while the sub-subleading term is constant.
The matching region is determined by the requirements that the subleading term in Eq. (\[QS14\]) is much less than the leading term, but much greater than the rest of terms. These yield an approximate condition $\ln^2 a \ll- x+\frac{\pi}{2}\,a\ll a^{2/3}$, so that the matching region expands as $\tau \to \infty$.
We compared the asymptotic solution with numerical simulations. Finding logarithmic corrections numerically usually requires going to very long times. Instead, we directly solved Eq. (\[3\]) in the new variables, which enabled us to reach $\tau \sim 20$, that is $t \sim 5 \cdot 10^8$. Eq. (\[3\]) was solved on the interval $x \in (0, 9)$ subject to the Neumann boundary conditions. The initial condition was $u (x, \tau=0) =
2 \exp\left[-3\,(x^2+0.1)^{1/2}\right]$, so that $c=2$ and $k=3$. The system length was large enough for the solution to enter the asymptotic regime before the expanding “core" reaches the boundary $x=9$. Fig. 1 compares $a=\left(2 u(0,\tau)\right)^{1/2}$ found numerically with the prediction of Eq. (\[MT3\]). At long times the agreement is excellent. We also verified other attributes of the asymptotic solution.
In separate simulations that will be presented elsewhere [@KLM], evolution of the same initial condition was investigated in the framework of the [*full gasdynamic*]{} equations. Remarkably, the results essentially coincide, even for moderate Mach numbers, with those obtained with Eq. (\[1\]). This shows robustness of the reduced equation in the description of CCFs.
In conclusion, we have shown that hot bubbles in gases expand logarithmically slowly in the process of cooling. By constructing an asymptotic solution, that matches a “quasi-similarity" inner solution and a “glassy" outer solution, we have been able to see how logarithmic corrections enter dynamic scaling.
We are grateful to Y. Kurzweil for help with Fig. 1. This work was partially supported by the COE Visiting Research Scholar Program at YITP and by the Russian Foundation for Basic Research (grant No. 99-01-00123).
A.J. Bray, Adv. Phys. [**43**]{}, 357 (1994); A.-L. Barabási and H.E. Stanley, [*Fractal Concepts in Surface Growth*]{}, (Cambridge Univ. Press, Cambridge, 1995); N. Goldenfeld, [Lectures on Phase Transitions and the Renormalization Group]{} (Addison-Wesley, 1992). G.I. Barenblatt, [*Scaling, Self-similarity, and Intermediate Asymptotics*]{} (Cambridge Univ. Press, Cambridge, 1996). B. Meerson, Phys. Fluids A [**1**]{}, 887 (1989). D. Kaganovich, B. Meerson, A. Zigler, C. Cohen and J. Levin, Phys. Plasmas [**3**]{}, 631 (1996). A. Glasner, E. Livne and B. Meerson, Phys. Rev. Lett. [**78**]{}, 2112 (1997). Alternatively, one obtains a nonlinear diffusion equation for the gas density: $\partial_t \rho =
\partial_x (\rho^{-\nu-1} \partial_x \rho)$, with the effective diffusion coefficient [*decreasing*]{} with $\rho$. This equation appeared in a number of nonlinear diffusion problems where $\rho$ decays at $|x| \to \infty$ [@Rosenau]. In the hot bubble problem $\rho \to \infty$ at $|x| \to \infty$, and this difference results in quite a different dynamics. M.C. Cross and P.C. Hohenberg, Rev. Mod. Phys. [**65**]{}, 851 (1993). Ya. B. Zel’dovich and Yu. P. Raizer, [*The Physics of Shock Waves and High Temperature Hydrodynamic Phenomena*]{} (Academic, New York, 1967). Shrinking is possible in the problem of [*heating*]{} of a cold “cloud" by a hot and underdense peripheral gas. Some aspects of this regime were investigated in Ref. . In reality, the temperature should approach a [*finite*]{} value at large distances. Correspondingly, the solution we are interested in represents an [*intermediate*]{} asymptotics [@Barenblatt]. If $u (x,\tau)$ is a solution of Eq. (\[3\]), then $\lambda^2\,u (x/\lambda, \tau)$ is also a solution for any $\lambda>0$. This property enables one to restore the $k$-dependence in the final results. S. Wolfram, [*The Mathematica Book*]{}, 3rd edition (Cambridge Univ. Press, Cambridge, 1996). Y. Kurzweil, E. Livne and B. Meerson (unpublished). P. Rosenau, Phys. Rev. Lett. [**74**]{}, 1056 (1995).
|
---
abstract: 'This paper presents current noise characterization of circular pad Schottky barrier diodes with guard rings. The diodes were fabricated from undopped semi-insulating GaAs, SIU-GaAs, at the University of Glasgow. Current noise spectra were obtained for the detectors for two pad sizes, with reverse bias applied. Three measurements were also made on one of the detectors under forward bias. The noise spectra show an excess noise component, with a low frequency corner at less than 1kHz, and a flat region at higher frequencies. The magnitude of the white noise is approximately half that expected from shot noise theory for the given leakage currents. A fall in the magnitude of the noise was observed at 20kHz which is attributed to the dielectric relaxation time of the material.'
title: '**Noise Spectra of SIU-GaAs Pad Detectors With Guard Rings**'
---
[GLAS–PPE/95–06]{}
R.L. Bates S. D’Auria S.J. Gowdy V. O’Shea C. Raine K.M. Smith A. Longoni G. Bertuccio G. De Geronimo
*Presented by R.L. Bates at the $4^{th}$ workshop on GaAs detectors and related compounds*
*San Miniato (Italy) 19-21 March 1995*
Noise theory
============
A detailed discussion of noise theory is given in references 1 and 2.
For any quantity [*X(t)*]{} that exhibits noise, the noise power within a unit bandwidth or power spectral density [*S$_{X}$(f)*]{} is defined as $$\label{sdef}
S_{X}(f) = \lim_{T\to\infty} { {\overline{ 2|{\it X}(f)|^{2} }} \over{T} }$$ where [*X(f)*]{} is the Fourier transform of [*X(t)*]{}, and ${\overline{{|{\it X}(f)|}}}$ is an ensemble average. In the simplest case where the transitions that cause the noise are described by equation (\[equan3\]), where $\tau$ is the lifetime of the fluctuation causing interaction, the spectral density is given by equation (\[equan4\]). $$\label{equan3}
{-d \Delta X\over dt} ={ \Delta X \over \tau}$$ $$\label{equan4}
S_{X} (f) = {<(\Delta X)^{2}> 4\tau \over (1+(2\pi f \tau)^2)}$$ From simple statistical considerations $<$($\Delta~X$)$^{2}>$ can usually be found, for example in the case of number fluctuations it is given by Poisson statistics.
This equation is very general with the condition imposed that the interactions of the electrons are independent. For small fluctuations this is indeed true and thus the Lorentzian spectrum (equation (\[equan4\])) appears often. At low frequencies ($f\tau$ $<<$1) the spectrum is white, that is independent of frequency, while at high frequencies ($f\tau$ $>>$1) it varies as 1/$f^{2}$, and its half power point is at f=1/(2$\pi\tau$).
Equivalent Noise Generators
---------------------------
In a two terminal network, noise in a frequency interval $\Delta~f$ may be characterized using either an equivalent e.m.f. generator $\sqrt{S_{v}(f)}$ in series with the device or a current generator $\sqrt{S_{i}(f)}$ in parallel. An equivalent noise resistance, $R_{n}$, and an equivalent noise current, $I_{eq}$, may be defined $$\label{noiseres}
S_{v}(f) = 4kTR_{n}
\ , \ S_{i}(f) = 2eI_{eq}$$ where $T$ is the room temperature and $k$ is Boltzmann’s constant.
The equivalent circuit for a detector and amplifier system is shown in figure \[eqcir\] and the noise components for an FET amplifier input stage are given below. $$\label{serisenoi}
S_{v}(f) = N_{v}+ {A_{f}\over{f}}
\ , \
N_{v} = {3\over{2}}{4kT\over{g_{m}}}$$ $$\label{parallelnoi}
S_{i}(f) = N_{p} + {B_{f}\over{f}}
\ , \
N_{p} = 2e(I_{det}+I_{tran}) + {4kT\over{R_{f}}}$$ where $g_{m}$ is the transistor transconductance. The equivalent noise charge of a circuit with shaping time $\tau$, is then $$\label{enc}
ENC^{2} = S_{1} \ {1\over{\tau}} \ {C_{in}}^{2} \ N_{v} \
+ \ S_{2} \ \tau \ N_{p}
+ \ {\em O}({1\over{f}}) \$$ where $S_{1}$ and $S_{2}$ depend upon the type of shaping used. Thus an an optimum shaping time exists. If $S_{1}$ and $S_{2}$ are of similar magnitude and the leakage current term dominates, a short shaping time is required. The minimum obtainable equivalent noise charge may be limited by the [*1/f*]{} noise if this term is large.
Types of noise sources
----------------------
There are four noise source classifications in semiconductors: thermal, shot, generation-recombination, and modulation noise. The first three are well understood while the origin of the fourth with regard to semiconductors is less well so.
Thermal noise is a white noise source whose origin is based on fundamental thermodynamic physical laws. For a semiconductor of resistance [*R*]{} the spectral current and voltage noise densities are $$\label{equan5}
S_{i}={4kT\over{R}} \ \ \ S_{v}={4kTR}$$
Shot noise is due to the discreteness of the charge carriers and is related to the statistical nature of their injection into the semiconductor over a Schottky barrier. The origin of the spectral density may be found by considering Carson’s Theorem [@Ziel]. This states that for a diode with reverse current [*I*]{} which equals $e<n>$, where [*n*]{} is the spontaneously fluctuating number of electrons that cross the barrier per second, the spectral density is given by $$\label{equan6}
S_{i}(f) = 2e^{2}<n> = 2eI$$
It can be seen that the shot noise is linearly dependent on the frequency of the electrons crossing the barrier and quadratically dependent on the charge of the pulse that the electron creates in an external circuit. The spectrum is white because of the very short transit time of the carriers across the barrier, where the barrier includes the space charge region. For the barrier to show full shot noise at all frequencies two conditions must be met. The first is that when charge carriers are injected into the material, space-charge neutrality is re-established in a very short time, given by the material dielectric relaxation time $\epsilon\rho$ [@sze], where $\rho$ is the resistivity of the material. For SIU-GaAs this is of the order of $10^{-5}$s. The second is that each current pulse should be able to displace a charge equivalent to that of one electron in the external circuit, that is to say a low trap density is required at the metal-semiconductor interface. Generation-recombination noise can be understood by considering a semiconductor with a number of traps. The continuous trapping and de-trapping of the charge carriers causes a fluctuation in the number of carriers in the conduction and valence bands. The transitions are described by equation (\[equan3\]) and thus the noise spectral density is Lorentzian in nature, with a corner frequency given by $\tau _{n}$, the lifetime of the electrons in the conduction band. The current noise density was calculated by van Vliet [@boer] to be $$\label{equan7}
{S_{n} (f) ={ <(\Delta n)^{2}> 4 \tau_{n} \over (1+(2\pi f \tau_{n})^2)} }$$ where $<(\Delta n)^{2}> \propto I^{2} $. It should be noted that in observed spectra the corner frequency of this noise is dispersed, implying a wide distribution of lifetimes. If the current that flows through a device is due to generation then the expression for the noise present will be the generation-recombination one rather than the expression for the shot noise defined in equation (\[equan6\]). However in most Schottky junction devices the applied voltage will increase the injection rate and the noise will show a very close resemblance to shot noise.
At low frequencies an excess noise spectral density with a [*1/f*]{} amplitude dependence is observed in semiconductors, which is known as modulation noise. As stated earlier the exact cause of this noise is still not understood.
The Diodes
==========
The material used for the detectors was 200$\mu$m thick semi-insulating GaAs. The detector fabrication was performed at the University of Glasgow. The detectors were designed with a circular pad contact and guard ring on top of the substrate. The bottom contact was a uniform contact that spread to the edge of the substrate. Two geometries were used, one with a pad diameter of 2mm the other of 3mm; the width of the guard was 200$\mu$m and the pad guard separation was 10$\mu$m for both diameters. The pad and guard metallization layers were identical and rectifying in nature. The reverse contact had a different recipe but was also rectifying, however its saturation current was larger than the pad’s due to its larger surface area. In the following, the direction of the detector bias is quoted with respect to that of the pad. Thus reverse bias means that the pad is at a negative potential with respect to the back contact.
The [*I-V*]{} curves obtained for the diodes in reverse and forward bias are shown in figure \[figiv\].
[c]{}\
The purpose of the guard ring was to ensure that the current through the pad remained constant once in saturation. This was observed for all the samples, with the current that flowed through the pad, in the bias range 60 to 200 volts, increasing by only a few percent. For a given bias the current densities of both pads were the same and equalled 25nA/mm$^{2}$. The results showed that the saturation current density that flowed through the pad under forward bias was the same as that obtained in reverse bias.
Measurement Procedure
=====================
To measure the current noise of the diodes under an applied bias an amplifier designed by the Politecnico di Milano was used [@milano]. This had a large bandwidth ($\omega_{-3dB}\simeq 20kHz$), high gain ($\simeq 86dB$), low noise ($\sqrt{\overline{v_{ni}^{2}}}\simeq 1.
8nV / \sqrt{Hz} @ 1kHz $) and an adjustable input bias offset. The amplifier had a large input impedance ($>$1G$\Omega$) and a low output impedance ($<150\Omega$). This amplifier was used as an operational amplifier with a feedback resistor $R_{f}$ and a feedback capacitance $C_{f}$. The output of the amplifier was passed through the high impedance input of an AC coupled oscilloscope and out through the ’scope’s $50\Omega$ output. The ’scope thus acted as a low noise voltage amplifier. The output of the ’scope was sent to a HP signal analyser with $50\Omega$ input impedance. The feedback resistor was chosen to maximize the gain of the amplifier (large $R_{f}$), minimize the background noise due to the thermal noise of the feedback resistor (large $R_{f}$, see equation (\[equan5\])) and to make sure that the amplifier did not saturate (limits $R_{f}$).
Before the current noise density spectra of the diodes were measured the transfer function and the background noise of the amplifier were determined. The transfer function was required so that the measured voltage noise could be referred back to the input as a current noise. Background noise measurements were made to show that the noise contribution from the ’scope was negligible and that the overall background noise present in normal operation was close to the thermal noise of the feedback resistor.
Noise spectra were obtained for both diodes, at zero bias, under reverse bias, at the same leakage current for both diodes and under forward bias for the small diode at three leakage currents equivalent to those used for the reverse bias case.
Results
=======
In this section the current noise spectral densities of the diodes are shown and some tentative interpretations of the results are proposed.
At zero bias the measured noise is the thermal noise of both the feedback resistor and the diode. From equation (\[noiseres\]) the equivalent noise resistances of the diodes were found which with the use of the diode dimensions enabled resistivities for the material of 12.4M$\Omega$cm and 18.3M$\Omega$cm to be obtained. Both values are of the correct order of magnitude.
Figure \[61\] shows a typical noise spectrum. All the spectra are white at frequencies above the corner frequency of the excess noise. The value of the current noise density is less that that expected from equation (\[equan6\]). At a frequency of 10kHz the noise over the measured leakage current range is between 1.7 and 2.3 times less than the simple shot noise theory. At frequencies between 20kHz and 30kHz there is a corner in the white noise.
If the leakage current is due to the thermionic emission of carriers over the Schottky barrier then the noise should be shot noise. A simple model can be devised to explain the shot noise spectra obtained, with reference to equation (\[equan6\]) and figure \[simpshot\]. One can say that if an electron is trapped and at a later instant released then this electron will produce two pulses of charge whose sum is equal to [*e*]{}. If the electron is captured after producing a charge pulse equal to ${1\over{2}}~e
$ in the external circuit and then released, on average the frequency of charge pulses produced by the emitted electrons is doubled while the charge is halved. The shot noise spectral density is thus reduced by a factor of two. Shot noise should be white due to the fast transit time of the electrons across the high field region behind the reversed biased Schottky junction, however, this is not observed. The corner frequency at 30kHz can be explained with reference to the condition imposed that charge neutrality is obtained in a time given by the dielectric relaxation time. Assuming that the noise source is Lorentzian, the corner frequency corresponds to the reciprocal of this time constant. Thus a lifetime of the order of 10$^{-5}$s is required, which is equivalent to the dielectric relaxation time in SIU-GaAs.
If the major contribution to the reverse bias leakage current is the generation current, rather than the injection of carriers over the Schottky barrier then the noise is due to generation-recombination noise from the bulk and not thermionic emission at the barrier. The noise due to generation has a characteristic time constant, the lifetime of the generated electron in the conduction band, which is inversely proportional to the corner frequency of the spectrum. If the transport mechanism is said to be by relaxation processes then the lifetime, $\tau_{d}$ in equation (\[equan7\]), is the dielectric relaxation time. Although, in principle, the dependence of the measured spectral density on the leakage current should help separate the two causes of the white noise, no clear dependence was observed.
The final part of the diode noise spectrum is the low frequency excess noise. This is seen for measurements made on the small diode for currents up to 70nA (75V bias) as approximately [*1/f*]{} noise which meets the white noise at a frequency of 500Hz. At 10Hz the excess noise is an order of magnitude greater than the white noise component of the spectrum. This is also observed in the large diode for currents up to 170nA (50V bias).
At larger currents (above 71nA for the small diode) a second low frequency noise component appears which has a much steeper frequency dependence, $1/f^{\alpha}$ where $\alpha$ is close to 3. At 10Hz the excess noise is now almost two orders of magnitude larger than the white noise. (figure \[steepnoi\]).
The magnitude of the excess noise is seen to increase with bias towards the ‘breakdown’ of the diode. Here the corner frequency of the excess noise extends to much higher frequencies; 1kHz @ 230V, and $>$100kHz @ 240V for the small diode. The magnitude of the noise increases dramatically, to reach 3$\times$10$^{-21}$ A$^{2}$/Hz at 10Hz at 230V for the small diode (see figures \[excess1\] and \[excess2\]). At a bias of 200V the electric field extends across the sample to the forward biased contact which causes charge injection and thus the dramatic increase in the measured noise which occurred between 230V and 240V.
Noise spectra for the forward biased diode are similar to those obtained in reverse bias as shown in figure \[figrf75\]. The white noise component is approximately the same, the high frequency corner is present, but no low frequency excess noise is present in the forward biased case. The reason why no excess noise is seen for the forward biased diode is not clear.
Conclusions and Future Work
===========================
The noise measurements show three distinct regions in the spectra, the first being the low frequency, $1/f$, excess noise which has a corner frequency of 500Hz for bias voltages below 75V. Above this the excess noise has a second component which is steeper, being almost $1/f^{3}$ in nature, and has a corner frequency of 50Hz which increase with bias. As the diode approaches breakdown the corner frequency and the magnitude of the excess noise increases dramatically due to injection at the forward biased contact. The second feature of the spectra is the lower than expected value of the shot noise, being about half that of simple theory. This is due to trapping of the charge carriers in the bulk which reduces the amplitude of the individual pulses that the carriers produce while increasing their frequency. The third region is the corner frequency of the white noise between 20-30kHz. This is noted to be close to the reciprocal of the dielectric relaxation time for SIU-GaAs.
Further noise measurements need to be made on more samples using different types of contacts to see if there is any correlation between the noise of the sample and the properties of the material and fabrication. Diodes that have been irradiated also need to be tested to improve the present understanding of the effects of irradiation. The technique can also be extended to examine strip detectors made from GaAs. The biasing structure used at present utilizes a punch-through mechanism via a MSM diode. There is a possibility that this may introduce more noise and it is important to understand if this is true.
Acknowledgements
================
The authors wish to express their appreciation to A. Meikle, F. Doherty and F. McDevitt of the University of Glasgow for their fabrication skills.
[99]{} Van der Ziel: [*Fluctuation phenomena in semiconductors*]{}\
Butterworths Scientific Pub., London, (1959). M.J. Buckingham: [*Noise in electronic devices and systems*]{}\
John Wiley and Sons, London, (1983) S. M. Sze: [*Physics of semiconductor devices*]{}\
J. Wiley, New York, (1991). K. W. Boer: [*Survey of Semiconductor Physics*]{}\
Van Nostrand Reinhold, New York, (1990). A. Longoni et al.: [*To be published.*]{}\
Department of Physics, Politecnico di Milano, Milano, Italy.
|
---
abstract: 'We present measurements of the velocity distribution of calcium atoms in an atomic beam generated using a dual-stage laser back-ablation apparatus. Distributions are measured using a velocity selective Doppler time-of-flight technique. They are Boltzmann-like with rms velocities corresponding to temperatures above the melting point for calcium. Contrary to a recent report in the literature, this method does not generate a sub-thermal atomic beam.'
author:
- 'A. Denning, A. Booth, S. Lee, M. Ammonson, and S. D. Bergeson'
title: 'Velocity distribution measurements in atomic beams generated using laser induced back-ablation'
---
Introduction
============
A recent publication reported a sub-thermal atomic beam generated by laser induced back-ablation [@alti05]. In this method, a thin film that has been deposited onto a transparent substrate is ablated by laser illumination through the substrate. Atomic beams generated this way can be collimated or focused depending on the geometry of the ablation laser [@kallen89; @bullock99]. This has obvious advantages for spectroscopy and atomic physics experiments.
The sub-thermal velocities reported in Ref. [@alti05] are surprising because there is no known mechanism to explain this. Ejected plumes from laser ablation sources have been studied extensively [@tallents80; @albritton86; @dreyfus86; @wang91; @kools91; @rupp95; @chichkov96; @zhigilei97; @hansen97; @toftmann00]. While the details of ablation using ns-duration lasers are complicated, the general physical principles are well known. Laser light is absorbed by a target. The laser pulse energy raises the local temperature to the melting point and atoms evaporate from the surface. The temperature of the ablated material is determined by the thermal conductivity of the target material and the laser pulse duration. For very high intensity lasers, atoms in the ablated plume can be reach high ionization states. The electron transport properties can be nonlocal. In high density plasmas, significant recombination can also occur. Depending on the density and experimental conditions, shock fronts can also form in the expanding plasma.
Apparently, there are no velocity distribution measurements for atomic beams generated via back-ablation. Early work on back-ablated beams [@kallen89] concentrated on angular distributions and assumed the ion temperature was equal to the target melting temperature. Somewhat more recent work interferometrically measured the plume edge velocity in high-intensity picosecond laser back-ablated plasmas [@bullock99]. The recent work of Ref. [@alti05] used a time-of-flight laser deflection method to deduce the beam velocity.
In this paper we report new measurements of the velocity distribution in a calcium atomic beam generated by a dual-stage back-ablation atomic beam source. We measure the distribution using a velocity-selective Doppler time-of-flight method, similar to previously published work (see, for example, Ref. [@dreyfus86]). We find that at moderate intensities of the back-ablation laser, the distribution is Boltzmann-like. At the lowest intensities, when the calcium film is not completely ablated, the rms velocity distribution never falls below a thermal distribution at the melting temperature.
Experiment
==========
A schematic diagram of our experiment is shown in Fig. \[fig:schematic\]. The two-stage back-ablation source is similar to Ref. [@kallen89]. The front surface of a transparent sapphire substrate is coated by a thin film of calcium using regular laser ablation. This thin film is then ablated off of the substrate by illuminating the thin film through the back of the sapphire disk using another high intensity laser beam.
![\[fig:schematic\] A Schematic diagram of our dual-stage laser ablation experiment. An Nd:YAG laser beam is divided into two beams. One is used to ablate a thin calcium film onto a sapphire substrate. The other is used to generate the collimated calcium beam using back-ablation. A single-frequency probe laser at 423 nm excites specific velocity classes in the atomic beam, depending on the laser frequency. The ablation targets and associated optics are placed inside a vacuum system that is maintained using a turbo-molecular pump. ](schemat3.ps){width="3.2in"}
A 532 nm Nd:YAG laser with a 3 ns duration pulse is used for ablation. The laser pulse is divided into two beams. One is focused using a 90 mm focal length lens onto a calcium target. The target is mounted on a stepper motor. It is rotated and advanced in such a way that each laser shot ablates a new target region. Target is advanced back and forth repeatedly and the experiment is run for several hours and the surface of the ablated target is not smooth.
The ablated metal deposits a thin film on a nearby rotating sapphire disk. The thickness of the film can be adjusted by attenuating the ablation laser or by controlling the rotation speed of the sapphire disk. The disk has a 10 cm diameter and is rotated at a frequency of $\sim 0.1$ Hz. The calcium target is located 1.5 cm from the disk, and the ablated plume coats an area of approximately 3 cm diameter near the outer edge of the disk. Contrary to Ref. [@kallen89], the sapphire disk is continuously loaded. The thickness of the calcium film is determined by the angular velocity of the disk and the number of laser shots that coat the disk with ablated atoms while the disk covers an arc equivalent in length to the width of the ablating beam. The Nd:YAG laser operates at 10 Hz. The calcium film thickness results from approximately 30 laser shots as the disk rotates through the ablated plume. The second ablation laser beam is weakly focused to a Gaussian waist of 3 mm at the sapphire disk and back-illuminates the calcium thin film. At full power (30 mJ), the calcium film is completely ablated.
A probe laser beam at 423 nm crosses the calcium beam at an angle of 53$^{\rm o}$ at a distance $d$ from the sapphire disk (see Fig. \[fig:schematic\]). The probe laser beam is generated by frequency doubling an injection-locked ti:sapphire laser, and is described elsewhere [@cummings03]. The bandwidth of this laser is less than 1 MHz, and the laser frequency can be swept across the Doppler-broadened $4s^2 \; ^1S_0 \rightarrow 4s4p \; ^1P_1^{\rm o}$ calcium resonance line. Not shown in this figure is a calcium vapor cell in which we measure the calcium resonance transition using saturated absorption spectroscopy. This measurement provides an important check on the initial frequency of the probe laser beam.
The fluorescence signal is collected using a lens, measured using a 1P28 photomultiplier tube (PMT), and digitized and averaged using a fast oscilloscope. Special care is taken to ensure the linearity of the PMT response and that the detection bandwidth is high enough not to compromise the fluorescence signal. By imaging the intersection of the atomic and laser beams onto the PMT photocathode through an aperture, we optimize the fluorescence signal strength and minimize the scattered Nd:YAG laser light.
The wavelength of the probe laser beam is monitored using a wavemeter. For each probe laser wavelength setting we measure the laser-induced fluorescence and average over 100 laser shots. Typical fluorescence curves are shown in Fig. \[fig:signal\]. As we increase the detuning of the probe laser beam from resonance, the laser beam interacts with faster atoms and the peak of the fluorescence signal occurs earlier in time.
![\[fig:signal\] Measured (top panel) and simulated (bottom panel) laser-induced fluorescence signals from atoms in the atomic beam for a range of different probe laser beam detunings and $d=7.8$ cm. As the probe laser beam frequency detuning increases, the laser interacts with faster atoms and the fluorescence signal peak occurs earlier in time. The inset to each plot is the velocity distribution in the atomic beam as determined by the fluorescence signal.](flsum1d.eps "fig:"){width="3.2in"}\
![\[fig:signal\] Measured (top panel) and simulated (bottom panel) laser-induced fluorescence signals from atoms in the atomic beam for a range of different probe laser beam detunings and $d=7.8$ cm. As the probe laser beam frequency detuning increases, the laser interacts with faster atoms and the fluorescence signal peak occurs earlier in time. The inset to each plot is the velocity distribution in the atomic beam as determined by the fluorescence signal.](simsig1.eps "fig:"){width="3.2in"}
Analysis
========
An atom in the atomic beam will fluoresce when it is Doppler-shifted into resonance with the probe laser beam. The probability of fluorescing depends on both the atomic line shape and the velocity distribution. For an individual atom moving at some velocity $v$, the fluorescence signal will be a Doppler-shifted Lorentzian:
$$\begin{aligned}
{\cal L} & = & \frac{\gamma/2\pi}{[\nu_L - \nu_0(1 + v\;\cos\theta/c)]^2 + \gamma^2/4} \\
& = & \frac{\gamma/2\pi}{(\Delta - v\;\cos\theta/\lambda)^2 + \gamma^2/4}, \label{eqn:lorentz}\end{aligned}$$
where $\gamma$ is the full-width at half-maximum (FWHM), $\nu_L$ is the laser frequency, $\nu_0$ is the resonance frequency in the rest frame of the atom, $v$ is the atom velocity, $c$ is the speed of light, $\Delta = \nu_L - \nu_0$ is the laser detuning, $\theta$ is the angle between the atomic beam direction and the probe laser wave beam direction, and $\lambda$ is the transition wavelength. For the 423 nm resonance transition in calcium, the FWHM is 35 MHz. Other factors, such as power broadening of the atomic transition and divergence of the atomic beam, contribute additional width to the atomic transition. In our experiment, we set the laser detuning $\Delta$ and measure the fluorescence as a function of time.
We have performed a simulation of our experiment. At a time $t=0$, atoms with a particular velocity $v$ are launched from the sapphire disk. At a time $t_0 = d/v$ the atoms encounter the probe laser beam. If the width of the probe laser beam is $w$, the atoms spend a time $\tau = w/v$ in the laser beam. For simplicity, we assume that atoms scatter photons at a rate given by Eq. \[eqn:lorentz\] only while they are in the beam. This is a simplification to the Gaussian spatial profile of the probe laser beam. However, the intensity of the probe laser is approximately 10$\times$ the saturation intensity, making this flat-top approximation somewhat more realistic. The fluorescence signal from each velocity class is multiplied by the probability of finding that velocity in the Maxwell-Boltzmann distribution, $\exp(-v^2/2v_{th}^2)$. We launch atoms with a range of velocities and sum their fluorescence signal to simulate the total signal at a particular laser detuning $\Delta$.
Typical fluorescence data for $d=7.8$ cm are shown in the top panel of Fig. \[fig:signal\] for a range of laser detunings. As the laser detuning increases, the probe laser beam interacts with faster atoms and the fluorescence signal peaks at earlier times. The inset to each plot shows the derived velocity distribution. These distributions are determined from the peak of the fluorescence signal at each laser detuning. The velocity can be determined either from the wavemeter reading or from the time of flight. These methods agree to within the uncertainties of the wavemeter data.
The bottom panel of Fig. \[fig:signal\] shows the result of our simulation for $d=7.8$ cm. As mentioned previously, beam divergence and power broadening contribute to the width of the fluorescence signal at a given laser detuning. We approximate those effects by increasing the Lorentzian linewidth from the natural linewidth of 35 MHz to a power-broadened linewidth of 150 MHz. The probe laser beam is approximately 50 mW at 423 nm with a Gaussian waist around 1.5 mm. Based on our measurements of the fluorescence signals for different values of $d$, the atomic beam divergence appears to be less than 15 mrad. This is consistent with earlier studies of atomic beams generated by back-ablation [@kallen89; @bullock99].
The general agreement between the experiment and the simulation in Fig. \[fig:signal\] indicates that the velocity distribution in the atomic beam is generally Maxwellian. The thermal velocity $v_{th}=\sqrt{k_BT/m}$ in the simulation is 3000 m/s, corresponding to a temperature of $T = 44,000$ K $= 3.8$ eV. We have repeated these measurements for lower back-ablation laser intensity. We find that the rms velocity decreases, but so does the ablation efficiency. For our lowest intensity, the rms velocity appears to be 1000 m/s, corresponding to a temperature of $4,000$ K, well above the melting temperature. In this limit, the back-ablation does not completely ablate the calcium film, meaning that the thickness of the film increases over time.
These data show that for all back-ablation intensities, the longitudinal velocity in the beam corresponds to temperatures well above the melting temperature of the ablated metal. While the details of the model and the distributions can be argued, we see no evidence for a sub-thermal atomic beam as reported recently [@alti05].
Discussion
==========
Velocity-selective Doppler time-of-flight methods provide direct information about the velocity distribution in atomic beams. In particular, it can be used to determine both the speed and direction of atoms that cross the probe laser beam. In a compact beam apparatus, such as we use, this can be important.
![\[fig:bounce\] (color) Measurements of the velocity distribution in an un-optimized experimental setup. Top: False color plot of the fluorescence signal versus time and probe laser detuning. Anomalously slow signals appear at small positive and negative probe laser beam detunings. Anomalous signal also appears at large negative detunings (lower left portion of the image). These signals are due to a poor experimental design (see text) and are dramatically reduced in a more carefully configured experiment. Bottom: The total fluorescence signal summed over all laser detunings, from -2.5 GHz to +10 GHz.](pcolor1.eps "fig:"){width="3.2in"}\
![\[fig:bounce\] (color) Measurements of the velocity distribution in an un-optimized experimental setup. Top: False color plot of the fluorescence signal versus time and probe laser detuning. Anomalously slow signals appear at small positive and negative probe laser beam detunings. Anomalous signal also appears at large negative detunings (lower left portion of the image). These signals are due to a poor experimental design (see text) and are dramatically reduced in a more carefully configured experiment. Bottom: The total fluorescence signal summed over all laser detunings, from -2.5 GHz to +10 GHz.](tsignal1.eps "fig:"){width="3.2in"}
In an early iteration of our experiment, we found features of the velocity distribution that did not agree with a Maxwellian distribution. We measured anomalous fluorescence signals at negative and also small positive values of the laser detuning $\Delta$. A representation of that data is shown in Fig. \[fig:bounce\]. As we studied the data and apparatus more carefully, we found evidence of interaction between atoms from the solid calcium target ablation and the thin film back-ablation. It is possible that atoms moving in the backwards direction are generated by the transmitted back-ablation beam interacting with calcium atoms deposited on the beam block. This could explain the weak and very fast signal component at very early times at large negative detunings. The apparently slow atoms seen at small positive detunings could come from the interaction of the forwards and backwards traveling atomic beams, or from the interaction between atoms in the two different ablation sources. We find that in a more carefully designed experiment, such as shown in Fig. \[fig:schematic\], in which the atomic beam path is clear from obstruction, the anomalous signal is significantly reduced.
Without a velocity-selective probe, it is impossible to accurately characterize the velocity profile of the beam. If we mistakenly assume that the late peak at 0.5 ms in the bottom panel of Fig. \[fig:bounce\] corresponds to atoms generated by the back ablation laser at the disk, we derive an atomic velocity near 80 m/s and a temperature of 30 K. However, more careful measurements show that such a conclusion is in error.
Conclusion
==========
We report measurements of longitudinal velocity distributions of calcium atoms in an atomic beam generated using a two-stage back-ablation experiment. We use a Doppler-selective time-of-flight method to determine the distributions. Fluorescence signals are reproduced in a simple simulation assuming a Maxwellian distribution at a temperature well above the melting temperature. We find no evidence for sub-thermal velocities as recently reported in the literature. Moreover, we point out that there is no know physical mechanism for producing a sub-thermal atomic beam by laser ablation.
This work was supported in part by the Research Corporation, Brigham Young University, and by the National Science Foundation (PHY-0601699).
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|
---
abstract: 'We present a numerical study of Rayleigh-Bénard convection disturbed by a longitudinal wind. Our results show that under the action of the wind, the vertical heat flux through the cell initially decreases, due to the mechanism of plumes-sweeping, and then increases again when turbulent forced convection dominates over the buoyancy. As a result, the Nusselt number is a non-monotonic function of the shear Reynolds number. We provide a simple model that captures with good accuracy all the dynamical regimes observed. We expect that our findings can lead the way to a more fundamental understanding of the of the complex interplay between mean-wind and plumes ejection in the Rayleigh-Bénard phenomenology.'
author:
- Andrea Scagliarini
- Ármann Gylfason
- Federico Toschi
title: |
Heat flux scaling in turbulent Rayleigh-Bénard convection\
with an imposed longitudinal wind
---
Thermal convection plays an important role in many geophysical, environmental and industrial flows, such as in the Earth’s mantle, in the atmosphere, in the oceans, to name but a few relevant examples. In particular, the idealized case of Rayleigh-Bénard (RB) convection occurring in a layer of fluid confined between two differentially heated parallel plates under a constant gravitational field has been extensively studied [@kadanoff; @ahlersrev; @lohsexia; @schumachilla]. However, in several real-life situations, the picture can be much more complex with horizontal winds perturbing natural convection. In the atmosphere, for instance, this competition plays a crucial role in the formation of thermoconvective storms [@bluestein]. On the other side buoyancy effects can be relevant in a number of industrial processes based on forced convection, such as coiled heat exchangers [@steenhoven]. Similarly, a combination of forced and natural convection is present in indoor ventilation applications [@linden_review; @bailon12; @shishkina12].
![(top panel) Snapshot of the temperature field for the pure RB case ($Re_{\tau}=0$) at $Ra=1.3 \times 10^7$. (bottom) Snapshot of the temperature field for the $Ra=1.3 \times 10^7$ and $Re_{\tau}=92$. Notice that, unlike figure \[fig0a\], where a buoyant plume detaching from the bottom boundary layer can easily enter the bulk up to the top plate while, here plumes are considerably distorted in the direction of the wind.](./figure0a-crop.pdf "fig:"){width=".73\hsize"} \[fig0\] ![(top panel) Snapshot of the temperature field for the pure RB case ($Re_{\tau}=0$) at $Ra=1.3 \times 10^7$. (bottom) Snapshot of the temperature field for the $Ra=1.3 \times 10^7$ and $Re_{\tau}=92$. Notice that, unlike figure \[fig0a\], where a buoyant plume detaching from the bottom boundary layer can easily enter the bulk up to the top plate while, here plumes are considerably distorted in the direction of the wind.](./figure0b-crop.pdf "fig:"){width=".73\hsize"}
According to the standard picture at the basis of the existing models for the scaling laws of the heat flux [@shraiman; @GL00; @GL04], the RB system is characterized by the multi-scale coupling between large-scale circulation (mean wind) and detaching thermal structures from the boundary layers at the walls (plumes). Besides the above mentioned motivations, applying a mean wind to a natural convection setup can shed light on the effectes of bulk flow on the boundary layer dynamics, thus helping to better understand one of the most intriguing feature of RB convection.\
In this Letter we report on a numerical study of RB convection with an imposed constant horizontal pressure gradient, orthogonal to gravity, that induces the wind (the so-called Poiseuille-Rayleigh-Bénard (PRB) flow setup [@gage; @carriere]). We show that the heat transfer from the walls can be dominated by either the buoyancy or by the “forced” convection and that the interplay of the two mechanisms gives rise to a non-trivial dependence of the Nusselt number, $Nu$, on the parameter space that is spanned by the Rayleigh, $Ra$, and shear Reynolds numbers, $Re_{\tau}$ (quantifying, respectively, the intensity of buoyancy and of the pressure gradient relatively to viscous forces).\
Our main result consists in the observation that, taken a standard RB system as reference, $Nu$ initially decreases and then, when the dynamics is completely dominated by the forced convection regime, it increases again with $Re_{\tau}$. A phenomenological explanation for this behaviour is provided together with discussions on the possible implications for the modelling of the $Nu \; vs \; Ra$ relation in pure natural convection setup.
The equations of motion for the fluid velocity, ${\mathbf{u}}$, and temperature, $T$, are: $$\label{eq:NS}
\partial_t {\mathbf{u}}+ \mathbf{u}\cdot \nabla{\mathbf{u}}= -\frac{1}{\rho}\nabla P +
\nu \nabla^2 {\mathbf{u}}+ \alpha\mathbf{g}T + \mathbf{f}$$ $$\label{eq:temp}
\partial_t T + \mathbf{u}\cdot \nabla T = \kappa \nabla^2 T,$$ in addition to the incompressibility condition, $\nabla \cdot
{\mathbf{u}}$. The properties of the fluid are $\rho$ the (assumed constant) fluid density, $\nu$ the kinematic viscosity, $\alpha$ the thermal expansion coefficient, and $\kappa$ the thermal diffusivity. $P$ is the pressure field, $\mathbf{g}=g \hat{z}$ the gravity and $\mathbf{f}$ a forcing term of the form $\mathbf{f} = (F/\rho) \hat{x}
\equiv \tilde{F} \hat{x}$ ($\hat{x}$ is the direction parallel to the walls, or stream-wise direction). Equations (\[eq:NS\]) and (\[eq:temp\]) are evolved using a 3d thermal lattice Boltzmann algorithm [@succi; @wolf-gladrow] with two probability densities (for density/momentum and for temperature, respectively). As mentioned in the introduction, to characterize the dynamics we need two parameters: the Rayleigh number, $Ra$, quantifying the strength of buoyancy (with respect to viscous forces), $$Ra= \frac{\alpha g \Delta H^3}{\nu \kappa},$$ (where $\Delta = T_{hot} - T_{cold}$ is the temperature drop across the cell and $H$ the cell height), and the shear Reynolds number $Re_{\tau}$, $$Re_{\tau} = {H \over {2\nu}}\sqrt{{\tilde{F} H} \over 2}.$$ We performed several runs (in a computational box of size $256 \times 128
\times 128$, uniform grid; see figure \[fig0\] for snapshots of the temperature field in the simulation cell)), exploring the two dimensional parameter space $(Ra, Re_{\tau})$, within the ranges $Ra \in [0; 1.3 \times 10^7]$ and $Re_{\tau} \in [0; 205]$; the Prandtl number $Pr=\nu/\kappa$ is kept fixed and equal to one.
A typical key question in RB studies is how the dimensionless heat flux through the cell, $Nu$, varies as a function of the Rayleigh number: $$\label{eq:nudef}
Nu(z) = \frac{\overline{u_z T}(z) - \kappa \partial_z
\overline{T}(z)}{\kappa \frac{\Delta}{H}} = const \equiv Nu$$ with $Ra$; here and hereafter the overline indicates a spatial (over planes $z=const$) and temporal (over the statistically stationary state) average. The second and third equalities (which state that $Nu$ is constant with $z$) follow from taking the average of equation (\[eq:temp\]). In our setup in addition to buoyancy there is the longitudinal pressure gradient which affects the heat flux. We therefore focus on the dependence of $Nu$ on the two-dimensional parameter space $(Ra, Re_{\tau})$; in figure \[fig1\] we plot $Nu$ as a function of $Re_{\tau}$ for various fixed $Ra$. We find that, for moderate/high $Ra$, the effect of the lateral wind is to quench the buoyancy driven convection, and thus $Nu$ decreases with $Re_{\tau}$. For very low $Ra$, below the critical Rayleigh number $Ra_c$, the dynamics of the flow is instead completely dominated by the forced convection and thus $Nu$ increases with the $Re_{\tau}$ (In Fig. \[fig1\] we show the $Ra=0$ case). Correspondingly, at increasing $Re_{\tau}$ the mean temperature profiles (see figure \[fig2\]) show a bending in the bulk and a decrease of the gradient in the boundary layer.
-0.55cm ![Nusselt number as a function of $Re_{\tau}$ for $Ra=0,
8.125 \times 10^5, 6.5 \times 10^6, 1.3 \times 10^7$ (from bottom to top, symbols respectively $\blacktriangledown$, $\blacktriangle$, $\blacksquare$, $\bullet$). Notice the non-monotonic behaviour for $Ra>0$: for low-to-moderate $Re_{\tau}$ a decrease of $Nu$ is observed due to plumes-sweeping by the mean wind; at higher $Re_{\tau}$, when bouyancy is basically irrelevant with respect to the channel flow, $Nu$ increases again, as expected for turbulent forced convection. The solid lines are the predictions from equation (\[eq:model\]) obtained using $A_1=2.1$ and $A_2=0.045$, values that have been chosen to best fit the case $Ra =6.5 \times 10^6$ ($\blacksquare$ symbols). The yellow band around the solid lines show the prediction of the model for a variation of the parameters $A_1$ and $A_2$ of $\pm 10\%$. The dashed lines represent the fall-off for small $Re_{\tau}$ provided by equation (\[eq:Ret4\]), with $A_3 = 1$ for all the three $Ra$. The reader is referred to the text for the details. (Inset) Shear Reynolds number $Re_{\tau}^{\ast}$ corresponding to the crossover to the $Nu \sim Re_{\tau}^{-3/2}$ regime as function of Rayleigh number. The dashed line is the $Re_{\tau}^{\ast} \sim Ra^{1/4}$ power law, equation (\[eq:crossover\]).[]{data-label="fig1"}](./figure1.eps "fig:")
-0.55cm ![Normalized temperature profiles $\Theta(z) \equiv
\frac{\overline{T}(z) - T_m}{\Delta}$ ($T_m=(T_{hot}+T_{cold})/2$ being the mean temperature) for various $Re_{\tau}$ at a fixed $Ra=6.5 \times 10^6$. Notice the bending of the profile in the bulk in comparison with the usual ’thermal short-cut’ for $Re_{\tau} = 0$ (i.e., no lateral wind).[]{data-label="fig2"}](./figure2.eps "fig:")
Our interpretation of these observations is that, for small $Re_{\tau}$/high $Ra$ (i.e. in the natural convection dominated regime), the wind acts essentially sweeping away thermal plumes (which are mixed and lose their coherence closer to the walls) and hence the heat flux is depleted. Increasing $Re_{\tau}$ more and more we eventually reach a state where buoyancy becomes irrelevant. Here, $Nu$ starts to increase again by resuspension of temperature puffs in the bulk due to bursts from the wall emerging because of the turbulent channel flow. To give an indication of the validity of such a conjecture we have measured the following quantity: $$\label{eq:aniso}
\phi_{\ell}(z) \stackrel{def}{=} \frac{\overline{(\delta_{\ell}
u_z)^2}}{\overline{(\delta_{\ell} u_x)^2}},$$ where $\delta_{\ell} u_i \equiv u_i(x+\ell,y,z;t)- u_i(x,y,z;t)$. The observable (\[eq:aniso\]) is the ratio of a generalized second order transverse over longitudinal structure function and, as such, it serves as a sort of scale-dependent anisotropy indicator: a large value of $\phi_{\ell}(z)$ means a coherent motion in the wall-normal direction. In figure \[fig3\] we plot $\phi_{\ell}(z)$ on a large scale ($\ell \approx H$) and on a scale of the order of the thermal boundary layer thickness ($\ell = \lambda_{\theta}$), which gives an estimate of a characteristic size of plumes, for natural convection ($Ra=6.5 \times 10^6$, $Re_{\tau}=0$) and for a case with the wind ($Ra=6.5 \times 10^6$, $Re_{\tau}=205$). For the pure RB case $\phi_{\ell \approx H}(z)$ grows to large values in the bulk, due to the thermal wind, while $\phi_{\ell = \lambda_{\theta}}(z)$ goes to the isotropic value $\phi \approx 2$ in the bulk and it is larger than $\phi_{\ell \approx H }(z)$ close to the wall, pointing out the presence of detaching plumes. The same quantity $\phi_{\ell =
\lambda_{\theta}}(z)$ in the wall-proximal region is significantly smaller for $Re_{\tau}=205$, indicating the depletion of plumes ejection.
With this picture in mind we are now going to build a model to recover the numerical findings. Our argument goes as follows. As shown in figure \[fig2\], under the action of the lateral wind the temperature profile ceases to be flat in the bulk. This permits us to write a first order closure for the turbulent heat flux of the kind: $$\label{eq:closure}
\overline{u_z T} = - \kappa_T \partial_z \overline{T},$$
-0.55cm ![The anisotropy indicator defined in Eqn. (\[eq:aniso\]) as function of the cell height for pure RB ($Re_{\tau}=0$) and for $Re_{\tau}=205$ and for two separation in $x$, at $Ra = 6.5 \times
10^6$.[]{data-label="fig3"}](./figure3.eps "fig:")
where $\kappa_T$ is a turbulent diffusivity. When writing (\[eq:closure\]), where $\kappa_T$ is constant with $z$, we are implicitly restricting ourselves to the bulk region (where the mean temperature gradient is basically constant); we are allowed to do that by (\[eq:nudef\]), i.e. the constancy of the heat flux through planes parallel to the walls. The Nusselt number will assume the form $$\label{eq:nusselt}
Nu \sim \left(1 + \frac{\kappa_T}{\kappa} \right)\frac{\left| \partial_z \overline{T} \right|}{(\Delta/H)}.$$ We consider that two types of structures contribute to turbulent diffusion, namely buoyant plumes ($\kappa_T^{(P)}$) and bursts ($\kappa_T^{(B)}$, triggered by the turbulent channel flow), so that we may write $$\kappa_T = \kappa_T^{(P)}+ \kappa_T^{(B)}.$$ As previously discussed we attribute the heat flux reduction to the sweeping of plumes by the wind; we model this saying that the plume looses its coherence (or else, it releases its heat content) after travelling a distance from the wall of the order of the kinetic boundary layer thickness ($\lambda_u$), i.e. we suggest that we can adopt a Prandtl mixing length ($\ell_m$) theory kind of approach, using $$\ell_m \sim \lambda_u;$$ the latter relation should be interpreted as a scaling (or proportionality) relation rather than an order of magnitude. The characteristic velocity of a rising plume reaching a height $\sim
\ell_m$ can be estimated as $u \sim \sqrt{\alpha g \Delta \ell_m}$ [@footnote1], hence the contribution to the turbulent diffusion will be $$\kappa_T^{(P)} = \sqrt{\alpha g \Delta} \lambda_u^{3/2}$$ and assuming a laminar boundary layer of Blasius type of thickness [@landau] $$\label{eq:blasius}
\lambda_u \sim \frac{H}{Re_{\tau}}$$ we get $$\label{eq:Kplumes}
\kappa_T^{(P)}\sim \sqrt{\alpha g \Delta} \frac{H^{3/2}}{Re_{\tau}^{3/2}}.$$ For a turbulent burst one can also assume that $\ell_m \sim
\lambda_u$, but the expression for the characteristic advecting velocity requires some more care. In a pure forced convection setup (or in our case when the wind is dominant) there is no buoyancy, so we cannot use the expression of the free-fall velocity; instead convection is driven by turbulent fluctuations from the wall. Invoking again the mixing length theory for a first order closure for the velocity we can write $u_z \sim \ell_m \partial_z \overline{U}_x$, whence $$\kappa_T^{(B)} \sim \ell_m^2 \partial_z \overline{U}_x \sim \lambda_u^2 \partial_z \overline{U}_x;$$ estimating the shear as $\partial_z U_x \sim U_c/\lambda_u$ ($U_c$ being the centreline velocity) we get $$\kappa_T^{(B)}\sim \lambda_u U_c.$$ In the limited range of $Re_{\tau}$ that we span it is reasonable to assume that the friction coefficient goes as $C_f \sim
Re_{\tau}^{-2}$, hence that $U_c$ scale as $U_c \sim (\nu / H )Re_{\tau}^2$ [@footnote2]. Inserting this scaling law together with the relation (\[eq:blasius\]) inside the expression for $\kappa_T^{(B)}
$ we obtain $$\label{eq:Kbursts}
\kappa_T^{(B)} \sim \nu \cdot Re_{\tau}.$$ Putting the expressions (\[eq:Kplumes\]) and (\[eq:Kbursts\]) inside equation (\[eq:nusselt\]) we end up with $$Nu -1 \sim \left( A_1\frac{\sqrt{\alpha g \Delta}
H^{3/2}}{\kappa Re_{\tau}^{3/2}} + A_2 \frac{\nu}{\kappa} Re_{\tau} \right)
\frac{\left| \partial_z \overline{T} \right|}{(\Delta/H)},$$ which can be recast, introducing the dimensionless numbers $Ra$ and $Pr$, into the following form: $$\label{eq:model}
Nu -1\sim \left( A_1\frac{Ra^{1/2}Pr^{1/2}}{Re_{\tau}^{3/2}} + A_2 Pr
Re_{\tau} \right)
\frac{\left| \partial_z \overline{T} \right|}{(\Delta/H)},$$ where $A_1$ and $A_2$ are two free parameters of the model. Some comments on equation (\[eq:model\]) are in order. Firstly, it reproduces the non-monotonic dependence of the heat flux, $Nu$, on the applied wind, $Re_{\tau}$, and it turns out to be in fair agreement with the numerical data (see figure \[fig1\]). Secondly, it provides an argument for the scaling $Nu \sim Re_{\tau}$ for the case of pure forced convection ($Ra=0$, see figure \[fig1\]). It is interesting to note that, for very small $Re_{\tau}$, our model would give a scaling $Nu \sim Ra^{1/2}$, i.e. what expected for Kraichnan’s [*ultimate regime*]{} of convection [@ahlersrev].
The phenomenology behind it suggests that the lower heat flux observed in the standard RB convection (with respect to $Ra^{1/2}$) may be seen as the result of a negative feedback of the shear, due to the large scale circulation, on the plumes detaching from the boundary layer. Indeed, if we imagine the Nusselt number to follow a Kraichnan scaling on an [*effective*]{} Rayleigh $Ra_{eff}$, renormalized by turbulent viscosity and thermal diffusivity (behaving as (\[eq:Kbursts\])), that is $$Ra_{eff} = \frac{Ra}{(\nu_T/\nu)(\kappa_T/\kappa)},$$ we end up with the following relation $$\label{eq:effsca}
Nu \sim Ra_{eff}^{1/2} \equiv \frac{Ra^{1/2}}{Re_{\tau}}.$$ If we now insert into (\[eq:effsca\]) the ultimate regime scaling for Reynolds $Re_{\tau} \sim Ra^{1/4}$ [@footnote3], we obtain $$Nu \sim Ra^{1/4},$$ a well known scaling, predicted theoretically and found in a vast number of experiments (see [@GL00] and references therein). Let us, finally, remark that equation (\[eq:model\]) should not be expected to be valid for $Re_{\tau} \rightarrow 0$, since in this case the mean temperature gradient is zero and a closure like (\[eq:closure\]) does not apply [@footnote4]. In particular we detect a region where the sweeping mechanism is not yet effective and $Nu$ decreases slowly with $Re_{\tau}$; we denote the shear Reynolds number at which the crossover between such region and the $Nu \sim
Re_{\tau}^{-3/2}$ regime takes place as $Re_{\tau}^{\ast}$ and we argue that such crossover can be determined under the condition that the characteristic velocity of a rising plume, $U^{(RB)} \sim
\sqrt{\alpha g \Delta H}$, be of the same order of the centreline velocity of the Poiseuille flow, $U^{(P)} \sim \tilde{F}
H^2/\nu$. Equating these two latter relations we have $$\sqrt{\alpha g \Delta H} \sim \frac{ \tilde{F} H^2}{\nu},$$ which gives, in dimensionless form and introducing the crossover Reynolds, $$\label{eq:crossover}
Re_{\tau}^{\ast} \sim Ra^{1/4}.$$ This results is compared with the numerical data in the inset of figure \[fig1\]. For $Re_{\tau} < Re_{\tau}^{\ast}$ the longitudinal flow is still laminar (notice that the Nusselt number for $Ra=0$ remains equal to one) and represents just a small disturbance to the buoyant circulation. The initial fall-off of $Nu$ vs $Re_{\tau}$ can be captured by looking at the conservation equation for the total energy, which can be derived from (\[eq:NS\]) and (\[eq:temp\]) to be [@shraiman] $$\label{eq:enecons}
\varepsilon = (Nu-1)Ra + 8 Re_{\tau}^2 \langle u_x \rangle,$$ where $ \langle \cdots \rangle$ denotes an average over the entire volume, $\varepsilon = \langle (\partial_i u_j)^2 \rangle$ is the kinetic energy dissipation rate and we set $Pr=1$. It is clear that $\langle u_x \rangle \sim U^{(P)} \sim Re_{\tau}^2$ ($\langle U^{(RB)}
\rangle \sim 0$). Since $U^{(RB)} \gg U^{(P)}$ the longitudinal wind perturbs the RB dynamics only slightly so that $\varepsilon \approx
\varepsilon^{(RB)}$ [@footnote5]. From (\[eq:enecons\]) we therefore derive $$\label{eq:Ret4}
Nu \approx Nu_0(Ra) - (A_3/Ra) Re_{\tau}^4,$$ where $Nu_0$ is the Nusselt number for $Re_{\tau}=0$, i.e. pure RB, and $A_3$ is an order one constant. Equation (\[eq:Ret4\]) is plotted in figure \[fig1\] for three different $Ra$ showing good agreement with the numerics up to the expected crossover shear Reynolds $Re_{\tau}^{\ast}$.
We have performed direct numerical simulation of Rayleigh-Bénard convection with an imposed longitudinal pressure gradient inducing a mean wind. We found that the Nusselt number has a non-monotonic dependence on the shear Reynolds number based on the applied pressure drop: to an initial decrease (justifiable in terms of a mechanism of sweeping of plumes by the longitudinal wind) an increase follows, when the dynamics is dominated by the turbulent “forced convection” regime. Based on these empirical concepts, we provided a correlation which proved able to recover the numerical findings with reasonable accuracy. The observations and the modelling give a hint that, in standard RB convection, the shear due to the large scale circulation may act back onto the boundary layer against the ejection of plumes to the bulk (thus being a possible mechanism for the depletion of heat transfer respect to the ultimate state of turbulent convection). Our work is a first attempt to look directly at the effect of disturbing in a controlled manner the dynamics of the boundary layer in such a way to give an insight of its role in natural convection. A possible follow-up of the present study is to use a perturbation other than a simple Poiseuille flow.
[*Acknowledgements*]{}. We thank R. Benzi, P. Roche, R.P.J. Kunnen, F. Zonta and P. Ripesi for useful discussions and L. Bouhlali for careful reading of the manuscript. AS and AG acknowledge financial support from the Icelandic Research Fund. AS acknowledges FT and the Department of Mathematics and Computer Science of the Eindhoven University of Technology for the hospitality.
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We approximate that the plume still feels an acceleration proportional to the temperature difference $\Delta$, i.e. that the bending of the thermal short-cut is small.
L.D. Landau and E.M Lifshitz, [*Fluid Mechanics*]{}. Pergamon Press (1959).
More precisely our simulations suggest something closer to $U_c \sim Re_{\tau}^{1.9}$.
The scaling is $Re \sim Ra^{1/2}$ and then, since $Re_{\tau} \sim Re^{1/2}$, we get $Re_{\tau} \sim Ra^{1/4}$.
In principle there would be an extra dependence of $Nu$ on $Re_{\tau}$ in equation (\[eq:model\]) stemming from $\left|\partial_z \overline{T} \right|/(\Delta/H) \sim
f(Re_{\tau})$, which, however, turns out to be a subdominant correction, only becoming relevant for very low $Re_{\tau}$.
We assume that the energy dissipation rate is not affected by the longitudinal wind, at least in its boundary layer contributions which are dominant in this regime (our numerical simulations confirm this picture).
|
---
abstract: 'We exploit symmetries to give short proofs for two prominent formula families of QBF proof complexity. On the one hand, we employ symmetry breakers. On the other hand, we enrich the (relatively weak) QBF resolution calculus Q-Res with the symmetry rule and obtain separations to powerful QBF calculi.'
address:
- 'Institute for Algebra, J. Kepler University Linz, Austria'
- 'Institute for Formal Models and Verification, J. Kepler University Linz, Austria'
author:
- Manuel Kauers
- Martina Seidl
bibliography:
- 'refs.bib'
title: Short Proofs for Some Symmetric Quantified Boolean Formulas
---
Automated Theorem Proving ,Proof Complexity ,QBF
Introduction {#sec:0}
============
A Quantified Boolean Formula (QBF) is a formula of the form $P.\phi$, where $\phi$ is a propositional formula, say in the variables $x_1,\dots,x_n$, and $P$ is a quantifier prefix $P=Q_1x_1Q_2x_2\cdots Q_nx_n$ with $Q_i\in\{\forall,\exists\}$. From QBF proof complexity, it is well-known that the shortest proof of certain QBFs may have exponential size in a resolution-based calculus [@DBLP:books/daglib/0075409; @DBLP:conf/stacs/BeyersdorffCJ15]. We consider here two families of QBFs (cf. Section \[sec:1\]) which play a prominent role in QBF proof complexity for separating various calculi. We make the observation that short proofs can be obtained if we take into account the symmetries of the formulas. In Section \[sec:2\], we do so by using symmetry breakers. In Section \[sec:3\], we enrich the oldest variant of the resolution calculus for QBF, Q-Res [@DBLP:journals/iandc/BuningKF95], by a *symmetry rule*, generalizing an idea reported in [@DBLP:journals/acta/Krishnamurthy85; @DBLP:journals/dam/Urquhart99] for SAT. In both cases, it turns out that the proof sizes for both families of formulas shrinks from exponential to linear. As consequences, we obtain separation results between Q-Res with the symmetry rule and powerful proof systems like IR-calc [@DBLP:conf/stacs/BeyersdorffCJ15] and LQU$^+$ [@DBLP:conf/sat/BalabanovWJ14] (cf. Section \[sec:6\]).
Let us recall some basic facts and fix some notation. We only consider QBFs $P.\phi$ where $\phi$ is in conjunctive normal form (CNF), i.e., $\phi$ is a conjunction of clauses, each clause being a disjunction of literals, each literal being a variable or a negated variable, i.e., if $x$ is a variable, $x$ and $\bar x$ are literals. We also view clauses as sets of literals. The prefix $P = Q_1x_1\ldots Q_nx_n$ imposes an order $<_P$ on its variables: $x_i <_P x_j$ if $i < j$. The Q-Res calculus [@DBLP:journals/iandc/BuningKF95] applies the following rules on a QBF $P.\phi$:
1. Any clause of $\phi$ can be derived.
2. From the already derived clauses $C\lor x$ and $C'\lor\bar x$ with existentially quantified variable $x$ and $C,C'$ such that $C\cup C'$ is not a tautology, the clause $C\lor C'$ can be derived.
3. Let $C\lor l$ be an already derived clause where $l$ is a universal literal, $\bar l \not\in C$ and all existential literals $k \in C$ are such that $k <_P l$. Then the clause $C$ can be derived.
In the following, we do not mention the application of the axiom rule A explicitly. We write and for the application of R and U. A refutation of a QBF $P.\phi$ is the consecutive application of the resolution rule R and the universal reduction rule U until the empty clause is derived. Q-Res is sound and complete.
Finally, let us recall the notion of (syntactic) symmetries for QBFs. A bijective map $\sigma$ from the set $\{x_1,\dots,x_n,\bar x_1,\dots,\bar x_n\}$ of literals to itself is called admissible for a prefix $P=Q_1x_1\dots Q_nx_n$ if $\overline{\sigma(x)}\leftrightarrow\sigma(\bar x)$ for all $x\in\{x_1,\dots,x_n\}$ and for all $i,j\in\{1,\dots,n\}$, we have $\sigma(x_i)\in\{x_j,\bar x_j\}$ only if $x_i$ and $x_j$ belong to the same quantifier block, i.e., $Q_{\min(i,j)}=\cdots=Q_{\max(i,j)}$. An admissible function $\sigma$ is called a symmetry for a QBF $P.\phi$ with $\phi$ in CNF if applying $\sigma$ to all literals in $\phi$ maps $\phi$ to itself (possibly up to reordering clauses and literals).
Formula Families {#sec:1}
================
We consider the following two families of formulas.
\[def:hkb\] For $n\in\set N$, the formula $\kbkf_n$ is defined by the prefix $$\exists x_1y_1\forall a_1
\exists x_2y_2\forall a_2\dots
\exists x_ny_n\forall a_n
\exists z_1\dots z_n$$ and the following clauses:
- $C_1=(\bar x_1\lor\bar y_1)$
- for $j=1,\dots,n-1$:
$C_{2j} =(x_j\lor\bar a_j\lor\bar x_{j+1}\lor\bar y_{j+1})$\
$C_{2j+1}=(y_j\lor a_j\lor\bar x_{j+1}\lor\bar y_{j+1})$.
- $C_{2n} = (x_n\lor \bar a_n\lor\bar z_1\lor \ldots \bar z_n)$,\
$C_{2n+1} = (y_n\lor a_n\lor\bar z_1\lor \ldots \bar z_n)$
- for $j=1,\dots,n$:
$B_{2j-1}=(a_j\lor z_j)$ and $B_{2j}=(\bar a_j\lor z_j)$.
For every $n\in\set N$, the formula $\kbkf_n$ is false, and it is known [@DBLP:books/daglib/0075409] that any Q-Res refutation needs a number of steps which is at least exponential in $n$.
For $n\in\set N$ with $n > 1$, the formula $\parity_n$ is defined by the prefix $$\exists x_1\dots x_n\forall a_1a_2\exists y_2\dots y_n$$ and the following clauses:
- $A_2=(\bar x_1\lor\bar x_2\lor\bar y_2 \lor a_1 \lor a_2)$\
$B_2=(\bar x_1\lor x_2\lor y_2 \lor a_1 \lor a_2)$\
$C_2=(x_1\lor\bar x_2\lor y_2 \lor a_1 \lor a_2)$\
$D_2=(x_1\lor x_2\lor\bar y_2 \lor a_1 \lor a_2)$
- for $j=3,\dots,n$:
$A_j=(\bar y_{j-1}\lor\bar x_j\lor\bar y_j \lor a_1 \lor a_2)$\
$B_j=(\bar y_{j-1}\lor x_j\lor y_j \lor a_1 \lor a_2)$\
$C_j=(y_{j-1}\lor\bar x_j\lor y_j \lor a_1 \lor a_2)$\
$D_j=(y_{j-1}\lor x_j\lor\bar y_j \lor a_1 \lor a_2)$
- $E_1=(a_1 \lor a_2 \lor y_n)$ and $E_2=(\bar a_1 \lor \bar a_2 \lor\bar y_n)$
- for $i=2,\dots,n$, $A'_i, B'_i, C'_i, D'_i$ are obtained from $A_i, B_i, C_i, D_i$ by replacing $a_1 \lor a_2$ by $\bar a_1 \lor \bar a_2$.
$\parity_n$ is a variant of the $\qparity_n$ family [@DBLP:conf/stacs/BeyersdorffCJ15] which encodes $\exists x_1\dots x_n\forall z. z\not=x_1\oplus\cdots\oplus x_n$, where $\oplus$ stands for exclusive or. Obviously all these formulas are false. Refuting $\qparity_n$ needs an exponential number of steps in the calculus Q-Res, but not in the stronger calculus LQU$^+$. We use $\parity_n$ instead of $\qparity_n$ because for this family, also LQU$^+$ needs exponentially many steps [@DBLP:conf/stacs/BeyersdorffCJ15]. This will be used in Section \[sec:6\].
Symmetry Breakers {#sec:2}
=================
Let $S$ be a set of symmetries for a QBF $P.\phi$. A symmetry breaker is a certain Boolean formula $\psi$ such that when $P.\phi$ is true, so is $P.(\phi\land\psi)$. Writing $P=Q_1x_1\cdots Q_nx_n$, it was shown in [@audemard2007efficient; @DBLP:journals/corr/abs-1802-03993] that $$\psi=\bigwedge_{\vbox{\hbox to0pt{\hss$\scriptstyle i=1$\hss}\kern-3pt\hbox to0pt{\hss$\scriptstyle Q_i=\exists$\hss}}}^n
\ \ \bigwedge_{\sigma\in S}
\biggl(\Bigl(\bigwedge_{j<i} (x_j\leftrightarrow\sigma(x_j))\Bigr)\rightarrow(x_i\rightarrow\sigma(x_i))\biggr)$$ is a symmetry breaker.
For the formulas $\kbkf_n$ (Def. \[def:hkb\]), we have for every $i=1,\dots,n$ the symmetry $\sigma_i=(x_i\ y_i)(\bar x_i\ \bar y_i)(a_i\ \bar a_i)$ which exchanges the variables $x_i, y_i$, the literals $\bar x_i,\bar y_i$, and the literals $a_i,\bar a_i$. Therefore, $$\psi_n = (\bar x_1\lor y_1)\land\cdots\land(\bar x_n\lor y_n)$$ is a symmetry breaker for $\kbkf_n$.
For $n\in\set N$, write $\kbkf_n$ as $P_n.\phi_n$, and let $\psi_n$ be the symmetry breaker from above. Then $P_n.(\phi_n\land\psi_n)$ has a refutation proof with no more than $4n$ steps.
The proof proceeds as follows.
- .
-
- for $j=1,\dots,n-1$, do
.
.
Then $U_{n-1}=(\bar a_1\lor\dots\lor\bar a_{n-1}\lor\bar x_n)$.
-
- .
-
- for $j=1,\dots,n$, do
.
Then $W_0:=V_n=(\bar a_1\lor\dots\lor\bar a_n)$.
-
- for $j=1,\dots,n$, do
.
$W_n$ is the empty clause.
For the formulas $\parity_n$, the argument is similar. In this case, we have the symmetries $\sigma_1 = (x_1\ x_2)(\bar x_1\ \bar x_2)$ and $$\sigma_i=(x_i\ \bar x_i)(a_1\ \bar a_1)(a_2\ \bar a_2)(y_i\ \bar y_i)\cdots(y_n\ \bar y_n)$$ for every $i=2,\dots,n$. There are some further symmetries which we will not need. The symmetries $\sigma_1,\dots,\sigma_n$ give rise to the symmetry breaker $$\psi_n=(\bar x_1 \lor x_2) \land \bar x_2\land\dots\land \bar x_n$$ for $\parity_n$.
For $n\in\set N$ with $n > 1$, write $\parity_n$ as $P_n.\phi_n$, and let $\psi_n$ be the symmetry breaker from above. Then $P_n.(\phi_n\land\psi_n)$ has a refutation proof with no more than $2n+1$ steps.
The proof proceeds as follows.
- .
-
- .
-
- for $j=3,\dots,n$, do
.
-
- for $j = 3,\dots,n$, do
.
-
- .
-
- .
The Symmetry Rule {#sec:3}
=================
As an alternative to using symmetry breakers, we can enrich the calculus Q-Res as introduced in Section \[sec:0\] to the calculus Q-Res+S by adding the following rule, which allows us to exploit symmetries of the input formula $P.\phi$ within the proof.
1. From an already derived clause $C$ and a symmetry $\sigma$ of $P.\phi$, the clause $\sigma(C)$ can be derived.
Several variants of this rule have been proposed for SAT in [@DBLP:journals/acta/Krishnamurthy85; @DBLP:journals/dam/Urquhart99], but to our knowledge it has not yet been considered in the context of QBF. However, it is easy to see that the rule also works for QBF.
Let $P.\phi$ be a QBF, and suppose that $C$ is a clause which can be derived from $\phi$ using the rules S, R, U. Then it can also be derived using only the rules R, U.
Suppose otherwise. Then there are clauses which can be derived with S, R, U but not with R, U alone. Let $C$ be such a clause, and consider a derivation of $C$ with a minimal number of applications of S. The rule S is used at least once during the derivation. Consider its earliest application, suppose this application derives $\sigma(D)$ from the clause $D$. If we can show that $\sigma(D)$ can also be derived using only R and U, then we can eliminate this first application of S in the derivation of $C$ and obtain a contradiction to the assumed minimality.
To show that $\sigma(D)$ can be derived using only R and U, observe first that $D$ was derived only using R and U. For an admissible function $\sigma$, we have $\overline{\sigma(x)}\leftrightarrow\sigma(\bar x)$ for every variable $x$. Therefore, if a clause $E$ can be derived by R from two clauses $E_1$ and $E_2$, we can derive $\sigma(E)$ by R from $\sigma(E_1)$ and $\sigma(E_2)$. Furthermore, an admissible function cannot permute literals across quantifier blocks, which implies that if $F$ can be derived by U from $F_1$, then $\sigma(F)$ can be derived by U from $\sigma(F_1)$. Finally, when $\sigma$ is a symmetry of $\phi$ and $G$ is a clause of $\phi$, then also $\sigma(G)$ is a clause of $\phi$. By combining these three observations, it follows that applying $\sigma$ to all clauses appearing in the derivation of $D$ yields a derivation of $\sigma(D)$. This completes the proof.
According to the previous proposition, with S we cannot derive any clause that we cannot also derive without. Therefore, soundness of Q-Res+S follows from soundness of Q-Res. Next, we illustrate that Q-Res+S allows for shorter proofs than Q-Res. For the application of S, we write .
For every $n\in\set N$, the formula $\kbkf_n$ can be refuted by no more than $5n$ applications of S, R, U.
We proceed as follows by using the symmetries of the form $\sigma_i=(x_i\ y_i)(\bar x_i\ \bar y_i)(a_i\ \bar a_i)$ for $i=1,\dots,n$.
- set $U_{n+1}=C_{2n+1}$.
-
- for $j=n,\dots,1$, do
.
-
- set $W_n:=U_1=(y_n\lor a_1\lor\dots\lor a_n)$.
-
- for $j=n,\dots,2$, do
.\
.\
.\
.
-
- .
-
- .
-
- .
-
- .
For every $n\in\set N$ with $n > 1$, the formula $\parity_n$ can be refuted by no more than $3n+2$ applications of S, R, U.
Recall from Section \[sec:3\] that $\parity_n$ has the symmetries $\sigma_1=(x_1\ x_2)(\bar x_1\ \bar x_2)$ and $\sigma_i=(x_i\ \bar x_i)(a_1\ \bar a_1)(a_2\ \bar a_2)(y_i\ \bar y_i)\cdots(y_n\ \bar y_n)$ for $i > 1$.
- .
-
- for $j=n-1,\dots,3$, do
.-5pt
-
- .
-
- .
-
- for $j=n,\dots,2$, do
.\
.
-
- .
-
- .
-
- .
Consequences {#sec:6}
============
From recent results, it is known that plain Q-Res is rather weak (for a fine-grained comparison of QBF proof systems see [@DBLP:conf/stacs/BeyersdorffCJ15]). Both, the expansion-based proof system IR-calc and the CDCL-based proof system LQU$^+$ are strictly stronger than Q-Res. The addition of the symmetry rule changes the situation. While the $\parity_n$ formulas are hard for LQU$^+$ and the $\kbkf_n$ formulas are hard for IR-calc, we have shown that both are easy for Q-Res+S. Now one may ask if Q-Res+S is strictly stronger than IR-calc or LQU$^+$. The answer is clearly “no”. For $\kbkf_n$, the application of the symmetry rule can be hindered by introducing $n$ universally quantified variables $b_i$ which are placed between $x_i$ and $y_i$ in the prefix. Further, each clause $C_{2j}$ changes to $C_{2j} \vee b_j$. For this modified formula, LQU$^+$ can still find a short proof, but Q-Res+S can only apply R and U, hence it falls back to Q-Res which does not exhibit short proofs for $\kbkf_n$. In a similar way, $\parity_n$ can be modified such that these formulas remain simple for IR-calc, but become hard for Q-Res+S.
Q-Res+S and IR-calc are incomparable, and so are Q-Res+S and LQU$^+$.
For the future, the effects of adding S to more powerful proof systems than Q-Res remain to be investigated.
**Acknowledgements.** Parts of this work were supported by the Austrian Science Fund (FWF) under grant numbers NFN S11408-N23 (RiSE), Y464-N18, and SFB F5004.
|
---
abstract: 'We devise a new high order local absorbing boundary condition (ABC) for radiating problems and scattering of [ time-harmonic acoustic waves]{} from obstacles of arbitrary shape. By introducing an artificial boundary $S$ enclosing the scatterer, the original unbounded domain $\Omega$ is decomposed into a bounded computational domain $\Omega^{-}$ and an exterior unbounded domain $\Omega^{+}$. Then, we define interface conditions at the artificial boundary $S$, from truncated versions of the well-known Wilcox and Karp farfield expansion representations of the exact solution in the exterior region $\Omega^{+}$. As a result, we obtain a new local absorbing boundary condition (ABC) for a bounded problem on $\Omega^{-}$, which effectively accounts for the outgoing behavior of the scattered field. Contrary to the low order absorbing conditions previously defined, the order of the error induced by this ABC can easily match the order of the numerical method in $\Omega^{-}$. We accomplish this by simply adding as many terms as needed to the truncated farfield expansions of Wilcox or Karp. The convergence of these expansions guarantees that the order of approximation of the new ABC can be increased arbitrarily without having to enlarge the radius of the artificial boundary. We include numerical results in two and three dimensions which demonstrate the improved accuracy and simplicity of this new formulation when compared to other absorbing boundary conditions.'
address:
- 'Department of Mathematics, Brigham Young University, Provo, UT'
- 'Department of Pediatrics - Cardiology, Baylor College of Medicine, Houston, TX'
author:
- Vianey Villamizar
- Sebastian Acosta
- Blake Dastrup
bibliography:
- 'AcoBib.bib'
title: |
**High order local absorbing boundary conditions for acoustic\
waves in terms of farfield expansions**
---
Acoustic scattering ,Nonreflecting boundary condition ,High order absorbing boundary condition ,Helmholtz equation,Farfield pattern
Introduction {#Section.Intro}
============
Equations modeling wave phenomena in fields such as geophysics, oceanography, and acoustics among others, are normally defined on unbounded domains. Due to the complexity of the corresponding boundary value problems (BVP), in general, an explicit analytical technique cannot be found. Therefore, they are treated by numerical methods. Major challenges appear when numerically solving wave problems defined in these unbounded regions using volume discretization methods. One of them consists of the appropriate definition of absorbing boundary conditions (ABC) on artificial boundaries such that the solution of the new bounded problem approximates to a reasonable degree the solution of the original unbounded problem in their common domain. That is why the definition of ABCs for wave propagation problems in unbounded domain plays a key role in computation.
Historically two main approaches were initially followed in the evolution of ABCs, as described by Givoli in [@GivoliReview2]. First, low order local ABCs were constructed. Undoubtedly, one of the most important ABCs in this category was introduced by Bayliss-Gunzburger-Turkel in their celebrated paper [@Bayliss01]. This condition is denoted as BGT in the ABC literature. Other well-known conditions in this category were introduced by Engquist-Majda [@Engquist01], Feng [@Feng] and Li-Cendes [@Li-Cendes]. Some of them became references for many that followed thereafter. Several years later in the late 1980s and early 1990s, exact non-local ABCs made their appearance. Since their definitions are based on Dirichlet-to-Neumann (DtN) maps, they are called DtN absorbing boundary conditions. The pioneer work in their formulations and implementations was performed by Keller-Givoli [@Keller01; @Givoli-Keller1990] and Grote-Keller [@Grote-Keller01]. The main virtue of the DtN absorbing conditions is that they approximate the field at the artificial boundary almost exactly. Therefore, the accuracy of the numerical computation depends almost entirely on the accuracy of the numerical method employed for the computation at the interior points.
The BGT absorbing condition consists of a sequence of differential operators applied at the artificial boundary (a circle or a sphere of radius $R$) which progressively annihilate the first terms of a farfield expansion of the outgoing wave valid in the exterior of the artificial boundary. We call the first of these operators BGT$_1$. In three dimensions, it provides an accuracy of $O(1/R^{3})$ and involves a first order normal derivative. The next condition in this sequence, BGT$_2$ has $O(1/R^{5})$ accuracy and includes a second order normal derivative in its definition. They are called BGT operators of order one and order two, respectively. The drawback of the BGT and of the other ABCs in the first category is that to increase the order of the approximation at the boundary, the order of the derivatives present in their definitions also needs to be increased. This leads to impractical ABCs due to the difficulty found in their implementations beyond the first two orders. There is also a downside for the DtN-ABCs stemming from the fact the computation of the field at any boundary point involves all the other boundary points which leads to partially dense matrices at the final stage of the numerical computation.
The above disadvantages are overcome by the introduction of [*high order local*]{} ABCs without high order derivatives. According to [@GivoliReview2], they are sequences of ABCs of increasing accuracy which are also practically implementable for an arbitrarily high order. Several ABCs have been formulated within this category in recent years. A detailed description of some of them is found in [@GivoliReview2]. A common feature of all these high order local ABCs is the presence of auxiliary variables which avoid the occurrence of high derivatives (beyond order two) in the ABC’s formulation. Probably, the best known of all these high order local conditions was formulated by Hagstrom-Hariharan [@Hagstrom98] which we denote as HH. They start representing the outgoing solution by an infinite series in inverse powers of $\frac{1}{R}$, where $R$ is the radius of a circular or spherical artificial boundary. Analogous to the BGT formulation, the key idea in this formulation is the construction of a sequence of operators that approximately annihilate the residual of the preceding term in the sequence. As a result, a sequence of conditions in the form of recurrence formulas for a set of unknown auxiliary variables is obtained. The expression for the first auxiliary variable coincides with BGT$_1$. Similarly, combining the formulas for the first two auxiliary variables, the HH absorbing condition reduces to BGT$_2$. Actually, Zarmi [@ZarmiThesis] proves that HH is equivalent to BGT for all orders. The difference between these two formulations is that HH does not involve high derivatives owing to the use of the auxiliary variables. Thus, HH can be implemented for arbitrarily high order. The three-dimensional (3D) HH can be considered an exact ABC since it is obtained from an exact representation of the solution in the exterior of the artificial boundary. However, the two-dimensional (2D) HH is only asymptotic because it is obtained from an asymptotic expansion of the exact representation of the solution. Recently, Zarmi-Turkel [@Zarmi-Turkel] generalized the HH construction of local high order ABCs. They developed an annihilating technique that can be applied to rather general series representation of the solution in the exterior of the computational domain. As a result, they were able to reobtain HH and derive new high order local ABCs such as a high order extension of Li-Cendes ABC [@Li-Cendes].
Our construction of high order local ABCs proceeds in the opposite way of the previous ABCs discussed above. Instead of defining local differential operators which progressively annihilate the first terms of a series representation of the solution in the exterior of the artificial boundary, we use a truncated version of the series representations directly to define the ABC without defining special differential operators at the boundary. As a consequence, the derivation of the absorbing condition is extremely simple. Moreover, the order of the error induced by this ABC can be easily improved by simply adding as many terms as needed to the truncated farfield expansions The series representations employed are Karp’s farfield expansion [@Karp] in 2D, and Wilcox’s farfield expansion [@Wil-1956] in 3D. They are exact representations of the outgoing wave outside the circular or spherical artificial boundaries of radius $R$, respectively. Therefore, the resulting ABC which we call Karp’s double farfield expansion (KDFE) and Wilcox farfield expansion (WFE), respectively, can be considered exact ABCs. The exact character of KDFE represents an improvement over HH in 2D, which is only asymptotically valid. Instead of having unknown auxiliary functions as part of the new condition, we simply consider as unknowns the original angular functions appearing in Wilcox’s or Karp’s farfield expansions. To determine these angular functions, we use the recurrence formulas derived from Wilcox’s or Karp’s theorems which do not disturb the local character of the ABC. A relevant feature of the farfield expansions approach is that the coefficient (angular function) of their leading term is the farfield pattern of the propagating wave. Thus, no additional computation is required to obtain an approximation for this important profile. For comparison purposes, we also obtain a farfield expansion ABC from the asymptotic farfield expansion of Karp’s exact series. We call it Karp’s single farfield expansion (KSFE) absorbing boundary condition.
An important consideration is that the formulation of these absorbing boundary conditions depends on existing knowledge of an exact or asymptotic series representation for the outgoing waves of the problem being studied. This limits the use of Karp and Wilcox farfield expansions ABCs to problems in the entire plane or space, respectively. As a consequence, problems involving straight infinite boundaries as waveguide problems, half-plane, or quarter-plane cannot be modeled by these ABCs. For these type of problems, the most popular method to formulate ABCs is the perfectly matched layer (PML) introduced by Berenger [@Berenger01]. However, a class of high order absorbing boundary conditions has also been employed by several authors. For instance, Hagstrom, Mar-Or, and Givoli [@H-MO-Givoli2008] obtained high order local ABCs for two-dimensional waveguide problems modeled by the wave equation. This ABC was first formulated for the wave equation by Hagstrom-Warburton [@H-W] which in turn is based on a modification of the Higdon ABCs [@Higdon1987]. More recently, Rabinovich and et al. [@Rab-Giv-Bec-2010] adapted Hagstrom-Warburton ABC to time-harmonic problems in a waveguide and a quarter-plane modeled by the Helmholtz equation.
The outline of the succeeding sections is as follows. In Section \[Section.Formulation\], details about the expansions KDFE, KSFE, and WFE are given. Also, the relationships between lower orders of KSFE absorbing boundary condition, BGT$_1$, and BGT$_2$ are established. Then in Section \[Section:NumMethd\], the numerical method is described in the 2D case for KDFE. In particular, the discrete equations at the boundary are carefully derived. This is followed by an analysis of the structure of the matrices defining the ultimate linear systems for KSFE, KDFE, and DtN boundary value problems, respectively. Finally, numerical results for scattering and radiating problems, from circular and complexly shaped obstacles in 2D, and also from spherical obstacles in 3D, employing the novel farfield expansions ABCs, are reported in Section \[Section.Numerics2D\].
High order local absorbing conditions from farfield expansions {#Section.Formulation}
===============================================================
We start this Section by considering the scattering problem of a time-harmonic incident wave, ${u_{\rm inc}}$, from a single obstacle in two or three dimensions. This scatterer is an impenetrable obstacle that occupies a simply connected bounded region with boundary $\Gamma$. The open unbounded region in the exterior of $\Gamma$ is denoted as $\Omega$. This region $\Omega$ is occupied by a homogeneous and isotropic medium. Both the incident field ${u_{\rm inc}}$ and the scattered field ${u_{\rm sc}}$ satisfy the Helmholtz equation in $\Omega$. For simplicity, we assume a Dirichlet boundary condition (soft obstacle) on $\Gamma$. However, the analysis in this article can be easily extended to Neumann or Robin boundary conditions, and to a bounded penetrable scatterer with inhomogeneous and anisotropic properties. Then, ${u_{\rm sc}}$ solves the following boundary value problem (BVP): $$\begin{aligned}
&& \Delta {u_{\rm sc}}+ k^2 {u_{\rm sc}}= f \quad\qquad \text{in $\Omega$}, \label{BVPsc1} \\
&& {u_{\rm sc}}= - {u_{\rm inc}}\qquad\qquad \text{on $\Gamma$,} \label{BVPsc2} \\
&& \lim_{r \rightarrow \infty} r^{(\delta-1)/2} \left( \partial_{r} {u_{\rm sc}}- \mathrm{i} k {u_{\rm sc}}\right) = 0.\label{BVPsc3}\end{aligned}$$ [The wave number $k$ and the source $f$ may vary in space]{}. Equation (\[BVPsc3\]) is known as the Sommerfeld radiation condition where $r = |\textbf{x}|$ and $\delta=2$ or 3 for two or three dimensions, respectively. It implies that ${u_{\rm sc}}$ is an outgoing wave. This boundary value problem is well-posed under classical and weak formulations [@ColtonKress02; @Nedelec01; @McLean2000].
As pointed out in the introduction, the unbounded BVP (\[BVPsc1\])-(\[BVPsc3\]) needs to be transformed into a bounded BVP before a numerical solution can be sought. This is typically done by introducing an artificial boundary $S$ enclosing the obstacle followed by defining an appropriate absorbing boundary condition (ABC) on $S$. We choose a circular or spherical artificial boundary for the two- or three-dimensional scenarios, respectively. As a result, the region $\Omega$ is divided into two open regions. The region $\Omega^{-},$ bounded internally by the obstacle boundary $\Gamma$ and externally by the artificial boundary $S$ (a circle or a sphere of radius $r=R$), and the open unbounded connected region $\Omega^{+}= \Omega \setminus {\overline}{\Omega^{-}}$. [We assume that the source $f$ has its support in $\Omega^{-}$, and the wave number $k$ is constant in $\Omega^{+}$]{}. An appropriate ABC should induce no or little spurious reflections from the artificial boundary $S$ in order to maintain a good accuracy for the numerical solution inside $\Omega^{-}$.
As an intermediate step before constructing our high order local ABC in the next sections, we consider the following equivalent interface problem to the original BVP (\[BVPsc1\])-(\[BVPsc3\]) for ${u_{\rm sc}}^{-} = {u_{\rm sc}}|_{\Omega^{-}}$ and ${u_{\rm sc}}^{+} = {u_{\rm sc}}|_{\Omega^{+}}$: $$\begin{aligned}
&& \Delta {u_{\rm sc}}^{-} + k^2 {u_{\rm sc}}^{-} = f, \quad\qquad \text{in $\Omega^{-}$}, \label{BVPInterf1} \\
&& \Delta {u_{\rm sc}}^{+} + k^2 {u_{\rm sc}}^{+} = 0, \quad\qquad \text{in $\Omega^{+}$}, \label{BVPInterf2} \\
&& {u_{\rm sc}}^{-} = - {u_{\rm inc}}, \qquad\qquad\, \text{on $\Gamma$,} \label{BVPInterf3}\end{aligned}$$ with the interface and Sommerfeld conditions: $$\begin{aligned}
&& {u_{\rm sc}}^{-} = {u_{\rm sc}}^{+}, \qquad\qquad\quad \text{on $S$,}\label{BVPInterf4} \\
&&\partial_{\nu} {u_{\rm sc}}^{-} = \partial_{\nu} {u_{\rm sc}}^{+} \quad\qquad\,\,\,\,\text{on $S$}, \label{BVPInterf5} \\
&& \lim_{r \rightarrow \infty} r^{(\delta-1)/2}\left( \partial_{r} {u_{\rm sc}}^{+} - \mathrm{i} k {u_{\rm sc}}^{+} \right) = 0, \label{BVPInterf6}\end{aligned}$$ where $\partial_{\nu}$ denotes the derivative in the outer normal direction on $S$. The original scattering problem (\[BVPsc1\])-(\[BVPsc3\]), and the interface problem (\[BVPInterf1\])-(\[BVPInterf6\]) are equivalent as shown in [@JCP2010 Thm 1] or [@McLean2000 Lemma 4.19]. As a consequence, by simply requiring the Cauchy data to match at the artificial boundary $S$, all higher order derivatives also match at the interface. [This matching condition at the artificial boundary will ultimately lead to a bounded BVP in $\Omega^{-}$ whose numerical solution approximates to a reasonable degree the solution of the original unbounded problem in $\Omega^{-}$. This bounded BVP is constructed by realizing that there is an analytical representation of the solution ${u_{\rm sc}}^{+}$ for the portion of the interface problem defined in $\Omega^{+}$. By matching, at the artificial boundary $S$, this analytical solution with the solution ${u_{\rm sc}}^{-}$ defined in the interior region $\Omega^{-}$, the bounded BVP sought in $\Omega^{-}$ is finally obtained. The numerical solution of this bounded BVP in $\Omega^{-}$ is the main subject of this work. Furthermore, once this numerical solution for ${u_{\rm sc}}^{-}$ is obtained, the analytical representation for ${u_{\rm sc}}^{+}$ can be evaluated in $\Omega^{+}$. Details of the derivation of the bounded BVP in $\Omega^{-}$ are given in the sections below. Moreover, since the problem in $\Omega^{-}$ is to be solved numerically, we can consider a rather general source term $f$ and a variable wave number $k$ inside $\Omega^{-}$. However, for sake of simplicity, from now on we assume $f=0$ and $k$ constant.]{}
Karp’s double farfield expansion (KDFE) absorbing boundary condition in 2D {#Section.ABC2D}
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Here, we consider the outgoing field ${u_{\rm sc}}^{+}$ satisfying the 2D Helmholtz equation exterior to a circle $r=R$ and the Sommerfeld radiation condition (\[BVPsc3\]) for $\delta = 2$. Our derivation of the new exact absorbing boundary condition is based on a well-known representation of outgoing solutions of the Helmholtz equation in 2D by two infinite series in powers of $1/kr$. This representation is provided by the following theorem due to Karp.
\[Thm.Karp\] Let ${u_{\rm sc}}^{+}$ be an outgoing solution of the two-dimensional Helmholtz equation in the exterior region to a circle $r=R$. Then, ${u_{\rm sc}}^{+}$ can be represented by a convergent expansion $$\begin{aligned}
{u_{\rm sc}}^{+} (r,\theta) = H_0(kr) \sum_{l=0}^{\infty} \frac{F_l(\theta)}{(kr)^l} + H_1(kr) \sum_{l=0}^{\infty} \frac{G_l(\theta)}{(kr)^l}, \qquad \mbox{for}\,\, r> R. \label{KarpExp}\end{aligned}$$ This series is uniformly and absolutely convergent for $r>R$ and can be differentiated term by term with respect to $r$ and $\theta$ any number of times.
Here, $r$ and $\theta$ are polar coordinates. The functions $H_0$ and $H_1$ are Hankel functions of first kind of order 0 and 1, respectively. Karp also claimed that the terms $F_l$ and $G_l$ ($l=1,2,\dots$) can be computed recursively from $F_0$ and $G_0$. To accomplish this, he suggested the substitution of the expansion (\[KarpExp\]) into Helmholtz equation in polar coordinates and the use of the identities: $H_0'(z)=-H_1(z)$ and $H_1'(z)= H_0(z) - \frac{1}{z}H_1(z)$. In fact, by doing this and requiring the coefficients of $H_0$ and $H_1$ to vanish, we derive a recurrence formula for the coefficients $F_l$ and $G_l$ of the expansion (\[KarpExp\]). This result is stated in the following corollary.
\[KarpRecurrence\] The coefficients $F_l(\theta)$ and $G_l(\theta)$ ($l>1$) of the expansion (\[KarpExp\]), can be determined from $F_0(\theta)$ and $G_0(\theta)$ by the recursion formulas $$\begin{aligned}
& 2 l G_{l}(\theta) = (l-1)^2 F_{l-1}(\theta) + d^2_{\theta} F_{l-1}(\theta) , \qquad && \text{for $l=1,2, \dots$} \label{Recurrence1}\\
& 2 l F_{l}(\theta) = - l^2 G_{l-1}(\theta) - d^2_{\theta} G_{l-1}(\theta),
\qquad && \text{for $l=1,2, \dots$}. \label{Recurrence2}\end{aligned}$$
[As discussed in the previous section, we use the semi-analytical representation of ${u_{\rm sc}}^{+}$ given by (\[KarpExp\]) and the matching conditions (\[BVPInterf4\])-(\[BVPInterf5\]), at the interface $S$, to obtain an approximation $u \approx {u_{\rm sc}}^{-}$ that satisfies the following bounded BVP in the region ${\Omega^{-}}$:]{} $$\begin{aligned}
&& \Delta u + k^2 u = f, \quad\qquad \text{in $\Omega^{-}$}, \label{BVPBd1} \\
&& u = - {u_{\rm inc}}, \qquad\qquad\, \text{on $\Gamma$,} \label{BVPBd2} \\
&& u(R,\theta)=H_0(kR) \sum_{l=0}^{L-1} \frac{F_l(\theta)}{(kR)^l} + H_1(kR)\sum_{l=0}^{L-1} \frac{G_l(\theta)}{(kR)^l},\label{BVPBd3} \\
&&\partial_{r} u(R,\theta) = \partial_{r}\left( H_0(kr) \sum_{l=0}^{L-1} \frac{F_l(\theta)}{(kr)^l} + H_1(kr)\sum_{l=0}^{L-1} \frac{G_l(\theta)}{(kr)^l}\right) \bigg|_{r=R},\label{BVPBd4} \end{aligned}$$ where $R$ is the radius of the circular artificial boundary $S$. This problem is not complete until enough conditions at the artificial boundary $S$, for the two families of unknown angular functions $F_l$ and $G_l$ of Karp’s expansion, are specified. Clearly, extra conditions to determine $F_l$ and $G_l$ for $l=1,\dots L-1$ are provided by the recurrence formulas (\[Recurrence1\]) and (\[Recurrence2\]). To apply these recurrence formulas, $F_0$ and $G_0$ need to be known. [The boundary conditions $(\ref{BVPBd3})$ and $(\ref{BVPBd4})$ may be used to determine $u$ and $F_0$ at the boundary $S$. Therefore, we are still short by another condition to determine $G_0$ at $S$. Now, ${u_{\rm sc}}$ has a second order partial derivative which is continuous with respect to $r$ at $r=R$. Thus, a natural condition to add at $r=R,$ to our new bounded problem (\[BVPBd1\])-(\[BVPBd4\]) supplemented with (\[Recurrence1\])-(\[Recurrence2\]), is $\partial_{\nu}^2 {u_{\rm sc}}^{-} = \partial_{\nu}^2 {u_{\rm sc}}^{+} $ which can be fully written in terms of $u$ as]{} $$\begin{aligned}
\partial_{r}^2 u(R,\theta) = \partial_{r}^2\left( H_0(kr)\sum_{l=0}^{L-1} \frac{F_l(\theta)}{(kr)^l} + H_1(kr)\sum_{l=0}^{L-1} \frac{G_l(\theta)}{(kr)^l}\right) \bigg|_{r=R}, \label{BVPBd5}\end{aligned}$$ where the second radial derivative $\partial_{r}^2 u$ may also be expressed in terms of $\partial_{r} u$ and $\partial_{\theta}^2 u$ using the Helmholtz equation itself.
[Summarizing, we approximate the solution of the interface problem (\[BVPInterf1\])-(\[BVPInterf6\]) in the region $\Omega^{-}$ by the solution of the bounded BVP consisting of (\[BVPBd1\])-(\[BVPBd5\]) and (\[Recurrence1\])-(\[Recurrence2\]). The equations (\[BVPBd3\])-(\[BVPBd5\]) for the *double* family of farfield functions $F_l$ and $G_l$, supplemented by the recurrence formulas (\[Recurrence1\])-(\[Recurrence2\]), constitute our novel *Karp’s Double Farfield Expansion* $(\text{KDFE}_{L})$ absorbing boundary conditions with $L$ terms.]{}
Karp’s single farfield expansion (KSFE) absorbing boundary condition in 2D {#Section.ABC2DAsymp}
--------------------------------------------------------------------------
It is possible to approximate the two-family expansion (\[KarpExp\]) with a one-family expansion by means of an asymptotic approximation for large values of $kr$. A similar procedure was employed in [@Bayliss01]. The Hankel functions $H_{0}(z)$ and $H_{1}(z)$ admit the following approximations [@HandMathFunct §9.2], $$\begin{aligned}
& H_{0}(z) = \frac{e^{i z}}{\sqrt{z}} \sum_{l=0}^{L-1} \frac{C_{0,l}}{z^l} + O(|z|^{-L}) \qquad \text{and} \qquad H_{1}(z) = \frac{e^{i z}}{\sqrt{z}} \sum_{l=0}^{L-1} \frac{C_{1,l}}{z^l} + O(|z|^{-L}) \label{Eqn:Hpower}\end{aligned}$$ valid for $z \in \mathbb{C}$ with $| \text{arg}(z) | < \pi$ as $|z| \to \infty$. Therefore, after multiplication of the power series of (\[KarpExp\]) with these approximations for $H_{0}(kr)$ and $H_{1}(kr)$, re-arranging terms of same powers, and neglecting the terms $O(|kr|^{-L})$, we can combine the two families of angular functions $F_l$ and $G_l$ into one family $f_l$. As a result, a new asymptotic series representation of the outgoing wave (\[BVP2D3\_Asymp\] ) is obtained. Moreover, the application of the 2D Helmholtz operator to the new asymptotic expansion renders a recursive formula (\[BVP2D5\_Asymp\]) for the functions $f_{l}$. [Thus, in virtue of the approximation (\[Eqn:Hpower\]) and the matching at the artificial boundary $S$ described in the previous section, we obtain a new absorbing boundary condition for the problem (\[BVPBd1\])-(\[BVPBd2\]) given by]{} $$\begin{aligned}
&& u(R,\theta)=\frac{e^{ikR}}{\sqrt{kR}}\sum_{l=0}^{L-1} \frac{f_l(\theta)}{(kR)^l}\label{BVP2D3_Asymp} \\
&&\partial_{r} u(R,\theta) = \frac{e^{i k R}}{\sqrt{k R}} \sum_{l=0}^{L-1} \left( ik - \left( l +\tfrac{1}{2} \right)/R \right) \frac{f_l(\theta)}{(kR)^l}, \label{BVP2D4_Asymp}\\
&& 2 i l f_{l}(\theta) = \left( l - \tfrac{1}{2} \right)^2 f_{l-1}(\theta) + \partial_{\theta}^{2} f_{l-1}(\theta), \qquad l \geq 1. \label{BVP2D5_Asymp}\end{aligned}$$ We call the boundary condition defined by (\[BVP2D3\_Asymp\])-(\[BVP2D5\_Asymp\]) with a *single* family of farfield functions $f_{l}$ the *Karp’s Single Farfield Expansion* $(\text{KSFE}_{L})$ absorbing boundary condition with $L$ terms. As we see in the numerical results in Section \[Section.Numerics2D\], both the KSFE$_{L}$ and KDFE$_{L}$ render similar results as the number of terms $L$ increases, for moderate to large values of $kR$. But, KSFE$_{L}$ exhibits a slower convergence behavior. However, we warn that (as discussed in [@Karp]) the approximations (\[Eqn:Hpower\]) cannot be convergent for fixed $|z|$ as $L \to \infty$, because the Hankel functions possess a branch cut on the negative real axis which prevents them to be expanded by any Laurent series. Thus the number $L$ should be chosen judiciously, especially for small values of $kR$.
### Relationship between KSFE and BGT absorbing conditions
First, we consider the relationship between the BVPs corresponding to KSFE$_{1}$ and BGT$_{1}$ (the first order ABC from [@Bayliss01]). More precisely, we consider $u_{1}$ solving a BVP corresponding to the KSFE$_1$ condition (KSFE$_1$-BVP): $$\begin{aligned}
&& \Delta u_1+ k^2 u_1= 0, \qquad\qquad\qquad\quad\qquad \text{in $\Omega^{-}$}, \label{BVP1_1} \\
&& u_1 = - {u_{\rm inc}}, \quad\qquad\qquad\qquad\qquad\qquad\, \text{on $\Gamma$,} \label{BVP1_2} \\
&& u_1(R,\theta)= e^{ikR}\frac{f_0(\theta)}{(kR)^{1/2}}\label{BVP1_3} \\
&&\partial_{r} u_1(R,\theta) = \partial_{r}\left(e^{ikr}\frac{f_0(\theta)}{(kr)^{1/2}} \right) \bigg|_{r=R}=
e^{ikR} \frac{f_{0}(\theta)}{(kR)^{1/2}} \left( ik - \frac{1}{2R} \right),\label{BVP1_4} \end{aligned}$$ and $U_{1}$ solving a BVP corresponding to the BGT$_1$ condition (BGT$_1$-BVP): $$\begin{aligned}
&& \Delta U_1+ k^2 U_1= 0, \qquad\qquad\qquad\quad\qquad \text{in $\Omega^{-}$}, \label{BGT1_1} \\
&& U_1 = - {u_{\rm inc}}, \quad\qquad\qquad\qquad\qquad\qquad\, \text{on $\Gamma$,} \label{BGT1_2} \\
&& \partial_{r}U_1(R,\theta) + \frac{1}{2R}U_1(R,\theta) - ikU_1(R,\theta) = 0.\label{BGT1_3} \end{aligned}$$ It is clear from combining (\[BVP1\_3\]) and (\[BVP1\_4\]) that a solution $u_{1}$ of (\[BVP1\_1\])-(\[BVP1\_4\]) also satisfies the BVP (\[BGT1\_1\])-(\[BGT1\_3\]). Conversely, if $U_1$ is a solution of (\[BGT1\_1\])-(\[BGT1\_3\]), then by defining $f_0(\theta)=U_1(R,\theta)(kR)^{1/2}e^{-ikR}$, we immediately show that $U_1$ is a also a solution of (\[BVP1\_1\])-(\[BVP1\_4\]). Furthermore, the BVP (\[BGT1\_1\])-(\[BGT1\_3\]) has a unique solution as shown in [@Bayliss01]. As a consequence, the BVPs defined by the BGT$_1$ and KSFE$_1$ conditions have the same unique solution, which we state in the form of a theorem.
\[Equiv1\] The boundary value problems (\[BVP1\_1\])-(\[BVP1\_4\]) and (\[BGT1\_1\])-(\[BGT1\_3\]) are equivalent and they have a unique solution.
Secondly, we analyze if the BVPs corresponding to KSFE$_2$ and BGT$_2$ are equivalent. The KSFE$_2$-BVP consists of finding a function $u_2$ satisfying Helmholtz equation in $\Omega^{-}$, Dirichlet boundary condition on $\Gamma$, and the following absorbing boundary condition on $S$: $$\begin{aligned}
&& u_2(R,\theta)= \frac{e^{ikR}}{(kR)^{1/2}}\left( f_0(\theta)+ \frac{f_1(\theta)}{kR}\right)\label{BVP2_3} \\
&&\partial_{r} u_2(R,\theta) = \frac{e^{ikR}}{(kR)^{1/2}} \left( \left(ik - \frac{1}{2R} \right)f_{0}(\theta) + \left( ik - \frac{3}{2R} \right) \frac{f_{1}(\theta)}{kR} \right), \label{BVP2_4} \\
&& 2 i f_1(\theta) = \frac{1}{4} f_0(\theta) + f_0''(\theta).\label{BVP2_5}\end{aligned}$$ Similarly, the BGT$_2$-BVP consists of finding a function $U_2$ satisfying Helmholtz equation in $\Omega^{-}$, Dirichlet boundary condition on $\Gamma$, and the following absorbing boundary condition on $S$: $$\partial_{r}U_2 = \frac{( 2(kR)^2 + 3ikR - 3/4) U_2 + \partial^{2}_{\theta}U_2}{2R(1 - ikR)}
,\label{BGT2ABC}$$ Next, we will prove the following statement about the relationship between the BVPs corresponding to KSFE$_2$, KSFE$_3$, and BGT$_2$.
\[Non-Equiv\]
1. A solution $u_2$ of KSFE$_2$-BVP satisfies BGT$_2$-BVP only up to $O( R^{-7/2})$ at the artificial boundary $S$.
2. A solution $u_3$ of KSFE$_3$-BVP satisfies BGT$_2$-BVP up to $O(R^{-9/2})$ at the artificial boundary $S$.
We will prove statement (a) by showing that when $U_{2}$ is replaced by $u_{2}$ in (\[BGT2ABC\]), then the left hand side (lhs) of (\[BGT2ABC\]) is equal to its right hand side (rhs) up to $O(R^{-3/2})$. To obtain the expression for the lhs, we replace $\partial_r U_2$ in (\[BGT2ABC\]) with $\partial_r u_2$ and use (\[BVP2\_4\]). This leads to $$\text{lhs} = \left[ \left( ik - \frac{1}{2R} \right) f_{0} + \left( ik - \frac{3}{2R} \right) \frac{f_{1}}{kR} \right] \frac{e^{ikR}}{(kR)^{1/2}}. \label{lhs2}$$ On the other hand, replacing $U_2$ by $u_2$ defined by (\[BVP2\_3\]) into the rhs of (\[BGT2ABC\]), we obtain, $$\text{rhs} = \frac{1}{2R \left( 1 - ikR \right)} \left[ \left( 2(kR)^2 + 3ikR - \frac{3}{4} \right)\left( f_{0} + \frac{f_{1}}{kR} \right) + f''_{0} + \frac{f''_{1}}{kR} \right] \frac{e^{ikR}}{(kR)^{1/2}}. \label{rhs2}$$ Now, using the recurrence formula (\[BVP2\_5\]) in (\[lhs2\])-(\[rhs2\]), we obtain, $$\left( 1- ikR \right) \left( \text{lhs} - \text{rhs} \right) = \frac{ik e^{ikR}}{(kR)^{5/2}} \left( \frac{9}{16} f_{0} + \frac{5}{2} f''_{0} + f''''_{0} \right). \label{diff}$$ Hence, division by $(1-ikR)$ renders the statement (a).
A similar procedure leads to the proof of statement (b). First, we consider BVP defining the absorbing condition KSFE$_3$ which consists of finding a function $u_3$ satisfying Helmholtz equation in $\Omega^{-}$, Dirichlet boundary condition on $\Gamma$, and the following absorbing boundary condition on $S$: $$\begin{aligned}
&& u_3(R,\theta)= \frac{e^{ikR}}{(kR)^{1/2}}\left(f_0(\theta) + \frac{f_1(\theta)}{kR} + \frac{f_2(\theta)}{(kR)^{2}}\right), \label{BVP3_3} \\
&& \partial_{r} u_3(R,\theta) = \frac{e^{ikR}}{(kR)^{1/2}} \left( \left( ik - \frac{1}{2R} \right) f_{0}(\theta) + \left( ik - \frac{3}{2R} \right) \frac{f_{1}(\theta)}{kR} + \left( ik - \frac{5}{2R} \right) \frac{f_{2}(\theta)}{(kR)^2} \right),\label{BVP3_4} \\
&& 2i f_1(\theta) = \frac{1}{4}f_0(\theta) + f_0''(\theta), \label{BVP3_5} \\
&& 4i f_2(\theta) = \frac{9}{4}f_1(\theta) + f_1''(\theta). \label{BVP3_6}\end{aligned}$$ When replacing $U_3$ with $u_3$, then lhs of (\[BGT2ABC\]) becomes equal to $$\text{lhs} = \frac{e^{ikR}}{(kR)^{1/2}} \left( \left( ik - \frac{1}{2R} \right) f_{0}(\theta) + \left( ik - \frac{3}{2R} \right) \frac{f_{1}(\theta)}{kR} + \left( ik - \frac{5}{2R} \right) \frac{f_{2}(\theta)}{(kR)^2} \right). \label{lhs3}$$
Similarly, substituting $u_3$ into the rhs of (\[BGT2ABC\]) leads to $$2R (1 - ikR) \, \text{rhs} = \left( 2 (kR)^2 + 3ikR - 3/4 \right) \left( f_0 + \frac{f_1}{kR} + \frac{f_{2}}{(kR)^2} \right) + f''_0 + \frac{f''_1}{kR} + \frac{f''_2}{(kR)^2}. \label{rhs3}$$ Then, using the recurrence formulas (\[BVP3\_5\])-(\[BVP3\_6\]), we obtain that $(1 - ikR)\left( \text{lhs} - \text{rhs} \right) = O (R^{-7/2})$. Finally, the statement (b) is proved by dividing both sides by $(1 - ikR)$.
It was shown in [@Bayliss01] that a solution $U_2$ of the BGT$_2$-BVP approximates the exact solution of (\[BVPsc1\])-(\[BVPsc3\]) to $O (R^{-9/2} )$ when $R\rightarrow\infty$. From our previous results, we conclude that BGT$_2$-BVP and KSFE$_2$-BVP are not equivalent. Since a solution of KSFE$_2$-BVP satisfies BGT$_2$ to $O(R^{-7/2})$, a solution of KSFE$_2$-BVP will be a poorer approximation to the exact solution than $U_2$. However, the solution of KSFE$_3$-BVP satisfies BGT$_2$ to $O (R^{-9/2})$ also. It means that the solutions of BGT$_2$-BVP and KSFE$_3$-BVP approximate the exact solution at a comparable rate. This behavior is confirmed in our numerical experiments in Section \[Section.Numerics2D\].
Wilcox’s farfield expansion absorbing boundary condition in 3D {#Section.ABC3D}
--------------------------------------------------------------
For the 3D case ($\delta=3$), we also use a representation of outgoing waves by an infinite series in powers of $1/{kr}$. This representation is provided by a well-known theorem due to Atkinson and Wilcox, which is stated here for completeness.
\[Thm.Wilcox\] Let ${u_{\rm sc}}^{+}$ be an outgoing solution of the three-dimensional Helmholtz equation in the exterior region to a sphere of radius $r=R$. Then, ${u_{\rm sc}}^{+}$ can be represented by a convergent expansion $$\begin{aligned}
{u_{\rm sc}}^{+}(r,\theta,\phi) = \frac{e^{ikr}}{kr}\sum_{l=0}^{\infty} \frac{F_l(\theta,\phi)}{(kr)^l} \qquad\qquad \text{for $r> R$}.\label{WilcoxExp}\end{aligned}$$ This series is uniformly and absolutely convergent for $r>R$, $\theta$, and $\phi$. It can be differentiated term by term with respect to $r$, $\theta$, and $\phi$ any number of times and the resulting series all converge absolutely and uniformly. Moreover the coefficients $F_l$ ($l \ge 1$) can be determined by the recursion formula, $$\begin{aligned}
2 i l F_l(\theta,\phi) = l(l-1) F_{l-1}(\theta,\phi) + \Delta_{{\mathbb{S}}} F_{l-1}(\theta,\phi), \qquad l \geq 1. \label{AW-Recursive}\end{aligned}$$
Here, $r$, $\theta$, and $\phi$ are spherical coordinates and $\Delta_{{\mathbb{S}}}$ is the Laplace-Beltrami operator in the angular coordinates $\theta$ and $\phi$. See [@Bayliss01].
[Following an analogous procedure to the one employed in Section \[Section.ABC2D\] for the 2D case, we use a truncated version of the series (\[WilcoxExp\]) defined in $\Omega^{+}$ to match the solution in $\Omega^{-}$ through the interface conditions (\[BVPInterf4\])-(\[BVPInterf5\]).]{} This yields an approximation $u \approx {u_{\rm sc}}^{-}$ that is defined to be the solution of the following BVP in the region $\Omega^{-}$: $$\begin{aligned}
&& \Delta u + k^2 u = 0, \quad\qquad \text{in $\Omega^{-}$}, \label{BVP3D1} \\
&& u = - {u_{\rm inc}}, \qquad\qquad\, \text{on $\Gamma$,} \label{BVP3D2} \\
&& u(R,\theta,\phi)=\frac{e^{ikR}}{kR}\sum_{l=0}^{L-1} \frac{F_l(\theta,\phi)}{(kR)^l}\label{BVP3D3} \\
&&\partial_{r} u(R,\theta,\phi) = \frac{e^{i k R}}{k R} \sum_{l=0}^{L-1} \left( ik - \frac{l+1}{R} \right) \frac{F_{l}(\theta,\phi)}{(kR)^l}, \label{BVP3D4}\\
&& 2ilF_l(\theta,\phi) = l(l-1) F_{l-1}(\theta,\phi) + \Delta_{{\mathbb{S}}} F_{l-1}(\theta,\phi), \qquad l \geq 1. \label{BVP3D5}\end{aligned}$$ The equations (\[BVP3D3\])-(\[BVP3D5\]) form the absorbing boundary condition with $L$ terms which we call [*Wilcox farfield expansion absorbing boundary condition*]{} and denote WFE$_L$. We also denote the BVP (\[BVP3D1\])-(\[BVP3D5\]) as WFE$_L$-BVP. [Contrary to the 2D case, there is only one family of unknown angular functions $F_l$ in this case. Hence, we only need the interface conditions (\[BVPInterf4\])-(\[BVPInterf5\]) plus the recurrence formula (\[AW-Recursive\]) to define the new farfield expansion ABC at the artificial boundary $S$.]{}
The WFE-BVP (\[BVP3D1\])-(\[BVP3D5\]) can also be posed in weak form which is essential for the finite element methods. First we define the following (affine) spaces to deal with the Dirichlet boundary conditions (\[BVP3D2\]), $$\begin{aligned}
&& H^{1}_{\rm \Gamma , Dir}(\Omega^{-}) = \left\{ \text{$u \in H^{1}(\Omega^{-})$ : $u = - {u_{\rm inc}}$ on $\Gamma$} \right\}, \\
&& H^{1}_{\rm \Gamma,0}(\Omega^{-}) = \left\{ \text{$u \in H^{1}(\Omega^{-})$ : $u = 0$ on $\Gamma$} \right\}.\end{aligned}$$ We require the solution $(u,F_{0},F_{1},...,F_{L-1})$ to satisfy $u \in H^{1}_{\rm \Gamma , Dir}(\Omega^{-})$, $F_{l} \in H^{1}(S)$ for $l=0,...,L-2$, $F_{L-1} \in H^{0}(S)$, and $$\begin{aligned}
&& - {\langle}\nabla u , \nabla v {\rangle}_{\Omega} + k^2 {\langle}u , v {\rangle}_{\Omega} + \frac{e^{ikR}}{kR}\sum_{l=0}^{L-1} \frac{ik - (l+1)/R}{(kR)^l} {\langle}F_{l}, v {\rangle}_{S} = 0, \quad \text{for all $v \in H^{1}_{\rm \Gamma,0}(\Omega^{-})$}, \label{BVP3Dvar1} \\
&& {\langle}u , v_0 {\rangle}_{S} = \frac{e^{ikR}}{kR} \sum_{l=0}^{L-1} \frac{1}{(kR)^l} {\langle}F_{l} , v_0 {\rangle}_{S}, \qquad \text{for all $v_0 \in H^{0}(S)$}, \label{BVP3Dvar2} \\
&& 2 i l {\langle}F_{l} , v_l {\rangle}_{S} = l(l-1) {\langle}F_{l-1} , v_l {\rangle}_{S} - {\langle}\nabla_{S} F_{l-1}, \nabla_{S} v_l{\rangle}_{S}, \qquad \text{for all $v_l \in H^{1}(S)$, $l \geq 1$}. \label{BVP3Dvar3}\end{aligned}$$ where the symbol $\nabla_{S}$ represent the gradient in the geometry of the sphere $S$, and the functions $F_{l}$, originally defined on the unit-sphere, can be seen as defined on the sphere $S$ of radius $R$ by writing the argument as $\hat{x} = x/R$ where $x \in S$ and $R$ is fixed.
Numerical method {#Section:NumMethd}
================
We start this section describing how to obtain a numerical approximation of the solution for the acoustic scattering of a plane wave from a circular shaped obstacle of radius $r=r_0$ using Karp farfield expansions as ABC. As discussed in previous sections, our approach consists of numerically solving the KDFE$_L$-BVP defined by (\[BVPBd1\])-(\[BVPBd2\]) with the ABC given by (\[BVPBd3\])-(\[BVPBd5\]) supplemented by the recurrence formulas (\[Recurrence1\])-(\[Recurrence2\]). For this particular circular scatterer, polar coordinates $(r,\theta)$ is the natural choice of coordinate system. However, we will extend the discussion to more general obstacle shapes in generalized curvilinear coordinates in the next section. The numerical method chosen is based on a centered second order finite difference. The number of grid points in the radial direction is $N$ and in the angular direction is $m+1.$ Therefore, the step sizes in the radial and angular directions are $\Delta r =(R-r_0)/(N-1)$ and $\Delta\theta = 2\pi/m,$ respectively. Also, $r_i = (i-1)\Delta r,$ $\theta_j=(j-1)\Delta \theta$ and $u_{i,j}=u(r_i,\theta_j),$ where $i=1,\dots N$ and $j=1,\dots, m+1.$ Since the pairs $(r_i,\theta_1)$ and $(r_i,\theta_{m+1})$ represent the same physical point, $u_{i,1}=u_{i,m+1},$ for $i=1,\dots N.$ The discretization of the governing equations varies according to the location of the grid points. The areas of interest within the numerical domain and its boundaries are: the obstacle boundary $\Gamma$, the interior of the domain $\Omega^{-},$ and the artificial boundary $S$.
At the obstacle boundary $u=-{u_{\rm inc}}$ holds. Then, we start constructing the corresponding linear system $A{\bf U}={\bf b}$ by simply including the negative value of ${u_{\rm inc}}$ at each boundary grid point in the forcing vector ${\bf b}$, and let the corresponding entry in the coefficient matrix $A$ equal unity. This results in an identity matrix of size $m\times m$ in the upper left-hand corner of the matrix $A$.
At the interior points $(r_i,\theta_j)$ ($i=2,\dots N-2$, $j=1,\dots m$) in $\Omega^{-},$ we discretize Helmholtz equation to obtain $$\begin{aligned}
&&\alpha_i^{+}{u_{i+1,j}}+ \alpha_i^{-}{u_{i-1,j}}+ \alpha_i{u_{i,j}}+ \beta_i{u_{i,j+1}}+ \beta_i{u_{i,j-1}}=0,\label{intequ}\\
&&\alpha_i^{+} = \tfrac{1}{\Delta r^2} + \tfrac{1}{(2\Delta r) r_i},\qquad
\alpha_i^{-} = \tfrac{1}{\Delta r^2} - \tfrac{1}{(2\Delta r) r_i},\nonumber\\
&&\alpha_i = k^2 - \tfrac{2}{\Delta r^2} - \tfrac{2}{\Delta \theta^2 r_i^2},\qquad
\beta_i = \tfrac{1}{\Delta \theta^2 r_i^2}.\nonumber\end{aligned}$$ This discrete equation renders $(N-3) m$ new rows to the sparse matrix $A$ with a total of $5(N-3)m$ non-zero entries.
At the interior points $(r_{N-1},\theta_j)$ with $j=1,\dots m,$ we replace the $u_{N,j}$ term by $H_0(kR) \sum_{l=0}^{L-1} \frac{F_{l,j}}{(kR)^l} + H_1(kR)\sum_{l=0}^{L-1} \frac{G_{l,j}}{(kR)^l}$ from the farfield absorbing condition. This leads to the following discrete equation: $$\begin{aligned}
&&\alpha_{N-1}^{+}\sum_{l=0}^{L-1}\frac{H_0(kR)}{(kR)^l} F_{l,j} + \alpha_{N-1}^{+}\sum_{l=0}^{L-1}\frac{H_1(kR)}{(kR)^l} G_{l,j}\nonumber\\
&&\quad+ \alpha_{N-1}^{-}u_{N-2,j} + \alpha_{N-1}u_{N-1,j}+ \beta_{N-1}u_{N-1,j+1} + \beta_{N-1}u_{N-1,j-1}=0.\label{interioreq}\end{aligned}$$ This equation adds $ m$ new rows to the matrix $A$ with a total of $2(L+2)m$ nonzero entries.
Also at the artificial boundary points $(r_{N},\theta_j)$ with $j=1,\dots m,$ the discrete equation (\[intequ\]) is written as $$\begin{aligned}
&&\alpha_N^{+} u_{N+1,j}+ \alpha_N^{-} u_{N-1,j} + \alpha_N u_{N,j}+ \beta_{N} u_{N,j+1}+ \beta_N u_{N,j-1} =0.
\label{AB}\end{aligned}$$
Now, consider the discretization of equation (\[BVPBd4\]) using centered finite difference, [*i.e.*]{} $$u_{N+1,j} = u_{N-1,j}
-2\Delta r \sum_{l=0}^{L-1}A_l(kR)F_{l,j} -
2\Delta r \sum_{l=0}^{L-1}B_l(kR)G_{l,j}, \label{1deriv}$$ where $$A_l(kR)=\frac{kH_1(kR)}{(kR)^l} + \frac{klH_0(kR)}{(kR)^{l+1}}\quad\mbox{and}\quad
B_l(kR)=\frac{k(l+1)H_1(kR)}{(kR)^{l+1}} - \frac{kH_0(kR)}{(kR)^{l}}. \nonumber$$
Substitution of $u_{N+1,j}$, $u_{N,j+1}$, and $u_{N,j-1}$ into (\[AB\]) using the previous expression and Karp’s expansion, respectively, leads to the following set of $m$ equations for $j=1,\dots m$, $$\begin{aligned}
&&(\alpha_N^{+} +\alpha_N^{-} )u_{N-1,j} + \sum_{l=0}^{L-1}C_l(kR) F_{l,j} +\sum_{l=0}^{L-1}D_l(kR) G_{l,j}+ \label{uneq}\\
&&\beta_{N}\sum_{l=0}^{L-1}\frac{H_0(kR)}{(kR)^l} F_{l,j+1} + \beta_{N}\sum_{l=0}^{L-1}\frac{H_1(kR)}{(kR)^l} G_{l,j+1}
+ \beta_{N}\sum_{l=0}^{L-1}\frac{H_0(kR)}{(kR)^l} F_{l,j-1} + \beta_{N}\sum_{l=0}^{L-1}\frac{H_1(kR)}{(kR)^l} G_{l,j-1}=0,
\nonumber\end{aligned}$$ where the coefficients $C_l$ and $D_l$ are given by $$\begin{aligned}
C_l(kR) = -\alpha_N^{+}2\Delta r k \,A_l(kR)
+ \alpha_N \frac{H_0(kR)}{(kR)^l} \quad \mbox{and}\quad
D_l(kR) = -\alpha_N^{+} 2\Delta r k \,B_l(kR) + \alpha_N \frac{H_1(kR)}{(kR)^l}.\nonumber\end{aligned}$$ The number of non-zero entries for these set of equation is $nz = (6L+1)m$.
Another set of $m$ equations is obtained from the discretization of the continuity condition on the second derivative combined with (\[1deriv\]), and Karp farfield expansion, $$\begin{aligned}
\frac{2}{\Delta r^2}u_{N-1,j} + \sum_{l=0}^{L-1}M_l(kR)F_{l,j} + \sum_{l=0}^{L-1}N_l(kR)G_{l,j}=0,\end{aligned}$$ where $$\begin{aligned}
&&M_l(kR) =- \frac{2}{\Delta r} k\, A_l(kR) +
k^2\left(\frac{H_0(kR)}{(kR)^l} - (2l+1) \frac{H_1(kR)}{(kR)^{l+1}}- \frac{l(l+1)H_0(kR)}
{(kR)^{l+2}}\right) - \frac{2H_0(kR)}{\Delta r^2(kR)^l} \nonumber\\
&& N_l(kR) = - \frac{2}{\Delta r} k \,B_l(kR)+
k^2\left(-(2l+1)\frac{H_0(kR)}{(kR)^{l+1}} +\frac{H_1(kR)}{(kR)^{l}}- \frac{(l+1)(l+2)H_1(kR)}
{(kR)^{l+2}}\right) - \frac{2H_1(kR)}{\Delta r^2(kR)^l} \nonumber\end{aligned}$$ The total number of nonzero entries for these equations is $(2L+1)m$.
Finally, each one of the recurrence formulas (\[Recurrence1\])-(\[Recurrence2\]) contribute with $(L-1)m$ new equations. They are given by $$\begin{aligned}
&&2lG_{l,j} = \left((l-1)^2 -\frac{2}{\Delta \theta^2}\right)F_{l-1,j} + \frac{1}{\Delta \theta^2}F_{l-1,j+1} + \frac{1}{\Delta \theta^2}F_{l-1,j-1}, \label{RecurrGF1} \\
&&2lF_{l,j} = \left(-l^2 +\frac{2}{\Delta \theta^2}\right)G_{l-1,j} - \frac{1}{\Delta \theta^2}G_{l-1,j+1} - \frac{1}{\Delta \theta^2}G_{l-1,j-1} \label{RecurrGF2}\end{aligned}$$ for $l=1,\dots L-1$ and $j=1,\dots m$. The number of nonzero entries is four for each $j$ and for each recurrence formula.
The above discrete equations are written as a linear system of equations $A{\bf U} = {\bf b}$. The matrix $A$ structure depends on how the unknown vector ${\bf U}$ is ordered. We chose ${\bf U}$ as follows: $$\begin{aligned}
&&{\bf U}=
\Big[ \overbrace{u_{1,1} ...u_{1,m}}^{\text{at boundary }} ~ ~
\overbrace{u_{2,1} ...u_{2,m}...
u_{N-1,1}...u_{N-1,m}}^{\text{at interior grid points}} ~ ~\nonumber\\
&&\qquad\qquad\qquad\overbrace{F_{0,1}...F_{0,m}\, G_{0,1}...G_{0,m}...\,F_{L-1,1}...F_{L-1,m}\,G_{L-1,1}...G_{L-1,m}}^{\text{at artificial boundary}}
\Big]^{T} \label{VectorU}\end{aligned}$$ From the previous discrete equations (\[intequ\])-(\[VectorU\]), (\[uneq\])-(\[RecurrGF2\]), it can be seen that the matrix $A$ has dimension $(N-1+2L)m\times(N-1+2L)m$. Furthermore, adding the $m$ non-zero entries corresponding to the upper left-hand corner subdiagonal matrix of $A$ to the non-zero entries of the discrete equations (\[intequ\])-(\[VectorU\]) and (\[uneq\])-(\[RecurrGF2\]), it can be shown that the non-zero entries of $A$ are $nz=(5N-16)m+18Lm$.
A completely analogous work can be performed for the discretization of KSFE-BVPs. However, the BVP defined by (\[BVPBd1\])-(\[BVPBd2\]) with the KSFE$_L$ condition (\[BVP2D3\_Asymp\])-(\[BVP2D5\_Asymp\]) has only one family of unknown farfield angular coefficients $f_l(\theta)$ ($l=0,\dots L-1$). As a consequence, the matrix $A$ corresponding to its discrete equations has dimension $(N-1+L)m\times(N-1+L)m$. Moreover, it can be shown that its number of non-zero entries is $nz=(5N-13)m+8Lm$. For purpose of comparison, we also consider the discretization of the Dirichlet-to-Neumann boundary value problem (DtN-BVP) derived by Keller and Givoli [@Keller01]. For this BVP the matrix $A$, obtained by employing a second order centered finite difference method, has dimension $Nm\times Nm$ and its non-zero entries are $nz=(5N-8)m+m^2$. A relevant feature of the matrices $A$ for the KDFE-BVP and KSFE-BVP is that they do not have full blocks as found in the case of DtN-BVP. In fact, the number of non-zero entries for the DtN-BVP matrix is $O(m^2)$ against $O(Lm)$ for the KDFE-BVP and KSFE-BVP matrices, respectively. Now, the number L of terms in the farfield expansion is always much smaller than m (nodes in the angular direction). As a consequence, the non-zeros of the matrices associated to the KDFE and KSFE boundary value problems are considerable less than those of the matrix corresponding to the DtN-BVP for the same problem. This is a key property for the computational efficiency of the numerical technique proposed in this work. Furthermore, this is why higher order local ABCs are preferred over global exact ABCs such as DtN.
Applications of farfield ABCs {#Problems}
=============================
Scattering from a circular obstacle {#ScattCircle}
-----------------------------------
To illustrate the computational advantage of the exact farfield expansions ABCs over the DtN-ABC, we consider the acoustic scattering of a plane wave propagating along the positive $x$-axis from a circular obstacle of radius $r_0=1$. We place the artificial boundary at $R=2$ and select a frequency $k=2\pi$ for the incident wave. Then, we apply the centered finite difference scheme described in Section \[Section:NumMethd\] for the KDFE-BVP. For purpose of comparison, we also apply it with its respective modifications to KSFE-BVP and DtN-BVP. The points per wavelength in each case is $PPW=20$. The number of terms employed for KDFE$_L$ is $L=3$ and for the KSFE$_L$ is $L=8$. These choices of $L$ made possible that the three numerical solutions approximate the exact solution at the artificial boundary with about the same relative error of $3.8 \times 10^{-3}$ in the L$^2$-norm. In Fig. \[SpyGraph\], the structure of their respective matrices are depicted. Although the matrix corresponding to the DtN-ABC has the smallest dimension, it has more than one and a half times as many non-zero entries as the farfield expansions ABCs. As the number of point per wavelength increases, this difference is even bigger since the number of points for the DtN-ABC is $O(m^2)$ while for the farfield ABCs is only $O(Lm)$.
[0.32]{} ![Comparison of the matrix structure for: a) KDFE$_3$, b) DtN, and c) KSFE$_8$ with $R=2$ and $PPW =20$.[]{data-label="SpyGraph"}](SpyKFDE20.pdf "fig:"){width="\textwidth"}
[0.32]{} ![Comparison of the matrix structure for: a) KDFE$_3$, b) DtN, and c) KSFE$_8$ with $R=2$ and $PPW =20$.[]{data-label="SpyGraph"}](SpyDtN20.pdf "fig:"){width="\textwidth"}
[0.32]{} ![Comparison of the matrix structure for: a) KDFE$_3$, b) DtN, and c) KSFE$_8$ with $R=2$ and $PPW =20$.[]{data-label="SpyGraph"}](SpyKSFE20.pdf "fig:"){width="\textwidth"}
It is timely to comment on the numerical difficulties that can be faced when solving three-dimensional problems modeled by the farfield ABCs. Using our finite difference technique will lead to sparse but very large matrices at the discrete level. Therefore, a direct solver may not be a feasible choice as the mesh is refined. Iterative methods become an imperative choice. Among such methods are Krylov subspace iterative methods, multigrid and domain decomposition methods. However, their applications to the resulting sparse matrices experience difficulties because these matrices are known to be non-Hermitian and poorly conditioned. Efforts have been made to develop good preconditioners and parallelizable methods tailored to these wave scattering problems modeled by the Helmholtz equation [@Kechroud-Soulaimani2004; @Erlanga2008]. We intend to explore some of these new techniques along with the application of the farfield absorbing boundary conditions to complex 3D problems in future work.
Scattering from a spherical obstacle. Axisymmetric case {#ScattSphere}
-------------------------------------------------------
In this section, we formulate the BVP corresponding to scattering from a spherical obstacle of an incident plane wave ${u_{\rm inc}}= e^{ikz}$ propagating along the positive z-axis. The mathematical model including our novel Wilcox farfield ABC (WFE-ABC) consists of the BVP (\[BVP3D1\])-(\[BVP3D5\]) in spherical coordinates $(r,\theta,\phi)$. This problem is axisymmetric about the $z-$axis. Therefore, the governing Helmholtz equation for the approximation $u$ of the scattered field ${u_{\rm sc}}$ is independent of the angle $\phi$. As a consequence, it reduces to $$\frac{\partial ^2u}{\partial r^2} + \frac{2}{r}\frac{\partial u}{\partial r} +
\frac{1}{r^2\sin \theta}\frac{\partial}{\partial\theta}\left( \sin\theta\frac{\partial u}{\partial \theta} \right) + k^2 u =0, \quad \mbox{in}\,\, \Omega^{-}. \label{Helmholtzsphsymm}$$ Obviously, this equation is singular at the poles when $\theta = 0,\pi .$ However, there is not such singularity at these angular values for Helmholtz equation in cartesian coordinates. The singularity arises by the introduction of spherical coordinates. It can be shown [@SadikuBook] that equation (\[Helmholtzsphsymm\]) reduces to $\frac{\partial u}{\partial \theta}(r,\theta)=0,$ when $\theta=0,\pi$. The angular coefficients $F_l$ of the Wilcox farfield expansion are also independent of $\phi$. As in the two-dimensional case, we employ a second order centered finite difference scheme as our numerical method to obtain the approximate solution to this scattering problem. Due to the analogy between the KSFE and the WFE absorbing boundary conditions for this axisymmetric case, the discretization of the equations and the structure of the matrix obtained after applying a centered finite difference approximation to the equations defining this BVP are similar to those of KSFE-BVP. In Section \[orderconverg3D\], numerical results for this problem are presented.
Radiation and scattering from complexly shaped obstacles in two-dimensions {#ScattComplex}
---------------------------------------------------------------------------
Since most real applications deal with obstacle of arbitrary shape, in this section, we consider scattering problems for arbitrary shaped scatterers using the farfield absorbing boundary conditions. In order to do this, we introduce generalized curvilinear coordinates such that the physical scatterer boundaries correspond to coordinate lines. These type of coordinates, called boundary conforming coordinates [@Steinberg], are generated by invertible transformations $
T:\thinspace \mathcal{D^{\prime}}\rightarrow \mathcal{D}$, from a rectangular computational domain ${\cal D^{\prime}}$ with coordinates $(\xi,\eta)$ to the physical domain ${\cal D}$ with coordinates $(x,y)=(x(\xi,\eta),y(\xi,\eta))$. A common practice in elliptic grid generation is to implicitly define the transformation $T$ as the numerical solution to a Dirichlet boundary value problem governed by a system of quasi-linear elliptic equations for the physical coordinates $x$ and $y.$ Following this approach, the authors Acosta and Villamizar [@JCP2010] introduced the [*elliptic-polar grids*]{} as the solution to the following quasi-linear elliptic system of equations: $$\begin{aligned}
&& \alpha x_{\xi \xi }-2\beta x_{\xi \eta}+\gamma x_{\eta \eta } +
\frac{1}{2} \alpha_{\xi}x_{\xi} + \frac{1}{2}
\gamma_{\eta}x_{\eta} = 0, \label{Elliptic3} \\
&& \alpha y_{\xi \xi }-2\beta y_{\xi \eta}+\gamma y_{\eta \eta } +
\frac{1}{2} \alpha_{\xi}y_{\xi} + \frac{1}{2} \gamma_{\eta}y_{\eta} = 0.
\label{Elliptic4}\end{aligned}$$ The symbols $\alpha$, $\beta$, and $\gamma$, represent the scale metric factors of the coordinates transformation $T$, respectively. These are defined as $$\alpha =x_{\eta }^{2}+y_{\eta }^{2}, \qquad
\beta =x_{\xi }x_{\eta }+y_{\xi }y_{\eta }, \qquad \gamma=x_{\xi
}^{2}+y_{\xi }^{2}.$$ In this work, we adopt the elliptic-polar coordinates in the presence of complexly shaped obstacles. Before we attempt a numerical solution to our BVP with the farfield expansions ABCs in these coordinates, we express the governing equations in terms of them. For instance, the two-dimensional Helmholtz equation transforms into $$\begin{aligned}
\frac{1}{J^2} \Bigg[\alpha u_{\xi \xi
}-2\beta u_{\xi \eta}+\gamma u_{\eta \eta } +
\frac{1}{2}
\Big(\alpha_{\xi}\,u_{\xi}+\gamma_{\eta}\,u_{\eta}\Big)\Bigg]
+ k^2 u = 0, \label{HelmElliptic}\end{aligned}$$ where the symbol $J$ corresponds to the jacobian of the transformation $T$. Once the farfield expansions ABC equations are also expressed in terms of elliptic-polar coordinates, we transform all of these continuous equations into discrete ones using centered second order finite difference schemes. This process is described in detail in [@JCP2010]. Then, the corresponding linear system is derived in much the same way as we did above for polar coordinates. Numerical results for several complexly shaped obstacles are discussed in Section \[Section.Numerics2D\].
Farfield Pattern definition and its accurate numerical computation {#NumericalFFP}
==================================================================
In scattering problems, an important property to be determined is the scattered field far from the obstacles. The geometry and physical properties of the scatterers are closely related to it. In Section 4.2.1 of [@MartinBook], Martin defines the farfield pattern (FFP) as the angular function present in the dominant term of the asymptotic expansions for the scattered wave when $r\rightarrow\infty$. For instance in 2D, the farfield pattern is the coefficient $f_0(\theta)$ of KSFE, $$u(r,\theta)=
\frac{e^{ikr}}{(kr)^{1/2}}f_0(\theta)+O\left(1/(kr)^{3/2}\right). \label{KSFE}$$ Following Bruno and Hyde [@McKaySIAM], we now describe how the FFP can be efficiently calculated from the approximation of the scattered wave at the artificial boundary.
If $r>R$, where $R$ is the radius of the artificial circular boundary enclosing the obstacle, then, the scattered wave can be represented as the following complex Fourier series, $$u(r,\theta)= \sum_{q=-\infty}^{\infty} c_q(r) e^{i
q\theta}=\sum_{q=-\infty}^{\infty} b_q H_q^{(1)}(kr)e^{i q\theta}, \qquad \text{where} \quad b_q=\dfrac{c_q(r)}{H_q^{(1)}(kr)}.
\label{FourierSeries}$$ Using the asymptotic expansion of the Hankel function $H_q^{(1)}(kr)$ when $r\rightarrow\infty$, equation (\[FourierSeries\]) transforms into $$u(r,\theta) =
\frac{e^{ikr}}{(kr)^{1/2}}\left(\sqrt{\frac{2}{\pi
}}e^{-i\pi/4}\sum_{q=-\infty}^{\infty} b_q (-i)^q e^{i
q\theta}\right) + \mathcal{O}\left(1/{(kr)^{3/2}}\right). \label{Asymp}$$ By comparing (\[Asymp\]) with (\[KSFE\]) the following expression for $f_0(\theta)$ is derived $$f_0(\theta)=\sqrt{\frac{2}{\pi}}e^{-i\pi/4}\sum_{q=-\infty}^{\infty} b_q (-i)^q e^{i q\theta}.$$ Thus, the FFP can be determined once the coefficients $b_q$ have been calculated. But as pointed out above, the coefficients $b_q$ can be determined from the coefficients $c_q(r)$ for $r$ fixed. Likewise, approximated values of $c_q(R)$ can be obtained from the scattered field approximation at the artificial boundary $r=R$, [*i.e.,*]{} $u_{N,j}$ for $j=1,\dots m$. In fact as stated by Kress [@Kress], approximations ${\hat
c}_q$ to the coefficients $c_q(R)$, at the fictitious infinite boundary can be obtained by considering the discrete Fourier transform vector ${\hat c}_q$ ($q=-m/2 ,\dots m/2 -1$) of the vector $u_{N,j}$, interpolating the points $\left(\theta_j,u_{N,j}\right)$ for $j=1,\dots
m\, (m\,\, \mbox{even})$. More precisely, $${\hat c}_q=\frac{1}{m}\sum_{j=1}^{m-1} u_{N,j}
e^{-i q\theta_j},\qquad\mbox{ for}\quad q=-m/2,\dots m/2-1.
\label{FCoeffs}$$ These finite series can be directly evaluated, or a FFT algorithm can be used to compute them. The importance of the above derivation is that a semi-analytical formula $$f_0(\theta)=\sqrt{\frac{2}{\pi
}}e^{-i\pi/4}\sum_{q=-m/2}^{m/2-1} {\hat b}_q (-i)^q e^{i
q\theta}, \label{AnalyticalSCS}$$ approximating the FFP for arbitrary shaped obstacles, can be obtained from the numerical approximation of the scattered far field, where ${\hat b}_q={\hat c}_q/H^{(1)}_q(kR)$. This formula is extremely accurate as shown in [@McKaySIAM]. The error in the approximation of $f_0(\theta)$ using (\[AnalyticalSCS\]) depends almost entirely upon the error made in the approximation of the coefficients ${\hat b}_q$.
Numerical Results {#Section.Numerics2D}
=================
In this Section, [we present numerical evidences of the advantages of using the exact farfield expansions ABCs, when dealing with acoustic scattering and radiating problems, compared with other commonly used ABCs.]{} First, we numerically solve bounded problems with farfield expansions ABCs as defined in Sections \[Section.ABC2D\]-\[Section.ABC3D\]. Then, we show that these numerical solutions indeed converge to the exact solutions of the original unbounded BVPs. As described in Section \[Section:NumMethd\], the numerical method employed consists of familiar second order centered finite difference discretizations for Helmholtz equation in polar, spherical, and generalized curvilinear coordinates. This numerical method is completed with the discrete equations of the farfield expansions ABCs on the artificial boundary $S$. Our numerical results contain two sources of error. The first one is the error introduced by the finite difference scheme employed to discretize the Helmholtz equation in the computational domain $\Omega^{-}$. This error can be diminished by refining the finite difference mesh as we increase the number of points per wavelength. The second source of error is due to the truncation of the farfield expansion series. This error can be diminished by increasing the number $L$ of terms in the absorbing conditions KSFE$_L$, KDFE$_L$, or WFE$_L$. For example, if a finite difference scheme for a two-dimensional problem in polar coordinates leads to a second order convergence, then the order of the total error introduced by combining the finite difference scheme with the proposed absorbing boundary conditions is given by $$\begin{aligned}
\text{error} = O \left( h^2 \right) + O\left( (kR)^{-L} \right), \label{TotalError}\end{aligned}$$ where $h = r_{0} \Delta \theta = \Delta r$ is the mesh refinement parameter and $L$ is the number of terms in the farfield expansions absorbing boundary conditions. Then for the total error to exhibit second order convergence with mesh refinement, it is necessary to choose $L = O \left( \log( 1/h) \right)$. Therefore, there is no need of large increments of $L$ to improve the order of convergence. In practice, we can expect that a moderately large (but fixed) choice of $L$ will be sufficient to reveal the order of convergence of the finite difference method for a reasonable range of mesh refinements. We construct our numerical experiments, reported later in this section, according to this fact.
First, we present numerical results for the scattering of a plane wave ${u_{\rm inc}}= e^{ikx}$ from a circular obstacle in 2D. As a reference point, we also display the results from the use of the DtN nonreflecting condition [@Keller01; @Givoli05; @Grote-Keller01; @GivoliReview]. Since this latter condition is considered *exact*, it serves as a reference point to gauge the error introduced by the finite difference scheme alone. For all ABCs, we use a second order centered finite difference scheme in the interior of the domain $\Omega^{-}.$ In our experiments for the circular scatterer, we are able to obtain second order convergence of the numerical solution to the exact solution. In fact, we found that the numerical solution obtained from KDFE$_L$ and KSFE$_{L}$, for appropriate number of terms $L$, are comparable to the approximation obtained from the DtN absorbing boundary condition. However, the advantage over the DtN-ABC formulation is that the farfield expansions ABCs are local while the former is not.
Secondly, we numerically solve the scattering from complexly shaped scatterers, using the exact farfield expansions ABCs. As a result, the farfield patterns (FFP) for several obstacles of arbitrary shape are obtained. Then we present our results, with the new ABCs, for an exterior radiating problem obtained from two sources conveniently located inside a domain bounded by complexly shaped curves. [For these specials radiating problems, it is possible to obtain analytical solutions.]{} Then, by comparing the numerical approximations and the exact solutions, we determine the order of convergence for several non-separable geometries, as we do in the circular case.
Finally, results for a spherical scatterer are presented. The numerical method is analogous to the one employed in the two-dimensional case for the KDFE$_L$-BVP: a centered finite difference of second order in $\Omega^{-}$ and WFE-ABC on the artificial boundary. Again, a second order convergence is reached by using few terms in the WFE farfield expansions.
Scattering from a circular obstacle. Comparison against exact solution and order of convergence {#orderconverg2D}
-----------------------------------------------------------------------------------------------
First, we point out that approximated solutions of the scattering problem obtained for the BVP corresponding to KSFE$_1$ are identical to the numerical solutions obtained for the BVP corresponding to BGT$_1$. This is a numerical evidence of the equivalence between these two problems, as proved in Theorem \[Equiv1\].
Another important result is the convergence of the numerical solutions of KDFE$_L$ and KSFE$_L$ boundary value problems to the exact solution as the number $L$ of terms in the farfield expansion is increased for a sufficiently small $h$. In particular, this is shown in Fig. [\[fig:ScattFields\]]{}. However, the KSFE$_L$ numerical solutions only converge asymptotically as $kR \to \infty$ when $L$ is fixed. Indeed when $kR$ is fixed, [*e.g.*]{} $kR=\pi/2$, and $L$ grows then the numerical solution unavoidably diverges as explained at the end of subsection \[Section.ABC2DAsymp\]. This fact is also discussed in more detail in the Conclusions Section \[Section.Conclusions\]. Actually, Fig. [\[fig:ScattFields\]]{} (left panel) shows the appearance of unphysical oscillations in the farfield pattern for KSFE$_{9}$ which become larger as $L$ increases.
[0.5]{} ![Convergence of solutions of KSFE$_L$- and KDFE$_L$-BVPs ($h$ fixed and $L$ increasing) to the exact solution of scattering of a plane wave from a circular scatterer of radius $r_0=1$ along the artificial boundary with radius $R=1.05.$[]{data-label="fig:ScattFields"}](KSFEConverg.pdf "fig:"){width="\textwidth"}
[0.5]{} ![Convergence of solutions of KSFE$_L$- and KDFE$_L$-BVPs ($h$ fixed and $L$ increasing) to the exact solution of scattering of a plane wave from a circular scatterer of radius $r_0=1$ along the artificial boundary with radius $R=1.05.$[]{data-label="fig:ScattFields"}](KDFEConverg.pdf "fig:"){width="\textwidth"}
The relevant data used in these problems is the following: wavenumber $k=2\pi$, radius of the circular obstacle is $r_0 =1$, and radius of artificial boundary $R=1.05$. We define the grid such that the number of points per wavelength in all experiments is PPW = 30 in the angular direction and $N=21$ points in the radial direction. This is an extreme problem where the artificial boundary radius has been chosen almost equal to the radius of the circular scatterer. So, the domain of computation is very small. Even in this extreme situation, it is observed how well the numerical solution of KDFE$_7$-BVP approximates the exact solution at the artificial boundary with a $L_2$-norm relative error equal to $3.44 \times 10^{-4}$ with only seven terms in the farfield expansion. Similarly, the numerical solution of KSFE$_{11}$-BVP at the artificial boundary also approximates the exact solution with a $L_2$-norm relative error equal to $3.73 \times 10^{-4}$ with eleven terms in the expansion. This illustrates the slower convergence of the numerical solutions of KSFE$_L$-BVP when compared with the sequence of solutions obtained from KDFE$_L$-BVP.
[0.5]{} ![Comparison of L$_2$-norm relative error of the Farfield Pattern among DtN, BGT$_2$, KSFE$_L$, and KDFE$_L$ for $L=2,4,8,10$. The data in use is $r_0=1$, $R=2$, and $k=2\pi$.[]{data-label="fig:ConvergencePPW"}](PPW_NT2MB.pdf "fig:"){width="\textwidth"}
[0.5]{} ![Comparison of L$_2$-norm relative error of the Farfield Pattern among DtN, BGT$_2$, KSFE$_L$, and KDFE$_L$ for $L=2,4,8,10$. The data in use is $r_0=1$, $R=2$, and $k=2\pi$.[]{data-label="fig:ConvergencePPW"}](PPW_NT4B.pdf "fig:"){width="\textwidth"}
[0.5]{} ![Comparison of L$_2$-norm relative error of the Farfield Pattern among DtN, BGT$_2$, KSFE$_L$, and KDFE$_L$ for $L=2,4,8,10$. The data in use is $r_0=1$, $R=2$, and $k=2\pi$.[]{data-label="fig:ConvergencePPW"}](PPW_NT8B.pdf "fig:"){width="\textwidth"}
[0.5]{} ![Comparison of L$_2$-norm relative error of the Farfield Pattern among DtN, BGT$_2$, KSFE$_L$, and KDFE$_L$ for $L=2,4,8,10$. The data in use is $r_0=1$, $R=2$, and $k=2\pi$.[]{data-label="fig:ConvergencePPW"}](PPW_NT10B.pdf "fig:"){width="\textwidth"}
In our next set of numerical experiments, we analyze the performance of the second order finite difference method for the scattering from a circular scatterer using the following ABC: BGT$_2$, DtN, KSFE$_L$, and KDFE$_L$ ($L= 2,4,8,10$). By comparing the numerical farfield pattern (FFP) with the one obtained from the exact solution, we obtain the L$^2$-norm relative error. The formula employed to compute the FFP, for all types of ABCs from the numerical solution of the scattered field at the artificial boundary, is the formula (\[AnalyticalSCS\]) described in Section \[Section:NumMethd\]. The results of these experiments are illustrated in Fig. \[fig:ConvergencePPW\]. The common data in these numerical simulations is the following: frequency $k=2\pi$, radius of the circular obstacle is $r_0 =1$, and radius of artificial boundary $R=2$. In all our experiments, the error reported is the $L^2$-norm relative error. The grid is systematically refined, as $L$ is kept fixed, to discover the rate of convergence. For $L=2$ (top left corner subgraph), it is observed that the rate of convergence for three of the four types of ABC is close to zero, while the approximation to the exact solution of the numerical solution corresponding to DtN improves as the grid is refined. The subgraph at the top right corner reveals that the numerical solution of KDFE$_4$-BVP has almost the same rate of convergence than the one corresponding to DtN-BVP. From the subgraph in the lower row left corner, we conclude that the numerical solution of KSFE$_8$-BVP also converges at almost the same rate as the one for KDFE$_4$ and DtN boundary value problems. Finally for ten terms in both farfield expansions, the rate of convergence for the ABC: KSFE, KDFE, and DtN is basically the same.
The previous discussion illustrated in Fig. \[fig:ConvergencePPW\] is appropriately summarized by a single graph depicted in Fig. \[fig:OrderConv\]. This figure clearly shows the second order convergence of the three methods using: DtN, KDFE$_L$ ($L\ge 5$) and KSFE$_{L}$, ($L\ge 9$) while BGT$_2$-BVP order of convergence is around $3.8 \times 10^{-1}$. The set of grids employed to obtain Fig. \[fig:OrderConv\] consist of PPW = 30, 40, 50, 60, and 70, respectively. As a particular case of the quadratic convergence of KDFE$_L$-BVP ($L\ge 5$), we show the convergence of the numerical solution of KDFE$_5$-BVP in Table \[table:1\]. The grids are ordered from less to more refine. Furthermore, Fig. \[LeastSqKDFE5\] shows the line obtained from the least squares approximation of the orders between progressively finer grids. The slope of this line is 1.99948, which confirms the quadratic order of convergence for the numerical solution of KFDE$_5$-BVP using the technique proposed in this work.
![Comparison of order of convergence of FFP approximation for various ABCs versus the number of terms in the farfield expansion. The data in use is $r_0=1$, $k=2\pi$, $R=2$, and PPW = 30, 40, 50, 60, 70.[]{data-label="fig:OrderConv"}](OrderofConvergenceB.pdf){width="0.6\linewidth"}
---------------- ---------------- ------------------------------------- ----------------------- ----------------
PPW Grid size $h= r_{0} \Delta \theta = \Delta r$ $L^2$-norm Rel. Error Observed order
\[0.5ex\] $30$ $30\times 190$ $0.03324$ $1.64\times 10^{-3}$ $ $
$40$ $40\times 253$ $0.02493$ $9.19\times 10^{-4}$ $ 2.02$
$50$ $50\times 316$ $0.01995$ $5.87\times 10^{-4}$ $ 2.00$
$60$ $60\times 378$ $0.01667$ $4.10\times 10^{-4}$ $ 1.99$
$70$ $70\times 441$ $0.01428$ $3.04\times 10^{-4}$ $ 1.95$
\[1ex\]
---------------- ---------------- ------------------------------------- ----------------------- ----------------
: Order of convergence of FFP approximation using KDFE$_5$-BVP[]{data-label="table:1"}
![Least squares fitting line for the data in Table \[table:1\][]{data-label="LeastSqKDFE5"}](LeastSqKDFE5.pdf){width="0.5\linewidth"}
We observe that the numerical solution of KSFE-BVP also exhibits a second order convergence to the exact solution, although it requires more terms than the solution of KDFE-BVP to converge at the same rate. In fact as shown in Fig. \[fig:OrderConv\], nine terms or more are required in the farfield expansion of KSFE-ABC to reach second order convergence while only four or more terms are required in the farfield expansion of KDFE-ABC. Moreover, these numerical experiments provide numerical evidence of the non-equivalence between KSFE$_2$-ABC and BGT$_2$ as established in Theorem \[Non-Equiv\].
Scattering and radiation from complexly shaped obstacles {#ComplexObst}
--------------------------------------------------------
Our results in Section \[orderconverg2D\] for a circular shaped scatterer reveals the high precision that can be achieved by using the farfield expansions as ABCs with the appropriate number of terms and reasonable set of grids. As pointed out above, the accuracy of the overall numerical method is limited by the accuracy of the numerical method employed in the interior of the domain for relatively small number of terms, L, of the farfield ABCs.
[0.5]{} ![Total field and corresponding FFP for scattering from complexly shaped obstacles on elliptic-polar grids using KSFE$_5$-ABC with $k=2\pi$, $R=3$, and PPW=50.[]{data-label="fig:ComplexObst"}](ScattStarField5_50.jpg "fig:"){height="70.00000%"}
[0.5]{} ![Total field and corresponding FFP for scattering from complexly shaped obstacles on elliptic-polar grids using KSFE$_5$-ABC with $k=2\pi$, $R=3$, and PPW=50.[]{data-label="fig:ComplexObst"}](FFPStar.pdf "fig:"){height="70.00000%"}
[0.5]{} ![Total field and corresponding FFP for scattering from complexly shaped obstacles on elliptic-polar grids using KSFE$_5$-ABC with $k=2\pi$, $R=3$, and PPW=50.[]{data-label="fig:ComplexObst"}](ScattEpiField5_50.jpg "fig:"){height="70.00000%"}
[0.5]{} ![Total field and corresponding FFP for scattering from complexly shaped obstacles on elliptic-polar grids using KSFE$_5$-ABC with $k=2\pi$, $R=3$, and PPW=50.[]{data-label="fig:ComplexObst"}](FFPEpi.pdf "fig:"){height="70.00000%"}
[0.5]{} ![Total field and corresponding FFP for scattering from complexly shaped obstacles on elliptic-polar grids using KSFE$_5$-ABC with $k=2\pi$, $R=3$, and PPW=50.[]{data-label="fig:ComplexObst"}](ScattAstr3Field5_50.jpg "fig:"){height="70.00000%"}
[0.5]{} ![Total field and corresponding FFP for scattering from complexly shaped obstacles on elliptic-polar grids using KSFE$_5$-ABC with $k=2\pi$, $R=3$, and PPW=50.[]{data-label="fig:ComplexObst"}](FFP3Astroid.pdf "fig:"){height="70.00000%"}
In this section, we take advantage of this fact to numerically solve more realistic scattering problems. In fact, we find numerical solutions for acoustic scattering problems from obstacles with complexly shaped bounding curves such as a star, epicycloid, and astroid. We choose as the artificial boundary a circle of radius R = 3 and the frequency $k =2\pi$. As described in Section \[ScattComplex\], the differential equations defining these BVPs are written in terms of generalized curvilinear coordinates that Acosta and Villamizar derived in [@JCP2010]. The corresponding grids for these curvilinear coordinates were obtained from an elliptic grid generator and they were named elliptic-polar grids. Following the circular scatterer case, we use a second order centered finite difference method as our numerical technique for the interior points. A detailed account of the discretized equations in curvilinear coordinates are also found at [@JCP2010]. We employ KSFE$_5$ as our farfield expansion combined with PPW =50 (points per wavelength). The results are illustrated in Fig. \[fig:ComplexObst\] where the total field and its corresponding FFP are shown for each one of these obstacles. The parametric equations of these bounding curves are given by $$\begin{aligned}
\mbox{Star:}\quad x(\theta)=
0.2(4+\cos(5\theta))\cos(\theta)\qquad
y(\theta)=0.2(4+\cos(5\theta))\sin(\theta),\qquad 0\leq \theta\leq
2\pi. \label{star}\end{aligned}$$ $$\begin{aligned}
\label{epicycloid}
\mbox{Epicycloid:}\quad& &x(\theta)= ((5\sin(-(\theta+5\pi/4))-
\sin(-5(\theta+5\pi/4)))\cos(\pi/4)-\nonumber\\\nonumber
& &\qquad (5\cos(-(\theta+5\pi/4))-
\cos(-5(\theta+5\pi/4)))\sin(\pi/4))1/6\\
& &y(\theta)= ((5\sin(-(\theta+5\pi/4))-
\sin(-5(\theta+5\pi/4)))\sin(\pi/4)-\\\nonumber
& & \qquad(5\cos(-(\theta+5\pi/4))-
\cos(-5(\theta+5\pi/4)))\cos(\pi/4))1/6,\nonumber
\qquad 0\leq \theta\leq 2\pi. \end{aligned}$$ $$\begin{aligned}
\mbox{Astroid:}\quad& &x(\theta)= \left(2\cos(\theta-\pi/3)+
\cos(2(\theta-\pi/3))\right)\cos(\pi/3)/3-\nonumber\\
& &\qquad\left(2\sin(\theta-\pi/3)+
\sin(2(\theta-\pi/3))\right)\sin(\pi/3)/3\\
& &y(\theta)= \left(2\cos(\theta-\pi/3)+
\cos(2(\theta-\pi/3))\right)\sin(\pi/3)/3-\nonumber\\\nonumber
& &\qquad\left(2\sin(\theta-\pi/3)+
\sin(2(\theta-\pi/3))\right)\cos(\pi/3)/3,
\qquad 0\leq \theta\leq 2\pi. \label{astroid}\end{aligned}$$ For the experiments corresponding to the graphs shown in Fig. \[fig:ComplexObst\], using relatively fine grids with $PPW=50$, we did not find significant changes in the numerical solution by increasing the number of terms in the KSFE$_L$ condition up to $L=12$ terms.
Next, we discuss the numerical results for radiating problems defined in the exterior region $\Omega$ bounded internally by an arbitrary simple closed curve $\Gamma$. These BVPs consist of Helmholtz equation, Sommerfeldt radiation condition, and a Dirichlet condition on the complexly shaped bounding curve $\Gamma$. By imposing an appropriate boundary condition on $\Gamma$, we can easily prescribe a solution for each one of these BVPs. In fact, consider the function $u$ defined in the exterior region $\Omega$ from the superposition of two sources which are located inside the closed region bounded by $\Gamma$. More precisely, $u$ is given in terms of Hankel functions of first kind of order zero as $$u({\bf{x}}) = H_0^{(1)}(k r_{1}({\bf x})) + H_0^{(1)}(k r_{2}({\bf x}) ), \qquad {\bf x}\in \Omega\label{SourceSoln}$$ where $r_1 = |{\bf{x}}-{\bf x}_1|$, and $r_2 = |{\bf x}-{\bf x}_2|$ with ${\bf x}_1$ and ${\bf x}_2$ inside the region bounded by $\Gamma$. Clearly, the function $u$ satisfies Helmholtz equation in $\Omega$ since $H_0^{(1)}(k r_{i})$ does for $i=1,2$. It also satisfies the Sommerfeld radiation condition. Thus if we also impose the values of $u$ at the boundary $\Gamma$ (superposition of the two sources) as the boundary condition on $\Gamma$, the function $u$ defined by (\[SourceSoln\]) satisfies the radiating problem just defined, regardless of the shape of the bounding curve $\Gamma$.
Starting with the previously superimposed boundary condition on the bounding curve $\Gamma$, it is possible to obtain a numerical solution. First, we transform the unbounded radiating problem into a bounded one by introducing the KDFE-ABC or KSFE-ABC on a circular artificial boundary ($r=R$). Then, we apply the proposed numerical technique in generalized curvilinear coordinates in the region $\Omega^{-},$ bounded internally by $\Gamma$ and externally by the circle of radius $R$ to obtain the numerical solution sought.
[0.35]{} ![Numerical computation of a radiating field from two sources using KSFE$_{10}$-ABC, $k=2\pi$, $R=2$, and PPW=80. Order of convergence of FFP approximation for PPW = 60,65,70,75,80 for complex bounding curves.[]{data-label="fig:ComplexObstSources"}](SourceFieldCircle.pdf "fig:"){width="\linewidth" height="5.5cm"}
[0.3]{} ![Numerical computation of a radiating field from two sources using KSFE$_{10}$-ABC, $k=2\pi$, $R=2$, and PPW=80. Order of convergence of FFP approximation for PPW = 60,65,70,75,80 for complex bounding curves.[]{data-label="fig:ComplexObstSources"}](SourceFFPCircle.pdf "fig:"){width="\linewidth" height="5.2cm"}
[0.3]{} ![Numerical computation of a radiating field from two sources using KSFE$_{10}$-ABC, $k=2\pi$, $R=2$, and PPW=80. Order of convergence of FFP approximation for PPW = 60,65,70,75,80 for complex bounding curves.[]{data-label="fig:ComplexObstSources"}](SourceOrderConvCircle.pdf "fig:"){width="\linewidth" height="5cm"}
[0.35]{} ![Numerical computation of a radiating field from two sources using KSFE$_{10}$-ABC, $k=2\pi$, $R=2$, and PPW=80. Order of convergence of FFP approximation for PPW = 60,65,70,75,80 for complex bounding curves.[]{data-label="fig:ComplexObstSources"}](SourceFieldEpi.pdf "fig:"){width="\linewidth" height="5.5cm"}
[0.3]{} ![Numerical computation of a radiating field from two sources using KSFE$_{10}$-ABC, $k=2\pi$, $R=2$, and PPW=80. Order of convergence of FFP approximation for PPW = 60,65,70,75,80 for complex bounding curves.[]{data-label="fig:ComplexObstSources"}](SourceFFPEpi.pdf "fig:"){width="\linewidth" height="5.2cm"}
[0.3]{} ![Numerical computation of a radiating field from two sources using KSFE$_{10}$-ABC, $k=2\pi$, $R=2$, and PPW=80. Order of convergence of FFP approximation for PPW = 60,65,70,75,80 for complex bounding curves.[]{data-label="fig:ComplexObstSources"}](SourceOrderConvEpi.pdf "fig:"){width="\linewidth" height="5cm"}
[0.35]{} ![Numerical computation of a radiating field from two sources using KSFE$_{10}$-ABC, $k=2\pi$, $R=2$, and PPW=80. Order of convergence of FFP approximation for PPW = 60,65,70,75,80 for complex bounding curves.[]{data-label="fig:ComplexObstSources"}](SourceFieldStar.pdf "fig:"){width="\linewidth" height="5.5cm"}
[0.3]{} ![Numerical computation of a radiating field from two sources using KSFE$_{10}$-ABC, $k=2\pi$, $R=2$, and PPW=80. Order of convergence of FFP approximation for PPW = 60,65,70,75,80 for complex bounding curves.[]{data-label="fig:ComplexObstSources"}](SourceFFPStar.pdf "fig:"){width="\linewidth" height="5.2cm"}
[0.3]{} ![Numerical computation of a radiating field from two sources using KSFE$_{10}$-ABC, $k=2\pi$, $R=2$, and PPW=80. Order of convergence of FFP approximation for PPW = 60,65,70,75,80 for complex bounding curves.[]{data-label="fig:ComplexObstSources"}](SourceOrderConvStar.pdf "fig:"){width="\linewidth" height="5cm"}
The relevant data employed in our numerical experiments is the following: artificial boundary $R=2$, frequency $k=2\pi$, number of terms in the KSFE expansion $L=10$, location of sources ${\bf x}_1 =(0,1/2)$ and ${\bf x}_2 =(0,-1/2)$, set of grid points PPW $= 60, 65, 70, 75, 80$. We show that these numerical solutions indeed converge to the exact prescribed solution (\[SourceSoln\]) of the original radiating BVP. This is illustrated in Fig. \[fig:ComplexObstSources\] where the known radiating field from the two sources is numerically approximated in three different regions $\Omega^{-}$ which are internally bounded by three different curves. They are a circle of radius $r_0=1$, the epicycloid boundary curve defined in (\[epicycloid\]), and the star curve defined in (\[star\]). The relative $L^2$-norm error between the FFP of the prescribed solution and the approximated solution is computed for each different grid. Then, the order of convergence is estimated based on these errors. As seen in Fig. \[fig:ComplexObstSources\], we are able to prove quadratic convergence for the circle and for the star bounding curves. However, for the epicycloid we can only get 1.5 as order of convergence for the same set of grid points and number of terms in the farfield expansion $L$. This is due to the difficulty of generating conforming smooth grids in the neighborhood of the epicycloid singularities.
Scattering from a spherical obstacle. The axisymmetric case. Numerical approximation and order of convergence {#orderconverg3D}
-------------------------------------------------------------------------------------------------------------
In this section, we discuss the results for the scattering from a spherical obstacle modeled by WFE$_L$-BVP as described in Section \[ScattSphere\].
[0.37]{} ![Numerical results for scattering from a spherical scatterer using Wilcox farfield ABC: cross-sections of the total field for arbitrary $\theta$, farfield pattern, and order of convergence for two different frequencies $k=2\pi,4\pi$.[]{data-label="fig:3DSpericalScattering"}](TotField3Dk2pi.jpg "fig:"){width="\linewidth" height="5.5cm"}
[0.37]{} ![Numerical results for scattering from a spherical scatterer using Wilcox farfield ABC: cross-sections of the total field for arbitrary $\theta$, farfield pattern, and order of convergence for two different frequencies $k=2\pi,4\pi$.[]{data-label="fig:3DSpericalScattering"}](FFPk2pi3D.pdf "fig:"){width="\linewidth" height="5.5cm"}
[0.25]{} ![Numerical results for scattering from a spherical scatterer using Wilcox farfield ABC: cross-sections of the total field for arbitrary $\theta$, farfield pattern, and order of convergence for two different frequencies $k=2\pi,4\pi$.[]{data-label="fig:3DSpericalScattering"}](FFPOofC3D.pdf "fig:"){width="\linewidth" height="4.cm"}
[0.37]{} ![Numerical results for scattering from a spherical scatterer using Wilcox farfield ABC: cross-sections of the total field for arbitrary $\theta$, farfield pattern, and order of convergence for two different frequencies $k=2\pi,4\pi$.[]{data-label="fig:3DSpericalScattering"}](TotField3Dk4pi.jpg "fig:"){width="\linewidth" height="5.5cm"}
[0.37]{} ![Numerical results for scattering from a spherical scatterer using Wilcox farfield ABC: cross-sections of the total field for arbitrary $\theta$, farfield pattern, and order of convergence for two different frequencies $k=2\pi,4\pi$.[]{data-label="fig:3DSpericalScattering"}](FFPk4pi3D.pdf "fig:"){width="\linewidth" height="5.5cm"}
[0.25]{} ![Numerical results for scattering from a spherical scatterer using Wilcox farfield ABC: cross-sections of the total field for arbitrary $\theta$, farfield pattern, and order of convergence for two different frequencies $k=2\pi,4\pi$.[]{data-label="fig:3DSpericalScattering"}](FFPk4piOofC3D.pdf "fig:"){width="\linewidth" height="4.cm"}
In Fig. \[fig:3DSpericalScattering\], cross-sections of the total field for an arbitrary angle $\theta$ are depicted. The middle graphs corresponds to the approximation of the farfield pattern of this scattering problem. These graphs were extended to the interval $[0,2\pi]$ by taking the mirror image of the solution in $[0,\pi]$. Finally, the rightmost graphs show the second order convergence of the numerical solution to the exact solution when Wilcox farfield expansions ABC are employed. The data employed to generate the graphs in the top row of Fig. \[fig:3DSpericalScattering\] is: $k=2\pi$, $R=3$, terms in WFE$_L$, $L = 8$, and set of grid points used to achieve the second order convergence, PPW = 25, 30, 35, 40, 45. Similarly, the bottom row graphs were obtained using: $k=4\pi$, $R=3$, terms in WFE$_L$, $L = 8$, and set of grid points used to achieve the second order convergence, PPW = 30, 35, 40, 45, 50.
These results reveals the high accuracy that can be achieved using the exact Wilcox farfield expansions in the 3D case. As we showed in the 2D case, the accuracy of the numerical solutions depends only on the order of approximation of the numerical method employed in the interior of $\Omega^{-}$ when enough terms in the exact farfield expansions ABCs are used.
Concluding remarks {#Section.Conclusions}
==================
We have derived exact local ABCs for acoustic waves in two-dimensions (KDFE), and in three-dimensions (WFE). We have constructed them directly from Karp’s and Atkinson-Wilcox’s farfield expansions, respectively. A previous attempt by Zarmi and Turkel [@Zarmi-Turkel] to derive a high order local ABC from Karp’s expansion was partially successful. However, they were able to obtain other high order local conditions using an annihilating technique more general than the procedure used to obtain HH-ABC.
[0.33]{}
[0.33]{}
[0.33]{}
Some of the attributes of the novel farfield ABCs have been highlighted in various sections of this article. Among the most relevant attributes we find the exact character of these absorbing conditions according to [@GivoliReview2]. This means the error between the solutions of KDFE$_L$-BVP and WFE$_L$-BVP, and the solutions of their corresponding original unbounded problems approaches zero when $L\rightarrow\infty$ and the radius $R$ of the artificial boundary is held fixed. Although, it is not possible to prove this exact property merely from numerical experiments, it is still possible to determine this behavior for moderately large values of $L$. A discussion on this convergence properties follows in the next paragraphs.
As we pointed out earlier, possibly the most well-known higher order local absorbing boundary condition in two-dimensions is due to Hagstrom and Hariharan [@Hagstrom98] which we denote as HH-ABC. The advantage of KDFE sequence of ABCs over the HH counterpart is that the former leads to convergence of the numerical approximation to the exact solution for a fixed value of $R$, while the HH-ABC only converges asymptotically as $R$ increases. In addition to KDFE and WFE farfield ABCs, we also derived KSFE in Section \[Section.ABC2DAsymp\], which is a farfield expansion obtained from a classical asymptotic expansion of Karp’s series. This asymptotic expansion is the same employed in the derivation of the BGT and HH absorbing conditions in two-dimensions.
In Fig. \[FFEConvergence3\] the convergence properties of the numerical FFP obtained form KSFE-BVP and KDFE-BVP are compared. The physical problem is the same scattering problem studied in Section \[orderconverg2D\] and illustrated in Fig. \[fig:ScattFields\]. However, instead of describing the approximation of the outgoing wave at the artificial boundary, we describe the approximation of the farfield pattern for different values of $kR$ which are obtained for a fixed $R=1.05$ combined with appropriate values of $k$.
Notice that the convergence of the solutions of KSFE$_L$-BVP is conditioned by the value of the frequency $k$ and radius $R$ of the artificial boundary. More precisely, for $kR=\pi/2$ and $kR=\pi$, the FFP approximation of KSFE$_L$-BVP begins to diverge from the exact FFP for $L\ge 4$ and $L\ge 7$, respectively. However for $kR=2\pi$, the FFP of KSFE$_L$-BVP converges to the exact FFP when $L$ increases, for $1\le L\le 15$. Furthermore, this approximation is as good as the one obtained using the exact DtN boundary condition for $10\le L\le 15$. However, as we continue increasing the number of terms $L$, the solutions of KSFE$_L$-BVP will eventually diverge. This behavior parallels the one established by a rigorous proof given by Schmidt and Heier [@Kersten] for the convergence properties of the solution obtained using Feng’s absorbing boundary conditions. Feng’s condition arises from an asymptotic expansion of the exact DtN boundary condition for large $R$.
In practical terms, the use of KSFE-ABC is advisable only if the product $kR$ is large enough which is also applicable to any absorbing condition obtained from an asymptotic expansion of series representation of the outgoing waves. The application of KSFE$_L$ is still useful in many physical problems where $kR$ is sufficiently large since it takes only a few terms to reach the same order of convergence than the one obtained from DtN-ABC. On the other hand, the exact character of KDFE-BVP is clearly shown in Fig. \[FFEConvergence3\]. In fact, it only takes four terms of Karp’s expansion ($L=4$) to reach the same level of convergence of the solution of the DtN-BVP when $kr=\pi/2$. This level is maintained until $L=15$. Similar behavior is observed for the other two values of the product $kR$. In all these experiments $R=1.05$ and the frequency $k$ was chosen according to the desired value of $kR$. We also employed the same grid in all these experiments.
A non-asymptotic version of BGT$_2$ can be obtained by constructing a second order operator that annihilates the terms of $O \left( 1\right)$ in Karp’s expansion. This was the approach followed by Grote and Keller [@Grote-Keller01] to obtain the second order differential operator, $$L_0u = \partial_r u - k \left( \frac{H_0'(kr)}{H_0(kr)}u - \left(\frac{H_0'(kr)}{H_0(kr)} -\frac{H_1'(kr)}{H_1(kr)}\right)\partial_{\theta}^2 u\right)\label{BGTH2}$$ An alternative derivation of (\[BGTH2\]) was given by Li and Cendes [@Li-Cendes] by requiring that the first two terms of the exact solution of normal modes of Helmholtz equation in cylindrical coordinates were annihilated. All these authors and more recently Turkel, Farhat, and Hetmaniuk [@Turkel-Farhat2004] used the differential operator (\[BGTH2\]) at the artificial boundary as an ABC for the scattering of a plane wave from a circular obstacle. They noticed the superior accuracy of the solution obtained with this condition compared with the one obtained from the absorbing boundary conditions BGT$_L$ (for $L=1,2$), for low values of the frequency $k$. In particular, Turkel et al. [@Turkel-Farhat2004] showed that for a frequency $k=0.01$ and radius $R=5$ (artificial boundary) the $L^2$-norm relative error at the artificial boundary is about 50 times better using (\[BGTH2\]) over BGT$_2$. These results can be considered as a low order version of the results illustrated in Fig. \[FFEConvergence3\] for the high order local KSFE and KDFE absorbing boundary conditions. Zarmi and Turkel [@Zarmi-Turkel] also arrived to the same conclusion by comparing their higher order version of Li and Cendes’ operator in 2D with the higher order versions of HH operators obtained from the asymptotic Karp’s expansion.
We would like to highlight two other valuable attributes of the farfield ABCs. First, the farfield pattern is the coefficient of the leading term of the farfield expansion. This leading coefficient (angular function) is one of the unknowns of the linear system to be solved to obtain the approximation of the exact solution. So, there is no additional computation afterward to obtain the FFP. In most of our experiments, we decided to use the FFP approximation formula obtained in Section \[NumericalFFP\] for comparison purposes. Secondly, by increasing the parameter $L$ (number of terms in the expansion), the error introduced by KDFE$_L$ and WFE$_L$ can easily be reduced and made negligible compared with the error from the numerical method in the interior domain $\Omega^{-}$.
There are numerous directions in which the application of farfield ABCs can be extended. Some of those on which we are currently working or plan to work are the following:
1. [The combined formulation of high order finite difference (or finite element), for the discretization of the Helmholtz equation in $\Omega^{-}$, with the novel exact local farfield ABCs. This will show the high accuracy that can be achieved by simply increasing the number of terms in the ABCs expansion, using relatively coarse grids. For this purpose, we plan to explore several high order compact finite difference schemes that have been recently developed [@Britt-Tsynkov-Turkel2010; @Turkel-Gordon2013] and others well-established found in [@Lele1992].]{}
2. The extension of the formulation of our ABC to the wave equation (time-domain). This extension is clearly feasible in 3D since the time-domain analogue of the Wilcox expansion is available [@Bayliss-TurkelWave1980; @Grote03] due to the Fourier duality between $\partial_{t}$ and $ik$, and between $e^{ikr}$ and time shift. This is also valid for the KSFE-ABC in 2D. However, for the KDFE absorbing condition in 2D, Karp’s expansion has no closed-form transformation to the time domain due to the complexity of the terms $H_{0}(kr)$ and $H_{1}(kr)$. Such a transformation would lead to nonlocal operators in the time variable similar to the ones discussed in [@Alpert-Greengard-Hagstrom2000].
3. Construction of exact local farfield ABCs for multiple scattering of time-harmonic waves. The farfield expansions of Wilcox and Karp allow the evaluation of the scattered field semi-analytically at any point outside the artificial boundary. This property is fundamental in the multiple scattering setting for the introduction of artificial sub-boundaries enclosing obstacles disjointly *à la* Grote-Kirsch [@Grote01].
Acknowledgments {#acknowledgments .unnumbered}
===============
The first and third authors acknowledge the support provided by the Office of Research and Creative Activities (ORCA) of Brigham Young University. Thanks are also due to the referees for their useful suggestions.
|
---
abstract: 'We report a novel crossover behavior in the long-range-ordered phase of a prototypical spin-$1/2$ Heisenberg antiferromagnetic ladder compound $\mathrm{(C_7H_{10}N)_2CuBr_4}$. The staggered order was previously evidenced from a continuous and symmetric splitting of $^{14}$N NMR spectral lines on lowering temperature below $T_c\simeq 330$ mK, with a saturation towards $\simeq 150$ mK. Unexpectedly, the split lines begin to further separate away below $T^*\sim 100$ mK while the line width and shape remain completely invariable. This crossover behavior is further corroborated by the NMR relaxation rate $T_1^{-1}$ measurements. A very strong suppression reflecting the ordering, $T_1^{-1}\sim T^{5.5}$, observed above $T^*$, is replaced by $T_1^{-1}\sim T$ below $T^*$. These original NMR features are indicative of unconventional nature of the crossover, which may arise from a unique arrangement of the ladders into a spatially anisotropic and frustrated coupling network.'
author:
- 'M. Jeong (정민기)'
- 'H. Mayaffre'
- 'C. Berthier'
- 'D. Schmidiger'
- 'A. Zheludev'
- 'M. Horvatić'
bibliography:
- 'bib\_QSpinSystems.bib'
title: Novel Crossover in Coupled Spin Ladders
---
[UTF8]{}[mj]{}
Spin ladders in a magnetic field are a paradigmatic model in quantum magnetism and many-body physics [@Giamarchi99PRB; @Giamarchi]. For instance, a spin-$1/2$ Heisenberg antiferromagnetic (AFM) ladder in a field between the two critical values, $H_{c1}$ and $H_{c2}$, hosts as the ground state a Tomonaga-Luttinger liquid (TLL), a state universal to interacting quantum particles in one dimension (1D) with gapless excitations [@Giamarchi; @Haldane81JPC; @Haldane81PRL]. When the ladders are embedded in real material, a weak residual coupling between them comes into play at sufficiently low temperatures, and this dimensional crossover manifests itself as a second-order phase transition into a canted $XY$ ordered phase [@Giamarchi99PRB]. This 3D long-range-ordered (LRO) phase is described as a Bose-Einstein condensate (BEC) of magnons [@Giamarchi99PRB; @Nikuni00PRL; @Giamarchi; @Giamarchi08NPhys; @Bouillot11PRB]. The transition between the two canonical quantum phases, 1D TLL and 3D BEC, has been successfully demonstrated with a metal-organic spin-ladder compound $\mathrm{(C_5H_{12}N)_2CuBr_4}$, known as BPCB, by NMR [@Klanjsek08PRL] and neutron diffraction [@Thielemann09PRB]. The same class of transition has been observed since then in an increasing number of quasi-1D spin systems of magnetic insulators including bond-alternating AFM chains [@Willenberg15PRB], and also in ultracold atoms trapped in an array [@Vogler14PRL].
![Crystal structure of DIMPY where orange balls in green tetrahedra represent spin-$1/2$ Cu ions of $\mathrm{CuBr}_4$ units and solid lines represent predominant exchange pathways forming a ladder-like network. Broken lines represent much weaker couplings between the ladders. (a) View presenting the ladders side by side. (b) View along the ladder direction showing the coupling of the ladders along the $b$ and $c$ axes.[]{data-label="exchange"}](exchange){width="50.00000%"}
{width="100.00000%"}
Recently, another metal-organic spin-ladder compound $\mathrm{(C_7H_{10}N)_2CuBr_4}$, known as DIMPY, has attracted much attention [@Shapiro07JACS; @White10PRB; @Hong10PRL; @Schmidiger11PRB; @Ninios12PRL; @Schmidiger12PRL] as a unique example for a strong-leg regime, i.e., $J_\mathrm{leg}/J_\mathrm{rung}=1.7$ where $J_\mathrm{leg}=16.5$ K (see Fig. \[exchange\] for crystal structure) [@Shapiro07JACS; @White10PRB; @Hong10PRL; @Schmidiger11PRB; @Ninios12PRL; @Schmidiger12PRL] with experimentally accessible $H_{c1}\simeq 2.5$ T and $H_{c2}\simeq 29$ T. The single-ladder (1D) Hamiltonian of DIMPY has been thoroughly determined by using inelastic neutron scattering in conjunction with the density matrix renormalization group (DMRG) calculations and bulk measurements [@White10PRB; @Hong10PRL; @Schmidiger11PRB; @Ninios12PRL; @Schmidiger12PRL]. The low-energy excitations in a single-ladder limit, probed via inelastic neutron scattering [@Ninios12PRL; @Schmidiger12PRL; @Schmidiger13PRL; @Povarov15PRB] and NMR relaxation [@Jeong13PRL; @Jeong16PRL], were shown to agree with the TLL predictions [@Giamarchi99PRB; @Giamarchi]. Moreover, specific-heat anomalies [@Ninios12PRL; @Schmidiger12PRL] observed typically around $T_c\sim 300$ mK in a magnetic field $H>H_{c1}$ were shown to correspond to an onset of a staggered LRO due to weak interladder couplings [@Jeong13PRL]. Therefore, DIMPY, together with a weak-leg ladder representative compound BPCB ($J_\mathrm{leg}/J_\mathrm{rung}=0.28$) [@Bouillot11PRB; @Klanjsek08PRL; @Ruegg08PRL], is considered to provide a complete experimental toolkit for exploring the physics of coupled spin ladders in a field [@Bouillot11Geneve; @Schmidiger13PRB; @Schmidiger14ETH; @Jeong16PRL; @Steinigeweg16PRL].
We report in this Letter a new set of NMR observation on DIMPY which defies the standard paradigm [@Giamarchi99PRB] of the coupled spin ladders in a field. We discover an unexpected crossover taking place around $T^*\sim 100$ mK, where upon cooling the size of the seemingly saturated ordered moments grows again and the low-energy excitations change the nature. We present the original NMR signatures of the crossover and discuss a possible origin in light of the recent theory [@Furuya16PRB].
A single-crystal sample was directly put into a $^3$He-$^4$He mixture of a dilution refrigerator to ensure a good thermal contact. $^{14}$N (nuclear spin value $I=1$) NMR experiments were performed using a standard pulsed spin-echo technique. The spectrum was obtained by performing a Fourier transform of the spin echo signal that follows an excitation and refocusing NMR pulses. NMR spin-lattice relaxation rate, $T_1^{-1}$, was obtained by a saturation-recovery method, using the theoretical relaxation function for $I=1$ nuclei, $M(t)/M_0=1-0.25\exp (-(t/T_1)^\alpha) - 0.75\exp (-(3t/T_1)^\alpha)$, where $M(t)$ is nuclear magnetization, $t$ is a time interval between the saturation pulse and the echo pulses, and $M_0$ the nuclear magnetization in equilibrium ($t\rightarrow \infty$). The stretch exponent $\alpha$ was introduced to describe distribution of $T_1^{-1}$ values. The saturation of nuclear magnetization was achieved by using a single pulse as long as $10\sim 20\,\mu s$ to reduce the excitation power so that unwanted heating effects were avoided.
Figure \[splitting\](a) and (b) show the $^{14}$N NMR line shape as a function of temperature in an applied field of 9.0 and 15.0 T, respectively. In both fields, a spectral line at high temperatures becomes broadened as temperature is lowered, and then splits symmetrically into two lines across $T_c\simeq 330$ mK [@Jeong13PRL]. This splitting reflects the growth of the staggered transverse ($\perp H$) moments, i.e., the order parameter (OP). Figure \[splitting\](c) plots temperature evolution of the splitting which tends to saturate as temperature approaches 150 mK. However, as temperature is further lowered across $T^*\sim 100$ mK, the split lines begin to separate further away symmetrically. At the lowest measured temperature of 40 mK the splitting becomes 33 kHz, which is 50 % larger than the 22 kHz observed at $\sim 150$ mK.
The NMR lines have a Gaussian shape over the measured temperature and field ranges, except close to $T_c$ where the line shape can be decomposed into two superimposed Gaussians (Fig. \[splitting\](a) and (b)). The line widths at high temperatures above 400 mK are 4.4 and 6.9 kHz in 9 and 15 T, respectively, meaning that the line broadening scales with the field and is thus of magnetic origin. When temperature is lowered across $T_c$, the line broadens on top of the splitting which is a hallmark of magnetic ordering transition. On the other hand, the line shape and width remain almost completely intact across $T^*$. Figure \[splitting\](d) plots the line width normalized by the high temperature value as a function of temperature. The overall spectral features are practically indistinguishable between 9 and 15 T.
{width="80.00000%"}
The crossover behavior in the spectrum across $T^*$ is further corroborated by the relaxation rate measurements. Figure \[T1ex\](a) shows $T_1^{-1}$ as a function of temperature in 9.0 T and 15.0 T. Note that $T_1^{-1}$ probes Cu$^{2+}$ electron spin fluctuations in the low energy limit. At high temperatures in the TLL regime, $T_1^{-1}$ increases with lowering temperature by 1D quantum-critical fluctuations [@Jeong13PRL]. As temperature further approaches $T_c$, the $T_1^{-1}$ increases even more rapidly by the addition of thermal-critical fluctuations, which is another hallmark of magnetic ordering transition. Then, a very strong suppression of $T_1^{-1}$, by more than two orders of magnitude, follows the peak at $T_c$ as temperature is lowered below 300 mK. In the temperature range where the OP is apparently saturated, we find $T_1^{-1}\sim T^{5.5}$. A similar suppression has been observed in other quasi-low-dimensional quantum magnets below the ordering transition [@Mayaffre00PRL]. However, as temperature is further lowered across $T^*$, the $T_1^{-1}$ begins to bend out from the strong suppression, and roughly follows $T_1^{-1}\propto T$ behavior. In addition, $T_1^{-1}$ in 9 T is roughly twice larger than the one in 15 T in this regime.
Figure \[T1ex\](b) plots the stretch exponent $\alpha$ used to fit the nuclear magnetization recovery curves shown in Fig. \[T1ex\](c) by the theoretical relaxation function. This exponent is indicative of local magnetic inhomogeneity. For instance, $\alpha$ remains practically 1 above $T_c$, indicating a homogeneous magnetic environment. When temperature is lowered below $T_c$, $\alpha$ drops down to $0.6\sim 0.7$, which indicates the development of local dynamic inhomogeneity or distribution of $T_1^{-1}$ values. Although the temperature dependence of $T_1^{-1}$ changes across $T^*$, this is not accompanied by any noticeable modification of $\alpha$.
Let us now discuss the consequences of the observed NMR signatures of the crossover between the two low temperature regimes, above and below $\sim$100 mK, which we label as LRO I and LRO II, respectively (see Fig. \[splitting\](c) and \[T1ex\](a)). Since the width of the NMR lines does not increase nor $T_1^{-1}$ shows a peak across $T^*$, one can rule out a symmetry-breaking, continuous phase transition accompanied by critical fluctuations. In addition, quadrupolar splitting (not shown), which probes crystalline electric field gradient, does not change over the whole temperature range, pointing to the absence of any structural change. Indeed, at such low temperatures phonon modes are likely to be completely quenched. We thus simply associate the increasing NMR line separation below $T^*$ with additional size growth of the OP. The enhanced $T_1^{-1}$ at high temperatures in the TLL and thermal-critical regimes are suppressed below $T_c$ as the fluctuations associated with the magnetic moments are suppressed and the long range order develops. Regarding the emergence of the $T_1^{-1}\propto T$ behavior below $T^*$, there are two possible cases: one is that the fluctuation spectrum itself changes qualitatively, while the other is that the temperature-linear behavior is intrinsic to the ordered phases but only revealed once the enhanced fluctuations are quenched out.
We note that our NMR observations have certain correspondence to the recent theory on spatially anisotropic Heisenberg antiferromagnets, with DIMPY as a specific example [@Furuya16PRB]. The theory bases the argument on a uniquely anisotropic interladder coupling network of DIMPY. Naively, from the distances between the magnetic $\mathrm{Cu}^{2+}$ ions, one could expect that the interladder exchange interactions would be stronger along the $c$ direction than the $b$ direction (Fig. \[exchange\]). Moreover, presuming the interactions are antiferromagnetic along the $b$ axis, they would be frustrated such that the effective coupling strength is even further reduced. Thus, depending on the energy scale defined by the temperature, the magnetic lattice of DIMPY may be considered as a 1D ladder, a 2D net of coupled ladders, and a 3D stack. Taking into account this hierarchical coupling strengths, the analytical and numerical results predict on cooling a transition from the TLL into a quasi-2D regime of the ordered phase, which in fact does present a full 3D coherence, but has reduced OP because of strong fluctuations. This is followed by a crossover to a “true” 3D regime, where these fluctuations are frozen and the OP is thus bigger. In this crossover, the OP monotonically increases on cooling [@Furuya16PRB], which apparently corresponds to our experimentally observed NMR line splitting in the LRO II regime. However, the theoretically predicted temperature dependence of the OP does not really present a plateau for the quasi-2D regime, and is thus somewhat different from the one observed in the LRO I regime.
On the other hand, the NMR line shape is in sharp contrast to what is expected from the above theory. The main characteristic of the predicted quasi-2D ordered regime is that the phase (orientation) of the local OP value is only very weakly correlated between the planes, which connects to a Berezinskii-Kosterlitz-Thouless phase [@Berezinskii1SJETP71; @Kosterlitz73JPC] in the limit of an isolated plane. In the above theory [@Furuya16PRB], presuming a clean system without impurities, very weak interplane coupling and thus the strong phase fluctuations give rise to a small OP in the quasi-2D regime. Real compounds always have impurities which act as the pinning centers, so that the *local* OP phase is expected to be pinned in different directions throughout the sample. As the NMR hyperfine coupling is strongly sensitive to the orientation of the OP, different local orientations correspond to different NMR line positions. Thus, in the quasi-2D regime having relatively uncorrelated planes, one expects a broadly distributed NMR spectrum. In the 3D regime at lower temperature, the OP is growing because the 3D coupling suppresses the strong phase fluctuation. In real compounds, this corresponds to the local OP phase becoming homogeneously locked to the same value throughout the sample, meaning that the NMR line width should shrink to its normal value (as above $T_c$). In contrast to these prediction, the NMR lines in DIMPY do not broaden nor change their shape upon entering the ordered phases, and we therefore conclude that the OP has the same average orientation throughout the whole sample, in both LRO I and LRO II regimes.
The absence of the temperature dependence of the NMR line widths suggests an alternative scenario: some (small) anisotropy defining a preferential direction in real materials, particularly in crystals of low symmetry as DIMPY, may play a role. This would fix the OP phase and thus ensure a full 3D coherence at all temperature, leading to a temperature-independent NMR line width. Furthermore, depending whether the temperature is higher or lower than the energy scale of this anisotropy, the OP phase fluctuations will be either strong or frozen to the optimal direction; the corresponding average OP value will be thus reduced at higher temperature and will grow towards its full size at lower temperature, as observed by the NMR line splitting. Moreover, there is an obvious candidate for the anisotropy in DIMPY: the local crystal symmetry allows uniform Dzyaloshinkii-Moriya interactions, whose coupling strength is estimated to be as large as 310 mK, of the same size as the $T_c$ value [@Ozerov15PRB; @Glazkov15PRB]. Inclusion of such anisotropic terms into the theory is challenging, and beyond the scope of this work.
Unfortunately, the anisotropy scenario may have difficulty in explaining the low temperature $T_1^{-1}$ data. Across $T^*$, the accompanying freezing of spin fluctuations would lead to the quenching of $T_1^{-1}$ relaxation rate. However, $T_1^{-1}$ is not quenched at low temperature, but is rather maintained or enhanced. One possible explanation is that the low temperature relaxation is due to *accidentally* concomitant setting in of the impurity relaxation. This may indeed be field dependent as observed, as stronger field reduces impurity spin fluctuations. Another possibility is that the crossover comprises “quasi-critical” enhanced fluctuations which could give rise to the observed $T_1^{-1}$ behavior (though the field dependence would be subject to the actual model for the crossover). On the other hand, it is also interesting to note that the $T_1^{-1}\propto T$ behavior found in the LRO II is precisely the one that has been, on general grounds, theoretically predicted for a 3D BEC phase of weakly-coupled spin-$1/2$ Heisenberg AFM ladders [@Giamarchi99PRB].
To conclude, the magnetized spin-ladder compound DIMPY displays a novel type of crossover with temperature in the LRO phase. We have shown that on cooling across the crossover the seemingly saturated order parameter amplitude grows again and, moreover, the low-energy excitations become strongly modified so that NMR relaxation rate becomes linear in temperature. These original observations have certain correspondence to the recent theory on spatially anisotropic Heisenberg antiferromagnets showing the crossover between the quasi-2D and the 3D ordered phases [@Furuya16PRB]. Admittedly, the correspondence is not complete, which calls for future theoretical and experimental studies. We hope that our new finding will help to elucidate the intriguing manifestation of effective dimensionality and frustration, and their interplay.
We acknowledge fruitful discussions with T. Giamarchi, S. C. Furuya and N. Laflorencie. This work was supported by the French ANR project BOLODISS (Grant No. ANR-14-CE32-0018). M.J. is grateful for the support of European Commission through Marie Sk[ł]{}odowska-Curie Action COFUND (EPFL Fellows), and Swiss National Science Foundation through Korean-Swiss Science and Technology Programme.
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abstract: 'We introduce a class of F-theory vacua which may be viewed as a specialization of the so-called $E_6$ fibration, and construct a weak coupling limit associated with such vacua which we view as the ‘square’ of the Sen limit. We find that while Sen’s limit is naturally viewed as an orientifold theory, the universal tadpole relation which equates the D3 charge between the associated F-theory compactification and the limit we construct suggests that perhaps the limiting theory is in fact an oriented theory compactified on the base of the F-theory elliptic fibration.'
address:
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School of Mathematical Sciences\
Shanghai Jiao Tong University\
800 Dongchuan Road, Shanghai, China
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Department of Mathematics\
Dongbei University of Economics and Finance\
217 Jianshan St, Shahekou District, Dalian, China
author:
- James Fullwood
- Dongxu Wang
bibliography:
- 'SSL2.bib'
title: On the Sen limit squared
---
Introduction
============
F-theory compactified on an elliptic Calabi-Yau $(n+1)$-fold $Y\to X$ is essentially a type-IIB compactification on the base $X$ with varying axio-dilaton [@VFT]. The $\text{SL}_2({{\mathbb{Z}}})$-invariant value of the axio-dilaton at a point $p\in X$ is then interpreted as the complex structure modulus of the elliptic fiber over $p$. If the Calabi-Yau $(n+1)$-fold $Y$ is a Weierstrass fibration, i.e., if it globally given by an equation of the form $$y^2z=x^3+fxz^2+gz^3,$$ Sen constructed in a systematic way how to identify F-theory compactified on $Y$ with a weakly-coupled orientifold theory compactified on a Calabi-Yau $n$-fold which is a two-sheeted cover of $X$ [@Sen]. The identification was established via an explicit deformation of the total space $Y$ of the elliptic $(n+1)$-fold to a degenerate Calabi-Yau in which all the fibers are singular, which signals weak coupling everywhere on the base. This systematic procedure via deformation of a Weierstrass equation for establishing a duality between F-theory and orientifold type-IIB theories is then referred to as the *Sen limit*.
Since every elliptic fibration is birational to a fibration in Weierstrass form, for a while Sen’s limit was viewed as being applicable to a general F-theory compactification. But as pointed out in [@AE1][@AE2], birational transformations of elliptic fibrations don’t preserve singular fibers, and since the singular fibers of an elliptic fibration play a crucial role in the physics of F-theory, one is not investigating the general case by considering fibrations in Weierstrass form whose total space is smooth. Moreover, the Weierstrass form of a smooth elliptic fibration admits a total space which is in general singular, so in dealing with Weierstrass models of elliptic fibrations not initially given by a global Weierstrass equation, one must either resolve singularities or take up the issue of defining F-theory on singular elliptic fibrations [@EYSU5][@TFSt][@CSSF][@TGPMil].
In light of this, in [@AE2] Aluffi and Esole initiated a program of moving beyond Weierstrass models in F-theory, where they considered families of smooth elliptic $n$-folds not given by global Weierstrass equations, and constructed orientifold limits associated with such fibrations by generalizing the method first employed by Sen. Further investigations into non-Weierstrass fibrations and weak coupling limits involving the Tate form of Weierstrass fibrations were explored in [@DWHBUV][@ESTF][@EFY][@CCG][@GTSGUT]. A comparison of the D3 charge in F-theory and its limit poses consistency conditions which take the form of an identity relating the Euler characteristic of the elliptic $(n+1)$-fold associated with the F-theory compactification and the Euler characteristics of D-branes which arise in the limit. For a general deformation of an elliptic Calabi-Yau, such consistency conditions – referred to as *tadpole relations* – will not hold, so constructing a consistent limit is a non-trivial affair. Moreover, what is interesting from a purely mathematical perspective is that when the tadpole relations *do* hold between an F-theory compactification and its limit, in all known examples the associated identity between Euler characteristics turns out to be the dimension-zero component of a much more general identity which holds at the level of Chern classes, and moreover, the identities hold without any dimensional constraints on the base of the fibration and without any Calabi-Yau hypothesis on the total space [@AE1][@AE2][@EFY][@EJY]. Such Chern class identities were then referred to as *universal tadpole relations*, and a purely mathematical explanation for the existence of such identities was given in [@FTR2].\
In this note, we introduce a class of F-theory vacua that, while not given by a global Weierstrass equation, its discriminant is given by an equation which may be smoothly deformed to the *square* of the discriminant of a smooth Weierstrass fibration. In particular, we construct a weak coupling limit associated with such fibrations in such a way that the limiting discriminant is the square of the limiting discriminant in Sen’s limit, and as such, the the branes arising in the limit are all doubles of the branes arising in Sen’s limit. The main difference from Sen’s limit however, is we find that the universal tadpole relation associated with the limit holds only once a factor of two that appears when viewing charges from the perspective of an orientifold theory is canceled. So while an orientifold interpretation of the limit may exist, the form of the universal tadpole relation suggests to us that the limit we construct may exist more naturally as an oriented theory on the base of the orientifold projection, i.e., on the ‘square’ of the orientifold.\
*Acknowledgements*. JF would like to thank Mboyo Esole and Matt Young for useful discussions.
Sen’s limit {#SL}
===========
The $F$-theory vacua considered by Sen in constructing his limit correspond precisely to smooth Weierstrass models, i.e., elliptic fibrations associated with the Weierstrass equation $$\label{we}
y^2z=x^3+fxz^2+gz^3.$$ For equation (\[we\]) to define an elliptic fibration, let $X$ be a smooth compact Fano $n$-fold over ${{\mathbb{C}}}$, so that its anti-canonical bundle $\mathscr{O}(-K_X)$ is ample. We then consider the vector bundle[^1] $$\mathscr{E}=\mathscr{O}\oplus \mathscr{O}(-K_X)^2\oplus \mathscr{O}(-K_X)^3,$$ and denote by $\pi:\mathbb{P}(\mathscr{E})\to X$ the projective bundle of *lines* in $\mathscr{E}$. The tautological bundle on $\mathbb{P}(\mathscr{E})$ will then be denoted by $\mathscr{O}(-1)$. We then associate with equation (\[we\]) a section of $\mathscr{O}(3)\otimes \pi^*\mathscr{O}(-K_X)^6$ by taking $x$ to be a section of $\mathscr{O}(1)\otimes \pi^*\mathscr{O}(-K_X)^2$, $y$ to be a section of $\mathscr{O}(1)\otimes \pi^*\mathscr{O}(-K_X)^3$, $z$ to be a section of $\mathscr{O}(1)$, $f$ to be a section of $\pi^*\mathscr{O}(-K_X)^4$ and $g$ to be a section of $\pi^*\mathscr{O}(-K_X)^6$. With these prescriptions, equation (\[we\]) then defines a codimension 1 embedding $W\hookrightarrow \mathbb{P}(\mathscr{E})$, and composing the embedding with the bundle projection $\pi:\mathbb{P}(\mathscr{E})\to X$ endows $Y$ with the structure of an elliptic $(n+1)$-fold $Y\to X$, where the fiber over a point $p$ in $X$ is given by $$W_p:(y^2z=x^3+f(p)xz^2+g(p)z^3)\subset \mathbb{P}^2.$$ For $W$ to be smooth we require that the hypersurfaces $$F:(f=0)\subset X \quad \text{and} \quad G:(g=0)\subset X$$ are both smooth and intersect transversally. The singular fibers of $W\to X$ then lie over the discriminant hypersurface $$\Delta_W:(4f^3+27g^2=0)\subset X.$$ Over a general point of $\Delta_W$, the fiber of $W\to X$ will be a nodal cubic, and over the codimension 2 smooth complete intersection $$C:(f=g=0)\subset X$$ the fibers enhance to cuspidal cubics. From B.5.8 in [@IT] along with the adjunction formula $$c(W)=\frac{(1+H)(1+H-2K_X)(1+H-3K_X)(3H-6K_X)}{1+3H-6K_X}\pi^*c(X),$$ where $H$ denotes the first Chern class of the bundle $\mathscr{O}(1)\to \mathbb{P}(\mathscr{E})$. It immediately follows that the first Chern class of $W$ is zero, so that the total space of the fibration is an anti-canonical hypersurface in $\mathbb{P}(\mathscr{E})$.
From the type-IIB viewpoint, the axio-dilaton field on $X$ is given by $$\tau=C_{(0)}+i\frac{1}{g_s},$$ where $C_{(0)}$ is the axion RR-scalar field and $g_s$ is the string coupling constant. From the $F$-theory viewpoint the axio-dilaton $\tau$ is then identified with the complex structure modulus of the elliptic fibers of $W\to X$. A weak coupling limit then corresponds to deforming $W$ in such a way that the complex structure modulus approaches $i\infty$ everywhere over $X$. Sen’s limit is then constructed by perturbing $f$ and $g$ in equation (\[we\]) in terms of a complex deformation parameter $t$ in such a way that the central fiber of the associated family is a degenerate fibration in which all the fibers are singular (so that $\tau$ approaches $i\infty$). In particular, the family corresponding to Sen’s limit is given by $$\mathscr{W}:(y^2z=x^3+ (-3h^2+t\eta)xz^2+(-2h^3+th\eta+t^2\psi)z^3)\subset \mathbb{P}(\mathscr{E})\times D,$$ where $D$ denotes an open disk centered about $0\in {{\mathbb{C}}}$, and $h$, $\eta$ and $\psi$ are general sections of $\pi^*\mathscr{O}(-K_X)^2$, $\pi^*\mathscr{O}(-K_X)^4$ and $\pi^*\mathscr{O}(-K_X)^6$ respectively. The central fiber is then given by $$W_0:(y^2z=x^2+(-3h^2)xz^2+(-2h^3)z^3)\subset \mathbb{P}(\mathscr{E}),$$ whose fibers are nodal for $h\neq 0$ and cuspidal for $h=0$. The auxiliary $n$-fold corresponding to the orientifold theory is then given by $$Z:(\zeta^2=h)\subset \mathscr{O}(-K_X),$$ where $\zeta$ is a general section $\mathscr{O}(-K_X)$. It then follows that $Z$ is a double cover of $X$ ramified over $O:(h=0)\subset X$, which we identify with the orientifold plane. The tangent bundle of the total space of the anti-canonical bundle $p:\mathscr{O}(-K_X)\to X$ fits into the exact sequence $$0\to p^*\mathscr{O}(-K_X)\to T\mathscr{O}(-K_X)\to p^*TX\to 0,$$ thus by adjunction we have $$c(Z)=p^*\left(\frac{c(\mathscr{O}(-K_X))c(TX)}{c(\mathscr{O}(-2K_X))}\right)\cap [Z].$$ It immediately follows that $c_1(Z)=0$, so $Z$ is in fact a Calabi-Yau $n$-fold.
To locate the D-branes associated with the limit, we take the flat limit as $t\to 0$ of the corresponding family of discriminants $$\mathscr{D}_W:(4(-3h^2+t\eta)^3+27(-2h^3+th\eta+t^2\psi)^2=0)\subset X\times D,$$ which yields $$\Delta_{W_0}:(h^2(\eta^2+12h\psi)=0)\subset X.$$ We then pullback the flat limit $\Delta_0$ of $\mathscr{D}_W$ as $t\to 0$ to $Z$ to obtain the D-brane spectrum of the orientifold limit, which is given by $$\widetilde{\Delta_{W_0}}:(\zeta^4(\eta^2+12\zeta^2\psi)=0)\subset Z.$$ The D-brane given by $$D:(\eta^2+12\zeta^2\psi=0)\subset Z$$ has singularities reminiscent of those of a Whitney umbrella, and as such was referred to as a ‘Whitney brane’ in [@AE1][@CDM]. Comparing the D3 tadpole condition between $F$-theory and type-IIB then predicts that $2\chi(W)$ coincides with $4\chi(O)+\chi(D)$ ($\chi$ denotes topological Euler characteristic with compact support), but it was shown in the literature that for such a relation to hold a certain contribution is needed coming from the singularities of $D$ [@CDM]. In particular, it was shown that such tadpole relations hold precisely when the pinch-point singularities of $D$ are subtracted from the charge of a small resolution $\overline{D}$ of $D$, thus yielding the tadpole relation $$\label{tr}
2\chi(W)=4\chi(O)+\chi(\overline{D})-\chi(S),$$ where $S$ is the pinch locus of $D$, namely $$S:(\zeta=\eta=\psi=0)\subset Z.$$ It was then shown by Aluffi and Esole that the tadpole relation (\[tr\]) is in fact a consequence of a much more general relation at the level of Chern classes, which they referred to as a *universal tadpole relation* [@AE2]. More precisely, let $\psi:W\to X$ denote the projection of the elliptic fibration corresponding to the F-theory vacua, then the tadpole relation (\[tr\]) is just the dimension-zero component of the universal tadpole relation $$\label{utr}
2\psi_*c(W)=4c(O)+c^{(\infty)}(D),$$ where $c^{(\infty)}(D)$ denotes a certain characteristic class of the singular variety $D$ whose component of dimension zero coincides with $\chi(\overline{D})-\chi(S)$ as appearing on the RHS of (\[tr\]) (see §5 of [@AE1] for a the precise definition of $c^{(\infty)}(D)$). We can also write formula is in terms of the components of the limiting discriminant $\Delta_{W_0}$. In particular, denote by $\underline{D}$ the singular component of $\Delta_{W_0}$ given by $$\underline{D}:(\eta^2+12h\psi=0)\subset X,$$ and denote its singular locus by $\underline{S}$, which is a smooth codimenison 3 complete intersection given by $$\underline{S}:(h=\eta=\psi=0)\subset X.$$ Then it turns out that $c^{(\infty)}(D)=2(c_{\text{SM}}(\underline{D})-c(\underline{S}))$ (again see [@AE1] §5 for details), thus may be rewritten as $$\label{utr2}
2\psi_*c(W)=4c(O)+2(c_{\text{SM}}(\underline{D})-c(\underline{S})),$$ where $c_{\text{SM}}(\underline{D})$ denotes the Chern-Schwartz-MacPherson class of $\underline{D}$ [^2]. This form of the universal tadpole relation for Sen’s limit will be of use when we make a connection with the limit constructed in §\[SRL\].
A specialization of the $E_6$ fibration {#NC}
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Let $X$ be a smooth compact Fano $n$-fold over ${{\mathbb{C}}}$ as in §\[SL\]. We now introduce a class of $F$-theory vacua over $X$ from which we will construct a weak coupling limit which may be viewed as the square of Sen’s limit. For this, we use the equation $$\label{ne}
x^3+y^3+eyz^2+fxz^2+gz^3=0.$$ We then consider the vector bundle $$\mathscr{E}=\mathscr{O}\oplus \mathscr{O}(-K_X)\oplus \mathscr{O}(-K_X),$$ and denote by $\pi:\mathbb{P}(\mathscr{E})\to X$ the projective bundle of lines in $\mathscr{E}$, with tautological bundle $\mathscr{O}(-1)\to \mathbb{P}(\mathscr{E})$. For equation (\[ne\]) to define the zero-locus of a well-defined section of a line bundle on $\mathbb{P}(\mathscr{E})$, we take $x$ and $y$ both to be sections of $\mathscr{O}(1)\otimes \pi^*\mathscr{O}(-K_X)$, $z$ to be a section of $\mathscr{O}(1)$, both $e$ and $f$ to be a section of $\pi^*\mathscr{O}(-K_X)^2$, and $g$ to be a section of $\pi^*\mathscr{O}(-K_X)^3$. With these prescriptions, the LHS of equation (\[ne\]) yields a well-defined section of $\mathscr{O}(3)\otimes \pi^*\mathscr{O}(-K_X)^3$, whose zero locus defines a codimension 1 embedding $Y\hookrightarrow \mathbb{P}(\mathscr{E})$. Composing the embedding $Y\hookrightarrow \mathbb{P}(\mathscr{E})$ with the bundle projection $\pi:\mathbb{P}(\mathscr{E})\to X$ yields the projection $\varphi:Y\to X$, which endows the total space $Y$ with the structure of an elliptic fibration. The fiber over a point $p\in X$ is then given by $$Y_p:(x^3+y^3+e(p)yz^2+f(p)xz^2+g(p)z^3=0)\subset \mathbb{P}^2.$$ The Chern class of $Y$ is given by $$c(Y)=\frac{(1+H)(1+H-K_X)^2(3H-3K_X)}{1+3H-3K_X}\pi^*c(X),$$ where again $H$ denotes the first Chern class of $\mathscr{O}(1)\to \mathbb{P}(\mathscr{E})$. It immediately follows that $c_1(Y)=0$, so that $Y$ is an anti-canonical hypersurface in $\mathbb{P}(\mathscr{E})$.
The connection with Sen’s limit comes from the fact that by setting $e=0$, the discriminant of $\varphi:Y\to X$ becomes the square of the discriminant of Weierstrass fibrations (which will be crucial in the next section, when we construct a weak coupling limit). In particular, we have that the Weierstrass form for $Y$ is given by $$y^2z+x^3+Fxz^2+Gz^3=y^2z+x^3+(-3ef)xz^2+\left(f^3+\frac{27}{4}g^2\right)z^3=0,$$ so that $F=-3ef$ and $G=e^3+f^3+\frac{27}{4}g^2$. The discriminant of $\varphi:Y\to X$ is then given by $$\Delta_Y:(4F^3+27G^2=0)\subset X.$$ We note that when $e=0$, $\Delta_Y$ takes the form $(4f^3+27g^2)^2$, which is the square of the discriminant of Weierstrass fibrations $\Delta_W$. The $j$-invariant viewed as a function on $X\setminus \Delta_Y$ then takes the form $$j=1728\frac{e^3f^3}{4e^3f^3-(e^3+f^3+\frac{27}{4}g^2)^2}.$$ As in the case of Weierstrass fibrations, in order to ensure that the total space of the fibration $Y$ is smooth we assume that the hypersurfaces $$E:(e=0)\subset X \quad \text{and} \quad F:(f=0)\subset X \quad \text{and} \quad G:(g=0)\subset X$$ are all smooth and intersect each other transversally. The fibers over a general point of $\Delta_Y$ are nodal cubics which enhance to cuspidal cubics over $e=0$, while over the smooth codimension 3 complete intersection $$C:(e=f=g=0)\subset X$$ the fibers enhance to a bouquet of three 2-spheres intersecting at a point.
\[r1\] The fibration $\varphi:Y\to X$ may be seen as a special case of the so-called $E_6$ fibration which was studied in detail in [@AE2], which is given by $$Y_{E_6}:(x^3+y^3= dxyz+exz^2+fyz^2 + gz^3)\subset \mathbb{P}(\mathscr{E}).$$ As such, we see that the fibration $\varphi:Y\to X$ corresponds to setting $d=0$ in the equation for $Y_{E_6}$. Moreover, since $Y_{E_6}$ may also be realized as the zero-locus of a section of $\mathscr{O}(3)\otimes \pi^*\mathscr{O}(-K_X)^3$, $Y_{E_6}$ has the same divisor class as $Y$ in $\mathbb{P}(\mathscr{E})$, thus they share the same Chern classes, Euler characteristic, Chern numbers etc. This is interesting in light of the fact that the $E_6$ fibration admits six topologically distinct singular fibers, while the fibration $\varphi:Y\to X$ admits only three.
Sen’s limit squared {#SRL}
===================
We now define a weak coupling limit associated with $\varphi:Y\to X$ by essentially deforming $e$ to 0, and perturbing $f$ and $g$ in the equation for $Y$ in precisely the same way as in Sen’s limit. In particular, we take $$e=t\epsilon, \quad f=-3\zeta^2+t\vartheta, \quad \text{and} \quad g=-2\zeta^3+t\zeta\vartheta+t^2\phi,$$ where $\zeta$, $\vartheta$ and $\phi$ are all sections of the square roots of the line bundles for which $h$, $\eta$ and $\psi$ denoted sections of in Sen’s limit respectively, and $t$ as before is a complex deformation parameter varying over a disk $D$ about the origin in ${{\mathbb{C}}}$. Such prescriptions then give rise to a family $$\mathscr{Y}:(x^3+y^3+t\epsilon yz^2+(-3\zeta^2+t\vartheta)xz^2+(-2\zeta^3+th\vartheta+t^2\phi)z^3=0)\subset \mathbb{P}(\mathscr{E})\times D,$$ whose central fiber corresponds to the weak coupling limit, since in the limit $t\to 0$ the $j$-invariant approaches $\infty$ everywhere over the base. The limiting discriminant then takes the form $$\label{ld}
\Delta_{Y_0}:(\zeta^4(\vartheta^2+12\zeta\phi)^2=0)\subset X,$$ which is essentially the same as the limiting discriminant in Sen’s limit but with all components squared.
At this point, one may define the auxiliary $n$-fold required for an orientifold compactification in precisely the same way as in Sen’s limit, i.e., by defining a double cover of $X$ given by $$Z:(\zeta^2=h)\subset \mathscr{O}(-K_X),$$ where $\zeta$ and $h$ now denote pull backs to $\mathscr{O}(-K_X)$ of sections of $\mathscr{O}(-K_X)$ and $\mathscr{O}(-K_X)^2$ respectively, so that the orientifold plane $O$ is given by $h=0$. But since the limiting discriminant $\Delta_{Y_0}$ appears with a factor of $\zeta^4$ rather than $h^2$ (as in the case of Sen’s limit), its pullback to $Z$ doesn’t change the form of its equation in any way. As such, it is perhaps more natural to view the limiting type-IIB theory as an oriented theory on the base $X$, rather than an orientifolded theory on $Z$. From this perspective, the D-brane spectrum associated with the limit then consists of a stack of 4 branes supported on the smooth locus $$\mathcal{D}:(\zeta=0)\subset X,$$ and a double-brane supported on $$D:(\vartheta^2+12\zeta\phi=0)\subset X.$$ The D-branes supported on $D$ then admit singularities along the codimension 3 locus $$S:(\vartheta=\zeta=\phi=0)\subset X.$$
In the next section we derive the universal tadpole relation associated with this limit, which takes the form $$\label{utr3}
\varphi_*c(Y)=4c(\mathcal{D})+2(c_{\text{SM}}(D)-c(S)).$$ We point out that is very similar to the universal tadpole relation associated with Sen’s limit, namely $$\label{utr4}
2\psi_*c(W)=4c(O)+2(c_{\text{SM}}(\underline{D})-c(\underline{S})).$$ In particular, while the RHS of both equations are identical in form, the LHS of equation differs from the LHS of by a factor of 2, which perhaps further suggests that while the tadpole relation corresponds to an orientifolded theory, the relation in fact corresponds to an oriented one.
Universal tadpole relation {#UTR}
==========================
We now derive the universal tadpole relation . We first note that since the divisor class of the total space of the fibration $\varphi:Y\to X$ in $\mathbb{P}(\mathscr{E})$ is the same as the $E_6$ fibration mentioned in Remark \[r1\], they share the same Chern classes. And since the pushforward to its base of the Chern class of an $E_6$ fibration was computed in Theorem 4.3 in [@AE2], we already know $\varphi_*c(Y)$, which is given by $$\label{l0}
\varphi_*c(Y)=4c(G),$$ where $G$ is the smooth hypersurface in $X$ given by $g=0$ (recall that $g$ is the coefficient of $z^3$ in the defining equation for $Y$). Moreover, viewing Lemma 4.4 in [@AE1] from a more general perspective, its conclusion says that if $V$ is a singular hypersurface in $X$ given by a general equation of the form $$V:(x^2+12yz=0)\subset X,$$ and if we denote by $\mathcal{Y}$ and $\mathcal{S}$ the subvarieties of $X$ corresponding to the equations $y=0$ and $x=y=z=0$ respectively, then $$\label{l1}
2c(\mathcal{Y})+c_{\text{SM}}(V)-c(\mathcal{S})=2c(\mathcal{G}),$$ where $\mathcal{G}$ is a smooth hypersurface in $X$ whose divisor class is $3[\mathcal{Y}]$. As such, from the definition of $\mathcal{D}$, $D$ and $S$ in the previous section, in equation we may replace $\mathcal{Y}$ by $\mathcal{D}$, $V$ by $D$ and $\mathcal{S}$ by $S$ to arrive at the equation $$2c(\mathcal{D})+c_{\text{SM}}(D)-c(S)=2c(G),$$ since $[G]=3[\mathcal{D}]$. Putting this equation together with equation then yields $$\label{l2}
\varphi_*c(Y)=4c(\mathcal{D})+2(c_{\text{SM}}(D)-c(S)),$$ which we view as the universal tadpole relation associated with the limit constructed in the previous section §\[SRL\]. Note that the limiting discriminant associated with the limit is of the form $$\zeta^4(\vartheta^2+12\zeta\phi)^2=0,$$ so that the RHS of is precisely the sum of Chern classes of branes supported on the irreducible components of the limiting discriminant weighted with the appropriate multiplicities (together with the negative contribution coming from the singularities $S$ of $D$, as in Sen’s limit). If the limit was then viewed as an oreintifold theory compactified on a double cover of $X$ given by $\zeta^2-h=0$, then the pullback to the orientifold of the limiting discriminant wouldn’t change its form at all, thus we’d expect a similar relation to but with $\mathcal{D}$ replaced by the orientifold plane $O$ and a factor of 2 appearing on the LHS, as in the universal tadpole relation associated with Sen’s limit, which we recall is given by $$2\psi_*c(W)=4c(O)+2(c_{\text{SM}}(\underline{D})-c(\underline{S})).$$
Perhaps there is a way to still view the limit as an orientifold theory by adding fluxes or more carefully investigating the nature of the D3 charge in the context at hand, but an oriented theory on the base seems more natural, at least from the form of equation .
[^1]: A super-script on a line bundle such as $\mathscr{L}^k$ will always be used to denote the $k$th tensor power of the line bundle for any positive integer $k$.
[^2]: A nice introduction to Chern-Schwartz-MacPherson classes is given in [@CSVA].
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abstract: 'The [COVID-19]{} pandemic demands the rapid identification of drug-repurpusing candidates. In the past decade, network medicine had developed a framework consisting of a series of quantitative approaches and predictive tools to study host-pathogen interactions, unveil the molecular mechanisms of the infection, identify comorbidities as well as rapidly detect drug repurpusing candidates. Here, we adapt the network-based toolset to COVID-19, recovering the primary pulmonary manifestations of the virus in the lung as well as observed comorbidities associated with cardiovascular diseases. We predict that the virus can manifest itself in other tissues, such as the reproductive system, and brain regions, moreover we predict neurological comorbidities. We build on these findings to deploy three network-based drug repurposing strategies, relying on network proximity, diffusion, and AI-based metrics, allowing to rank all approved drugs based on their likely efficacy for [COVID-19]{} patients, aggregate all predictions, and, thereby to arrive at $81$ promising repurposing candidates. We validate the accuracy of our predictions using drugs currently in clinical trials, and an expression-based validation of selected candidates suggests that these drugs, with known toxicities and side effects, could be moved to clinical trials rapidly.'
author:
- Deisy Morselli Gysi
- Ítalo Do Valle
- Marinka Zitnik
- Asher Ameli
- Xiao Gan
- Onur Varol
- Helia Sanchez
- Rebecca Marlene Baron
- Dina Ghiassian
- Joseph Loscalzo
- 'Albert-L[á]{}szl[ó]{} Barab[á]{}si'
date: April 2020
title: 'Network Medicine Framework for Identifying Drug Repurposing Opportunities for COVID-19'
---
Introduction
============
The speed and the disruptive nature of the [COVID-19]{} pandemic has taken both public health and biomedical research by surprise, demanding the rapid deployment of new interventions, the development, and testing of an effective cure and vaccine. Given the compressed timescales, the traditional methodologies relying on iterative development, experimental testing, clinical validation, and approval of new compounds are not feasible. A more realistic strategy relies on drug repurposing, requiring us to identify clinically approved drugs, with known toxicities and side effects, that may have a therapeutic effect in [COVID-19]{} patients.
In the past decade, network medicine has developed and validated a series of computational tools that help us identify drug repurposing opportunities [@Cheng2018; @Guney2016; @Zhou2020Network; @Cheng2019; @Zitnik2019; @Zitnik2018; @Casas2019]. Here we deploy these tools to analyze the molecular perturbations induced by the virus [SARS-CoV2]{}, causing a pathophenotype (disease) known as [COVID-19]{} (**Co**rona**vi**rus **Di**sease 20**19**), and to identify potential drug repurposing candidates. We start by characterizing the [COVID-19]{} disease module (\[fig:summary\_project\]A), representing the network neighborhood of the human interactome perturbed by [SARS-CoV2]{}, and its integrity in $56$ tissues, to identify the tissues and organs the virus could invade. We then explore multiple network-based strategies to prioritize existing drugs based on their ability to interact with their protein targets and, thereby, perturb the disease module: network proximity-based methods that use a graph theoretic repurposing strategy [@Guney2016]; diffusion-based methods to capture node similarity [@Cao2013]; and approaches relying on artificial intelligence network (AI-Net), that embed all available data to detect efficacy [@Zitnik2019; @Zitnik2018]. These three predictive approaches offer us twelve ranked lists, normally applied independently and validated on different datasets. Here, we combine them using a rank aggregation algorithm [@zitnik2018prioritizing], allowing to exploit their relative advantages and to obtain a final prioritized ranking of drug repurposing candidates that offers higher accuracy than any of the pipelines alone. After eliminating drugs based on toxicity, delivery, and appropriateness of their use in [COVID-19]{} patients, we selected $81$ approved drugs as candidates for drug repurposing. Finally, we integrate experimental data from *in vitro* models to help identify the network-based mechanism of action for selected compounds and offer further validation using existing gene expression data (\[fig:summary\_project\]B) [@Subramanian2017; @Lamb2006].
Results
=======
Mapping [SARS-CoV2]{} Targets to the Human Interactome
-------------------------------------------------------
[SARS-CoV2]{} infects human cells by hijacking the host’s translation mechanisms to generate $29$ viral proteins, which bind to multiple human proteins to initiate the molecular processes required for viral replication and additional host infection [@Fehr2015]. Gordon et al [@Gordon2020] expressed $26$ of the $29$ [SARS-CoV2]{} proteins and used affinity-purification followed by mass spectrometry to identify $332$ human proteins to which the viral proteins bind (\[tab:gordon\_data\_set\])[@Gordon2020]. We mapped these $332$ proteins to the human interactome, consisting of $18,508$ proteins and $332,749$ pairwise interactions between them (see Methods). Of the $332$ viral targets, $239$ proteins form a multiply connected subnetwork of viral targets (\[fig:LCC\_COVID\_Disease\_Module\]A), and $93$ viral targets do not interact with other targets, but only with other human proteins.
We find that $208$ viral targets form a large connected component (LCC) (\[fig:LCC\_COVID\_Disease\_Module\]A). To test whether the observed LCC could have emerged by chance, we randomly placed $332$ proteins in the interactome while matching the degrees of the original viral targets. The obtained random LCC of size $183.23 \pm 14.93$ proteins and the comparative [Z-Score]{}$=1.65$ indicates that the [SARS-CoV2]{} target-proteins aggregate in the same network vicinity [@Zhou2020Network; @Gulbahce2012], defining the location of the [COVID-19]{} disease module within the human interactome. Potential drug repurposing candidates must either target proteins within or in the network vicinity of this disease module.
### Tissue Specificity
Previous work indicates that the expression of a gene associated with a disease in a particular tissue is insufficient for a disease to be manifest in that tissue, but a statistically significant disease LCC for must be expressed [@Kitsak2016]. We, therefore, measured the statistical significance of the [COVID-19]{} LCC in $56$ tissues, using data from GTEx [@Lonsdale2013]. With GTEx median value $<$ 5, only $10,823$ ($58\%$) of the $18,406$ proteins in the interactome are expressed in lung [@Kitsak2016; @Lonsdale2013], while of the $332$ viral targets (\[fig:LCC\_COVID\_Disease\_Module\]C) $214$ ($64\%$) are expressed. We find that $182$ viral targets form a tissue specific LCC, and given the random expectation of $155.61 \pm 14.82$ for this LCC, we obtain a [Z-Score]{}$=1.78$ for the lung, larger than the [Z-Score]{}$=1.65$ of the LCC in the full-network. Overall, in $30$ tissues the LCC exceeds the [Z-Score]{} of the full-network, helping us to identify tissues where the virus-induced disease could be manifested (\[tab:TissueSpecificity\]). The list contains pulmonary and cardiovascular tissues, supporting the clinical observations that [COVID-19]{} manifests itself in the respiratory system [@Xu2020; @Yang2020], but infected patients often present significant cardiovascular involvement [@Huang2020; @Xu2020], and patients with underlying cardiovascular diseases show increased risk of death [@Zheng2020]. Interestingly, \[tab:TissueSpecificity\] indicates that the LCC is also expressed in the multiple brain regions, likely explaining the recently reported neurological manifestations [@Mao2020JAMA; @Eliezer2020SuddenCOVID-19.; @Pleasure2020TheFrontlines.] of the disease. We also observe multiple tissues related to the digestive system (colon, esophagus, pancreas) in this analysis, again consistent with clinical observations. Finally, equally unexpected is the fact that \[tab:TissueSpecificity\] indicates expression in multiple reproductive system tissues (vagina, uterus, testis, cervix, ovary), as well as spleen, potentially related to disruptions in the regulation of the immune system [@Qin2020; @Song2020DetectionPatients] (\[tab:TissueSpecificity\]).
### Predicting Disease Comorbidity
Pre-existing conditions worsen prognosis and recovery of [COVID-19]{} patients [@Grasselli2020]. Previous work has shown that the disease relevance of the human proteins targeted by the virus can predict the symptoms/signs and diseases caused by a pathogen [@Gulbahce2012], prompting us to identify diseases whose molecular mechanisms overlap with cellular processes targeted by [SARS-CoV2]{}, allowing us to predict potential comorbidity patterns [@Park2009; @Hidalgo2009; @Lee2008]. We retrieved $3,173$ disease-causing genes for $299$ diseases [@Menche2015], finding that $110$ of the $320$ proteins targeted by [SARS-CoV2]{} are implicated in disease; however, the overlap between [SARS-CoV2]{} targets and the pool of the disease genes is not statistically significant (Fisher’s exact test; FDR-BH [p$_{adj}$-value]{}$> 0.05$). We, therefore, evaluated the network-based overlap between the proteins associated with each of the $299$ diseases and the targets of [SARS-CoV2]{}, using the $S_{vb}$ metric[@Menche2015], where $S_{vb} < 0$ signals a network-based overlap between the [SARS-CoV2]{} viral targets $v$ and the gene pool associated with disease $b$. We find that $S_{vb} > 0$ for each disease, indicating that [SARS-CoV2]{} disease module does not directly overlap with any major disease module (\[fig:disease\_hist\] and \[tab:proximity\]). The diseases closest to the [COVID-19]{} proteins (smallest $S_{vb}$), include several cardiovascular diseases and cancer, whose comorbidity in [COVID-19]{} patients is well documented [@Chen2020; @Huang2020; @Wang2020] (\[fig:disease\_proximity\]). The same metric predicts comorbidity with neurological diseases, in line with our observation, that the viral targets are expressed in the brain (\[tab:TissueSpecificity\]).
In summary, we find that the [SARS-CoV2]{} targets do not overlap with disease genes associated with any major diseases, indicating that a potential [COVID-19]{} treatment can not be derived from the arsenal of therapies approved for specific diseases. These findings argue for a strategy that maps drug targets without regard to their localization within a particular disease module. However, the diseases modules closest to the [SARS-CoV2]{} viral targets are those with noted comorbidity for [COVID-19]{} infection, such as pulmonary and cardiovascular diseases, and cancer. We also find multiple network-based evidence linking the virus to the nervous system, a less explored comorbidity, consistent with the observations that many infected patients initially lose olfactory function and taste [@Giacomelli], and that $36\%$ of patients with severe infection requiring hospitalization have neurological manifestations [@Mao2020JAMA].
Identifying Drug Repurposing Candidates for [COVID-19]{}
--------------------------------------------------------
Traditional repurposing strategies focus on drugs that target the human proteins to which viral proteins bind [@Gordon2020], or on drugs previously approved for other pathogens. The network medicine approach described here is driven by the recognition that most approved drugs do not target directly disease proteins, but bind to proteins in their network vicinity [@Yildirim2007]. Hence our goal is to identify drug candidates that may or may not target the proteins to which the virus binds, but nevertheless have the potential to perturb the network vicinity of the virus disease module. To achieve this end, we utilized several network repurposing strategies: a network proximity strategy, identifying drugs whose targets are in the immediate network vicinity of the viral targets [@Guney2016]; a diffusion-based strategy [@Cao2013]; and an AI-Net based strategy that uses machine learning to combine multiple sources of evidence [@Zitnik2019; @Zitnik2018] (\[fig:summary\_project\]B). We test the predictive power of each method independently using a list of drugs under clinical trial for [COVID-19]{} and combine the evidence provided by each method, arriving at a ranked list of drug repurposing candidates derived from the complete list of drugs in DrugBank (see Methods).
### Proximity-based Ranking
Proximity-based methods allow us to measure the distance between two sets of nodes in a network, also determining the statistical significance for the observed proximity. Here we use proximity to explore the distance between the viral protein targets (approximating the [COVID-19]{} disease module), and (i) the targets of approved drugs; and (ii) the differentially expressed genes induced by each drug, arriving at three drug ranking lists.
- Pipeline P1: For each drug, we measured the network distance to the closest protein targeted by [COVID-19]{}, and applied a degree-preserving randomization procedure to assess its statistical significance, expecting a [Z-Score]{}$<$ 0 for proximal drugs. For example, chloroquine, a rheumatological and antimalarial drug currently in clinical trial for [COVID-19]{}, has [Z-Score]{}=$-1.82$, indicating its proximity to [SARS-CoV2]{} targets. In contrast, etanercept, another anti-inflammatory drug with no supported [COVID-19]{} relevance, has a [Z-Score]{}=$1.29$, indicating that the drug’s protein targets are far from the [SARS-CoV2]{} viral targets (\[fig:network\_proximity\_results\]A). We tested the proximity of $6,116$ drugs with at least one target in DrugBank, identifying $385$ drugs with [Z-Score]{}s$<-2$, and $1,201$ drugs with [Z-Score]{}s$<-1$, representing potential repurposing candidates (\[fig:network\_proximity\_results\]B).
- Pipeline P2: We computed the proximity [Z-Score]{} after disregarding for each drug the targets that are enzymes, carriers or transporters. These are proteins targeted by multiple drugs, and are often unrelated to the known pharmacological effects of the profiled drugs. Of the $5,550$ drugs obtained after the filtering, the metric identified $165$ drugs with [Z-Score]{}s$<-2$ and $541$ with [Z-Score]{}s$<-1$. Using this measure, chloroquine and hydroxychloroquine are less proximal to [COVID-19]{} targets, while ribavin, an antiviral drug in clinical trial, gain more proximity (\[fig:network\_proximity\_results\]C).
- Pipeline P3: The effect of a drug is rarely limited directly to the target proteins, but the drug can activate or repress biological cascades and biochemical pathways, that change the expression patterns of multiple proteins in the network neighborhood of the drug’s targets. DrugBank compiles $17,222$ differentially expressed genes (DEGs), linked to $793$ drugs in multiple cell lines. We measured the proximity between DEGs and [COVID-19]{} targets for $793$ drugs, finding $18$ drugs with [Z-Score]{}s $< - 2$, and $82$ drugs with [Z-Score]{}s $< -1$.
In summary, each of the pipelines P1-P3 offer a list of drug candidates ranked by the proximity [Z-Score]{} of the respective pipeline.
### Diffusion-based Methods
Diffusion State Distance (DSD) methods rank drugs based on the network similarity of their targets to [COVID-19]{} protein targets. The similarity of two nodes captures the overlap of two global (network-wide) states following the independent perturbation of the two nodes. We implemented three statistical measures that resulted in five ranking pipelines (see Methods).
- Pipeline D1: L1 norm (Manhattan distance) calculates similarity through the sum over the absolute value of differences between the elements of the two vectors, providing a symmetric measure whose lower values reflect higher similarity.
- Pipeline D2 : As the L1 norm may result in loss of information [@Aggarwal2001], we also implemented the Kullback-Leibler (KL) divergence [@kullback1951], which calculates the relative entropy of the vector representation of the two nodes, reporting the average asymmetric similarity value over the minimum pairwise similarity values (KL-min), and resulting in values between 0 and 1.
- Pipeline D3: We deployed the KL divergence measure, discussed above, but reporting the average similarity value over the median pairwise similarity (KL-median).
- Pipeline D4: We implemented Jensen-Shannon (JS) divergence [@Lin1991], a modified (symmetrized and smoothed) version of the KL divergence, reporting the average over the minimum value of pairwise similarities (JS-min).
- Pipeline D5: Similar to D4, but we report the average over the median value of pairwise similarities (JS-median).
We used these five metrics to rank $3,225$ drugs as potential treatments for [COVID-19]{}. Baricitinib, for example, is a rheumatological drug currently in trial for [COVID-19]{} and all diffusion-based pipelines rank it higher than tocilizumab, a drug also indicated for rheumatological and severe inflammatory diseases with no proven [COVID-19]{} relevance.
### AI-Net based Strategy
We adopted machine learning tools previously developed for drug repurposing using the protein-protein interaction network as input [@zitnik2018modeling; @Zitnik2019machine], resulting in the AI-Net pipeline that exploits the power of AI in a network context [@Zitnik2019machine] (see Methods).
The method learns how to represent ([*i.e.*]{}, embed) the multimodal graph into a compact, low-dimensional vector space such that the algebraic operations in the learned embedding space reflect the topology of the input network (\[fig:ai\_results\]A), and specifies a deep transformation function that maps drugs and diseases to points in the learned space, termed ‘drug and disease embeddings’. As diseases are not independent of each other and genes are often shared between distinct diseases, the method embeds diseases associated with similar genes close together in the embedding space. Similarly, the effects of drugs are not limited to proteins to which they directly bind, but effects spread throughout the protein-protein interaction network. To capture these effects, the method embeds closely together drugs whose target proteins have similar local neighborhoods in the underlying protein-protein interaction network.
We use the learned embeddings to generate four lists of candidate drugs for [COVID-19]{}, each ranked list containing $1,607$ treatment recommendations. To obtain the four rankings, we use four distinct decoders, which decode the structure of small network neighborhoods around a drug or a disease node from the learned embeddings.
- Pipeline A1: We search for drugs that are in the vicinity of the [COVID-19]{} disease module by calculating the cosine distance between [COVID-19]{} and all drugs in the decoded embedding space [@becht2019dimensionality]. The decoding is based on the $N=10$ nearest neighboring nodes in the embedding space, with a minimum distance between nodes of $D=0.25$.
- Pipeline A2: To prevent nodes in the decoding embedding space to pack together too closely, we choose $D=0.8$ and keep $N$ unchanged, pushing the structures apart into softer more general features, offering a better overarching view of the embedding space at the loss of the more detailed structure.
- Pipeline A3: Alternatively, to force the decoding to concentrate on the very local structure (to the detriment of the overall goal of the exercise), we choose $N=5$ to explore a smaller neighborhood while setting the minimum distance at a midrange point, $D=0.5$.
- Pipeline A4: Instead of focusing on the finer local structure, we specify the decoder such that it preserves the broad structure ($N=10$, $D=1$), offering a broader view of the embedding space at the loss of detailed structure.
By inspecting the $20$ highest ranked drug candidates offered by the AI-Net pipeline (\[tab:ai-ranking-AI3\]), we observe that several drugs in [COVID-19]{} clinical studies ([*e.g.*]{}, chloroquine, ritonavir). Other top-ranked drugs include anti-malarial medications and drugs used to treat autoimmune, pulmonary, and cardiovascular diseases.
Prioritizing Repurposing Candidates
-----------------------------------
The predictive pipelines discussed above offered altogether twelve rankings, each reflecting a different network-based criterion to estimate a drug’s likelihood to show efficacy in treating [COVID-19]{} patients. As they all start from the same list of drugs and drug-targets and operate on the same PPI network, the rankings provided by them are not expected to be fully independent. To quantify the similarity between them we measure the Kendall $\tau$ rank correlation of the rankings provided by each pipeline. We find that two of the target proximity-based pipelines, P1 and P2, show high correlation between each other, as do the four AI-Net pipelines (A1-A4), and the five diffusion-based pipelines (D1-D5). Yet, the correlations across the three basic methods are much lower, and P3, relying on gene expression patterns, is also somewhat uncorrelated with other pipelines, indicating that the different methods offer complementary ranking information (\[fig:correlation\_rankings\]A).
To evaluate the predictive power of the pipelines, we test their ability to recover the drugs currently in clinical trials as [COVID-19]{} treatment. For this purpose, we obtained a list of $67$ drugs currently undergoing clinical trials from ClinicalTrials.gov (\[tab:clinical\_trial\_list\]). We use the resulting list and the ranking predicted by each pipeline to compute the ROC (receiver operating characteristics) curves and the AUC (area under the curve) scores for model selection and performance analysis, measuring the quality of separation between positive and negative instances. As \[fig:correlation\_rankings\]B shows, the best individual ROC curves, of $0.86-0.87$, are obtained by the four AI-Net based methods. Note that the performance of the four AI-Net pipelines is largely indistinguishable, in line with the finding that the ranking lists provided by them are highly correlated (\[fig:correlation\_rankings\]A). The second-best performance, of $0.70$, is provided by the proximity method P3. Close behind is P1 with AUC $=0.68$, and we find that eliminating some drug targets in P2 decreases the AUC to $0.58$. As a group, the diffusion methods offer ROC between $0.55$-$0.56$. Their lower performance is somewhat unexpected, as diffusion-based methods should capture higher order correlations, compared to the proximity methods, thus one would expect a performance between the proximity-based and the AI-Net methods, which successfully integrate high order correlates.
Each method extracts its own network-based signal for prioritizing drugs. However, the scores of each method are biased differently, offering different rankings. We used a rank aggregation algorithm [@zitnik2018prioritizing] to combine the $12$ ranking lists, aiming to maximize the number of pairwise agreements between the final ranking and each input ranking. This objective, known as the Kemeny consensus, is NP-hard to compute[@bartholdi1989voting; @dwork2001rank]; hence, we used an algorithm to approximate it (see Methods). We first tested whether combining the ranking within each method class could improve the predictive power of the list provided by the individual pipelines (\[fig:correlation\_rankings\]C). The joint performance of the AI-Net group is $0.87$, the same as A3. We do observe, however, an improvement for the proximity pipelines in the joint ranking, increasing performance from $0.70$ for $0.72$. Interestingly, the combined diffusion pipelines have lower performance ($0.54$) than the best diffusion pipeline of $0.56$ observed for D1, D2, and D4. What is particularly encouraging, however, is that when we combine all $12$ pipelines, we obtain a ROC of $0.89$, the highest of any individual or combination-based pipelines, confirming that the individual pipelines offer complementary information that can be harnessed by the combined ranking. It is this combined list, therefore, that defines our final ranked list of predicted drugs for repurposing.
Finally, we manually inspected the joint ranking list, removing drugs with significant toxicities, eliminating those not appropriate, and removing lower-ranked members of the same drug class (with some exceptions). Through this process, we arrived at a list of $86$ drugs selected from the top $10\%$ of the total combined rank list, representing our final repurposing candidates for [COVID-19]{} (\[tab:drug\_repurposing\]). The selection contains drugs that are used for disorders of the respiratory (*e.g.*, theophylline, montelukast) and cardiovascular (*e.g.*, verapamil, atorvastatin) systems; antibiotics used to treat viral (*e.g.*, ribavirin, lopinavir), parasitic (*e.g.*, hydroxychloroquine, ivermectin, praziquantel), bacterial (*e.g.*, rifaximin, sulfanilamide), mycotic (*e.g.*,fluconazole), and mycobacterial (*e.g.*, isoniazid) infections; and immunomodulating/anti-inflammatory drugs (*e.g.*, interferon-$\beta$, auranofin, montelukast, colchicine); anti-proteasomal drugs (*e.g.*, bortezomib, carfilzomib); and a range of other less obvious drugs that warrant exploration (*e.g.*, aminoglutethimide, melatonin, levothyroxine, calcitriol, selegiline, deferoxamine, mitoxantrone, metformin, nintedanib, cinacalcet, and sildenafil, among others (\[tab:drug\_repurposing\]). Our final list includes $11$ previously proposed [@Gordon2020; @Zhou2020] potential drug-repurposing candidates for [COVID-19]{}, and $21$ drugs that are currently being tested in clinical trials (\[tab:drug\_repurposing\]).
### Validation Case Studies
The drug repurposing list provided in \[tab:drug\_repurposing\] ranks drugs based on their network-based relationship to the viral targets. However, for a drug to be effective, it may not be sufficient to be proximal—it also needs to induce the right perturbation in the cell, suppressing, for example, the expression of proteins the virus needs, and activating the expression of proteins essential for the cell function and survival that are suppressed by the virus. In this section we use expression data to understand how the drug affects the activity of proteins within the [COVID-19]{} disease module, offering insights about the mechanism of action of selected drugs.
**Connectivity Map:** We retrieved gene expression perturbation profiles for $59$ of the $81$ repurposing candidates from the Connectivity Map (CMap) database [@Subramanian2017; @Lamb2006], altogether including $5,291$ experimental instances (combination of different drugs, cell lines, doses, and time of treatment). To evaluate the degree to which each of these drugs modulate the activity of [COVID-19]{} targets, we measured the overlap between the perturbed genes and [COVID-19]{} targets. For example, for mitoxantrone, an antineoplastic drug (\[tab:drug\_repurposing\]), we find that $75$ ($22$%) of the [COVID-19]{} targets have a significant overlap with the $2,440$ genes highly perturbed by the drug ($3.33 \mu M$) in the lung cell line HCC515 (Fisher’s exact test, FDR-BH [p$_{adj}$-value]{} $< 0.05$) (\[fig:network\_drug\_cmap\]A). When evaluated across all experimental instances, we find that for $43$ of the $59$ drugs, there was a statistically significant overlap of the perturbed genes with the [COVID-19]{} targets (\[fig:network\_drug\_cmap\]B). For random selections of $59$ drugs from the pool of all drugs, only $13 \pm 7$ drugs on average have statistically significant overlap between perturbed genes and [COVID-19]{} targets (\[fig:cmap\_overlap\_random\_distribution\]), indicating that the repurposing candidates effectively perturb the network of the [COVID-19]{} disease module. We observed the highest number of perturbed [COVID-19]{} targets for carfilzomib (162, [p$_{adj}$-value]{} $=0.004$ , HA1E, $10.0 \mu M$), flutamide (162, [p$_{adj}$-value]{} $=0.003$, MCF7, $0.04 \mu M$), and bortezomib (162, [p$_{adj}$-value]{} $=0.02$, HA1E, $20.0 \mu M$). For cell lines derived from lung tissues (A549 and HCC515), the drugs with the highest overlap with [COVID-19]{} targets are mitoxantrone and ponatinib. These results can help us extract direct experimental evidence that the drug repurposing candidates selected by our methods modulate processes targeted by the virus, and offer mechanistic insights into the biological processes affected by these drugs. For example, we find that mitoxantrone (HUVEC, 10, $\mu M$, 24h) perturbs [COVID-19]{} targets related to cell cycle, viral life cycle, protein transport and organelle organization.
**Suppressing [COVID-19]{} Induced Expression:** We next asked whether the selected drugs can counteract the gene expression perturbations caused by the virus, *i.e.*, whether they down-regulate genes up-regulated by the virus or *vice versa*. For this analysis, we begin with the $120$ differentially expressed genes (DEGs) in the [SARS-CoV2]{} infected of the A549 cell line [@Blanco-Melo2020] and compare the list with the drug perturbation profiles. For example, bortezomib treatment of the cell line YAPC (20 $\mu M$) counteracts the effects of the [SARS-CoV2]{} infection for $65$ genes (\[fig:network\_drug\_cmap\]C), resulting in an inverted expression profile (Spearman correlation $\rho = -0.58$) (\[fig:network\_drug\_cmap\]C). We measured the Spearman correlation $\rho$ between the perturbations caused by the drug and perturbations caused by the virus in the A549 cell line, where negative correlation values indicate that the drug could counteract the effects of the infection. We find that $22$ of the $59$ drugs profiled in the Connectivity Map have negative correlation coefficients (Spearman $\rho < 0 $, FDR-BH [p$_{adj}$-value]{} $< 0.05$), indicating that they could beneficially modulate the effects of the virus infection. Again, for random selections of $59$ drugs from the pool of all drugs, only $3 \pm 2$ drugs on average have statistically significant negative correlation coefficients (\[fig:cmap\_cor\_random\_distribution\]), supporting, once again, the [COVID-19]{} relevance of the repurposing list. Among the $22$ drugs with significant perturbation overlap with both [COVID-19]{} targets and DEGs in [SARS-CoV2]{} infection, we find ivermectin and carfilzomib, each in clinical trial for [COVID-19]{} (\[tab:clinical\_trial\_list\]). Altogether, these results provide *in vitro* experimental support for the selected repurposing candidates as possible modulators of the biological processes targeted by the virus. It also indicates how network-based tools can utilize gene expression profiles to explore the potential efficacy of drugs.
Discussion
==========
In this study, we took advantage of recent advances in network medicine to define a list of $81$ drug repurposing candidates for the treatment of [COVID-19]{}, and, using *in vitro* data, we show that these drugs do affect biological processes targeted by the virus. The accuracy of our predictions will further improve as the input or validation data improve. For example, we relied on the results of Gordon et al (2020) [@Gordon2020], for the map of interactions between the virus and human proteins. There are, however, additional interactions not detected in the study [@Gordon2020]. For example, the ACE2 [@Zhang2020; @Hoffmann2020SARS-CoV-2Inhibitor] protein has been recently linked to initial viral association on airway epithelial cells, but in the current data set [@Gordon2020] no viral proteins target it.
Note that the utilized predictive pipelines select drugs that, by the virtue of the network-based relationship between their targets and the [SARS-CoV2]{} viral targets, are positioned to perturb effectively the [COVID-19]{} disease module. Some of the perturbations may block the virus’ ability to invade the host cells, or limit the molecular level disruption caused by the infection, potentially alleviating the disease symptoms and shortening the timeline of the disease. Others, however, may cause perturbations that aggravate the symptoms and the seriousness of the phenotype. Therefore, in ordinary circumstances, we would need molecular experiments to test the efficacy of these drugs for [COVID-19]{} infected cell lines (\[tab:drug\_repurposing\]). Yet, as many of these drugs have well-known side effects and toxicities, given the imminent need for a cure, it may be possible to move those drugs directly into clinical trials. While we are currently pursuing this possibility, releasing the list could offer opportunities for other groups, with appropriate resources and toolset, to move some of these drugs into screening or directly to rapid clinical trials. We are, of course, cognizant of the remote, yet real, possibility that these approved drugs with known side effects may exert unique toxicities in the setting of this novel infection, an outcome that can only be identified in clinical trial.
Our study focused on ranking the existing drugs based on their expected efficacy for [COVID-19]{} patients. This does not mean that drugs that did not make our final list could not have efficacy, or that they must be excluded from further consideration. As the input data improves, other, currently highly ranked drugs could move to a lower ranking, developing a case for experimental testing and clinical trial, and *vice versa*. The proposed methodology is general, allowing us to profile the potential efficacy of any drug or a family of drugs, whether or not they are included in our current reference list.
Normally, bioinformatic validation would be followed by experimental screening and potentially clinical validation before publication. We are currently pursuing these avenues, from screening in human cell lines to clinical trials. We feel, however, that given the strength of the bioinformatics validation and the obtained AUC, generating confidence in our methodologies, and the urgency of the [COVID-19]{} crisis, there is an imminent need for disclosure to offer rationale and guidance for upcoming clinical trials.
Methods
=======
Human Interactome, [SARS-CoV2]{} and Drug Targets
--------------------------------------------------
The human interactome was assembled from $21$ public databases that compile experimentally-derived protein-protein interactions (PPI) data: 1) binary PPIs, derived from high-throughput yeast-two hybrid (Y2H) expereriments (HI-Union [@Luck2019]), three-dimensional (3D) protein structures (Interactome3D [@Mosca2013], Instruct [@Meyer2013], Insider [@Meyer2018]) or literature curation (PINA [@Cowley2012], MINT [@Licata2012], LitBM17[@Luck2019], Interactome3D, Instruct, Insider, BioGrid [@Chatr-Aryamontri2017], HINT [@Das2012], HIPPIE [@Alanis-Lobato2017], APID [@Alonso-Lopez2019a], InWeb [@Li2016a]); 2) PPIs identified by affinity purification followed by mass spectrometry present in BioPlex2 [@Huttlin2017], QUBIC [@Hein2015], CoFrac [@Wan2015], HINT, HIPPIE, APID, LitBM17, InWeb; 3) kinase-substrate interactions from KinomeNetworkX [@Cheng2014b] and PhosphoSitePlus [@Hornbeck2015]; 4) signaling interactions from SignaLink [@Fazekas2013] and InnateDB [@Breuer2013]; and 5) regulatory interactions derived by the ENCODE consortium. We used the curated list of PSI-MI IDs provided by Alonso-López et al (2019)[@Alonso-Lopez2019a], for differentiating binary interactions among the several experimental methods present in the literature-curated databases. Specifically for InWeb, interactions with curation scores $< 0.175$ (75th percentile) were not considered. All proteins were mapped to their corresponding Entrez ID (NCBI) and the proteins that could not be mapped were removed. The final interactome used in our study contains $18,505$ proteins and $327,924$ interactions between them. We retrieved interactions between $26$ [SARS-CoV2]{} proteins and $332$ human proteins that were detected by Gordon, et al [@Gordon2020]. and drug-target information from the DrugBank database, containing $26,167$ interactions between $7,591$ drugs and their $4,187$ targets.
Tissue Specificity
------------------
We used the GTEx database [@Lonsdale2013], which contains the median gene expression from RNA-seq for $56$ different tissues, assuming that genes with a median count lower than $5$ are not expressed in that particular tissue. The LCC was calculated using a degree preserving approach [@Guney2016], preventing the repeated selection of the same high degree nodes by choosing $100$ degree bins in $1,000$ simulations.
Network Proximity
-----------------
Given $V$, the set of [COVID-19]{} virus targets, the set of drug targets, $T$, and $d(v,t)$, the shortest path length between nodes $v \in V$ and $t \in T$ in the network, we define [@Guney2016]
$$\label{eq:d_closest}
d_c (V,T) = \frac{1}{||T||} \sum_{t \in T} \text{min}_{v \in V} d(v,t) \text{.}$$
We also determined the expected distances between two randomly selected groups of proteins, matching the size and degrees of the original $V$ and $T$ sets. To avoid repeatedly selecting the same high degree nodes, we use degree-binning [@Guney2016] (see above). The mean $\mu_{d(V,T)}$ and standard deviation $\sigma_{d(V,T)}$ of the reference distribution allows us to convert the absolute distance $d_c$ to a relative distance $Z_{d_c}$, defined as
$$\label{eq:z_closest}
Z_{d_c} = \frac{d_c - \mu_{d_c (V,T)}}{\sigma_{d_c (V,T)}} \text{.}$$
Diffusion State Distance
------------------------
The diffusion state distance (DSD) [@Cao2013] algorithm uses a graph diffusion property to derive a similarity metric for pairs of nodes that takes into account how similarly they impact the rest of the network. We calculate the expected number of times $He(A,B)$ that a random walk starting at node $A$ visits node $B$, representing each node by the vector [@Cao2013]
$$\label{eq:dsd_he_vector}
He (V_i) = [He(V_i, V_1), He(V_i, V_2), He(V_i, V_3), ..., He(V_i, V_n)],$$
which describes how a perturbation initiated from that node impacts other nodes in the interactome. The similarity between nodes A and B is provided by the L1 norm of their corresponding vector representations,
$$\label{eq:dsd_l1norm}
DSD (A,B) = ||He(A) - He(B)|| \text{.}$$
Inspired by the DSD, we developed five new metrics to calculate the impact of drug targets $t$ on the [SARS-CoV2]{} targets $v$. The first (Pipeline D1) is defined as $$\label{eq:dsd_min}
I_{DSD}^{\min} = \frac{1}{|V|} \sum_{t \in T} \min_{v \in V} DSD(t,v)$$ where $DSD(s,t)$ represents the diffusion state distance between nodes $t$ and $v$. Since the L1 norm of two large vectors may result in loss of information [@Aggarwal2001], we also used the metric (Pipeline D2)
$$\label{eq:kl_min}
I_{KL}^{\min} = \sum_{t \in T} \min_{v \in V} KL(t,v)$$
and (Pipeline D3)
$$\label{eq:kl_med}
I_{KL}^{\text{med}} = \sum_{t \in T} \text{median}_{v \in V} KL(t,v)$$
where $KL$ is the Kullback-Leibler (KL) divergence between the vector representations of the nodes $t$ and $s$. Finally, to provide symmetric measures, we tested the measures (Pipeline D4)
$$\label{eq:js_min}
I_{JS}^{\min} = \sum_{t \in T} \min_{v \in V} JS(t,v)$$
and (Pipeline D5)
$$\label{eq:js_med}
I_{JS}^{\text{med}} = \sum_{t \in T} \text{median}_{v \in V} JS(t,v)$$
where JS is the Jensen Shannon (JS) divergence between the vector representations of nodes $t$ and $s$. All five measures consider $t \ne v$.
Graph convolutional networks
----------------------------
We designed a graph neural network for COVID-19 treatment recommendations based on a previously developed graph convolutional architecture [@zitnik2018modeling]. The multimodal graph is a heterogeneous graph $G = (\mathcal{V}, \mathcal{R})$ with $N$ nodes $v_i \in \mathcal{V}$ representing three distinct types of biomedical entities ([*i.e.*]{}, drugs, proteins, diseases), and labeled edges $(v_i, r, v_j) \in \mathcal{R}$ representing four semantically distinct types of edges $r$ between the entities ([*i.e.*]{}, protein-protein interactions, drug-target associations, disease-protein associations, and drug-disease treatments).
[**COVID-19 treatment recommendation task.**]{} We cast COVID-19 treatment recommendation as a link prediction problem on the multimodal graph. The task is to predict new edges between drug and disease nodes, so that a predicted link between a drug node $v_i$ and a disease node $v_j$ should indicate that drug $v_i$ is a promising treatment for disease $v_j$ ([*e.g.*]{}, COVID-19). Our graph neural network is an end-to-end trainable model for link prediction on the multimodal graph and has two main components: (1) an encoder: a graph convolutional network operating on $G$ and producing embeddings for nodes in $G$, and (2) a decoder: a model optimizing embeddings such that they are predictive of successful drug treatments.
[**Overview of graph neural architecture.**]{} The neural message passing encoder takes as input a graph $G$ and produces a node $d$-dimensional embedding $\mathbf{z}_i \in \mathbb{R}^d$ for every drug and disease node in the graph. We use the encoder [@zitnik2018modeling] that learns a message passing algorithm [@gilmer2017neural] and aggregation procedure to compute a function of the entire graph that transforms and propagates information across graph $G$. The graph convolutional operator takes into account the first-order neighborhood of a node and applies the same transformation across all locations in the graph. Successive application of these operations then effectively convolves information across the $K$-th order neighborhood (i.e., embedding of a node depends on all the nodes that are at most $K$ steps away), where $K$ is the number of successive operations of convolutional layers in the neural network model. The graph convolutional operator takes the form $$\mathbf{h}_i^{(k+1)} = \phi \bigg(\sum_{r} \sum_{j \in \mathcal{N}_r^i} \alpha_{r}^{ij} \mathbf{W}_r^{(k)} \mathbf{h}_j^{(k)} + \alpha_r^i \mathbf{h}_i^{(k)} \bigg),\label{eq:encoder}$$ where $\mathbf{h}_i^{(k)} \in \mathbb{R}^{d(k)}$ is the hidden state of node $v_i$ in the $k$-th layer of the neural network with $d^{(k)}$ being the dimensionality of this layer’s representation, $r$ is an edge type, matrix $\mathbf{W}_r^{(k)}$ is a edge-type specific parameter matrix, $\phi$ denotes a non-linear element-wise activation function (*i.e.*, a rectified linear unit), and $\alpha_r$ denote attention coefficients [@velivckovic2017graph]. To arrive at the final embedding $\mathbf{z}_i \in \mathbb{R}^d$ of node $v_i$, we compute its representation as: $\mathbf{z}_i = \mathbf{h}_i^{(K)}.$ Next, the decoder takes node embeddings and combines them to reconstruct labeled edges in $G$. In particular, decoder scores a $(v_i, r, v_j)$ triplet through a function $g$ whose goal is to assign a score $g(v_i, r, v_j)$ representing how likely it is that drugs $v_i$ will treat disease $v_j$ ([*i.e.*]{}, $r$ denotes a ‘treatment‘ relationship).
[**Training the graph neural network.**]{} During model training, we optimize model parameters using the max-margin loss functions to encourage the model to assign higher probabilities to successful drug indications $(v_i, r, v_j)$ than to random drug-disease pairs. We take an end-to-end optimization approach, that jointly optimize over all trainable parameters and propagates loss function gradients through both encoder and the decoder. To optimize the model, we train it for a maximum of 100 epochs (training iterations) using the Adam optimizer [@Kingma2014] with a learning rate of $0.001$. We initialize weights using the initialization described in [@Glorot2010]. To make the model comparable to other drug repurposing methodologies in this study, we do not integrate additional side information into node feature vectors; instead, we use one-hot indicator vectors [@hamilton2018embedding] as node features. In order for the model to generalize well to unobserved edges, we apply a regular dropout [@Srivastava2014] to hidden layer units (Eq. (\[eq:encoder\])). In practice, we use efficient sparse matrix multiplications, with complexity linear in the number of edges in $G$, to implement the model. We use a 2-layer neural architecture with $d_1=32$, $d_2=32$, $d_i=128$ hidden units in input, output, and intermediate layer, respectively, a dropout rate of $0.1$, and a max-margin of $0.1$. We use mini-batching [@hamilton2017inductive] by sampling triples from the multimodal graph. That is, we process multiple training mini-batches (mini-batches are of size $512$), each obtained by sampling only a fixed number of triplets, resulting in dynamic batches that change during training.
Expression perturbation profiles
--------------------------------
We retrieved drug perturbation profiles from the Connectivity Map (CMap) database [@Subramanian2017; @Lamb2006] using the Python package CMapPy [@Enache2019]. For each perturbation profile, we calculated the significance of the overlap of perturbed genes ($|{Z-Score}~| > 2$) and [SARS-CoV2]{} targets derived from Gordon, et. al., [@Gordon2020] using Fisher’s Exact Test. We also retrieved gene expression data of the cell line A549 after infection with [SARS-CoV2]{}[@Blanco-Melo2020]. The correlation between the perturbation scores provided in CMap and the gene expression fold change caused by [SARS-CoV2]{} infection was evaluated using the Spearman correlation coefficient. In both cases, we applied the Benjamini-Hochberg method for multiple testing correction (FDR $< 0.05$).
Rank aggregation
----------------
We used CRank algorithm [@zitnik2018prioritizing] to combine rankings returned by different methodologies into a single rank for each drug, which then determined the drug’s repurposing priority. The rank aggregation algorithm starts with ranked lists of drugs, $R_r$, each one arising from a different methodology $r$. Each ranked list is partitioned into equally sized groups, called bags. Each bag $i$ in ranked list $R_r$ has attached importance weight $K^i_r$ whose initial values are all equal.
CRank uses a two-stage iterative procedure to aggregate the individual rankings by taking into account uncertainty that is present across ranked lists. After initializing the aggregate ranking $R$ as a weighted average of ranked lists $R_\textrm{r}$, CRank alternates between the following two stages until no changes are observed in the aggregated ranking $R$. (1) First, it uses the current aggregated ranking $R$ to update the importance weights $K^i_r$ for each ranked list. For that purpose, the top-ranked drugs in $R$ serve as a temporary gold standard. Given bag $i$ and ranked list $R_r$, CRank updates importance weight $K_r^{i}$ based on how many drugs from the temporary gold standard appear in bag $i$ using the Bayes factors [@kass1995bayes; @casella2012assessing]. (2) Second, the ranked lists are re-aggregated based on the importance weights calculated in the previous stage. The updated importance weights are used to revise $R$ in which the new rank $R(C)$ of drug $C$ is expressed as: $R(C) = \sum_{r} \log K_r^{i_r(C)} R_r(C)$, where $K_r^{i_r(C)}$ indicates the importance weight of bag $i_r(C)$ of drug $C$ for ranking $r$, and $R_r(C)$ is the rank of $C$ according to $r$. By using an iterative approach, CRank allows for the importance of a ranking not to be predetermined and to vary across drugs.
The final output is a global ranked list $R$ of drugs that represents the collective opinion of the different repurposing methodologies. The Python source code implementation of CRank is available at <https://github.com/mims-harvard/crank>. In all experiments, we set the number of bags to 1,000, the size of the temporary gold standard to 0.5% of the total number of drugs in $R$, and the maximum number of iterations to 50. In all cases, the algorithm converged, in fewer than 20 iterations.
ROC curves
----------
We employed different methodologies to rank drug candidates. Since we lack ground-truth labels for drugs being effective against the disease, we rely on clinical trials to gather names of drugs currently in trial. We made an assumption that all the drugs tested in clinical trials are relevant and based on prior *in vitro* or *in vivo* observations. We used this information and the ranking of each method to compute ROC (Receiver Operating Characteristics) curves and AUC (area under the curve) scores for model selection and performance analysis. AUC score measures the quality of the separation between positive and negative instances. For the ranked list, we applied different thresholds to compute false-positive and true-positive rates to plot ROC. Scores of AUC range between 0 and 1, where 1 corresponds to perfect performance and 0.5 indicates the performance of a random classifier. Some methods fail to provide a ranking for each drug or to provide a fair comparison between methods, we assumed all the missing ranks should be listed at the bottom of the ranking. We use the Python package Scikit-learn [@Pedregosa2011] for computing AUC scores and plotting ROC curves.
For the ground-truth list, we consider the [ClinicalTrials.gov](ClinicalTrials.gov) website the primary source of ongoing trials of drugs fo [COVID-19]{}. We are cognizant of its limitations, primarily being one of time lags between the implementation of a trial and its appearance on the site. We also quantified the performance of models under different constraints: considering only drugs that have at least $N$ trials and considering only the evidence provided up to a certain date (\[fig:AUC\_time\]).
Authors Contribution
====================
A.L.B designed the study. A.A, D.M.G, M.Z, and X.G performed drug predictions. I.D.V analyzed disease comorbidities and drug validation. A.A, D.M.G, I.D.V, M.Z, O.V, and X.G analyzed the data. O.V. carried out ClinicalTrials.gov data analysis for model selection and performance analysis. J.L manually curated the drug candidates. A.L.B, D.M.G, and I.D.V wrote the paper with input from all authors. All authors read and approved the manuscript. D.G guided A.A with designing diffusion-based similarity implementations and H.S curated list of promising drugs for [COVID-19]{}.
Acknowledgments
===============
This work was supported, in part, by NIH grants HG007690, HL108630, and HL119145, and by AHA grant D700382 to J.L; A.L.B is supported by NIH grant 1P01HL132825, American Heart Association grant 151708, and ERC grant 810115-DYNASET.
We wish to thank Nicolette Lee and Grecia for providing support, Marc Santolini for suggestions in the diffusion-based methods.
Declaration of interests
========================
J.L. and A.L.B are co-scientific founder of Scipher Medicine, Inc., which applies network medicine strategies to biomarker development and personalized drug selection. A.L.B is the founder of Nomix Inc. and Foodome, Inc. that apply data science to health; O.V and D.M.G are scientific consultants for Nomix Inc. I.D.V is a scientific consultant for Foodome Inc.
![ **Network Medicine Approaches to Drug Repurposing.** **(A)** The physical interactions that we use as input in the network medicine framework: Virus-human protein interaction, capturing the human proteins to which the viral proteins can bind; human protein-protein interactions, defining the human interactome of $18,508$ proteins linked by $332,749$ pairwise physical interactions; and the drug-human protein interactions, capturing the human protein targets of each drug in DrugBank. **(B)** A schematic representation of the input data we use for the predictions, the three prediction methods and the resulting pipelines, and the outcomes provided by the analysis.[]{data-label="fig:summary_project"}](figures/fig1.pdf){width="80.00000%"}
![**The [COVID-19]{} Disease Module.** **(A)** Proteins targeted by [SARS-CoV2]{} are not distributed randomly in the human interactome, but form a large connected component (LCC) consisting of $208$ proteins, as well as multiple small subgraphs. We do not show the $93$ viral targets that do not interact with other viral targets. Proteins not expressed in the lung are shown in orange, indicating that almost all proteins in [SARS-CoV2]{} LCC are expressed in the lung, explaining the effectiveness of the virus in causing pulmonary infections. **(B)** The random expectation of the LCC size, indicating that the observed [COVID-19]{} LCC, whose size is indicated by the red arrow, is larger than expected by chance. **(C)** Similarly, the lung-based LCC is also greater than expected by chance.[]{data-label="fig:LCC_COVID_Disease_Module"}](figures/fig2.pdf){width="80.00000%"}
\[fig:disease\_polar\] ![**Disease Comorbidity**. We measured the network proximity between [COVID-19]{} targets and $299$ diseases. The figure represents each disease as a circle whose radius reflects the number of disease genes associated with it [@Menche2015]. The diseases closest to the center, whose names are marked, are expected to have higher comorbidity with the [COVID-19]{} outcome. The farther is a disease from the center, the more distant are its disease proteins from the [COVID-19]{} viral targets.[]{data-label="fig:disease_proximity"}](figures/fig3.pdf "fig:"){width="100.00000%"}
![**Using Proximity to Predict Repurposing Drugs:** **(A)** The local neighborhood of the human interactome showing the targets of the drug chloroquine and the reference drug, dextrotyroxine, and the proteins closest to them targeted by [COVID-19]{} viral proteins. **(B)** Distribution of proximity scores for $6,116$ drugs, capturing their distance to [SARS-CoV2]{} targets. The six lighter bars indicate the proximity of drugs currently tested in clinical trials for [COVID-19]{}. []{data-label="fig:network_proximity_results"}](figures/fig4.pdf){width="85.00000%"}
![ **Comparison of the Predictive Pipelines**. **(A)** Heatmap of the Kendall $\tau$ capturing the correlation between the ranking predicted by the $12$ drug repurposing pipelines. Methods using different approaches are not correlated, potentially prioritizing different drugs. **(B)** ROC Curves and AUC for each of the twelve pipelines used for drug repurposing, using as a gold standard the drugs under evaluation in clinical trial for treating [COVID-19]{} (\[tab:clinical\_trial\_list\]). **(C)** The performance of the overall cRank (all), which combines all pipelines into a final ranking list, is higher than the performance of each method individually (cRanks AIs, Ps and Ds).[]{data-label="fig:correlation_rankings"}](figures/fig5.pdf){width="80.00000%"}
![**Validation Using Gene Expression Data**. **(A)** Local region of the interactome showing the [COVID-19]{} targets. The drug mitoxantrone ($3.33 \mu M$, 24h) perturbs the gene expression of $75$ [COVID-19]{} targets (labeled proteins) in the lung cell line HCC515 (green and red colors represent down- and up-regulation, respectively). **(B)**: The comparison of bortezomib treatment (YAPC, 20 $\mu M$) and [SARS-CoV2]{} infection perturbation profiles shows a negative correlation (Spearman $\rho = -0.58$, FDR-BH [p$_{adj}$-value]{} $= 1.67 \times 10^{-7}$), indicating that the drug counteracts the effects of the infection for $65$ genes (orange dots). The straight line shows a linear fit between the two profiles and the respective confidence interval. Positive values represent upregulated expression and negative values represents down-regulated expression on both axes.[]{data-label="fig:network_drug_cmap"}](figures/fig6.pdf "fig:"){width="70.00000%"}\
{width="60.00000%"}
\[tab:drug\_repurposing\] {width="1.\textwidth"}
[75]{} \[1\][\#1]{} \[1\][`#1`]{} urlstyle \[1\][doi: \#1]{}
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Supplementary Material
======================
Rank Drug ID Drug name Current indications
------ --------- ----------------- ------------------------------------------------------------------------------------------------
1 DB01117 Atovaquone Hematologic cancer, Malaria
2 DB01201 Rifapentine Pulmonary tuberculosis
3 DB00608 Chloroquine Rheumatoid Arthritis, Malaria, Sarcoidosis
4 DB00834 Mifepristone Cushing’s disease, Meningioma, Brain cancer
5 DB00431 Lindane Pediculus capitis infestation
6 DB09029 Secukinumab Chronic small plaque psoriasis
7 DB11574 Elbasvir Hepatitis C
8 DB09065 Cobicistat HIV Infections
9 DB09054 Idelalisib Lymphocytic Leukemi
10 DB09102 Daclatasvir Hepatitis C
11 DB08880 Teriflunomide Multiple sclerosis, Lupus nephritis, Rheumatoid arthritis
12 DB11569 Ixekizumab Chronic small plaque psoriasis
13 DB01058 Praziquantel Schistosomiasis, Opisthorchiasis
14 DB00503 Ritonavir Acquired immunodeficiency syndrome, Hepatitis C, HIV-1 infection
15 DB13179 Troleandomycin
16 DB01222 Budesonide Ulcerative colitis, Liver cirrhosis, Lung diseases, Asthma
17 DB09212 Loxoprofen Rheumatoid Arthritis
18 DB00687 Fludrocortisone Dysautonomia, Mitral Valve Prolapse Syndrome, Parkinson’s disease, Peripheral motor neuropathy
19 DB08865 Crizotinib Lung cancer, Hematologic cancer
20 DB09101 Elvitegravir HIV Infections
: **Top-ranked drugs in the AI-based ranking (ranking ‘A3‘)**. Shown are top-20 drugs and conditions for which the drugs are indicated.
\[tab:ai-ranking-AI3\]
![**Distribution of the Proximity ($S_{vb}$) Between $299$ Diseases and [COVID-19]{} Targets**. $S_{vb}$ values repressent the network-based overlap between [SARS-CoV2]{} targets $v$ and the genes associated with each disease $b$. []{data-label="fig:disease_hist"}](figures/sab_virus_dist.png){width="80.00000%"}
![**Comparison of Diffusion-Based Measures on Ranking HIV Drugs**. Side by side boxplot of ranking distributions across $5$ different diffusion-based methods on HIV. The distribution of $22$ FDA approved HIV drugs is shown by red and each circle shows one distinct drug.[]{data-label="fig:dsd_hiv"}](figures/dsd_hiv.png){width="80.00000%"}
![**Comparison of Diffusion-Based Measures on Ranking Drugs Being Tested for [COVID-19]{}**. Side by side boxplot of ranking distributions across $5$ different diffusion-based methods on [COVID-19]{}. The distribution of $24$ distinct potential drugs is shown by red and each circle shows one distinct drug. The list of $24$ potential drugs was obtained from curation of Nature News (\[tab:labels\_dsd\_testing\]).[]{data-label="fig:dsd_results"}](figures/Asher_comparison_covid_intrials.png){width="80.00000%"}
![**Overview of AI-based Strategy for Drug Repurposing**. **(A)** Visualization of the learned embedding space. Every point represents a drug (in blue) or a disease (in orange). If a drug and a disease are embedded close together in this space, this means the underlying PPI networks of the drug and the disease are predictive of whether the drug can treat the disease. **(B)** Probability distributions of indications and non-indications learned by the AI model are well-separated, indicating the model can distinguish between successful and failed drug indications. **(C)** Predictive performance of the AI model on the held-out test set of drug indications. Higher values indicated better performance (AUROC, Area under the ROC curve; AUPRC, Area under the PR curve; MAP@50, Mean average precision at top 50.[]{data-label="fig:ai_results"}](figures/emb-figure-v2.pdf){width="90.00000%"}
![**Drug Repurposing Candidates Show Greater Overlap of Perturbed Genes and [COVID-19]{} Targets**. We find that of the $59$ repurposing candidates present in the Connectivity Map database $43$ have statistically significant overlap of perturbed genes and [COVID-19]{} targets. We created a reference distribution by randomly selecting $59$ drugs and measuring the same overlap in 500 iterations. On average, only $17 \pm 4.9$ of the $59$ randomly selected drugs have statistically significant overlap between perturbed genes and [COVID-19]{} targets.[]{data-label="fig:cmap_overlap_random_distribution"}](figures/cmap_overlap_covid_random_reference.png){width="50.00000%"}
![**Drug Repurposing Candidates Show Greater Anticorrelated Effects with [SARS-CoV2]{} infection**. We find that of the $59$ repurposing candidates present in the Connectivity Map database $22$ have have negative correlation coefficients (Spearman $\rho < 0 $, FDR-BH [p$_{adj}$-value]{} $< 0.05$) when comparing with [SARS-CoV2]{} infection perturbations. We created a reference distribution by randomly selecting $59$ drugs and measuring the Spearman correlation in 500 iterations. On average, only $3 \pm 2$ of the $59$ randomly selected drugs have statistically significant negative correlation coefficients.[]{data-label="fig:cmap_cor_random_distribution"}](figures/cmap_cor_covid_random_reference.png){width="50.00000%"}
![**AUC over time for each method**. We measured the performance of each ranking method by computing ROC (Receiver Operating Characteristics) curves and AUC (area underthe curve) values. True positives were obtained from ClinicalTrials.gov by retrieving the list of drugs currently undergoing clinical trials for [COVID-19]{} (\[tab:clinical\_trial\_list\]). Here, we quantified the performance of models considering only the evidence provided up to a certain date.[]{data-label="fig:AUC_time"}](figures/auc_compare_byTime.pdf){width="50.00000%"}
|
---
abstract: 'To estimate the diffusion constant $D$ of particles in a plasma, we develop a method that is based on the mean free path $\lambda$ for scatterings with momentum transfer $q \grtsim T$. Using this method, we estimate $\lambda$ and $D$ for squarks and quarks during the electroweak phase transition. Assuming that Debye and magnetic screening lengths provide suitable infrared cutoffs, our calculations yield $\lambda \lsim 18/T$ and $D \lsim 5/T$ for both squarks and quarks. Our estimate of $\lambda$ suggests that suppressions of charge transport due to decoherence of these strongly interacting particles during the electroweak phase transition are not severe and that these particles may contribute significantly to electroweak baryogenesis.'
address: 'California Institute of Technology, Pasadena, CA 91125 USA'
author:
- 'Hooman Davoudiasl[^1] and Eric Westphal[^2]'
title: Diffusion and Decoherence of Squarks and Quarks During the Electroweak Phase Transition
---
FEYNMAN
Introduction
============
In recent years, many scenarios incorporating extensions of the Minimal Standard Model (MSM) have been proposed for the generation of the Baryon Asymmetry of the Universe during the electroweak phase transition. A much studied mechanism uses the transport of some quantum number, via the CP violating interactions of certain particles with the expanding broken phase Higgs wall, into the unbroken phase during a first order electroweak phase transition. This transported quantum number, such as the axial top quark number, then biases the equilibrium in the direction of baryon number violation, ultimately generating a baryon asymmetry through weak sphaleron processes[@CKN; @HN]. Of the extensions of the MSM used in these scenarios, those based on supersymmetric models such as the Minimal Supersymmetric Standard Model (MSSM) are the best motivated. By including supersymmetric particles, extra CP violation can be naturally introduced into a mechanism for baryogenesis. It is argued that a strongly first order phase transition can be achieved if the broken phase right-handed stop mass is less than, or of the order of, the top quark mass [@Carena; @Riotto]. In this case, squarks with soft supersymmetry breaking masses $m_s \sim T$, where $T \sim 100$ GeV is the temperature of the plasma, could have an important role in the charge transport mechanism[@DRW]. However, to study charge transport via stops, one needs to have an estimate of their diffusion properties in a plasma at the electroweak scale.
In Refs. [@JPT; @Moore], a set of approximations in conjunction with the Boltzmann equation for quarks in the plasma of MSM particles were used to estimate the quark diffusion constant $D_q$. As strong interactions dominate the diffusion process, and since stops are strongly interacting particles, it has been assumed that the estimate $D_q \sim 6/T$ of Ref.[@JPT] is applicable to stops as well (the validity of this assumption is not [*a priori*]{} obvious because of the different statistics, masses, and couplings of squarks and quarks). This estimate is derived using an approximate gluon propagator with the thermal mass of the longitudinal gluons $m_g$ as an infrared cutoff and ignoring the different thermal properties of the transverse and longitudinal gluons. The use of the approximate gluon propagator can only yield the leading-logarithmic behavior and does not result in the correct leading $\alpha_s^2$ non-logarithmic contribution[@JPT; @Moore]. However, the diffusive process is expected to be dominated by the $t$-channel gluon exchange diagrams, and for these diagrams, the leading logarithm contribution is expected to be dominant[@JPT]. A more comprehensive treatment in Ref. [@Moore] gives $D_q \sim
3/T$. Again, this estimate is at the level of the leading logarithm.
To study the diffusion of particles in a plasma, one must consider scattering processes in which the momentum transfer $q$ is not small. For the $t$-channel processes we consider in this paper, we approximate this effect by an infrared regularization of $q$ such that $q \grtsim T$, since the typical momenta of the scattering particles are of order $T$. That is, we assume that such a transfer of momentum in the scattering process randomizes the momentum of the diffusing particle, as an approximation of the physics invovled. This momentum randomization approximation is implemented naturally by using as cutoffs the longitudinal and transverse thermal masses of the exchanged gluons which are comparable in magnitude to the temperature $T$ of the plasma. We note here that the thermal masses of gluons depend parametrically on the strong coupling constant $g_s$, and that these masses are comparable to the temperature only for realistic values of $g_s \sim 1$. If we take the limit in which $g_s \ll 1$, these masses will be small compared to the temeprature $T$, and cannot be used as cutoffs in our approach. In this paper, we use the above momentum randomization approximation to calculate the elastic[^3] mean free path $\lambda$ associated with the diffusive processes (for which $q \grtsim T$) and relate it to the diffusion constant $D$. Henceforth, the words “mean free path” refer to this diffusive mean free path.
In electroweak baryogenesis scenarios that use charge transport, the CP violating interactions of the charge carriers with an expanding Higgs wall eventually result in the generation of baryon number. However, within the Higgs wall, multiple scatterings in which the final momentum of the charge carrier differs significantly from its initial momentum wash out the asymmetry caused by CP violation and suppress baryogenesis. This effect is known as decoherence. These same processes also contribute to the diffusion of the particles within the plasma. Therefore, the diffusive mean free path $\lambda$ which we calculate is a relevant parameter for estimating the supression due to decoherence.
In this paper, we use the method described above to estimate the mean free path $\lambda_s$ and the diffusion constant $D_s$ of stops that have a soft supersymmetry breaking mass $m_s \sim T$ in the unbroken phase of the electroweak plasma. We use the same method to estimate the mean free path $\lambda_q$ and the diffusion constant $D_q$ for quarks and compare our values with those of Ref. [@JPT]. We find that our method reproduces the results of Ref. [@JPT] for the set of parameters used therein. In general, our results suggest that the values of $\lambda$ and $D$ of squarks are close to those of quarks.
In calculating $\lambda$ and $D$, we consider only strong interactions, for they dominate the diffusion of squarks and quarks in the plasma. In the case of squarks, we further assume that scatterings from and via the heavy gluinos and squarks do not contribute significantly, leaving only quarks and gluons as the dominant scatterers and mediators. As consideration of more scatterers can only decrease the calculated values of $\lambda$ and therefore $D$, the inclusion of only quark scatterers will yield an upper bound (and even a reasonable order of magnitude estimate) for the size of the squark diffusion constant $D_s$, in light of the results of Ref. [@Moore]. In computing $D_q$, to facilitate comparison, we follow Ref. [@JPT] and only consider $t$-channel quark-quark scattering.
As explained above, we implement our momentum randomization approximation using the longitudinal and transverse gluons thermal masses $m_g$ and $m_t$ in the plasma, referred to as Debye and magnetic masses, respectively, as physical infrared cutoffs for the exchanged gluon momentum. Since these masses depend on $g_s$, the momentum transfer $q$ in our calculations is parametrically of order $g_s T$, and the use of Hard Thermal Loop (HTL) propagator for the gluons is valid[@Weldon; @Braaten]. In the limit $g_s \ll 1$, this corresponds to small momentum transfer. However, we note that in our approximation, we demand $ q \grtsim T$. Thus, in the $g_s \ll 1$ limit, in our approach, we have to abandon the thermal masses as cutoffs, and choose a cutoff $q_{cut}$ parametrically independent of $g_s$, and such that $q_{cut} \grtsim T$. Then, in this technical sense, in our approach, the use of the HTL gluon propagator would not be justified, since the momentum carried by the exchanged gluon would not be parametrically soft, that is ${\cal O}(g_s T)$, as required by the HTL formalism[@Weldon; @Braaten]. Our method does not yield the leading logarithm behavior, as $g_s \to 0$, obtained in Refs. [@JPT; @Moore] for the $t$-channel processes considered therein. Nonetheless, for the physical values of $g_s$ at the electroweak phase transition, we obtain similar results, suggesting that our scheme reasonably approximates the physics. For realistic values of $g_s$, we use the gluon thermal masses as cutoffs in our momentum randomization approximation, since they naturally arise in the plasma. Then, parametrically, we are allowed to use the HTL propagator for the gluon. In this work, to incorporate thermal effects qualitatively, we approximate the effect of the HTL propagator by separating the gluon propagator into transverse and longitudinal parts that in general have different thermal masses. Whereas $m_g$ is calculable at one loop, $m_t$ is not calculable perturbatively and is unknown. Therefore, we will present our results for two representative values of $m_t$.
In the next section, we describe our method for calculating the diffusion constant of particles in the plasma. In Section III, we present our estimates for $\lambda_s$, $D_s$, $\lambda_q$, and $D_q$, followed by a discussion of our results. The appendix contains some information on the approximate thermal gluon propagator we use in our calculations.
Calculation of the Mean Free Path and the Diffusion Constant
============================================================
Let us consider a two body scattering process where the initial and final particles have 4-momenta $(p, k)$ and $(p^\prime, k^\prime)$, respectively. We refer to each particle by its 4-momentum for the rest of this section. The $p$-particle, whose diffusion constant we calculate, scatters from the $k$-particle. For processes relevant to the calculation of the diffusion constant $D$, the final state $p^\prime$-particle is of the same species as the initial $p$-particle. The $p$-particle, $k$-particle, $p^\prime$-particle, and $k^\prime$-particle have thermal distributions $\rho_p$, $\rho_k$, $\rho_{p^\prime}$, and $\rho_{k^\prime}$, respectively. The density per unit volume of a particle with 4-momentum $p$ is given by $\rho_p
\, d^3 p/(2\pi)^3$.
The transition probability for the above process per unit volume and per unit time is $$\eta = {d^3 p \over (2\pi)^3 \, 2 p^0}{d^3 k \over (2\pi)^3 \, 2 k^0}
{d^3 p^\prime \over (2\pi)^3 \, 2 p^{\prime 0}}
{d^3 k^\prime \over (2\pi)^3 \, 2 k^{\prime 0}}
(2 \pi)^4 \delta^{(4)}(p + k -
p^\prime - k^\prime) |{\cal M}|^2$$ $$\times \rho_p \, \rho_k\, (1 \pm \rho_{p^\prime}) \,
(1 \pm \rho_{k^\prime}),
\label{eta}$$ where $\cal M$ is the amplitude for the scattering, and $\pm$ is for final state bosons or fermions, respectively. Let $d\sigma$ be the differential cross section for this process, $$d\sigma = {d^3 p^\prime \over (2\pi)^3 \, 2 p^{\prime 0}}
{d^3 k^\prime \over (2\pi)^3 \, 2 k^{\prime 0}}
(2 \pi)^4 \delta^{(4)}(p + k - p^\prime - k^\prime) {|{\cal M}|^2
(1 \pm \rho_{p^{\prime}}) (1 \pm \rho_{k^{\prime}})
\over 4 \sqrt {(p\cdot k)^2 - m_p^2 \, m_k^2}}.
\label{dsig}$$ Comparing Eqs. (\[eta\]) and (\[dsig\]) yields $$\eta = \left[{d^3 p \over (2\pi)^3 \, p^0}\right]
\left[{d^3 k \over (2\pi)^3 \, k^0}\right]
\sqrt {(p\cdot k)^2 - m_p^2 \, m_k^2} \, \rho_p \, \rho_k \, d\sigma.
\label{etadsig}$$
To get the rate of collision per unit time $\eta^{(1)}$ of one $p$-particle in the plasma, we divide $\eta$ by $\rho_p \, d^3 p/(2\pi)^3$, the volume density of $p$-particles: $$\eta^{(1)} = {d^3 k \over (2\pi)^3 \, p^0 k^0 }
\sqrt {(p\cdot k)^2 - m_p^2 \, m_k^2} \, \rho_k \, d\sigma.
\label{eta1}$$ To calculate the mean free path associated with the above process, we need to find the total rate of collision per unit time $\eta_{tot}^{(1)}(p)$ for one particle with initial 4-momentum $p$ into any final state in the allowed phase space, using the total cross section $\sigma$. From Eqs. (\[dsig\]) and (\[eta1\]) we get $$\eta_{tot}^{(1)}(p) = {2 \over (4 \pi)^5} \int
{d^3 k \over p^0 k^0 }
\int {d^3 p^\prime d^3 k^\prime \over p^{\prime 0} k^{\prime 0}}
\delta^{(4)}(p + k - p^\prime - k^\prime)|{\cal M}|^2 \, \rho_k
(1 \pm \rho_{p^{\prime}}) (1 \pm \rho_{k^{\prime}}).
\label{eta1totp}$$ Note that each scatterer included will give an additive contribution of this form to $\eta_{tot}^{(1)}$.
The collision time $\tau(p)$, the length of time between two successive collisions for a $p$-particle, is the inverse of $\eta_{tot}^{(1)}(p)$ $$\tau(p) = {1 \over \eta_{tot}^{(1)}(p)}.
\label{taup}$$ The distance $l(p)$ such a $p$-particle travels between two collisions is then given by $l(p) = (|\vec p|/p^0) \, \tau(p)$. We finally get the mean free path $\lambda$ for the $p$-particle by taking the thermal average of $l(p)$, using the thermal distribution of the $p$-particles. We thus get $$\lambda = \left[\int \frac{d^3 p}{(2 \pi)^3} \, \rho_p
\right]^{- 1} \int \frac{d^3 p}{(2 \pi)^3} \, \rho_p {|\vec p| \over p^0 \,
\eta_{tot}^{(1)}(p)} \, ,
\label{lambda}$$ where we have used Eq. (\[taup\]). Note that this mean free path vanishes if the cross section suffers from infrared divergences. However, for diffusive processes, these divergences are suitably regulated and only processes with nontrivial momentum transfer contribute. The resulting mean free path (\[lambda\]) can then be related to the diffusion constant $D$ by the relation $$D = \frac{1}{3} \, \lambda \, {\bar v},
\label{D}$$ where ${\bar v}$ is the mean velocity of the diffusing particle.
Results and Discussion
======================
We begin this section by describing some of the thermal properties of gluons in the plasma and how we incorporate these properties into our calculations. Due to interactions with the plasma, gluons develop temperature dependent masses. The longitudinal gluons have a thermal Debye mass $m_g(T) = \sqrt {8
\pi \alpha_s} \, T$, where $\alpha_s = g_s^2/(4 \, \pi)$, at the 1-loop level and the transverse gluons have a non-perturbative thermal magnetic mass $m_t(T)$ that is zero at the 1-loop level and is expected to be ${\cal
O}(g_s^2 \, T)$. Thus, we may assume that the infrared screening of longitudinal gluons occurs at a momentum scale $m_g$, and the similar scale for the transverse gluons is likely lower. At the electroweak phase transition temperature $T_c \approx 100$ GeV, $\alpha_s \approx 0.1$ and $m_g
\approx 1.6 \, T$. Since the magnetic mass is unknown, the choice of transverse infrared momentum cutoff is rather arbitrary. However, because $g_s \approx 1$ at scale $T_c$, it is reasonable to assume that $m_t$ is of order $T$. In Table \[tbl1\], we take as two representative values $m_t =
T$ and $m_t = m_g$.
(25000,21000) (12500,7000)\[4\] (7500,3500)[$p$]{} (8500,4500)[(1,1)[1500]{}]{} (12500,7000)\[4\] (17500,3500)[$p'$]{} (15000,6000)[(1,-1)[1500]{}]{} (12500,7000)\[6\] (,)\[8000\] (7500,16500)[$k$]{} (8500,16000)[(1,-1)[1500]{}]{} (,)\[8000\] (17000,16500)[$k'$]{} (15000,14500)[(1,1)[1500]{}]{}
The amplitude for the $t$-channel squark-quark diagram of Fig. \[t-sq\] is $${\cal M}_{sq} = - 8 \pi \, \alpha_s \, T^a \, p^\mu D_{\mu \nu} \,
{\bar q} (k^\prime) \, T^a \gamma^\nu q (k),
\label{Msq}$$ where $T^a$ is a generator in the adjoint representation of the $SU(3)_c$ color gauge group, and $q$ is a quark spinor. We work in the Landau gauge where, as explained in the appendix, $(k^\prime - k)^\mu D_{\mu \nu} = 0$. Since $p + p^\prime =
2p + k - k^\prime$, ${\cal M}_{sq}$ does not depend on $p^\prime$ explicitly. We use the amplitude ${\cal M}_{sq}$ of Eq. (\[Msq\]) to estimate the squark mean free path $\lambda_s$ and diffusion constant $D_s$ from Eqs. (\[lambda\]) and (\[D\]). The amplitude is squared and summed over all $72$ species of quark scatterers ($3$ colors, $2$ spins, $6$ flavors, and antiparticles). We have numerically computed $\lambda_s$ and $D_s$ for supersymmetry breaking squark masses ranging from 50 GeV to 200 GeV and observed that their mass dependence is weak. Our results are presented in Table \[tbl1\], where the values of $\lambda_s$ and $D_s$ have been computed for $m_s = 100$ GeV.
(25000,21000) (12500,7000)\[8000\] (7500,3500)[$p$]{} (8500,4500)[(1,1)[1500]{}]{} (12500,7000)\[8000\] (17500,3500)[$p'$]{} (15000,6000)[(1,-1)[1500]{}]{} (12500,7000)\[6\] (,)\[8000\] (7500,16500)[$k$]{} (8500,16000)[(1,-1)[1500]{}]{} (,)\[8000\] (17000,16500)[$k'$]{} (15000,14500)[(1,1)[1500]{}]{}
In Ref. [@JPT], $D_q$ is computed using only the $t$-channel quark-quark scattering amplitude of Fig. \[t-qq\], given by $${\cal M}_{qq} = - 4 \pi \, \alpha_s \, {\bar q} (p') \, T^a \gamma^\mu q(p) \,
D_{\mu \nu} \, {\bar q} (k^\prime) \, T^a \gamma^\nu q(k).
\label{Mqq}$$
To compare our method with that of Ref. [@JPT], we have computed $D_q$, again numerically, using only ${\cal M}_{qq}$ (and again summing over $72$ species of scatterers). The entries in Table \[tbl1\] labeled “JPT” refer to the numbers we get for the set of parameters that are used in Ref.[@JPT], namely $\alpha_s = 1/7$ and $m_g = m_t = \sqrt{8 \pi \alpha_s} \,
T = 1.9 \, T$. We see that using the JPT parameters, we obtain the estimate $D_q \sim 6/T$ of Ref. [@JPT], where only the leading logarithmic contributions were considered. This suggests that our momentum randomization approximation, where the momentum transfer $q \grtsim T$, yields reasonable estimates for the diffusion constant. However, we also note that in Ref. [@JPT] the diffusion constant $D \sim 1/[g_s^4 \ln (g_s^{-1})]$, where the logarithmic dependence comes from the use of a $g_s$ dependent $m_g$ as the infrared regulator. In our approach, as mentioned before, for physical values of $g_s$, we can use $m_g$, and perhaps $m_t$, as the infrared cutoffs, but in general, the cutoff we choose is not a function of $g_s$, and does not vanish as $g_s \to 0$, for it has to be chosen to satisfy $q \grtsim T$. In this way, our computations will yield $D \sim 1/(g_s^4)$, which represents the $\alpha_s$ dependence of the amplitudes (\[Msq\]) and (\[Mqq\]). Therefore, the behavior of our results differs from those of Ref. [@JPT] by $1/\ln(g_s^{-1})$. However, for physical values of $g_s$, the logarithm is of order unity. Taking the result of Ref. [@JPT] as a fair estimate of $D_{q}$, we believe that our results are reliable up to factors of order unity. Note that we do not consider all the processes that contribute at this level: scatterings from on-shell gluons in the plasma also provide a substantial contribution. However, the results of Ref.[@Moore] suggest that the inclusion of these diagrams will not change the results by more than a factor of 2.
$\alpha_s$ $m_g$ $m_t$ $\lambda$ $D$ Remarks
----------------- ------------ ------------ ------------ ----------- ------- ---------
quark $1/10$ $1.6 \, T$ $T$ $13/T$ $4/T$
$1/10$ $1.6 \, T$ $1.6 \, T$ $24/T$ $8/T$
$1/7$ $1.9 \, T$ $1.9 \, T$ $18/T$ $6/T$ JPT
squark $1/10$ $1.6 \, T$ $T$ $12/T$ $3/T$
$m_s = 100$ GeV $1/10$ $1.6 \, T$ $1.6 \, T$ $18/T$ $5/T$
$1/7$ $1.9 \, T$ $1.9 \, T$ $14/T$ $4/T$ JPT
: Results for $\lambda$ and $D$ []{data-label="tbl1"}
Our calculations suggest that $\lambda_s \approx \lambda_q$ and $D_s \approx
D_q$, up to factors of order unity, and most likely to within $30\%$, and that the effects of different statistics, masses, and couplings on the values of $D$ and $\lambda$ for squarks and quarks are not strong. For $m_t = T < m_g$, we roughly get $\lambda \lsim 10/T$ and $D \lsim 3/T$. On the other hand, if $m_t = m_g$, our results increase by about a factor of 2.
Electroweak baryogenesis scenarios that use quarks or squarks for charge transport in a first order phase transition suffer from a suppression due to decoherence that is caused by multiple scatterings of these strongly interacting particles across the width of the expanding Higgs wall. A measure of the strength of decoherence is the ratio of the mean free path of the particles to the width $w$ of the wall[@DRW]. In Ref. [@Quiros], a 2-loop MSSM calculation of the Higgs wall profile gives $w \approx 25/T$. Using this value of $w$, our results yield $\lambda/w \lsim 1/2$, which suggests that although the effects of decoherence are not negligible, they are not severe.
Acknowledgements {#acknowledgements .unnumbered}
================
It is a pleasure to thank Ann Nelson, John Preskill, Krishna Rajagopal, Dam Son, Mark Wise, and Laurence Yaffe for insightful discussions. We would also like to thank Martin Gremm and Iain Stewart for their helpful comments. This work was supported in part by the U.S. Dept. of Energy under Grant No.DE-FG03-92-ER40701.
Thermal Gluon Propagator {#thermal-gluon-propagator .unnumbered}
========================
In this appendix, we give the expression we use for the approximate thermal gluon propagator in a plasma, taking the different properties of the longitudinal and transverse gluons into account as represented by their respective cutoffs $m_g$ and $m_t$. Let $n^\mu = (1, 0, 0,
0)$ be the 4-velocity of the plasma in the plasma frame. We denote the 4-momentum $q$ of the propagating gluon by $q^\mu=(q^0,\vec{q})$ in the plasma frame. The component of $n$ that is orthogonal to $q$ is given by ${\tilde n}$, where $${\tilde n}^\mu = n^\mu - \frac{(n \cdot q) q^\mu}{q^2}.
\label{ntil}$$ We define two projection operators $P_{\mu \nu}$ and $Q_{\mu \nu}$, where $$P_{\mu \nu} = g_{\mu \nu} - \frac{q_\mu q_\nu}{q^2} - \frac{{\tilde n}_\mu
{\tilde n}_\nu}{{\tilde n}^2}
\label{P}$$ and $$Q_{\mu \nu} =\frac{{\tilde n}_\mu {\tilde n}_\nu}{{\tilde n}^2}.
\label{Q}$$
The expression for the Landau gauge thermal gluon propagator in our approximation is then given by $$D^{(L)}_{\mu \nu} = - \left[\frac{1}{q_T^2} P_{\mu \nu} +
\frac{1}{q_L^2} Q_{\mu \nu}\right],
\label{Dapp}$$ where $q_T^2 = q^2 - m_t^2$ and $q_L^2 = q^2 - m_g^2$, and from $q^\mu P_{\mu \nu} = q^\mu Q_{\mu \nu} = 0$, we have $q^\mu D^{(L)}_{\mu \nu} = 0$.
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[^1]: E-mail address: hooman@theory.caltech.edu
[^2]: E-mail address: westphal@theory.caltech.edu
[^3]: Non-elastic processes in which the species of the particle changes do contribute to the mean free path; however, these are not considered to be diffusive processes here.
|
---
abstract: 'We study parity-even and parity-odd polarization observables for the process $p \, p \to l^{\pm} \, X$, where the lepton comes from the decay of a $W$-boson. By using the collinear twist-3 factorization approach, we consider the case when one proton is transversely polarized, while the other is either unpolarized or longitudinally polarized. These observables give access to two particular quark-gluon-quark correlation functions, which have a direct relation to transverse momentum dependent parton distributions. We present numerical estimates for RHIC kinematics. Measuring, for instance, the parity-even transverse single spin correlation would provide a crucial test of our current understanding of single spin asymmetries in the framework of QCD.'
author:
- |
Andreas Metz, Jian Zhou\
[*Department of Physics, Barton Hall, Temple University, Philadelphia, PA 19122, USA*]{}
title: |
**Transverse spin asymmetries for [$W$]{}-production in\
proton-proton collisions**
---
Introduction
============
It has long been recognized that production of $W$-bosons in hadronic collisions can provide new insights into the partonic structure of hadrons, with polarization observables being of particular interest. In this context the parity-odd longitudinal single spin asymmetry (SSA) in proton-proton scattering plays a very important role, both for leptonic as well as hadronic final states (see [@Bourrely:1990pz; @Bourrely:1993dd; @Weber:1993xm; @Kamal:1997fg; @Gehrmann:1997ez; @Bunce:2000uv; @Gluck:2000ek; @Nadolsky:2003fz; @Moretti:2005aa; @Arnold:2008zx; @Berger:2008jp; @deFlorian:2010aa] and references therein). A major aim of looking into this observable is to get new and complementary information on the quark helicity distributions inside the proton.
In the meantime, also a few studies for $W$-production with transversely polarized protons are available [@Brodsky:2002pr; @Schmidt:2003wi; @Kang:2009bp; @Kang:2010fu]. These papers mainly focus on a particular parity-even transverse single spin effect in $p \, p \to W^{\pm} \, X$ (with a subsequent decay of the $W^{\pm}$ into a lepton pair) that is related to the transverse momentum dependent Sivers function $f_{1T}^\perp$ [@Sivers:1989cc] in the polarized proton. Such an observable could, in principle, be measured at the Relativistic Heavy Ion Collider (RHIC) in Brookhaven. In order to have clean access to transverse momentum dependent parton distributions (TMDs) like the Sivers function, one has to reconstruct the $W$-boson in the experiment. However, what one measures is $p \, p \to l^{\pm} \, X$, and the detectors at RHIC do not allow to fully determine the momentum of the $W$.
The kinematics for inclusive production of a single lepton in proton-proton collisions coincides with the one for inclusive production of a jet or a hadron, for which mostly collinear factorization is used in the literature. In this Letter, we compute transverse spin observables for $p \, p \to l^{\pm} \, X$ in the collinear twist-3 formalism at the level of Born diagrams. The machinery of collinear twist-3 factorization was pioneered already in the early 1980’s [@Efremov:1981sh; @Ellis:1982wd], and in the meantime frequently applied to transverse spin effects in hard semi-inclusive reactions (see [@Efremov:1984ip; @Qiu:1991pp; @Eguchi:2006mc; @Zhou:2009jm] and references therein).
If one of the protons in $p \, p \to l^{\pm} \, X$ is transversely polarized, and the other is either unpolarized or longitudinally polarized, one can identify two parity-even and two parity-odd spin observables. We will discuss below that, in the collinear twist-3 approach, these four observables contain two specific twist-3 quark-gluon-quark correlation functions. One is the so-called ETQS (Efremov-Teryaev-Qiu-Sterman) matrix element [@Efremov:1981sh; @Efremov:1984ip; @Qiu:1991pp], which is related to a particular moment of the transverse momentum dependent Sivers function as shown in [@Boer:2003cm; @Ma:2003ut]. The second is related to the TMD $g_{1T}$ [@Zhou:2008mz; @Zhou:2009jm], where we use the TMD-notation of Refs. [@Mulders:1995dh; @Boer:1997nt; @Bacchetta:2006tn; @Arnold:2008kf].
In addition to the analytical results, we provide numerical estimates for typical RHIC kinematics $(\sqrt{s} = 500 \, \textrm{GeV})$. All the observables are peaked around $l_T \approx M_W/2$, with $l_T$ representing the transverse momentum of the lepton and $M_W$ the $W$-boson mass. In each case we predict clearly measurable effects. For the parity-even transverse SSA $A_{TU}^{e}$ our numerical results are very close to those obtained in Ref. [@Kang:2009bp] on the basis of factorization in terms of transverse momentum dependent parton correlators.
Before presenting our results we emphasize that measuring $A_{TU}^{e}$ would provide a crucial test of our present understanding of transverse SSAs in QCD. In particular, this means that such a measurement would test the same physics — the gluon exchange between the remnants of the hadrons and the active partons — which underlies the famous process-dependence of the Sivers function and of related time-reversal odd parton distributions [@Collins:2002kn]. In other words, experimental results for $A_{TU}^{e}$ in $p \, p \to l^{\pm} \, X$, even if analyzed in terms of collinear parton correlators, would check a crucial ingredient of TMD-factorization [@Collins:1981uk; @Ji:2004wu; @Collins:2004nx]. Such a check, in essence, can be considered to be as fundamental as measuring the sign of the Sivers asymmetry in the Drell-Yan process.
Analytical results
==================
We start by fixing the kinematical variables for the process $p \, p \to l^{\pm} \, X$, and assign 4-momenta to the particles according to $$p(P_a) + p(P_b) \to l^{\pm}(l) + X \,.$$ By means of these momenta we specify a coordinate system through $\hat{e}_z = \hat{P_a} = - \hat{P_b}$, $\hat{e}_x = \hat{l}_{T}$ (with $\vec{l}_{T}$ representing the transverse momentum of the jet), and $\hat{e}_y = \hat{e}_z \times \hat{e}_x$. Mandelstam variables are defined by $$\label{e:mandel_1}
s = (P_a + P_b)^2 \,, \qquad
t = (P_a - l)^2 \,, \qquad
u = (P_b - l)^2 \,,$$ while on the partonic level one has $$\label{e:mandel_2}
\hat{s} = (k_a + k_b)^2 = x_a x_b s\,, \qquad
\hat{t} = (k_a - l)^2 = x_a t\,, \qquad
\hat{u} = (k_b - l)^2 = x_b u \,,$$ where $k_a$ and $k_b$ denote the momentum of the active quark/antiquark in the protons; see also Fig. \[f:diagram\](a). The momentum fraction $x_a$ characterizes the (large) plus-momentum of the quark/antiquark in the proton moving along $\hat{e}_z$ through $k_a^+ = x_a P_a^+$.[^1] Likewise, one has $k_b^- = x_b P_b^-$. The relation $\hat{s} + \hat{t} + \hat{u} = 0$ implies $$\label{e:mom_rel}
x_a = - \frac{x_b u}{x_b s + t}
= \frac{x_b \sqrt{s} \, l_{T} \, e^{\eta}}
{x_b s - \sqrt{s} \, l_{T} \, e^{-\eta}} \,.$$ In the second step in (\[e:mom\_rel\]) we express $x_a$, for a given $\sqrt{s}$, through $l_{T} = |\vec{l}_{T}|$ and the pseudo-rapidity $\eta = - \ln \tan(\vartheta/2)$ of the lepton, since transverse momenta and (pseudo-)rapidities are commonly used to describe the kinematics of a final state particle in proton-proton collisions.
![Diagram (a): parton model representation for $p \, p \to l^{\pm} \, X$, where the lepton is produced in the decay of a $W$-boson. The final state (anti-)neutrino goes unobserved. Diagram (b): contribution from quark-gluon-quark correlation. This diagram, together with its Hermitian conjugate which is not displayed, needs to be taken into account when computing twist-3 observables.[]{data-label="f:diagram"}](FD_Wprod_1.eps "fig:"){width="7.5cm"} 1.0cm ![Diagram (a): parton model representation for $p \, p \to l^{\pm} \, X$, where the lepton is produced in the decay of a $W$-boson. The final state (anti-)neutrino goes unobserved. Diagram (b): contribution from quark-gluon-quark correlation. This diagram, together with its Hermitian conjugate which is not displayed, needs to be taken into account when computing twist-3 observables.[]{data-label="f:diagram"}](FD_Wprod_2.eps "fig:"){width="7.5cm"}
Next, we turn to the polarization observables for $p \, p \to l^{\pm} \, X$, which we compute in the collinear factorization framework. As already mentioned, we focus on the situation when one proton is transversely polarized, while the other is either unpolarized or longitudinally polarized. One finds the following expression for the cross section:[^2] $$\begin{aligned}
\label{e:master}
\lefteqn{l^0 \, \frac{d^3 \sigma}{d^3 l} =
\frac{\alpha_{em}^2}{12 \,s \sin^4 \vartheta_w} \, \sum_{a,b} \, |V_{ab}|^2
\int_{x_b^{min}}^1 \frac{dx_b}{x_a x_b} \, \frac{1}{x_b s + t} \,
\bigg\{ \, H^{ab} \, f_1^a(x_a) \, f_1^b(x_b)}
\nonumber \\
&&
+ \, 2 \pi M \, \varepsilon_T^{ij} l_{T}^i S_{aT}^j \, \tilde{H}^{ab} \,
\bigg[ \,
\bigg(T_{F}^a(x_a,x_a) - x_a \, \frac{d}{dx_a} \, T_{F}^a(x_a,x_a) \bigg)
+ K(\hat{s}) \, T_F^a(x_a,x_a) \bigg] f_1^b(x_b)
\nonumber \\
&&
+ \, 2M \, \vec{l}_T \cdot \vec{S}_{aT} \, \tilde{H}^{ab} \,
\bigg[ \,
\bigg(\tilde{g}^a(x_a) - x_a \, \frac{d}{dx_a} \, \tilde{g}^a(x_a) \bigg)
+ K(\hat{s}) \, \tilde{g}^a(x_a)
+ 2 x_a \, g_T^a(x_a) \bigg] f_1^b(x_b)
\nonumber \\
&&
- \, 2 \pi M \, \lambda_b \, \varepsilon_T^{ij} l_{T}^i S_{aT}^j \, \tilde{H}^{ab} \,
\bigg[ \,
\bigg(T_{F}^a(x_a,x_a) - x_a \, \frac{d}{dx_a} \, T_{F}^a(x_a,x_a) \bigg)
+ K(\hat{s}) \, T_F^a(x_a,x_a) \bigg] g_1^b(x_b)
\nonumber \\
&&
- \, 2M \, \lambda_b \, \vec{l}_T \cdot \vec{S}_{aT} \, \tilde{H}^{ab} \,
\bigg[ \,
\bigg(\tilde{g}^a(x_a) - x_a \, \frac{d}{dx_a} \, \tilde{g}^a(x_a) \bigg)
+ K(\hat{s}) \, \tilde{g}^a(x_a)
+ 2 x_a \, g_T^a(x_a) \bigg] g_1^b(x_b)
\nonumber \\
&&
+ \, \ldots \bigg\} \,,
\\
&& \textrm{with} \quad K(\hat{s}) = \frac{2 M_W^2 (\hat{s} - M_W^2 - \Gamma_W^2)}
{(\hat{s} - M_W^2)^2 + M_W^2 \Gamma_W^2} \,.
\nonumber\end{aligned}$$ In Eq. (\[e:master\]), $\vartheta_w$ is the weak mixing angle, $V_{ab}$ is a CKM matrix element, $M$ is the proton mass, $M_W$ is the $W$-mass and $\Gamma_W$ its decay width. We also use $\varepsilon_T^{ij} \equiv \varepsilon^{-+ij}$ with $\varepsilon^{0123} = 1$. The transverse spin vector of the proton moving along $\hat{e}_z$ is denoted by $\vec{S}_{aT}$, whereas $\lambda_b$ represents the helicity of the second proton. The lower limit of the $x_b$-integration is given by $x_b^{min}=-t/(s+u)$. One can project out the four spin-dependent components of the cross section in (\[e:master\]), in order, through $$\begin{aligned}
\label{e:sigma_tu_e}
\sigma_{TU}^e & = & \frac{1}{4}
\Big( \big[\sigma(\uparrow_{y},+) - \sigma(\downarrow_{y},+) \big]
+ \big[\sigma(\uparrow_{y},-) - \sigma(\downarrow_{y},-) \big] \Big) \,,
\\ \label{e:sigma_tu_o}
\sigma_{TU}^o & = & \frac{1}{4}
\Big( \big[\sigma(\uparrow_{x},+) - \sigma(\downarrow_{x},+) \big]
+ \big[\sigma(\uparrow_{x},-) - \sigma(\downarrow_{x},-) \big] \Big) \,,
\\ \label{e:sigma_tl_o}
\sigma_{TL}^o & = & \frac{1}{4}
\Big( \big[\sigma(\uparrow_{y},+) - \sigma(\downarrow_{y},+) \big]
- \big[\sigma(\uparrow_{y},-) - \sigma(\downarrow_{y},-) \big] \Big) \,,
\\ \label{e:sigma_tl_e}
\sigma_{TL}^e & = & \frac{1}{4}
\Big( \big[\sigma(\uparrow_{x},+) - \sigma(\downarrow_{x},+) \big]
- \big[\sigma(\uparrow_{x},-) - \sigma(\downarrow_{x},-) \big] \Big) \,.\end{aligned}$$ In these formulas, ’$\uparrow_{x/y}$’ (’$\downarrow_{x/y}$’) denotes transverse polarization along $\hat{e}_{x/y} \, (-\hat{e}_{x/y})$ for the proton moving in the $\hat{e}_z$-direction, whereas ’$+$’ and ’$-$’ represent the helicities of the second proton.
The dots in Eq. (\[e:master\]) indicate longitudinal single spin and double spin observables, as well as four possible correlations for double transverse polarization. In collinear factorization, the latter are at least twist-4 effects in the Standard Model. Note that double transverse polarization observables for $W$-production were also discussed in connection with potential physics beyond the Standard Model (see [@Rykov:1999ru; @Boer:2010mc] and references therein).
We computed the (twist-2) unpolarized cross section in the first line of (\[e:master\]) on the basis of diagram (a) in Fig. \[f:diagram\] by applying the collinear approximation to the momenta $k_a$ and $k_b$ of the active partons. The result contains the ordinary unpolarized quark distribution $f_1^a$ for a quark flavor $a$. The hard scattering coefficients $H^{ab}$ and $\tilde{H}^{ab}$ in Eq. (\[e:master\]), expressed through the partonic Mandelstam variables in (\[e:mandel\_2\]), read $$\label{e:hard}
H^{ab} = \frac{\hat{u}^2}{(\hat{s} - M_W^2)^2 + M_W^2 \Gamma_W^2} \,, \qquad
\tilde{H}^{ab} = \frac{1}{\hat{u}} \, H^{ab} \,, \quad
\textrm{for} \;\; ab = d\bar{u}, \; s\bar{u}, \; \bar{d}u, \; \bar{s}u \,.$$ In Eq. (\[e:hard\]), one has to replace $\hat{u}$ by $\hat{t}$ for $ab = \bar{u}d, \; \bar{u}s, \; u\bar{d}, \; u\bar{s}$.
The four cross sections in (\[e:sigma\_tu\_e\])–(\[e:sigma\_tl\_e\]) represent twist-3 observables. Calculational details for such observables in collinear factorization can be found in various papers; see, e.g., Refs. [@Qiu:1991pp; @Eguchi:2006mc; @Kouvaris:2006zy; @Yuan:2008it; @Kang:2008qh; @Zhou:2009jm]. We merely mention that one has to expand the hard scattering contributions around vanishing transverse parton momenta. While for twist-2 effects only the leading term of that expansion matters, in the case of twist-3 the second term is also relevant. In addition, the contribution from quark-gluon-quark correlations, as displayed in diagram (b) in Fig. \[f:diagram\], needs to be taken into consideration. The sum of all the terms can be written in a color gauge invariant form, which provides a consistency check of the calculation.
The quark-gluon-quark correlator showing up in $\sigma_{TU}^e$ and $\sigma_{TL}^o$ is the aforementioned ETQS matrix element $T_F^a(x,x)$ [@Efremov:1981sh; @Efremov:1984ip; @Qiu:1991pp]. The peculiar feature of this object is the vanishing gluon momentum — that’s why it is also called “soft gluon pole matrix element”. If the gluon momentum becomes soft one can hit the pole of a quark propagator in the hard part of the process, providing an imaginary part (nontrivial phase) which, quite generally, can lead to single spin effects [@Efremov:1981sh; @Efremov:1984ip; @Qiu:1991pp]. Note also that in our lowest order calculation no so-called soft fermion pole contribution (see [@Koike:2009ge] and references therein) emerges. For $\sigma_{TU}^o$ and $\sigma_{TL}^e$ another quark-gluon-quark matrix element — denoted as $\tilde{g}^a$; see, in particular, Refs. [@Eguchi:2006qz; @Zhou:2008mz; @Zhou:2009jm] — appears, together with the familiar twist-3 quark-quark correlator $g_T^a$ (and, in the case of $\sigma_{TL}^o$, together with the quark helicity distribution $g_1^a$).
We use the common definitions for $f_1$, $g_1$, and $g_T$. The quark-gluon-quark correlators $T_F$ and $\tilde{g}$ are specified according to[^3] $$\begin{aligned}
\label{e:defqgq_1}
- i \varepsilon_T^{ij} S_T^j \, T_F(x,x) & = &
\frac{1}{2M} \int \frac{d\xi^- d\zeta^-}{(2\pi)^2} \, e^{i x P^+ \xi^-} \,
\langle P,S_T | \bar{\psi}(0) \, \gamma^+ \,
ig F^{+i}(\zeta^-) \, \psi(\xi^-) | P,S_T \rangle \,,
\\ \label{e:defqgq_2}
S_T^i \, \tilde{g}(x) & = &
\frac{1}{2M} \int \frac{d\xi^-}{2\pi} \, e^{i x P^+ \xi^-}
\nonumber \\
&& \hspace{1.0cm} \mbox{} \times
\langle P,S_T | \bar{\psi}(0) \, \gamma_5 \gamma^+ \,
\bigg( i D_T^i - ig \int_0^\infty d\zeta^- F^{+i}(\zeta^-) \bigg) \,
\psi(\xi^-) | P,S_T \rangle \,,\end{aligned}$$ with $F^{\mu\nu}$ representing the gluon field strength tensor, and $D^{\mu} = \partial^{\mu} - i g A^{\mu}$ the covariant derivative. Equations (\[e:defqgq\_1\]) and (\[e:defqgq\_2\]) hold in the light-cone gauge $A^+ = 0$, while in a general gauge Wilson lines need to be inserted between the field operators.
It is important that $T_F$ and $\tilde{g}$ are related to moments of TMDs. To be explicit, one has [@Boer:2003cm; @Ma:2003ut; @Zhou:2008mz; @Zhou:2009jm] $$\begin{aligned}
\label{e:qgq_tmd_1}
\pi \, T_F(x,x) & = &
- \int d^2k_T \, \frac{\vec{k}_T^2}{2M^2} \,
f_{1T}^{\perp}(x,\vec{k}_T^2)\Big|_{DIS} \,,
\\ \label{e:qgq_tmd_2}
\tilde{g}(x) & = &
\int d^2k_T \, \frac{\vec{k}_T^2}{2M^2} \, g_{1T}(x,\vec{k}_T^2) \,,\end{aligned}$$ where we use the conventions of Refs. [@Mulders:1995dh; @Boer:1997nt; @Bacchetta:2006tn; @Arnold:2008kf] for the TMDs $f_{1T}^\perp$ and $g_{1T}$. In Eq. (\[e:qgq\_tmd\_1\]) we take into account that the Sivers function $f_{1T}^\perp$ depends on the process in which it is probed [@Collins:2002kn; @Brodsky:2002rv]. In order to make numerical estimates we will exploit the relations in (\[e:qgq\_tmd\_1\]), (\[e:qgq\_tmd\_2\]).
Finally, note that, due to the pure vector-axialvector coupling of the $W$-boson, no chiral-odd parton correlator shows up in any of the four spin correlations in (\[e:master\]), which makes those observables rather clean. The situation is different if one considers single lepton production from the decay of a virtual photon or of a $Z$-boson.
Numerical results
=================
Now we move on to discuss numerical results for the polarization observables by limiting ourselves to the transverse single spin effects. This means, we consider the two spin asymmetries $A_{TU}^e$ and $A_{TU}^o$, $$\label{e:asymm}
A_{TU}^e = \frac{\sigma_{TU}^e}{\sigma_{UU}} \,, \qquad
A_{TU}^o = \frac{\sigma_{TU}^o}{\sigma_{UU}} \,,$$ with $\sigma_{TU}^e$ and $\sigma_{TU}^o$ from Eq. (\[e:sigma\_tu\_e\]) and (\[e:sigma\_tu\_o\]), respectively, and $\sigma_{UU}$ denoting the unpolarized cross section. Note that the definition of $A_{TU}^e$ corresponds to the one of the transverse SSA $A_N$, which has been extensively studied in one-particle inclusive production for hadron-hadron collisions; see [@Adams:2003fx; @Adler:2005in; @Arsene:2008mi] for recent experimental results from RHIC.
To compute $\sigma_{UU}$ we use the unpolarized parton densities from the CTEQ6-parameterization [@Pumplin:2002vw]. For the ETQS matrix element we use the relation (\[e:qgq\_tmd\_1\]) between $T_F$ and the Sivers function, and take $f_{1T}^\perp$ from the recent fit provided in Ref. [@Anselmino:2008sga] on the basis of data from semi-inclusive DIS. (For experimental studies of the Sivers effect we refer to [@Airapetian:2004tw; @Alexakhin:2005iw], while extractions of the Sivers function from data can be found in [@Anselmino:2008sga; @Efremov:2004tp; @Anselmino:2005nn; @Vogelsang:2005cs; @Collins:2005ie; @Arnold:2008ap].) In the case of $A_{TU}^o$ one needs input for $g_T$ and $\tilde{g}$. For $g_T$ we resort to the frequently used Wandzura-Wilczek approximation [@Wandzura:1977qf] (see [@Accardi:2009au] for a recent study of the quality of this approximation) $$\label{e:appr_1}
g_T(x) \approx \int_x^1 \frac{dy}{y} \, g_1(y) \,,$$ whereas for $\tilde{g}$ we use (\[e:qgq\_tmd\_2\]) and a Wandzura-Wilczek-type approximation for the particular $k_T$-moment of $g_{1T}$ in (\[e:qgq\_tmd\_2\]) [@Metz:2008ib], leading to $$\label{e:appr_2}
\tilde{g}(x) \approx x \int_x^1 \frac{dy}{y} \, g_1(y) \,.$$ We mention that (\[e:appr\_2\]) and a corresponding relation between chiral-odd parton distributions were used in [@Kotzinian:2006dw; @Avakian:2007mv] in order to estimate certain spin asymmetries in semi-inclusive DIS. The comparison to data discussed in [@Avakian:2007mv] looks promising, though more experimental information is needed for a thorough test of approximate relations like the one in (\[e:appr\_2\]). Measuring the SSA $A_{TU}^o$ could provide such a test. The helicity distributions $g_1^a$ in (\[e:appr\_1\]) and (\[e:appr\_2\]) are taken from the DSSV-parameterization [@deFlorian:2008mr]. The transverse momentum of the lepton $l_T$ serves as the scale for the parton distributions.
The numerical estimates are for typical RHIC kinematics, i.e., $\sqrt{s} = 500 \, \textrm{GeV}$. We present the asymmetries either as function of $\eta$ for fixed $l_T$ or vice versa.
![$A_{TU}^e$ for $p \, p \to l^{\pm} \, X$ as a function of $\eta$ (left) and $l_T$ (right) for $\sqrt{s} = 500 \, \textrm{GeV}$. The solid line is for $l^-$-production, and the dashed line is for $l^+$-production.[]{data-label="f:atu_e"}](ATU_e_eta.eps "fig:"){width="7.5cm"} 1.0cm ![$A_{TU}^e$ for $p \, p \to l^{\pm} \, X$ as a function of $\eta$ (left) and $l_T$ (right) for $\sqrt{s} = 500 \, \textrm{GeV}$. The solid line is for $l^-$-production, and the dashed line is for $l^+$-production.[]{data-label="f:atu_e"}](ATU_e_lt.eps "fig:"){width="7.5cm"}
We start by discussing the parity-even asymmetry $A_{TU}^e$. As shown in the right plot in Fig. \[f:atu\_e\], this observable is peaked around $l_T \approx M_W/2$ — a feature that does not depend on the value of $\eta$. To be more precise, the peak is at $l_T = 41 \, \textrm{GeV}$, i.e., slightly above $M_W/2$. The peak in the polarized cross section $\sigma_{TU}^e$ gets enhanced in the asymmetry, because the unpolarized cross section drops rather fast when going beyond $l_T = M_W/2$. (As a side-remark we point out that the asymmetry in the peak region is completely dominated by the third term in the 2nd line in (\[e:master\]) containing the factor $K(\hat{s})$.) Nevertheless, in this kinematical region we expect $A_{TU}^e$ to be measurable. As discussed in the introduction, in this context it is important to recall that information on the sign of the asymmetry is already sufficient for a crucial test of our current understanding of transverse SSAs.
In particular in the peak region, the asymmetry is larger for $l^-$-production ($W^-$-production) than for $l^+$-production, which is partly due to the rather large Sivers function for $d$-quarks obtained in the fit of Ref. [@Anselmino:2008sga]. The $l^-$-asymmetry and $l^+$-asymmetry come with opposite sign because the Sivers function for $u$-quarks and $d$-quarks have an opposite sign. Note also that both asymmetries change sign as function of $l_T$. Therefore, whether the sign of the asymmetry can be measured unambiguously may critically depend on the $l_T$-resolution in the experiment.
As the $\eta$-dependence of $A_{TU}^e$ in left plot in Fig. \[f:atu\_e\] shows, the asymmetry is maximal in the positive $\eta$ range, when a large-$x$ parton from the polarized proton participates in the hard scattering. Obviously, by integrating over a suitable $\eta$-range one may optimize between magnitude of the asymmetry on the one hand and the size of the statistical error bars on the other. Moreover, it is worthwhile to mention that the contributions from the antiquark Sivers functions are not negligible in the backward region. (Here we refer to a corresponding discussion on the Sivers asymmetry in the Drell-Yan process for proton-proton collisions in [@Collins:2005rq], where the strong sensitivity to the Sivers function for antiquarks was already pointed out.)
![$A_{TU}^o$ for $p \, p \to l^{\pm} \, X$ as a function of $\eta$ (left) and $l_T$ (right) for $\sqrt{s} = 500 \, \textrm{GeV}$. The solid line is for $l^-$-production, and the dashed line is for $l^+$-production.[]{data-label="f:atu_o"}](ATU_o_eta.eps "fig:"){width="7.5cm"} 1.0cm ![$A_{TU}^o$ for $p \, p \to l^{\pm} \, X$ as a function of $\eta$ (left) and $l_T$ (right) for $\sqrt{s} = 500 \, \textrm{GeV}$. The solid line is for $l^-$-production, and the dashed line is for $l^+$-production.[]{data-label="f:atu_o"}](ATU_o_lt.eps "fig:"){width="7.5cm"}
It is also interesting that for both $l^+$-production and $l^-$-production the overall magnitude of $A_{TU}^e$ is very similar to the predictions presented in Ref. [@Kang:2009bp], where TMD-factorization was used.
Let us now turn to the parity-odd transverse SSA $A_{TU}^o$, which is displayed in Fig. \[f:atu\_o\]. Again, this asymmetry has a pronounced peak at $l_T = 41 \, \textrm{GeV}$, and it is largest for $l^+$-production (up to about $8 \, \%$). As outlined above, our prediction for $A_{TU}^o$ is based on the Wandzura-Wilczek-type approximation leading to (\[e:appr\_2\]), which probably represents the most uncertain part of our calculation. Nevertheless, the asymmetry should be within experimental reach. Like in the case of the parity-even SSA, also $A_{TU}^o$ is almost entirely determined by the $K(\hat{s})$-term in the 3rd line in (\[e:master\]). This implies that, due to the relation (\[e:qgq\_tmd\_2\]), it gives rather clean access to the TMD $g_{1T}$, which so far is experimentally unconstrained. Therefore, in any case, a measurement of $A_{TU}^o$ would provide very interesting new information.
Summary
=======
We have studied transverse spin asymmetries for the process $p \, p \to l^{\pm} \, X$, where the lepton is produced in the decay of a $W$-boson. If one of the protons is transversely polarized, and the other is either unpolarized or longitudinally polarized, there exist two parity-even and two parity-odd spin asymmetries. We computed these asymmetries in collinear twist-3 factorization at the level of Born diagrams. Moreover, for the two transverse single spin asymmetries $A_{TU}^e$ and $A_{TU}^o$ — defined through Eq. (\[e:asymm\]) and (\[e:sigma\_tu\_e\]), (\[e:sigma\_tu\_o\]) — we made numerical estimates for typical kinematics at RHIC ($\sqrt{s} = 500 \, \textrm{GeV}$). In the following we summarize our main results:
- The analytical results for all four spin-dependent cross sections are given by two particular quark-gluon-quark correlators, which have a direct relation to transverse momentum dependent parton distributions: the Sivers function $f_{1T}^\perp$ and the TMD $g_{1T}$; see Eqs. (\[e:qgq\_tmd\_1\]), (\[e:qgq\_tmd\_2\]). Measuring these observables could therefore provide new information on the structure of the proton that goes beyond the collinear parton model.
- The parity-even SSA $A_{TU}^e$ is largest for $l^-$-production (up to about $8 \, \%$), and it is peaked for transverse momenta $l_T$ of the lepton slightly above $M_W/2$. (Actually, all the asymmetries studied in this Letter are significant only in a relatively narrow region around $l_T \approx M_W/2$.) Measuring the sign of this asymmetry can, in essence, provide an as crucial test as measuring the sign of the Sivers asymmetry in Drell-Yan would do: it can test our present understanding of the underlying dynamics of transverse SSAs and at the same time check an important ingredient of TMD-factorization, namely the influence of the Wilson-line which is generated by the interaction between the active partons and the remnants of the protons. (For related work we refer to [@Collins:2002kn; @Brodsky:2002rv; @Efremov:2004tp; @Bacchetta:2007sz; @Kang:2009bp; @Anselmino:2009st; @Kang:2009sm].)
- To the best of our knowledge the parity-odd SSA $A_{TU}^o$ was never before explored in the literature. We find $A_{TU}^o$ to be largest for $l^+$-production (also up to about $8 \, \%$, like $A_{TU}^e$ for $l^-$-production). This observable is directly related to (a moment of) the TMD $g_{1T}$, for which at this time no experimental information exists.
In general, we believe that $W$-physics for polarized proton-proton collisions is very promising not only in the case of longitudinally polarized protons, but has also a considerable discovery potential for transverse polarization.\
[**Acknowledgments:**]{} We thank Zhongbo Kang, Jianwei Qiu, and Feng Yuan for helpful discussions. A.M. acknowledges the support of the NSF under Grant No. PHY-0855501.
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[^1]: For a generic 4-vector $v$, we define light-cone coordinates according to $v^{\pm} = (v^0 \pm v^3) / \sqrt{2}$ and $\vec{v}_T = (v^1,v^2)$.
[^2]: Polarization degrees are suppressed in the cross section formula (\[e:master\]).
[^3]: Note that in the literature different conventions for $T_F$ exist.
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addtoreset[equation]{}[section]{}
Imperial/TP/2013/JG/04\
1.5cm 1.5cm Aristomenis Donos$^1$ and Jerome P. Gauntlett$^2$\
.6cm .6cm
*$^1$DAMTP, University of Cambridge\
Cambridge, CB3 0WA, U.K.*
\
.6cm
*$^2$Blackett Laboratory, Imperial College\
London, SW7 2AZ, U.K.*
\
**Abstract**
> We introduce a new framework for constructing black hole solutions that are holographically dual to strongly coupled field theories with explicitly broken translation invariance. Using a classical gravitational theory with a continuous global symmetry leads to constructions that involve solving ODEs instead of PDEs. We study in detail $D=4$ Einstein-Maxwell theory coupled to a complex scalar field with a simple mass term. We construct black holes dual to metallic phases which exhibit a Drude-type peak in the optical conductivity, but there is no evidence of an intermediate scaling that has been reported in other holographic lattice constructions. We also construct black holes dual to insulating phases which exhibit a suppression of spectral weight at low frequencies. We show that the model also admits a novel $AdS_3\times\mathbb{R}$ solution.
Introduction
============
It is a remarkable fact that many phenomena observed in condensed matter systems are now known to have gravitational analogues via the AdS/CFT correspondence. One area of focus, where there has been significant recent progress, concerns the holographic description of physics associated with a “lattice". More specifically, there are are now several different constructions of black hole solutions that are holographically dual to strongly coupled systems which explicitly break translation invariance using a spatially periodic deformation [@Horowitz:2012ky; @Horowitz:2012gs; @Horowitz:2013jaa; @Donos:2012js; @Ling:2013nxa; @Chesler:2013qla].
One motivation for constructing such black holes arises in the context of studying the optical conductivity of strongly coupled systems at finite charge density. In the absence of a lattice the translation invariance of the system implies that there is a delta function peak at zero frequency, implying that the system is an ideal conductor. To extract more realistic metallic behaviour one can investigate the impact of a lattice. The first construction of electrically charged black holes describing holographic lattices was made in $D=4$ Einstein-Maxwell theory coupled to a real scalar field [@Horowitz:2012ky]. For the specific black holes that were constructed, it was shown that the system is in a metallic phase with the delta function peak smeared out into a Drude-type peak[^1]. This observed low frequency behaviour is consistent with the general analysis of conductivities that was made earlier in [@Hartnoll:2012rj] (see also [@Mahajan:2013cja]).
Moving away from the low-frequency regime, with the scale set by the chemical potential, a particularly striking conclusion of [@Horowitz:2012ky] was that the optical conductivity appears to exhibit a power-law behaviour at intermediate frequencies. More precisely the optical conductivity was seen to have the form $$\begin{aligned}
\label{scal}
|\sigma(\omega)|=B\omega^{-2/3}+C\,,\end{aligned}$$ where $B,C$ are frequency independent constants, and furthermore, the same behaviour was also seen for other lattices and other spacetime dimensions in [@Horowitz:2012gs; @Horowitz:2013jaa; @Ling:2013nxa]. Since an intermediate scaling of the optical conductivity for the high $T_c$ cuprates is seen with the same scaling exponent $-2/3$, albeit with $C=0$ and a frequency independent phase (e.g. [@2003Natur.425..271M; @2006AnPhy.321.1716V]), it is important to analyse this result in more detail. In fact for the holographic lattice that we construct in this paper we will not see such scaling behaviour. We will discuss the connection between our results and [@Horowitz:2012ky; @Horowitz:2012gs; @Horowitz:2013jaa; @Ling:2013nxa] at the end of the paper.
A more recent motivation for studying holographic lattices is that it provides a framework for investigating metal-insulator transitions within a holographic context [@Donos:2012js]. This is particularly interesting because there are many perplexing systems, such as the cuprates, where such transitions are observed and holographic techniques may provide important new insights. The strategy of [@Donos:2012js] is to construct black holes dual to holographic lattices that flow in the IR to metallic ground states and then to vary the strength and/or the periodicity of the lattice aiming to induce a transition to a new insulating phase. In [@Donos:2012js] this was achieved using $D=5$ electrically charged black holes dual to helical lattices. Furthermore, new zero temperature insulating ground states that break translation invariance were also found in [@Donos:2012js].
An important technical issue that arises in constructing black holes dual to lattices is that, in general, they require solving partial differential equations. For example, the holographic lattices that were constructed in [@Horowitz:2012gs; @Horowitz:2013jaa; @Horowitz:2012ky; @Ling:2013nxa] break translation invariance in one of the spatial dimensions and lead to a problem in PDEs in two variables; the one spatial direction as well as a radial direction. For the general setup where the translation invariance is broken in all of the spatial directions, time independent black holes in $D$ spacetime dimensions will typically depend on $D-2$ spatial variables as well as a radial variable, leading to PDEs in $D-1$ variables. For $D=4,5$ solving such PDEs numerically is an involved exercise. An interesting exception is the construction of the $D=5$ black holes dual to helical lattices [@Donos:2012js], where a Bianchi VII$_0$ symmetry was utilised to construct black holes by solving ODEs only.
In this paper we introduce a new framework for constructing holographic lattices that also involves just solving ODEs. The key idea is to break the translation invariance by exploiting a continuous global symmetry of the bulk classical gravitational theory. A simple theory that can be used to illustrate the idea, which is also the theory we will focus on in the paper, consists of Einstein-Maxwell theory coupled to a complex scalar field, $\phi$. The field $\phi$ is neutral with respect to the Maxwell field, and the model is taken to have a global $U(1)$ symmetry in addition to the $U(1)$ gauge-symmetry associated with the Maxwell field. For example, the Lagrangian density involving $\phi$ can take the form $$\begin{aligned}
{\cal L}(\phi)=\sqrt{-g}\left[-|\partial\phi|^2-V(|\phi|)\right]\,,\end{aligned}$$ leading to the following contribution to the bulk stress-tensor $$\begin{aligned}
\label{stten}
T_{\mu\nu}(\phi)=\partial_{(\mu}\phi\partial_{\nu)}\phi^*-\frac{1}{2}g_{\mu\nu}\left[|\partial\phi|^2+V(|\phi|)\right]\,.\end{aligned}$$ The breaking of the translation invariance in, say, the $x_1$ direction can be achieved using the ansatz $\phi=e^{ikx_1}\varphi(r)$ and it is clear from the form of the stress tensor given in that this can be combined with an ansatz for the metric and Maxwell fields that is dependent on the radial variable only[^2]. This construction shares some similarities with the construction of Q-balls [@Coleman:1985ki], which exploits a global symmetry and a time dependent phase to construct spherically symmetric solitons, and so we call them holographic Q-lattices.
It is worth noting that this particular Q-lattice, involving a single complex scalar field, can be viewed as arising from two real scalar fields, with the same mass, each with a periodic spatial dependence in the same direction that is shifted by an amount $\pi/2k$. In this sense it can be viewed as a simple generalisation of the lattice studied in [@Horowitz:2012gs]. More generally, this lattice construction can easily be extended to study the breaking of translation invariance in additional spatial directions by considering a model with a larger global symmetry. For example, one can use a model with additional complex scalar fields and with additional global $U(1)$ symmetries. One can also have larger global symmetry groups and/or use higher rank tensor fields instead of scalars. Such lattices will be studied in detail elsewhere.
The plan of the rest of the paper, including some of the key results, are as follows. In section \[bhs\] we study $D=4$ Einstein-Maxwell theory coupled to a complex scalar field with a simple mass term. We construct Q-lattice black holes that describe metallic phases which at zero temperature approach $AdS_2\times\mathbb{R}^2$ in the far IR. We numerically calculate the low temperature behaviour of the DC resistivity and extract the scaling behaviour that is predicted from [@Hartnoll:2012rj] using the memory matrix formalism. This comprises the first[^3] numerical confirmation of [@Hartnoll:2012rj] for fully back reacted black holes and complements the recent analytic results of [@Blake:2013owa] in the context of perturbative lattices. We also construct black holes that describe insulating phases, realising the first holographic metal-insulator transition for $d=3$ field theories. At low temperatures there is a transfer of spectral weight in the insulating phase and the real part of the optical conductivity develops a mid frequency hump. Some details of the conductivity calculation is presented in section \[condsec\], which includes some new technical material. Interestingly, the model that we analyse also admits an $AdS_3\times\mathbb{R}$ solution which we discuss in an appendix. We conclude with some final comments in section \[fincom\], including a discussion of the absence of intermediary scaling in the optical conductivity.
Black hole solutions {#bhs}
====================
We shall consider $D=4$ Einstein-Maxwell theory coupled to a complex field $\phi$ with action given by $$\begin{aligned}
\label{act}
S=\int d^4 x\sqrt{-g}\left[R+6-\frac{1}{4}F^2-|\partial\phi|^2 -m^2|\phi|^2\right]\,,\end{aligned}$$ where $F=d A$. We have set $16\pi G=1$ and also fixed the scale of the cosmological constant for convenience. The equations of motion can be written $$\begin{aligned}
\label{eqsmot}
R_{\mu\nu}&=g_{\mu\nu}(-3+\frac{m^2}{2}|\phi|^2)+\partial_{(\mu}\phi\partial_{\nu)}\phi^*+\tfrac{1}{2}\left( F^2_{\mu\nu}-\tfrac{1}{4}g_{\mu\nu}F^2\right)\,,{\notag \\}&\nabla_\mu F^{\mu\nu}=0,\qquad
(\nabla^2-m^2)\phi=0\,,\end{aligned}$$ and admit an $AdS_4$ vacuum solution, with unit radius, which is dual to a $d=3$ CFT. The CFT has two global abelian symmetries. The first arises from the gauge symmetry in the bulk and there is a corresponding conserved current which is dual to the bulk-gauge field $A$. The second arises from the global symmetry in the bulk, associated with multiplying $\phi$ by a constant phase, and there is not a corresponding conserved current[^4] in the CFT. The CFT also has a complex scalar operator with scaling dimension $\Delta= 3/2\pm (9/4+m^2)^{1/2}$ dual to the scalar field $\phi$. We want this to be a relevant operator in a unitary CFT and hence we take $-9/4\le m^2<0$.
The CFT at finite temperature $T$ and chemical potential $\mu$ can be holographically described by the standard electrically charged AdS-RN black solution given by $$\begin{aligned}
ds^2&=-Udt^2-U^{-1}dr^2+r^2\left(dx_1^2+dx_2^2\right)\,,{\notag \\}A&=\mu(1-\frac{r_+}{r})dt\,,\end{aligned}$$ with $\phi=0$ and $U=r^2-(r_+^2+\frac{\mu^2}{4})\frac{r_+}{r}+\frac{\mu^2r_+^2}{4r^2}$. The temperature is given by $T=(12r_+^2-\mu^2)/16\pi r_+$ and at $T= 0$ it approaches the following $AdS_2\times\mathbb{R}^2$ solution as $r\to r_+$: $$\begin{aligned}
\label{ads2}
ds^2&=\frac{1}{6}ds^2(AdS_2)+dx_1^2+dx_2^2\,,{\notag \\}F&=\frac{1}{\sqrt{3}}Vol(AdS_2)\,,\end{aligned}$$ where $ds^2(AdS_d)$ denotes the standard unit radius metric on $AdS_d$.
For the mass window $-9/4\le m^2<-3/2$ the scalar field $\phi$ violates the $AdS_2$ BF bound and hence the AdS-RN black hole solution will become unstable at some temperature, leading to a different $T=0$ ground state. In order to exclude this possibility, for most of the paper we will consider $$\begin{aligned}
m^2=-\frac{3}{2}\qquad \leftrightarrow\qquad \Delta=\frac{3+\sqrt 3}{2}\,.\end{aligned}$$ At the end of the paper we will comment on the case $m^2=-2$ and $\Delta=2$.
Black hole ansatz for the holographic $Q$-lattice
-------------------------------------------------
We are interested in describing the $d=3$ CFT with chemical potential $\mu$ and an explicit breaking of translation invariance in one of the spatial directions, which we take to be $x_1$. The ansatz we shall consider is given by $$\begin{aligned}
\label{ansatzbh}
ds^2&=-Udt^2+U^{-1}dr^2+e^{2V_1}dx_1^2+e^{2V_2}dx_2^2\,,{\notag \\}A&=adt\,,{\notag \\}\phi&=e^{ikx_1}\varphi\,,\end{aligned}$$ where $U, V_1, V_2, a$ and $\varphi$ are functions of the radial co-ordinate only and $k$ is a constant. Substituting this ansatz into we find that the equations of motion can be equivalently recast as four second order ODEs for $V_1, V_2, a, \varphi$ and one first order ODE for $U$. It is useful to note that this ansatz is invariant under the scaling $t\to ct, x_i\to c x_i,r\to c^{-1}r$ and $U\to c^{-2}U, e^{V_i}\to c^{-1}e^{V_i}, a\to c^{-1}a, k\to c^{-1}k$.
We will impose the following boundary conditions on the ODEs. We demand that we have a regular solution at the black hole event horizon at $r=r_+$, which leads to an expansion depending on six independent constants $r_+, V_{1+}, V_{2+},V_{22},a_+$ and $\varphi_+$. Specifically as $r\to r_+$ we have $$\begin{aligned}
U&=4\pi T(r-r_+)+\dots,{\notag \\}V_1&=V_{1+}+\left(1-\frac{4e^{-2V_{1+}}\varphi_+^2 k^2}{12-a_+^2-2\varphi_+^2 m^2}\right)V_{22}(r-r_+)\dots, {\notag \\}V_2&=V_{2+}+V_{22}(r-r_+)\dots, {\notag \\}a&=a_+(r-r_+)+\left(-1+\frac{2e^{-2V_{1+}}\varphi_+^2 k^2}{12-a_+^2-2\varphi_+^2 m^2}\right)a_+V_{22}(r-r_+)^2\dots,{\notag \\}\varphi&=\varphi_++\frac{4(m^2+e^{-2V_{1+}}k^2)}{12-a_+^2-2\varphi_+^2 m^2}\varphi_+V_{22}(r-r_+)\dots,\end{aligned}$$ where $T$ is the temperature of the black hole given by $$\begin{aligned}
T=(4\pi)^{-1}\frac{12-a_+^2-2\varphi_+^2 m^2}{4V_{22}}\,.\end{aligned}$$
At the UV boundary, $r\to\infty$, we demand that we approach $AdS_4$ with deformations corresponding to chemical potential $\mu$ and lattice deformation parameter $\lambda$. We find that, schematically, we can develop the expansion $$\begin{aligned}
\label{uvexp}
U&=r^2+\dots-\frac{M}{r}+\dots,{\notag \\}V_1&=\log r+\dots+\frac{V_v}{r^3}+\dots,{\notag \\}V_2&=\log r+\dots-\frac{V_v}{r^3}+\dots,{\notag \\}a&=\mu+\frac{q}{r}\dots,{\notag \\}\varphi&=\frac{\lambda}{r^{3-\Delta}}+\dots +\frac{\varphi_c}{r^{\Delta} }+\dots\,.\end{aligned}$$ This gives a UV expansion that depends on seven parameters $M, V_v, \mu,q,\lambda,\varphi_c$ and $k$.
Notice that for fixed $m^2$, the holographic Q-lattice is specified by three dimensionless quantities fixing the deformations in the UV: $T/\mu$, $\lambda/\mu^{3-\Delta}$ and $k/\mu$. We thus expect a three-parameter family of black holes. We have four second order ODEs and one first order ODE, and so a solution is specified by nine parameters. We have six parameters for the near horizon expansion plus another seven for the UV expansion. After subtracting one for the scaling symmetry that the system of ODEs possesses, we deduce that there is indeed, generically, a three-parameter family of black hole solutions. We also note that the scaling symmetry can be used to set $\mu=1$ if one wishes.
We will choose specific values in the two-dimensional space parameterised by $\lambda/\mu^{3-\Delta}$ and $k/\mu$, and then examine the behaviour as $T/\mu$ is lowered. In particular, we will see that there is a transition from metallic to insulating behaviour as we move in this two-dimensional space.
Black holes dual to the metallic phase
--------------------------------------
The CFT deformed by the Q-lattice will be in a metallic phase if the zero temperature limit of the black hole solutions interpolate between the lattice deformed $AdS_4$ in the UV and the stable $AdS_2\times \mathbb{R}^2$ solution in the IR. Indeed this will happen when the lattice deformation in the UV becomes an irrelevant deformation of the $AdS_2\times \mathbb{R}^2$ solution in the IR, and then the general arguments of [@Hartnoll:2012rj], based on the memory matrix formalism, show that the $T=0$ ground state must be metallic. In particular, at low temperatures, $T<<\mu$, the DC resistivity is expected to scale as[^5] $$\begin{aligned}
\label{dares}
\rho\sim \left(\frac{T}{\mu}\right)^{2\Delta(k)-2}\,,\end{aligned}$$ where $\Delta(k)$ is the smallest scaling dimension of the $k$-dependent irrelevant operators in the locally quantum critical theory arising in the IR. In addition to $k$, $\Delta(k)$ depends on other UV data, as we discuss below. Furthermore, there should be a Drude peak in the optical conductivity at small temperatures, which at $T=0$ becomes a delta-function at zero frequency.
To examine when this situation can arise we now analyse perturbations about the $AdS_2\times\mathbb{R}^2$ solution. Within our ansatz we consider $$\begin{aligned}
U&=6r^2(1+u_1 r^\delta),\quad V_1=v_{10}(1+v_{11}r^\delta),\quad V_2=v_{20}(1+v_{21}r^\delta),{\notag \\}a&=2\sqrt{3}r(1+a_1 r^\delta),\quad\phi=e^{ikx_1}\varphi_1 r^\delta\,.\end{aligned}$$ The corresponding perturbations are associated with operators with scaling dimension $\Delta=-\delta$ or $\Delta=\delta+1$ in the locally quantum critical IR theory captured by the $AdS_2\times\mathbb{R}^2$ solution. We find after substituting into equations of motion the exponents come in four pairs, satisfying $\delta_++\delta_-=-1$, with $\delta_+=0,0,1$ and a mode just involving the scalar field with $\delta_+=\delta_\varphi$, where $$\begin{aligned}
\label{deltexp}
\delta_\varphi
=-\frac{1}{2}+\frac{1}{2\sqrt 3}\sqrt{3+2m^2+2 e^{-2v_{10}}k^2}\,.\end{aligned}$$ There is also another additional single mode with $\delta_+=-1$ (corresponding to $r_+$ in below). What is most significant here is that the scalar field perturbation will be an irrelevant deformation in the IR (i.e. $\delta_\varphi>0$), provided that the lattice deformation in the IR satisfies $$\begin{aligned}
\label{kcond}
(e^{-v_{10}}k)^2>-m^2\,.\end{aligned}$$ In this case the dimension of the irrelevant operator in the locally quantum critical theory is given by $\Delta(k)=1+\delta_\varphi$ and we have $$\begin{aligned}
\label{dimk}
\Delta(k)=\frac{1}{2}+\frac{1}{2\sqrt 3}\sqrt{3+2m^2+2 e^{-2v_{10}}k^2}\,.\end{aligned}$$
When is satisfied we can use the two marginal modes with $\delta_+=0$ and the two irrelevant modes to construct domain walls interpolating between the lattice deformed $AdS_4$ in the UV and the $AdS_2\times\mathbb{R}^2$ solution in the IR. Specifically, we can develop the following IR expansion $$\begin{aligned}
\label{adsdwexp}
U&=6(r-r_+)^2(1-\frac{4}{3v_{10}}V_{+}(r-r_+)+\dots)\,,{\notag \\}V_1&=v_{10}(1+V_+(r-r_+)+\dots)\,,{\notag \\}V_2&=v_{20}(1+\frac{v_{10}}{v_{20}}V_+(r-r_+)+\dots)\,,{\notag \\}a&=\sqrt{12}(r-r_+)(1-v_{10}V_+\dots)\,,{\notag \\}\varphi&=\varphi_+(r-r_+)^{\delta_\varphi}+\dots\,.\end{aligned}$$ We have five IR parameters, $r_+, v_{10},v_{20},V_+,\varphi_+$ and hence when combined with the UV expansion and taking into the scaling symmetry, we expect, generically, a two parameter family of solutions which can be labelled by $\lambda/\mu^{3-\Delta}$ and $k/\mu$.
For the values of $\lambda/\mu^{3-\Delta}$, $k/\mu$ where these domain walls exist, we expect that they will arise as the zero temperature limit of lattice deformed black holes which will have, for very small $T/\mu$, DC resistivity scaling as in and a Drude peak in the optical conductivity for small $\omega/\mu$, of the form $$\begin{aligned}
\label{drude}
\sigma\sim \frac{K\tau}{1-i\omega \tau}\,,\end{aligned}$$ for constant $K,\tau$. It should be stressed that the value of $\Delta(k)$ appearing in the DC resistivity depends on the value of $v_{10}$ which is fixed by the details of domain wall solution, including all UV data. In effect the value of $v_{10}$ is renormalising the lattice momentum from $k$ in the UV to $e^{-v_{10}}k$ in the IR.
One might expect that this metallic scenario unfolds for large wavelength and small Q-lattice deformations of the AdS-RN black hole i.e. $\lambda/\mu^{3-\Delta}<<1$ and $k/\mu<<1$. As an illustrative example, we have numerically constructed Q-lattice black holes in the metallic phase with $\lambda/\mu=1/2$ and $k/\mu=1/\sqrt{2}$. By examining the properties of these solutions at very low temperatures, we find that they approach domain walls interpolating between $AdS_4$ in the UV and $AdS_2\times\mathbb{R}^2$ in the IR. In section \[condsec\] we describe the calculation of the optical conductivity; the results for the metallic phase black holes that we have constructed are presented in figure \[figone\].
1 em\
1 em\
In figure \[figone\](c) we see that the DC resistivity increases with temperature and hence we do indeed have a metallic phase. In figures \[figone\](a) and \[figone\](b) we have plotted the real an imaginary part of the optical conductivity, respectively, for four different temperatures. In particular, in \[figone\](a) we see the Drude-type peaks appearing, which get more pronounced as the temperature is lowered. By fitting[^6] to we obtain the values for $\tau\mu$ and $K/\mu$ given in table \[green\].
$T/\mu$ $\tau\mu$ $K/\mu$
--------- ----------- --------- --
0.1 20 0.37
0.0503 33 0.32
0.0154 113 0.26
0.00671 272 0.24
: Parameters after fitting to the Drude behaviour for small $\omega$, for the black holes in the metallic phase for lattice parameters $\lambda/\mu=1/2$ and $k/\mu=1/\sqrt{2}$. []{data-label="green"}
To observe the exact scaling behaviour $\rho\sim (T/\mu)^{2\Delta(k)-2}=(T/\mu)^{2\delta_\varphi}$, with $\Delta(k), \delta_\varphi$, as in , , as predicted by [@Hartnoll:2012rj], is not straightforward because the scaling only manifests itself when $T<<\mu$. We have constructed the black hole solutions down to temperatures $T/\mu\sim 2.5\times 10^{-7}$ and, as noted, we find that the black holes approach the $AdS_2\times\mathbb{R}^2$ solution. By identifying $v_{10}$ with $V_{1+}$ we deduce that $k\sim0.707$ gets renormalised to a value $e^{-v_{10}}k\sim 2.236$ and hence $\Delta(k)\sim 1.413$ corresponding to the scaling $\rho\sim (T/\mu)^{0.826}$. We have calculated the conductivity for temperatures down to $T/\mu\sim 7\times10^{-4}$ and from this deduced the DC resistivity. The scaling behaviour eventually manifests itself at these low temperatures as one can see from panel (c) of figure \[figone\]. Our results in \[figone\](c) are consistent with this scaling to the order of less than 1%. This is the first direct check of the prediction of [@Hartnoll:2012rj] for back-reacted holographic lattices[^7]. Note that for very large temperatures the resistivity should eventually approach unity, which is the constant value for the $AdS$-Schwarzschild black hole at zero momentum [@Herzog:2007ij].
We can also investigate the possibility that there is a scaling of the form , which has been reported for other models in the range $2\lesssim \omega\tau\lesssim8$ [@Horowitz:2012ky; @Horowitz:2012gs; @Horowitz:2013jaa; @Ling:2013nxa]. If this scaling is present then $1+\omega|\sigma|''/|\sigma|'=-2/3$. Our results are plotted in figure \[figone\](d) and, for example, from table \[green\] for $T/\mu=0.1$ the relevant range is $0.1\lesssim \omega/\mu\lesssim 0.4$, while for $T/\mu=0.00671$ it is $0.0073\lesssim \omega/\mu\lesssim 0.029$. Our results show that there is a strong temperature dependence and there is no evidence of a mid frequency scaling region. Note that $|\sigma|$ has a minimum at some value of $\omega$ and hence the function $1+\omega|\sigma|''/|\sigma|'$ will diverge at that point and, furthermore for larger values of $\omega$ it will be positive. Finally we note that for very large $\omega/\mu$ and fixed $T$, the conductivity should approach that of the AdS-RN black hole with $\sigma\to 1$ [@Herzog:2007ij].
Black holes dual to the insulating phase
----------------------------------------
The metallic phase discussed in the last subsection arises for a given UV lattice, specified by $\lambda/\mu^{3-\Delta}$ and $k/\mu$, whenever the $T=0$ ground state approaches $AdS_2\times\mathbb{R}^2$ in the far IR. In this section we will construct black holes where this does not occur and we will see that they exhibit insulating behaviour.
We focus on the specific values $\lambda/\mu^{3-\Delta}=2$ and $k/\mu=1/2^{3/2}$. The optical conductivity and the DC resistivities for these black holes are displayed in figure \[figins\].
\
1 em
The DC resistivity is increasing as we lower the temperature indicating that the system is in an insulating phase. Furthermore, for very low temperatures, for example $T/\mu\sim 0.00118$, we see that the real part of the optical conductivity reveals a suppression of spectral weight for small $\omega/\mu$, with the weight being transferred to a mid frequency hump. Very similar behaviour was seen for the helical lattice black holes dual to insulating phases in [@Donos:2012js].
Lowering the temperature further we might expect to find the $T=0$ ground states for this insulating phase. Actually this is not guaranteed as there are certainly situations in holography where black holes only exist down to a minimum temperature, for example [@Donos:2011ut]. For the insulating black holes with the above lattice parameters we have found an interesting feature at the low temperature $T_c/\mu\sim 2.8\times 10^{-5}$. Specifically we find that there appears to be a kink in the entropy density versus temperature curve, with $s'(T_c)=0$, which at first sight appears to represent a minimum temperature. However, closer detailed numerical investigation shows that there is another branch of insulating black holes at lower temperature, with broadly similar insulating behaviour. The simplest interpretation is that there is a first order transition at $T_c$. Assuming this to be the case, we have found that the low temperature branch exists at least down to the ultra low temperatures $T_c/\mu\sim 10^{-9}$. Furthermore, we find that the entropy density is going to zero and that the solutions are becoming singular. We are particularly interested in extracting the far IR behaviour of the $T=0$ black holes. However, in general, this is a non-trivial task unless some simplification represents itself in the numerical solutions, such as the functions approaching a power-law behaviour. We have not been able to find any evidence for such power law behaviour in the present setting.
It would be certainly interesting to explore these issues further. Note that we have considered other values for the UV lattice data, finding somewhat similar results, but a more comprehensive analysis of the behaviour for general values of $\lambda/\mu^{3-\Delta}$ and $k/\mu$ is left for future work. One point that is worth highlighting is that the model also possesses another fixed point solution that may play an important role in understanding the phase structure of the model. As we describe in the appendix there is a novel electrically neutral $AdS_3\times\mathbb{R}$ fixed point solution with a spectrum containing modes corresponding to both irrelevant and relevant operators. The presence of the relevant operator indicates that for generic lattice data it will not be possible to construct domain wall solutions interpolating between $AdS_4$ in the UV and $AdS_3\times\mathbb{R}$ in the IR. However, it is possible that a fine tuned domain wall solution exists for specific lattice data, which might correspond to an unstable RG flow providing a bifurcation between the metallic and insulating behaviours analogous to what was observed for the helical black hole lattices in [@Donos:2011ut].
Conductivity {#condsec}
============
In this section we explain how we calculate the conductivity for the black holes that we have constructed. Although the general idea is standard, the technical implementation in the presence of the lattice deformation warrants some discussion. We consider the following consistent linear perturbation about the black hole solutions $$\begin{aligned}
\label{eq:pert_ansatz1}
\delta g_{tx_1}&= \delta h_{tx_1}(t,r)\,,{\notag \\}\delta A_{x_1}&=\delta a_{x_1}(t,r)\,,{\notag \\}\delta\phi&=ie^{ikx_1}\delta\varphi(t,r)\,,\end{aligned}$$ where $\delta h_{tx_1}, \delta a_{x_1}$ and $\delta\varphi$ are all [*real*]{} functions of $(t,r)$ and we note the factor of $i$ in the last line. After substituting into the equations of motion we obtain real partial differential equations. We next allow for a time dependence of the form $e^{-i\omega t}$ by writing $$\begin{aligned}
\label{eq:pert_ansatz2}
\delta h_{tx_1}(t,r)&=e^{-i\omega t} \delta h_{tx_1}(r)\,,{\notag \\}\delta a_{x_1}(t,r)&=e^{-i\omega t}\delta a_{x_1}(r)\,,{\notag \\}\delta\varphi(t,r)&=e^{-i\omega t}\delta\varphi(r)\,,\end{aligned}$$ and we are lead to the following system of ODEs: $$\begin{aligned}
\label{eq:pert_eom}
&\delta a_{x_1}''+\left(U^{-2}\omega^2-U^{-1}a'^2\right)\delta a_{x_1}+\left(U^{-1}U'-V_1'+V_2'\right)\delta a_{x_1}'
+2i\frac{k}{\omega}a'\left(\varphi'\delta\varphi-\varphi\delta \varphi'\right)=0,{\notag \\}&\delta\varphi''
+\left(U^{-2}\omega^2-m^2U^{-1}-k^2 U^{-1}e^{-2 V_1}\right)\delta\varphi
+\left(U^{-1}U'+V_1'+V_2'\right)\delta\varphi'
-i k \omega U^{-2} e^{-2 V_1} \varphi \delta h_{tx_1}=0\,,{\notag \\}&\delta h_{tx_1}'+ a' \delta a_{x_1} -2 V_1'\delta h_{tx_1} -2 i \frac{k}{\omega} U \left(\varphi'\delta\varphi -\varphi\delta\varphi'\right)=0\,.\end{aligned}$$
At the black hole event horizon we impose purely ingoing boundary conditions with the perturbations behaving as $$\begin{aligned}
\label{irexp}
\delta a_{x_1}&=(r-r_+)^{-i\omega/4\pi T}\left(\delta a_{x_1}^{(+)}+\dots \right)\,,{\notag \\}\delta\varphi&=(r-r_+)^{-i\omega/4\pi T}\left(\delta\varphi^{(+)}+\dots \right)\,,{\notag \\}\delta h_{tx_1}&=(r-r_+)^{-i\omega/4\pi T}(\delta h_{tx_1}^{(+)}(r-r_+)+\dots)\,,\end{aligned}$$ where the dots refer to terms higher order in $(r-r_+)$. The regularity of this perturbation at the black horizon can be seen by using ingoing Eddington-Finklestein coordinates $(v,r)$ with $v=t+\log(r-r_+)^{4\pi T}$. Using the equations of motion we find that this expansion is fixed by two parameters $\delta a_{x_1}^{(+)}$, $\delta\varphi^{(+)}$ with $$\begin{aligned}
\delta h_{tx_1}^{(+)}=-\frac{a_+\delta a_{x_1}^{(+)}+2k\varphi_+\delta\varphi^{(+)}}{r_+^2(1-i\frac{\omega}{4\pi T})}\,.\end{aligned}$$
In the UV we impose that as $r\to\infty$: $$\begin{aligned}
\label{uvex}
\delta h_{tx_1}&={\delta h_{tx_1}^{(0)}}{r}^2+\dots\,,{\notag \\}\delta a_{x_1}&=\delta a_{x_1}^{(0)}+\frac{\delta a_{x_1}^{(1)}}{r}+\dots\,,{\notag \\}\delta\varphi&=\frac{\delta \varphi^{(0)}}{r^{3-\Delta}}+\dots +\frac{\delta \varphi^{(1)}}{r^{\Delta} }+\dots\,.\,.\end{aligned}$$ Now we are interested in a perturbation that switches on an electric field and then we want to read off the current to obtain the conductivity. One might be tempted to set $\delta h_{tx_1}^{(0)}=\delta \varphi^{(0)}=0$ but this over constrains the system. To see this we note that a solution to the ODEs is specified by five parameters. From the IR and UV expansions , we have a total of seven parameters. However, since the ODEs are linear we can scale one of the seven parameters to unity, leaving six. This means that we need to impose just one more constraint on the parameters. This constraint can be found as follows.
To ensure that we are extracting just the current-current correlator, we can use diffeomorphisms and gauge-transformations to demand that the perturbation satisfies, as $r\to\infty$, $$\begin{aligned}
\frac{1}{r^2}\left(\delta g_{\mu\nu}+{\cal L}_\zeta g_{\mu\nu}\right)&\to0\,,{\notag \\}\delta A+{\cal L}_\zeta A+d\Lambda&\to e^{-i\omega t} \mu_{x_1}dx_1\,,{\notag \\}r^{3-\Delta}\left(\delta\phi+{\cal L}_\zeta \phi\right)&\to 0\,,\end{aligned}$$ where $\zeta^\mu$ and $\Lambda$ are smooth and $\mu_{x_1}$ will be the source for the current. For our specific set-up we can take $\Lambda=0$ and the only non-vanishing component of $\zeta^\mu$ to be $\zeta^x=\epsilon e^{-i\omega t}$ where $\epsilon$ is a small parameter. From this we can deduce that we have $\mu_{x_1}=\delta a_{x_1}^{(0)}$ and that we should impose the condition $$\begin{aligned}
\label{constp}
\delta \varphi^{(0)}-i\frac{k\lambda}{\omega}\delta h_{tx_1}^{(0)}=0\,.\end{aligned}$$ The optical conductivity is then given by $$\begin{aligned}
\sigma(\omega)=-\frac{i}{\omega}\frac{\delta a_{x_1}^{(1)}}{\delta a_{x_1}^{(0)}}\,.\end{aligned}$$ The DC resistivity is given by $\rho=1/\sigma(0)$. It is worth mentioning that to calculate $\rho$ numerically, one needs to calculate the optical conductivity for $\omega<<T$.
Final comments {#fincom}
==============
We have studied holographic Q-lattices for Einstein-Maxwell theory coupled to a single complex scalar field in $D=4$ space-time dimensions. We have shown that the system exhibits both metallic and insulating phases. The metallic phase is governed by the electrically charged $AdS_2\times\mathbb{R}^2$ solution that appears in the IR region of the $T=0$ electrically charged AdS-RN solution. We showed in detail that the phase exhibits a Drude-type peak and furthermore, at low temperatures the DC resistivity exhibits a scaling behaviour confirming the prediction of [@Hartnoll:2012rj].
We have also constructed Q-lattice black holes in a new insulating phase down to very low temperatures. For temperatures lower than $T/\mu\sim 10^{-3}$ we see a transferral of spectral weight in the optical conductivity and the generation of a mid frequency hump. At temperatures $T/\mu\sim 2.8\times10^{-5}$ we have found evidence for a first order transition to another branch of insulating black holes. It would be interesting to investigate these further including trying to elucidate the ultimate IR ground states at $T=0$ which seem to have vanishing entropy density. A possibly related issue, is to further understand the role played by the neutral $AdS_3\times\mathbb{R}$ ground state that we have found and discussed in the appendix.
We focussed on the case where the mass of the complex scalar is given by $m^2=-3/2$ with $\Delta=(3+\sqrt{3})/2$ in the $d=3$ CFT, which saturates the $AdS_2\times\mathbb{R}^2$ BF bound, corresponding to a stable metallic phase. We have also made some numerical investigations into the case $m^2=-2$ with $\Delta=2$ in the $d=3$ CFT. We have constructed black holes with conductivities exhibiting metallic and insulating behaviours much as in figure \[figone\]. However, for this case the complex scalar violates the $AdS_2\times\mathbb{R}^2$ BF bound and hence, at least for the metallic black holes, one will find an additional new phase appearing at low temperatures[^8]. When there is no lattice deformation a possible ground state for this model was identified in [@Horowitz:2009ij]. It will be interesting to see how this is modified by the lattice deformation and also to investigate the impact on the insulating phase.
It is also natural to consider a more general class of models including a coupling of the scalar field to the gauge field and a more general potential than the simple mass term. We expect that within this more general class of models it will be possible to obtain the many novel IR ground states in explicit form [@Donos:2014uba]. It will be particularly interesting to explore interconnections with charge density waves [@Donos:2013gda] which should lead to close analogues of Mott insulating ground states. Such models can be studied in various spacetime dimensions.
For the Q-lattices that we have constructed for specific values of lattice strength $\lambda$ and wave-number $k$, for both $m^2=-3/2$ and $m^2=-2$, we find no evidence that the metallic phase has an intermediate scaling of the form . How can this be reconciled with the results reported in [@Horowitz:2012gs; @Horowitz:2013jaa; @Horowitz:2012ky; @Ling:2013nxa], where numerical evidence for this behaviour was found and moreover it was suggested that this might be a universal feature of holographic lattices? One possibility is that the numerical evidence found in those papers is actually misleading and in fact there is not a robust power-law behaviour for the lattices considered.
An interesting perspective is to consider the same model that we have in this paper, but with a family of lattice deformations, labelled by $\alpha$, given by $$\begin{aligned}
\phi=\sqrt{2}\lambda\left(\cos\alpha\cos k x_1+i \sin\alpha\sin kx_1\right)\frac{1}{r^{3-\Delta}}+\dots\end{aligned}$$ as $r\to\infty$. For $\alpha=\pi/4$ this gives the family Q-lattices that we discussed in this paper, while for $\alpha=0$ it gives the lattices discussed in [@Horowitz:2012gs] (who just considered $m^2=-2$). Notice that the strength of the lattice, $\lambda$, does not depend on $\alpha$ and also that for $\alpha\ne (2n+1)\pi/4$, for integer $n$, the metric will be co-homogeneity two and one will need to solve PDEs.
For this general family of lattices we can use the results of [@Hartnoll:2012rj] and also of [@Edalati:2010pn; @Donos:2013gda] to deduce the scaling behaviour of the DC resistivity in the metallic phase. In addition to the scalar mode with wave-number $k$, with dimension in the IR, one also needs to take into account[^9] longitudinal modes involving perturbations in $A_t, A_{x_1}$ and $g_{tt}, g_{x_1x_1},g_{tx_1}, g_{x_2x_2}$ and with wave-number $2k$ (corresponding to the fact that the scalar lattice sources them at least at quadratic order). From the analysis presented in [@Donos:2013gda] (in particular equation (2.17)), one can deduce that when $m^2\le -1/4$ the DC resistivity scaling will always be determined by the decoupled scalar mode in the IR. Interestingly for $-1/4<m^2< 0$, for certain windows of $k$, the scaling can be determined by the longitudinal modes. Note in particular, for the scalar lattice in [@Horowitz:2012ky] with $m^2=-2$ and $\alpha=0$, we are arguing that the DC resistivity scaling is actually governed by the scalar mode and not one of the longitudinal modes as was stated in [@Horowitz:2012ky]. Note that this work also claimed to see a numerical fit to a scaling governed by the longitudinal mode: we believe that the fitting was misleading and that continuing to lower temperatures will reveal the scaling behaviour that we are predicting.
It is also worth pointing out that we do not expect the black hole solutions will be substantially different as we vary $\alpha$ away from $\pi/4$, despite the fact that one is solving PDEs instead of ODEs as in this paper. While additional harmonics of the bulk fields will play a role, the higher harmonics are expected to be exponentially suppressed. In fact this was seen in the numerical work in [@Horowitz:2012ky]. Thus it is natural to expect that conductivity for non-zero $\omega$ is also not substantially different from what we have seen in this paper.
All of the constructions in this paper have just involved classical gravity. it is worth recalling, however, that there are good reasons to expect that there are no global symmetries in theories of quantum gravity (e.g. [@Banks:2010zn]). One point of view is that we are just studying a sector of a larger classical theory that does not have a global symmetry. Alternatively we can view the breaking of the continuous symmetry as a higher order effect in the large $N$ expansion. Within these contexts, or closely related ones, we think that top-down constructions should be possible.
Finally we point out that the holographic lattice constructions that we have discussed in this paper, where the translation symmetry is broken explicitly, can also be adapted to situations where the the symmetry is broken spontaneously.
Acknowledgements {#acknowledgements .unnumbered}
================
We thank Paul Chesler, Sean Hartnoll, Diego Hofman, Elias Kiritsis, Da-Wei Pang, Jorge Santos, Julian Sonner, David Tong and David Vegh for helpful conversations. The work is supported by STFC grant ST/J0003533/1 and also by the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013), ERC Grant agreement STG 279943, “Strongly Coupled Systems".
A novel $AdS_3\times\mathbb{R}$ solution
========================================
Provided that $m^2<0$ (equivalently, the operator dual to $\phi$ in the $d=3$ CFT dual to the $AdS_4$ vacuum is a relevant operator), the model admits an electrically neutral $AdS_3\times \mathbb{R}$ solution given by $$\begin{aligned}
ds^2&=\frac{1}{3}ds^2(AdS_3)+dx_1^2\,,{\notag \\}\phi&=\frac{6}{-m^2}e^{i\sqrt{-m^2}x_1}\,,\end{aligned}$$ with $A=0$.
To explore whether there are domain wall solutions which can connect this solution with $AdS_4$, we investigate the spectrum for this fixed point. Within our ansatz we can consider the perturbations given by $$\begin{aligned}
\label{exd}
U&=3r^2(1+u_1 r^\delta),\qquad V_1=v_{11}r^\delta,\qquad V_2=\log(r)+v_{21}r^\delta,{\notag \\}a&=a_1 r^{1+\delta},\qquad\phi=\left(\frac{6}{-m^2}\right)^{1/2}e^{i\sqrt{-m^2}x_1}\phi_1 r^\delta\,.\end{aligned}$$ These perturbations correspond to scaling dimension $\Delta=-\delta$ or $\Delta=\delta+2$ in the $d=2 $ CFT dual to the $AdS_3\times\mathbb{R}$ solution. We find that the exponents come in four pairs with $\delta_++\delta_-=-2$ and there is an unpaired mode with $\delta=-1$. The paired modes have $\delta_+$ values given by $0,-1$ and $$\begin{aligned}
\label{vals}
\delta_1=-1+\frac{1}{\sqrt{3}}\sqrt{9-2\sqrt{3}\sqrt{3-m^2}},\qquad \delta_2=-1+\frac{1}{\sqrt{3}}\sqrt{9+2\sqrt{3}\sqrt{3-m^2}}\,.\end{aligned}$$ We see that in the mass range $-9/4\le m^2<0$, which is relevant for trying to map onto $AdS_4$ in the UV, $\delta_1$ corresponds to a relevant operator (i.e. $\delta_1<0$) and $\delta_2$ corresponds to an irrelevant operator (i.e. $\delta_2>0$). Note that both of these deformations have $a_1=0$ in and do not involve the gauge-field.
A parameter count now reveals that, generically, because of the presence of the relevant operator, there will not be domain wall solutions interpolating between the lattice deformed $AdS_4$ in the UV and $AdS_3\times\mathbb{R}$ in the IR. However, there is the possibility that there is a fine-tuned domain wall solution. If this exists it might correspond to a bifurcating, unstable RG solution, separating the metallic and insulating behaviours, as in figure 2 of [@Donos:2012js].
More generally, we expect that there are closely related models where the $AdS_3\times\mathbb{R}$ geometry has irrelevant operators in the IR so that one can construct domain walls that interpolate from the Q-lattice deformed $AdS_4$ in the UV. Furthermore, changing the dimension of space-time and the number, $n$, of spatial directions where translation invariance is broken by the holographic Q-lattice will allow one to construct domain walls from $AdS_D$ in the UV and various $AdS_{D-n}\times\mathbb{R}^n$ in the IR. This will be explored in detail elsewhere.
[^1]: Drude-type physics has also been discussed in a holographic context in, for example, [@Karch:2007pd; @Hartnoll:2007ih; @Hartnoll:2007ip; @Hartnoll:2009ns; @Faulkner:2010zz; @Hartnoll:2012rj; @Liu:2012tr; @Vegh:2013sk; @Mahajan:2013cja; @Davison:2013jba; @Faulkner:2013bna].
[^2]: In the process of writing up this work, this possibility was also pointed out in a footnote in [@Blake:2013owa].
[^3]: We will comment on the results of [@Horowitz:2012ky] in section \[fincom\].
[^4]: A discussion of such global symmetries arising in a different holographic context appears in [@Amado:2013xya].
[^5]: Note that a different, non-standard, definition of $\Delta(k)$ is used in [@Hartnoll:2012rj; @Donos:2012js; @Blake:2013owa] for this expression.
[^6]: For $\omega<<T$ we make the four parameter fit: $1/\sigma= (a_1+a_2\omega^2)-i\omega(a_3+a_4\omega^2)$, for constants $a_i$, where we used $\sigma^*(\omega)=\sigma(-\omega)$, and we note that $a_1=(K\tau)^{-1}=\rho$ and $a_3=K^{-1}$.
[^7]: The recent analytic results on the scaling of the DC resistivity for perturbative lattices [@Blake:2013owa] also confirmed the prediction of [@Hartnoll:2012rj]. Note, though, that the order in perturbations that were considered do not include back reaction of the metric and, in particular, that length scales get renormalised from the UV to the IR. Analytic results for back-reacted Q lattice black holes will appear in [@Donos:2014uba].
[^8]: The same is true for the model considered in [@Horowitz:2012ky].
[^9]: Note that there will also be scalar modes with wave-number $nk$ and longitudinal modes with wave-number $2nk$, for $n>1$, but these will be more irrelevant in the IR and hence will not dominate the scaling of the DC resistivity.
|
---
abstract: 'The fundamental Hochschild cohomology class of the standard Podleś quantum sphere is expressed in terms of the spectral triple of Dbrowski and Sitarz by means of a residue formula.'
address:
- 'University of Glasgow, Dept. of Mathematics, University Gardens, Glasgow G12 8QW, Scotland'
- 'Universidad Michoacana de San Nicolás de Hidalgo, Instituto de Física y Matemáticas, Edificio C-3, Cd. Universitaria, 58040 Morelia, Michoacan, México '
author:
- Ulrich Krähmer
- Elmar Wagner
title: A residue formula for the fundamental Hochschild class on the Podleś sphere
---
Introduction
============
In the last decade many contributions have enhanced the understanding of how quantum groups and their homogeneous spaces can be studied in terms of spectral triples, see e.g. [@cp; @connes1; @triesteCP2; @trieste; @ds; @uli; @mnw; @nt; @voigt; @sw; @W] and the references therein. But still some basic questions remain untouched, e.g. in how far spectral triples generate the fundamental Hochschild cohomology class of the underlying algebra. This is what we investigate here for the standard Podleś quantum sphere [@Po].
To be more precise, the coordinate ring ${\mathcal{A}}={\mathcal{O}}(\mathrm{S}^2_q)$ is known to satisfy Poincaré duality in Hochschild (co)homology, that is, we have $$\label{pd}
H^i({\mathcal{A}},{\mathcal{M}}) \simeq
H_{\mathrm{dim}({\mathcal{A}})-i}({\mathcal{A}},\omega
\otimes_{\mathcal{A}}{\mathcal{M}}),$$ where ${\mathcal{M}}$ is an ${\mathcal{A}}$-bimodule and $\omega=H^{\mathrm{dim}({\mathcal{A}})}({\mathcal{A}},{\mathcal{A}}\otimes {\mathcal{A}}^\mathrm{op})$ [@uliisrael]. In the concrete case of the standard Podleś sphere we have $\mathrm{dim} ({\mathcal{A}})=2$ and $\omega \simeq {}_\sigma {\mathcal{A}}$, the bimodule obtained from ${\mathcal{A}}$ by deforming the canonical left ${\mathcal{A}}$-action to $a
\triangleright b:=\sigma (a)b$ for some noninner automorphism $\sigma \in
\mathrm{Aut}({\mathcal{A}})$ \[ibid.\]. Thus ${\mathcal{A}}$ is not a Calabi-Yau algebra (i.e. ${\mathcal{A}}\not\simeq \omega$) and hence the fundamental Hochschild homology class that corresponds under the isomorphism (\[pd\]) to $1 \in H^0({\mathcal{A}},{\mathcal{A}})$ belongs to $H_2({\mathcal{A}},{}_\sigma {\mathcal{A}})$.
So when switching to the dual picture of functionals on the canonical Hochschild complex, there is a cocycle $\varphi : {\mathcal{A}}^{\otimes 3} \rightarrow
\mathbb{C}$, $$\varphi (a_0a_1,a_2,a_3)
-\varphi (a_0,a_1a_2,a_3)
+\varphi (a_0,a_1,a_2a_3)
-\varphi (\sigma (a_3)a_0,a_1,a_2)=0$$ whose cohomology class in $(H_2({\mathcal{A}},{}_\sigma
{\mathcal{A}}))^* \simeq H^2({\mathcal{A}},({}_\sigma {\mathcal{A}})^*)$ is canonically determined by ${\mathcal{A}}$ (up to multiplication by invertible elements in the centre of ${\mathcal{A}}$ which in our example consists only of the scalars), and it is natural to ask whether this class that we refer to as the fundamental Hochschild cohomology class of ${\mathcal{A}}$ can be expressed in terms of a spectral triple over ${\mathcal{A}}$. Our main result is that this is indeed the case:
\[main\] Let $q \in (0,1)$, $({\mathcal{A}},{\mathcal{H}},D,\gamma)$ be the ${{\mathcal{U}}}_q(\mathfrak{su}(2))$-equivariant even spectral triple over the standard Podleś quantum sphere constructed by Dbrowski and Sitarz [@ds], $K$ be the standard group-like generator of ${{\mathcal{U}}}_q(\mathfrak{su}(2))$, and $a_0,a_1,a_2 \in {\mathcal{A}}$. Then we have:
1. The operator $
\gamma a_0[D,a_1][D,a_2]K^{-2}|D|^{-z}
$ is for ${\mathrm{Re}\,}z>2$ of trace class and the function $$\mathrm{tr}_{\mathcal{H}}(\gamma
a_0[D,a_1][D,a_2]K^{-2}|D|^{-z})$$ has a meromorphic continuation to $\{z \in \mathbb{C}\,|\,{\mathrm{Re}\,}z>1\}$ with a pole at $z=2$ of order at most 1.\
2. The residue $$\varphi
(a_0,a_1,a_2):=\underset{z=2}{\mathrm{Res}}\,
\mathrm{tr}_{\mathcal{H}}(\gamma a_0[D,a_1][D,a_2]K^{-2}|D|^{-z})$$ defines a Hochschild cocycle that represents the fundamental Hochschild cohomology class of ${\mathcal{A}}$.
Let us explain the context of the result. The pioneering papers on the noncommutative geometry of the Podleś sphere were [@mnw], where Masuda, Nakagami and Watanabe computed $HH_\bullet({\mathcal{A}}),HC_\bullet({\mathcal{A}})$ and the K-theory of the $C^*$-completion of ${\mathcal{A}}$, and [@ds], where Dbrowski and Sitarz found the spectral triple that we use here. Schmüdgen and the second author then gave a residue formula for a cyclic cocycle [@sw] that looks like the one from Theorem \[main\], only that $K^{-2}$ is replaced by $K^2$. However, Hadfield later computed the Hochschild and cyclic homology of ${\mathcal{A}}$ with coefficients in ${}_\sigma {\mathcal{A}}$ and deduced that the cocycle from [@sw] is trivial as a Hochschild cocycle [@tom]. Finally, the first author has recently used the cup and cap products between the Hochschild (co)homology groups $H^n({\mathcal{A}},{}_\sigma
{\mathcal{A}})$ [@uliarab] to produce the surprisingly simple formula $$\label{surpris}
\tilde\varphi(a_0,a_1,a_2)=\varepsilon (a_0)E(a_1) F(a_2)$$ for a nontrivial Hochschild 2-cocycle on ${\mathcal{A}}$. Here $\varepsilon$ is the counit of the quantum ${{\mathrm{SU}}}(2)$-group ${\mathcal{B}}:={\mathcal{O}}({{\mathrm{SU}}}_q(2))$ in which ${\mathcal{A}}$ is embedded as a subalgebra, and $E$ and $F$ are the standard twisted primitive generators of ${{\mathcal{U}}}_q(\mathfrak{su}(2))$, considered here as functionals on ${\mathcal{B}}$. What we achieve in Theorem \[main\] is to express (a multiple of) this cocycle in terms of the spectral triple by a residue formula.
The crucial result is Proposition \[reslemma\] in Section \[jetzelet\] from which it follows that $$\tau_\mu (a) :=
\frac{\underset{z=2|\mu|}{\mathrm{Res}}
\mathrm{tr}_{\mathcal{H}}(a K^{2 \mu} |D|^{-z})}
{\underset{z=2|\mu|}{\mathrm{Res}}\mathrm{tr}_{\mathcal{H}}(K^{2 \mu} |D|^{-z})}$$ defines for all $\mu \in \mathbb{R}$ a twisted trace on ${\mathcal{A}}$ that we can compute explicitly. In the notation of [@uliarab], these traces are given by:\
-------------------- ------------------------------------------- ---------------------------------------------------------------------
\[-8pt\] $\mu$ $\sigma$ $\tau_\mu$
\[-8pt\]
\[-8pt\] $< 0$ any $\int_{[1]}=\varepsilon$
\[-8pt\]
\[-8pt\] $0$ $\mathrm{id}$ $\ \int_{[1]} +
\frac{\ln q}{2(q^{-1}-q) \ln(q^{-1}-q)} \int_{[x_0]}\ $
\[-8pt\]
\[-8pt\] $\ > 0\ $ $\ A \mapsto A$, $B \mapsto q^{2 \mu}B\ $ $\int_{[1]}-\frac{1-q^{-2
\mu}}{q(1-q^{-2(\mu + 1)})}\int_{[x_0]}$
\[-8pt\]
-------------------- ------------------------------------------- ---------------------------------------------------------------------
\
Here $\sigma$ is the involved twisting automorphism and $A$, $B$ are certain generators of ${\mathcal{A}}$ (in the notation of [@uliarab], $A=-q^{-1}x_0$). From this fact it will be deduced in the final section that the multilinear functional defined in Theorem \[main\] is up to normalisation indeed cohomologous to the Hochschild cocycle (\[surpris\]) constructed in [@uliarab].
The remainder of the paper is divided into two sections: Section \[background\] contains background material taken mainly from [@ds; @sw]. The subsequent section discusses the meromorphic continuation of the zeta functions $\mathrm{tr}_{{\mathcal{H}}_\pm} (a K^{2 \mu}|D|^{-z})$, $a \in {\mathcal{A}}$, and how one can compute their residues by replacing certain algebra elements of ${\mathcal{A}}$ by simpler operators. The proof of Theorem \[main\] fills the final Subsection \[beweis\] of the paper.\
[^1]
Background
==========
The algebras {#stpodl}
-------------
We retain all notations and conventions used in [@sw]. In particular, we fix a deformation parameter $q \in
(0,1)$, and let ${\mathcal{A}}={\mathcal{O}}(\mathrm{S}^2_q)$ be the $*$-algebra (over $\mathbb{C}$) with generators $A=A^*$, $B$ and $B^*$ and defining relations $$B A=q^2 AB,\quad
A B^*=q^2 B^*A,\quad
B^*B=A-A^2,\quad
BB^*=q^2A-q^4A^2.$$
We consider ${\mathcal{A}}$ as a subalgebra of the quantised coordinate ring ${\mathcal{B}}={\mathcal{O}}({{\mathrm{SU}}}_q(2))$ which is the $*$-algebra generated by $a$, $b$, $c=-q^{-1}b^*$, $d=a^*$ satisfying the relations given e.g. in [@chef Eqs. (1) and (2) on p. 97]. The embedding is given by $B=ac$ and $A=-q^{-1}bc$, and it follows that $\varepsilon(A)=\varepsilon(B)=\varepsilon(B^*)=0$, where $\varepsilon$ denotes the counit of ${\mathcal{O}}({{\mathrm{SU}}}_q(2))$.
Note that it follows from the defining relations that the monomials $$\{A^nB^m,A^n{B^*}^m\,|\,n,m \ge 0\}$$ form a vector space basis of ${\mathcal{A}}$.
For the Hopf \*-algebra ${\mathcal{U}}={{\mathcal{U}}}_q(\mathfrak{su}(2))$, we use generators $K$, $K^{-1}$, $E$ and $F$ with involution $K^*=K$, $E^*=F$, defining relations $$KE=qEK,\quad KF=q^{-1}FK,\quad EF-FE= \frac{K^2-K^{-2}}{q-q^{-1}},$$ coproduct $$\Delta(K)=K\otimes K, \ \ \Delta(E)=E\otimes K+K^{-1}\otimes E,
\ \ \Delta(F)=F\otimes K+K^{-1}\otimes F,$$ and counit $\varepsilon(1-K)=\varepsilon(E)=\varepsilon(F)=0$.
There is a left ${\mathcal{U}}$-action on ${\mathcal{A}}$ satisfying $f{\triangleright}(ab)= (f_{(1)}{\triangleright}a)(f_{(2)}{\triangleright}b)$ and $f{\triangleright}1=\varepsilon(f)1$ for $f\in{\mathcal{U}}$ and $a,b\in{\mathcal{A}}$, that is, ${\mathcal{A}}$ is a left ${\mathcal{U}}$-module algebra. Here and in what follows, we use Sweedler’s notation $\Delta(f)=f_{(1)}\otimes f_{(2)}$. On the re-parametrised generators $$\label{xAB}
x_{-1}=(1+q^{-2})^{1/2}B, \quad x_0=1-(1+q^2)A, \quad x_{1}=-(1+q^2)^{1/2}B^*,$$ this action is given by $$K{\triangleright}x_i = q^i x_i,\quad E{\triangleright}x_i = (q+q^{-1}) x_{i+1}, \quad
F{\triangleright}x_i = (q+q^{-1}) x_{i-1},$$ where it is understood that $x_{2}=x_{-2}=0$.
The spectral triple
-------------------
Our calculations involve the spectral triple constructed by Dbrowski and Sitarz in [@ds]. For the reader’s convenience and to fix notation, we recall its definition.
First of all, the $*$-algebra ${\mathcal{A}}$ becomes represented by bounded operators on a Hilbert space ${\mathcal{H}}:={\mathcal{H}}_-\oplus {\mathcal{H}}_+$ with orthonormal basis $$v^l_{k,\pm} \in {\mathcal{H}}_\pm,\quad
l= \mbox{$\frac{1}{2}$}, \mbox{$\frac{3}{2}$}, \ldots,\quad
k=-l, -l+1,\ldots, l$$ where the generators $x_{-1}$, $x_0$, $x_1$ act by $$\label{pi}
x_i v^l_{k,\pm} =
\alpha^-_i(l,k)_\pm\hspace{1pt} v^{l-1}_{k+i,\pm} +
\alpha^0_i(l,k)_\pm\hspace{1pt} v^{l}_{k+i,\pm} +
\alpha^+_i(l,k)_\pm\hspace{1pt} v^{l+1}_{k+i,\pm}.$$ Here $\alpha^\nu_i(l,k)_\pm \in \mathbb{R}$ are coefficients that can be found e.g. in [@DDLW], where similar conventions are used. We will only need the formulas for $\alpha_0^\nu(l,k)_\pm$ which are given by $$\begin{aligned}
\alpha^-_0(l,k)_\pm
&= \frac{q^{k\pm 1/2}
[2]_q [l\!-\!k]_q^{1/2} [l\!+\!k]_q^{1/2}[l\!-\!1/2]_q^{1/2}
[l\!+\!1/2]_q^{1/2}}
{[2l\!-\!1]_q^{1/2} [2l]_q
[2l\!+\!1]_q^{1/2}},
\label{x0-}\\
\alpha^0_0(l,k)_\pm
&= [2l]_q^{-1}
\big([l\!-\!k\!+\!1]_q[l\!+\!k]_q -
q^{2} [l\!-\!k]_q[l\!+\!k\!+\!1]_q\big)
\beta_\pm (l),
\label{x00} \\
\alpha^+_0(l,k)_\pm
&= \alpha^-_0(l+1,k)_\pm \end{aligned}$$ with $$\label{qzahl}
[n]_q:=\frac{q^n-q^{-n}}{q-q^{-1}}$$ and $$\begin{aligned}
\label{beta+-}
\beta_\pm(l)
&= \frac{\pm q^{\mp 1} +
(q{\hspace{-1pt}}-{\hspace{-1pt}}q^{-1})([1/2]_q\hspace{1pt}[3/2]_q-
[l]_q[l\!+\!1]_q)
}{q[2l+2]_q}.\end{aligned}$$
We now define $${\mathrm{Dom}(D)}:= \mathrm{span}_\mathbb{C} \{v^l_{k,\pm}\ |\
l= \mbox{$\frac{1}{2}$}, \mbox{$\frac{3}{2}$}, \ldots,\ \,
k=-l, -l+1,\ldots, l \}$$ and on this domain an essentially self-adjoint operator $D$ by $$D v^l_{k,\pm} = [l+1/2]_q v^l_{k,\mp}.$$
In the sequel all operators we consider will be defined on this domain, leave it invariant, and be closable. By slight abuse of notation we will not distinguish between an operator defined on ${\mathrm{Dom}(D)}$ and its closure.
The $v^l_{k,\pm}$ are eigenvectors of $|D|$: $$|D| v^l_{k,\pm} = [l+1/2]_q v^l_{k,\pm}.$$ Furthermore, the spectral triple is even with grading $\gamma$ given by $$\gamma v^l_{k,\pm}=\pm v^l_{k,\pm}.$$
${\mathcal{U}}$-equivariance
----------------------------
The spectral triple is ${\mathcal{U}}$-equivariant in the sense of [@SitarzBCP]: On ${\mathrm{Dom}(D)}$ there is an action of ${\mathcal{U}}$ given by $$\label{K}
Kv^l_{k,\pm}=q^k v^l_{k,\pm},
\quad Ev^l_{k,\pm}=\alpha^l_k v^l_{k+1,\pm},
\quad Fv^l_{k,\pm}=\alpha^l_{k-1} v^l_{k-1,\pm},$$ where $\alpha^l_k:=([l-k]_q[l+k+1]_q)^{1/2}$, and we have on ${\mathrm{Dom}(D)}$ $$fa=(f_{(1)}{\triangleright}a)f_{(2)},\quad
fD=Df,\quad
f\gamma= \gamma f$$ for all $f \in {\mathcal{U}}$, $a \in {\mathcal{A}}$.
Note that the decompostion ${\mathcal{H}}={\mathcal{H}}_-\oplus {\mathcal{H}}_+$ reduces the representation of ${\mathcal{U}}$ and ${\mathcal{A}}$ on ${\mathcal{H}}$.
Results
=======
A family of $q$-zeta functions
------------------------------
Quantum group analogues of zeta functions were studied by several authors, in particular by Ueno and Nishizawa [@ueno], Cherednik [@cherednik], and Majid and Tomašić [@majid]. The ones we will consider here are given on a suitable domain by $$\zeta^\pm_T (z) := \mathrm{tr}_{{\mathcal{H}}_\pm}(T|D|^{-z})$$ for some possibly unbounded operator $T$ on ${\mathcal{H}}$. The most important case we need is $T=L^\beta K^\delta$ for $\beta,\delta \in\mathbb{R}$, where $$L v^l_{k,\pm}=q^l v^l_{k,\pm}$$ and thus $$L^\beta K^\delta v^l_{k,\pm}= q^{\beta l+\delta k}
v^l_{k,\pm}.$$
The resulting zeta functions differ slightly from those considered in [@cherednik; @ueno], and also from the one occuring in [@sw]. Yet the main argument leading to a meromorphic continuation of the functions to the whole complex plane given in [@ueno] can be applied in all these cases:
\[L137\] For all $\beta,\delta \in \mathbb{R}$, the function $$\zeta^\pm_{L^\beta K^\delta}(z)=
\sum_{l=\frac{1}{2},\frac{3}{2},\ldots}
^{\infty} \sum_{k=-l}^{l}
\frac{q^{\beta l+\delta k}}
{[l+1/2]^z_q},\quad
{\mathrm{Re}\,}z > -\beta+ |\delta|$$ admits a meromorphic continuation to the complex plane given by $$\zeta^\pm_{L^\beta K^\delta}(z)=
q^{\frac{\beta}{2}}
(q^{-\frac{\delta}{2}} {\hspace{-1pt}}+ {\hspace{-1pt}}q^{\frac{\delta}{2}})
(1 {\hspace{-1pt}}- {\hspace{-1pt}}q^2)^z
\sum_{j=0}^\infty
\frac{
\binom{z+j-1}{j} \, q^{2j} }
{(1 {\hspace{-1pt}}- {\hspace{-1pt}}q^{\beta-\delta + 2j + z })
(1 {\hspace{-1pt}}- {\hspace{-1pt}}q^{\beta+\delta + 2j + z })}$$ which is holomorphic except at $z=-\beta
\pm \delta,-\beta \pm \delta -2,\ldots$ and whose residue at $z=-\beta + |\delta|$ is given by $$\underset{z=-\beta + |\delta|}
{\mathrm{Res}}\;\zeta^\pm_{L^\beta K^\delta} (z)=
\left\{\begin{array}{ll}
\frac{q^{\frac{\beta-|\delta|}{2}}
\left(1-q^2\right)^{|\delta|-\beta}}{\left(q^{|\delta|}-1\right)
\ln (q)}{\hspace{1pt}},
\quad & \delta \neq 0,\\[8pt]
\frac{2q^{\frac{\beta}{2}}\ln(q^{-1}-q)}
{(1-q^2)^\beta (\ln q)^2}{\hspace{1pt}},
\quad & \delta = 0.
\end{array}\right.$$
The crucial step is to use the binomial series $$\label{bin}
(1-q^{2(n+1)})^{-z}=
\sum_{j=0}^\infty
\binom{z+j-1}{j} q^{2(n+1)j}$$ which holds for all $z\in\C$.
First, let $\delta = 0$. By summing over $k$, replacing $l=n+\frac{1}{2}$, inserting , and interchanging the order of the summations in the absolutely convergent series, we obtain for ${\mathrm{Re}\,}z > -\beta$ $$\begin{aligned}
\zeta^\pm_{L^\beta} (z)
&= 2(q^{-1}-q)^z \sum_{n = 0}^{\infty} (n+1)
q^{\beta (n+\frac{1}{2})} q^{(n+1)z} (1-q^{2(n+1)})^{-z}\\
&=
2 q^{-\frac{\beta}{2}}(q^{-1}-q)^z
\sum_{j=0}^\infty \sum_{n = 0}^{\infty} (n+1)
\binom{z{\hspace{1pt}}+{\hspace{1pt}}j{\hspace{1pt}}-{\hspace{1pt}}1}{j}
q^{(\beta+2j+z) (n+1)}. \end{aligned}$$ Using the identity $$\sum_{n = 0}^{\infty}
(n+1)
t^n=\frac{\mathrm{d}}{\mathrm{d}t}
\sum_{n=0}^{\infty} t^n = \frac{1}{(1-t)^2},$$ we can write the above sum as $$\zeta^\pm_{L^\beta} (z)=
2 q^{-\frac{\beta}{2}}(q^{-1}-q)^z
\sum_{j=0}^\infty
\left(\begin{array}{c} z+j-1 \\ j
\end{array}\right)
\frac{q^{\beta+2j+z}}{(1-q^{\beta+2j+z})^2},$$ and the right hand side is a meromorphic function with poles of second order in $z=-\beta,-\beta-2,-\beta-4,\ldots$
If $\delta \neq 0$, the sum over $k$ yields $$\sum_{k=-(n+\frac{1}{2})}^{n+\frac{1}{2}}
q^{\delta k} =
\frac{q^{-\delta (n+1)}-q^{\delta(n+1)}}
{q^{-\frac{\delta}{2}}-q^{\frac{\delta}{2}}}.$$ Similar to the above, we get for ${\mathrm{Re}\,}z > -\beta + |\delta|$ $$\begin{aligned}
&\zeta^\pm_{L^\beta K^\delta} (z) =\\
&
q^{-\frac{\beta}{2}}\frac{(q^{-1}{\hspace{-1pt}}-{\hspace{-1pt}}q)^z}{q^{-\frac{\delta}{2}}{\hspace{-1pt}}-{\hspace{-1pt}}q^{\frac{\delta}{2}}}
\sum_{j=0}^\infty \sum_{n = 0}^{\infty}
\!\binom{z \!+ \!j \!-\! 1}{j}
(q^{(\beta-\delta+2j+z) (n+1)} {\hspace{-1pt}}-{\hspace{-1pt}}q^{(\beta+\delta+2j+z) (n+1)}). \end{aligned}$$ The summation over $n$ gives $$\begin{aligned}
\sum_{n = 0}^{\infty} q^{(\beta-\delta+2j+z) (n+1)}{\hspace{-1pt}}-{\hspace{-1pt}}q^{(\beta+\delta+2j+z) (n+1)}=
\frac{ (q^{-\delta} - q^{\delta})q^{\beta+ 2j + z }}
{(1 {\hspace{-1pt}}- {\hspace{-1pt}}q^{\beta-\delta + 2j + z })(1 {\hspace{-1pt}}- {\hspace{-1pt}}q^{\beta+\delta + 2j + z })}. \end{aligned}$$ Inserting the last equation into the previous one yields the second formula of Lemma \[L137\] which defines a meromorphic function with poles at $z=-\beta {\hspace{-1pt}}\pm{\hspace{-1pt}}\delta,-{\hspace{-1pt}}\beta{\hspace{-1pt}}\pm{\hspace{-1pt}}\delta{\hspace{-1pt}}-{\hspace{-1pt}}2,\ldots$
When computing the residues at $z=-\beta+|\delta|$, we can ignore the sum over $j>0$ which is holomorphic in a neighbourhood of $-\beta+|\delta|$. Thus $$\underset{z=-\beta+|\delta|}{\mathrm{Res}}\;
\zeta^\pm_{L^\beta K^\delta} (z)=
\underset{z=-\beta+|\delta|}{\mathrm{Res}}\;
\frac{q^{\frac{\beta}{2}}(1-q^2)^z
(q^{-\frac{\delta}{2}}+q^{\frac{\delta}{2}})}
{(1-q^{\beta-\delta+z})(1-q^{\beta+\delta+z})}$$ which can be computed straightforwardly to yield the result.
A holomorphicity remark
-----------------------
Next we need to point out that $\zeta^\pm_{TL^\beta K^\delta}(z)$ is holomorphic for ${\mathrm{Re}\,}z > -\beta+
|\delta|$ whenever $T$ is bounded. Let us first introduce some notation that we will use throughout the rest of the paper in order to simplify statements and proofs:
We say that a set of complex numbers $$\{ \nu_{l,k}\;|\;l\in\mbox{$\frac{1}{2}$}\N,\ \, k=-l,\ldots l\}$$ is of order less than or equal to $q^\alpha$, $\alpha\in \R$, if there exists $C\in(0,\infty)$ such that $|\nu_{l,k}|\leq C q^{\alpha l}$ for all $k,l$. In this case we write $$\nu_{l,k}\precsim q^{\alpha l}.$$
We refrain from using the notation $\nu_{k,l}=\mathrm{O}(q^{\alpha l})$ to avoid confusion about the fact that the second parameter $k$ can take arbitrary values from $\{-l,\ldots,l\}$. Note that we have for all $\beta, \delta\in \R$ and $z\in\C$ $$\label{precsim}
q^{\beta l + \delta k} \precsim q^{(\beta-|\delta|)l},\ \
[l-k]_q[l+k]_q\precsim q^{-2l}, \ \
[\beta l+\delta]_q^{-z}\precsim q^{
|\beta| \,{\mathrm{Re}\,}(z)\,l} .$$
Now one easily observes:
\[Lq\] For all bounded operators $T$ on ${\mathcal{H}}$ and for all $\beta,\delta \in
\mathbb{R}$, the function $\zeta^\pm_{T{{L}}^\beta K^\delta}(z)$ is holomorphic on $\{z \in
\mathbb{C}\,|\,{\mathrm{Re}\,}z> -\beta+|\delta|\}$.
Since $L^\beta K^\delta |D|^{-r}$ is for $r \in \mathbb{R}$ positive and essentially self-adjoint, the summability of its eigenvalues verified in Lemma \[L137\] shows that it is of trace class if $r > - \beta +
|\delta|$. Therefore $TL^\beta K^\delta |D|^{-z}=
T|D|^{-\mathrm{i} s} L^\beta K^\delta
|D|^{-r}$, $z=r+\mathrm{i} s$, is a trace class operator.
If one fixes $\epsilon >
0$, then the infinite series defining $\mathrm{tr}_{{\mathcal{H}}_\pm}(T {{L}}^\beta K^\delta |D|^{-z})$ converges uniformly on $\{z \in \mathbb{C}\,|\,{\mathrm{Re}\,}z\geq
\epsilon-\beta+|\delta|\}$ since the geometric series $$\sum_{l \in \mathbb{N}}
q^{({\mathrm{Re}\,}(z)+\beta - |\delta|)l}$$ does so and we have for all bounded sequences $t^l_{k,\pm}:=\langle v^l_{k,\pm},T
v^l_{k,\pm} \rangle$ $$\frac{t^l_{k,\pm} q^{\beta l+\delta k}}{[l+1/2]_q^z}
\precsim
q^{({\mathrm{Re}\,}(z)+\beta-|\delta|)l}.$$
The partial sums of the series are clearly holomorphic functions and, by the above argument, converge uniformly on compact sets contained in $\{z \in \mathbb{C}\,|\,{\mathrm{Re}\,}z>
-\beta + |\delta |\}$. The result follows now from the Weierstra[ß]{} convergence theorem.
Approximating the generator
----------------------------
It is known [@nt] that the spectral triple we consider here violates Connes’ regularity condition, so the standard machinery of zeta functions and generalised pseudo-differential operators (see e.g. [@connesmosco; @higson]) can not be applied here. As a replacement of a pseudo-differential calculus, we approximate in the following lemma the generator $A \in {\mathcal{A}}$ on ${\mathcal{H}}$ by simpler operators. Similar ideas have been used in [@DDLW].
\[Mq\] There exists a bounded linear operator $A_0$ on ${\mathcal{H}}$ such that $$A={{M}}+ A_0{{L}},\quad \text{where}\ {{M}} := {L}^2K^2.$$
We have to prove that $A_0:=(A-{{M}}){{L}}^{-1}$ extends to a bounded operator on ${\mathcal{H}}$. Inserting and into this definition shows that it suffices to prove that the coefficients $$q^{-l}\alpha^\pm_0(l,k)_\pm\ \ \mbox{and} \ \
q^{-l}\left(\mbox{$\frac{1}{1+q^2}$}(1-\alpha_0^0(l,k)_\pm)-q^{2(l+k)}\right)$$ are bounded. Applying to gives $\alpha^\pm_0(l,k)_\pm\precsim
q^l$. Therefore we have $q^{-l}\alpha^\pm_0(l,k)_\pm\precsim 1$ which means that these coefficients are bounded.
Using and $\frac{1}{1-q^{4l+4}}-1\precsim q^{4l}$, we get from $$\beta_\pm(l) =
\frac{(q^{-1}{\hspace{-1pt}}-{\hspace{-1pt}}q)
[l]_q[l\!+\!1]_q}{q[2l+2]_q}+u_l =
\frac{1-q^{2l}-q^{2l+2}+
q^{4l+2}}{1-q^{4l+4}} + u_l=
1+v_l,$$ where $u_l,v_l\precsim
q^{2l}$. Similarly, we have $$\begin{aligned}
\frac{[l\!-\!k\!+\!1]_q[l\!+\!k]_q -
q^{2} [l\!-\!k]_q[l\!+\!k\!+\!1]_q}
{[2l]_q}
&= \frac{1-(1\!+\!q^2)q^{2l+2k}+(1\!+\!q^2)q^{2l}}{1-q^{4l}}\\[4pt]
&=1 -(1+q^2)q^{2l+2k} + w_{l,k},\end{aligned}$$ where $w_{l,k}\precsim q^{2l}$. Multiplying the last two equations and comparing with gives $$\alpha^0_0(l,k)_\pm = 1
-(1+q^2)q^{2l+2k} + x_{l,k}$$ with $x_{l,k}\precsim q^{2l}$. From this, we get $$q^{-l}\left(\mbox{$\frac{1}{1+q^2}$}
(1-\alpha_0^0(l,k)_\pm)-q^{2(l+k)}\right)=
q^{-l}x_{l,k}\precsim q^l\precsim 1$$ which finishes the proof.
Twisted traces as residues {#jetzelet}
--------------------------
We are now ready to prove the main technical result of the paper which expresses certain twisted traces of ${\mathcal{A}}$ as residues of zeta-functions:
\[reslemma\] The function $\zeta^\pm_{a K^{2 \mu}}(z)$, $a \in {\mathcal{A}}$, has a meromorphic continuation to $\{z \in
\mathbb{C}\,|\,{\mathrm{Re}\,}z> 2|\mu|-1 \}$. Its residues at $z=2|\mu|$ are (both for $+$ and $-$) given by\
----------------------------------------------------- -------------- -----------------------------------------------------------------------
\[-4pt\] $a$ $\mu$ $\underset{z=2|\mu|}
{\mathrm{Res}}\zeta^\pm_{a K^{2 \mu}}(z)$
\[12pt\]
\[-8pt\] $\ A^nB^m,\ A^nB^{*m},\ \ n \ge 0,\ m>0\ $ [any]{} $0$
\[-8pt\]
\[-8pt\] $A^n,\ \ n>0$ $< 0$ $0$
\[-8pt\]
\[-8pt\] $A^n,\ \ n>0$ $\ \geq 0\ $ $\frac{-q^\mu(q^{-1}-q)^{2\mu}}{(1-q^{2(n+\mu)}){\hspace{1pt}}\ln q}$
\[-8pt\]
\[-8pt\] $1$ $\neq 0$ $\ \frac{-q^{|\mu|}(q^{-1}-q)^{2|\mu|}}{(1-q^{2|\mu|})\,\ln (q)}\ $
\[-8pt\]
\[-8pt\] $1$ $0$ $\frac{2\ln(q^{-1}-q)}{(\ln q)^2}$
\[-8pt\]
----------------------------------------------------- -------------- -----------------------------------------------------------------------
Lemma \[Lq\] implies that the traces $\zeta^\pm_{aK^{2\mu}}$ exist and are holomorphic on $\{z \in\mathbb{C}\,|\,{\mathrm{Re}\,}z> 2|\mu| \}$ for all $a \in {\mathcal{A}}$.
Furthermore, $B$ and $B^*$ act as shift operators in the index $k$ of $v^l_{k,\pm}$. Hence the traces $\mathrm{tr}_{{\mathcal{H}}_\pm}(A^n B^mK^{2 \mu}|D|^{-z})$ and $\mathrm{tr}_{{\mathcal{H}}_\pm}(A^n B^{*m}K^{2 \mu}|D|^{-z})$ vanish whenever $m>0$, so we can use the trivial analytic continuation here. Thus it remains to prove the claim for $a=A^n$.
Applying first Lemma \[Mq\] and using then the fact that, by Lemma \[Lq\], $\mathrm{tr}_{{\mathcal{H}}_\pm}(A^{n-1}A_0{{M}}^m {{L}}K^{2\mu}|D|^{-z})$ is on $\{z \in\mathbb{C}\,|\,{\mathrm{Re}\,}z>
2|\mu|-1 \}$ holomorphic, we obtain for all $m \ge 0$, $n>0$ $$\begin{aligned}
& \underset{z=2|\mu|}{\mathrm{Res}}\,
\mathrm{tr}_{{\mathcal{H}}_\pm}(A^n {{M}}^mK^{2\mu}|D|^{-z})\\
&= \!\underset{z=2|\mu|}{\mathrm{Res}}\,
\mathrm{tr}_{{\mathcal{H}}_\pm}(A^{n-1}({{M}}+A_0{{L}})
{{M}}^mK^{2\mu}|D|^{-z})\\
&=\! \underset{z=2|\mu|}{\mathrm{Res}}
\mathrm{tr}_{{\mathcal{H}}_\pm}{\hspace{-1pt}}(A^{n-1} {{M}}^{m+1}K^{2\mu}|D|^{-z})
{\hspace{-1pt}}+\!\underset{z=2|\mu|}{\mathrm{Res}}
\mathrm{tr}_{{\mathcal{H}}_\pm}{\hspace{-1pt}}(A^{n-1}A_0{{M}}^m
{{L}}K^{2\mu}|D|^{-z})\\
&=\! \underset{z=2|\mu|}
{\mathrm{Res}}
\mathrm{tr}_{{\mathcal{H}}_\pm}(A^{n-1}
{{M}}^{m+1}K^{2\mu}|D|^{-z}). \end{aligned}$$ Recall that $M=L^2K^2$. An iterated application of the previous equation gives $$\begin{aligned}
\underset{z=2|\mu|}
{\mathrm{Res}}
\mathrm{tr}_{{\mathcal{H}}_\pm}(A^nK^{2\mu}|D|^{-z}) &=
\underset{z=2|\mu|}
{\mathrm{Res}}
\mathrm{tr}_{{\mathcal{H}}_\pm}({{M}}^nK^{2\mu}|D|^{-z})\\
&=\underset{z=2|\mu|}
{\mathrm{Res}}
\mathrm{tr}_{{\mathcal{H}}_\pm}({{L}}^{2n}K^{2(\mu+n)}|D|^{-z}).\end{aligned}$$ The result now reduces to Lemma \[L137\].
We remark here that the table in the introduction is obtained by comparing the values of the twisted traces $\int_{[1]}$ and $\int_{[x_0]}$ from [@uliarab] on the basis vectors $A^nB^m$ and $A^nB^{*m}$ with the residues of the last proposition.
Proof of Theorem \[main\] {#beweis}
-------------------------
Theorem \[main\] is an easy consequence of Proposition \[reslemma\]. As explained e.g. in [@sw], the operator $$\gamma a_0[D,a_1][D,a_2],\quad
a_0,a_1,a_2 \in {\mathcal{A}}$$ acts by multiplication with $$a_0(a_1 \triangleleft E)(a_2
\triangleleft F) \in {\mathcal{A}}$$ on ${\mathcal{H}}_+$ and by multiplication with $$-a_0(a_1 \triangleleft F)(a_2
\triangleleft E) \in {\mathcal{A}}$$ on ${\mathcal{H}}_-$. Here $\triangleleft$ denotes the standard right action of ${\mathcal{U}}\subset {\mathcal{B}}^\circ$ on ${\mathcal{B}}$ given by $$a \triangleleft f:=f(a_{(1)})f_{(2)}$$ Let us point out that unlike the left action $f \triangleright
a:=a_{(1)}f(a_{2})$, this right action does not leave ${\mathcal{A}}\subset {\mathcal{B}}$ invariant, but the products $(a_1 \triangleleft E)(a_2
\triangleleft F)$ and $(a_1
\triangleleft F)(a_2 \triangleleft E)$ belong to ${\mathcal{A}}$ again.
By the definition of $\triangleleft$ we have $$\varepsilon (a_0(a_1 \triangleleft E)(a_2
\triangleleft F))=\varepsilon
(a_0) E(a_1) F(a_2)$$ and $$\varepsilon (a_0(a_1 \triangleleft F)(a_2
\triangleleft E))=
\varepsilon (a_0) F(a_1) E(a_2),$$ and evaluation on an arbitrary cycle representing the fundamental Hochschild class in $H_2({\mathcal{A}},{}_\sigma {\mathcal{A}})$ (see the proof of the nontriviality of (\[surpris\]) in [@uliarab]) shows that the two functionals on $H_2({\mathcal{A}},{}_\sigma {\mathcal{A}})$ induced by these functionals on ${\mathcal{A}}^{\otimes 3}$ coincide up to a factor of $-q^{-2}$ (see also [@uliplural], where we carry this computation out with the help of the computer algebra system SINGULAR:PLURAL).
Thus, by Proposition \[reslemma\], the cocycle $$\begin{aligned}
\varphi
(a_0,a_1,a_2):=&\;
\underset{z=2}{\mathrm{Res}}\,
\mathrm{tr}_{\mathcal{H}}(\gamma
a_0[D,a_1][D,a_2]K^{-2}|D|^{-z})\\
=&\; \mbox{$\frac{q-q^{-1}}{\ln (q)}$} {\hspace{1pt}}\varepsilon (a_0) \big(E(a_1)F(a_2) - F(a_1)E(a_2)\big)\end{aligned}$$ is cohomologous to $\frac{q-q^{-3}}{\ln (q)}{\hspace{1pt}}\tilde \varphi$, where $\tilde \varphi$ denotes the fundamental cocycle from . This finishes the proof of Theorem \[main\]. $\Box$
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[^1]: The first author would like to thank Viktor Levandovskyy for his explanations of some of its functionalities of the computer algebra system SINGULAR:PLURAL [@plural] which is very capable in carrying out computations with algebras like the Podleś sphere, see [@uliplural] for a quick demonstration. This work was partially supported by the EPSRC fellowship EP/E/043267/1, the Polish Government Grant N201 1770 33, the Marie Curie PIRSES-GA-2008-230836 network, and the Mexican Government Grant PROMEP/103.5/09/4106, UMSNH-PTC-259.
|
---
abstract: 'We propose UDP, the first training-free parser for Universal Dependencies (UD). Our algorithm is based on PageRank and a small set of head attachment rules. It features two-step decoding to guarantee that function words are attached as leaf nodes. The parser requires no training, and it is competitive with a delexicalized transfer system. UDP offers a linguistically sound unsupervised alternative to cross-lingual parsing for UD, which can be used as a baseline for such systems. The parser has very few parameters and is distinctly robust to domain change across languages.'
author:
- |
Héctor Martínez Alonso$^\spadesuit$ Željko Agić$^\heartsuit$ Barbara Plank$^\clubsuit$ Anders Søgaard$^\diamondsuit$\
$^\spadesuit$Univ. Paris Diderot, Sorbonne Paris Cité – Alpage, INRIA, France\
$^\heartsuit$IT University of Copenhagen, Denmark\
$^\clubsuit$Center for Language and Cognition, University of Groningen, The Netherlands\
$^\diamondsuit$University of Copenhagen, Denmark\
[hector.martinez-alonso@inria.fr]{}
bibliography:
- 'upud.bib'
title: Parsing Universal Dependencies without training
---
Introduction
============
Grammar induction and unsupervised dependency parsing are active fields of research in natural language processing [@klein2004corpus; @gelling2012pascal]. However, many data-driven approaches struggle with learning relations that match the conventions of the test data, e.g., Klein and Manning reported the tendency of their DMV parser to make determiners the heads of German nouns, which would not be an error if the test data used a DP analysis [@abney1987english]. Even supervised transfer approaches [@mcdonald2011multi] suffer from target adaptation problems when facing word order differences.
The Universal Dependencies (UD) project [@UD12; @nivre2016universal] offers a dependency formalism that aims at providing a consistent representation across languages, while enforcing a few hard constraints. The arrival of such treebanks, expanded and improved on a regular basis, provides a new milestone for cross-lingual dependency parsing research [@mcdonald2013universal].
Furthermore, given that UD rests on a series of simple principles like the primacy of lexical heads, cf. for more details, we expect that such a formalism lends itself more naturally to a simple and linguistically sound rule-based approach to cross-lingual parsing. In this paper we present such an approach.
Our system is a dependency parser that requires no training, and relies solely on explicit part-of-speech (POS) constraints that UD imposes. In particular, UD prescribes that trees are single-rooted, and that function words like adpositions, auxiliaries, and determiners are always dependents of content words, while other formalisms might treat them as heads [@marneffe2014universal]. We ascribe our work to the viewpoints of about the incorporation of linguistic knowledge in language-independent systems.
#### Contributions
We introduce, to the best of our knowledge, the first unsupervised rule-based dependency parser for Universal Dependencies.
Our method goes substantially beyond the existing work on *rule-aided* unsupervised dependency parsing, specifically by:
- adapting the dependency head rules to UD-compliant POS relations,
- incorporating the UD restriction of function words being leaves,
- applying personalized PageRank to improve main predicate identification, and by
- making the parsing entirely free of language-specific parameters by estimating adposition attachment direction at runtime.
We evaluate our system on 32 languages[^1] in three setups, depending on the reliability of available POS tags, and compare to a multi-source delexicalized transfer system. In addition, we evaluate the systems’ sensitivity to domain change for a subset of UD languages for which domain information was retrievable. The results expose a solid and competitive system for all UD languages. Our unsupervised parser compares favorably to delexicalized parsing, while being more robust to domain change.
Related work {#sec:relatedwork}
============
#### Cross-lingual learning
Recent years have seen exciting developments in cross-lingual linguistic structure prediction based on transfer or projection of POS and dependencies [@das2011unsupervised; @mcdonald2011multi]. These works mainly use supervised learning and domain adaptation techniques for the target language.
The first group of approaches deals with annotation projection [@yarowsky2001inducing], whereby parallel corpora are used to transfer annotations between resource-rich source languages and low-resource target languages. Projection relies on the availability and quality of parallel corpora, source-side taggers and parsers, but also tokenizers, sentence aligners, and word aligners for sources and targets. were the first to project syntactic dependencies, and Tiedemann et al. improved on their projection algorithm. Current state of the art in cross-lingual dependency parsing involves leveraging parallel corpora for annotation projection [@ma2014unsupervised; @rasooli2015density].
The second group of approaches deals with transferring source parsing models to target languages. were the first to introduce the idea of delexicalization: removing lexical features by training and cross-lingually applying parsers solely on POS sequences. and independently extended the approach by using multiple sources, requiring uniform POS and dependency representations [@mcdonald2013universal].
Both model transfer and annotation projection rely on a large number of presumptions to derive their competitive parsing models. By and large, these presumptions are unrealistic and exclusive to a group of very closely related, resource-rich Indo-European languages. Agić et al. exposed some of these biases in their proposal for realistic cross-lingual tagging and parsing, as they emphasized the lack of perfect sentence- and word-splitting for truly low-resource languages. Further, introduced joint projection of POS and dependencies from multiple sources while sharing the outlook on bias removal in real-world multilingual processing.
#### Rule-based parsing
Cross-lingual methods, realistic or not, depend entirely on the availability of data: for the sources, for the targets, or most often for both sets of languages. Moreover, they typically do not exploit constraints placed on linguistic structures through a formalism, and they do so [*by design*]{}.
With the emergence of UD as the practical standard for multilingual POS and syntactic dependency annotation, we argue for an approach that takes a fresh angle on both aspects. Specifically, we propose a parser that i) requires [*no*]{} training data, and in contrast ii) critically [*relies*]{} on exploiting the UD constraints.
These two characteristics make our parser unsupervised. Data-driven unsupervised dependency parsing is now a well-established discipline [@klein2004corpus; @spitkovsky2010baby; @spitkovsky2010viterbi]. Still, the performance of these parsers falls far behind the approaches involving any sort of supervision.
Our work builds on the line of research on rule-aided unsupervised dependency parsing by and , and also relates to Søgaard’s work. Our parser, however, features two key differences:
- the usage of PageRank personalization [@lofgren2015efficient], and of
- two-step decoding to treat content and function words differently according to the UD formalism.
Through these differences, even without any training data, we parse nearly as well as a delexicalized transfer parser, and with increased stability to domain change.
Method {#sec:method}
======
Our approach does not use any training or unlabeled data. We have used the English treebank during development to assess the contribution of individual head rules, and to tune PageRank parameters (Sec. \[sec:pagerank\]) and function-word directionality (Sec. \[sec:directionality\]). Adposition direction is calculated on the fly at runtime. We refer henceforth to our UD parser as UDP.
PageRank setup {#sec:pagerank}
--------------
Our system uses the PageRank (PR) algorithm [@page1999pagerank] to estimate the relevance of the content words of a sentence. PR uses a random walk to estimate which nodes in the graph are more likely to be visited often, and thus, it gives higher rank to nodes with more incoming edges, as well as to nodes connected to those. Using PR to score word relevance requires an effective graph-building strategy. We have experimented with the strategies by , such as words being connected to adjacent words, but our system fares best strictly using the dependency rules in Table \[tab:rules\] to build the graph. UD trees are often very flat, and a highly connected graph yields a PR distribution that is closer to uniform, thereby removing some of the difference of word relevance.
We build a multigraph of all words in the sentence covered by the head-dependent rules in Table \[tab:rules\], giving each word an incoming edge for each eligible dependent, i.e., <span style="font-variant:small-caps;">adv</span> depends on <span style="font-variant:small-caps;">adj</span> and <span style="font-variant:small-caps;">verb</span>. This strategy does not always yield connected graphs, and we use a teleport probability of 0.05 to ensure PR convergence.
*Teleport probability* is the probability that, in any iteration of the PR calculation, the next active node is randomly chosen, instead of being one of the adjacent nodes of the current active node. See for more details on teleport probability, where the authors refer to one minus teleport probability as *damping factor*.
We chose this value incrementally in intervals of 0.01 during development until we found the smallest value that guaranteed PR convergence. A high teleport probability is undesirable, because the resulting stationary distribution can be almost uniform. We did not have to re-adjust this value when running on the actual test data.
The main idea behind our personalized PR approach is the observation that ranking is only relevant for content words.[^2] PR can incorporate a priori knowledge of the relevance of nodes by means of *personalization*, namely giving more weight to certain nodes.
Intuitively, the higher the rank of a word, the closer it should be to the root node, i.e., the main predicate of the sentence is the node that should have the highest PR, making it the dependent of the root node (Fig. \[fig:alg\], lines 4-5). We use PR personalization to give 5 times more weight (over an otherwise uniform distribution) to the node that is estimated to be main predicate, i.e., the first verb or the first content word if there are no verbs.
Head direction {#sec:directionality}
--------------
Head direction is an important trait in dependency syntax [@tesniere1959elements]. Indeed, the UD feature inventory contains a trait to distinguish the general adposition tag <span style="font-variant:small-caps;">adp</span> in pre- and post-positions.
Instead of relying on this feature from the treebanks, which is not always provided, we estimate the frequency of <span style="font-variant:small-caps;">adp-nominal</span> vs. <span style="font-variant:small-caps;">nominal-adp</span> bigrams.[^3] We calculate this estimation directly on input data at runtime to keep the system training-free. Moreover, it requires very few examples to converge (10-15 sentences). If a language has more <span style="font-variant:small-caps;">adp-nominal</span> bigrams, we consider all its <span style="font-variant:small-caps;">adp</span> to be prepositions (and thus dependent of elements at their right). Otherwise, we consider them postpositions.
For other function words, we have determined on the English dev data whether to make them strictly right- or left-attaching, or to allow either direction. There, <span style="font-variant:small-caps;">aux</span>, <span style="font-variant:small-caps;">det</span>, and <span style="font-variant:small-caps;">sconj</span> are right-attaching, while <span style="font-variant:small-caps;">conj</span> and <span style="font-variant:small-caps;">punct</span> are left-attaching. There are no direction constraints for the rest. Punctuation is a common source of parsing errors that has very little interest in this setup. While we do evaluate on all tokens including punctuation, we also apply a heuristic for the last token in a sentence; if it is a punctuation, we make it a dependent of the main predicate.
$H = \emptyset$; $D = \emptyset$ $C = \langle c_1, ... c_m\rangle$; $F = \langle f_1, ... f_m\rangle$ $h= root$ $h=$argmin$_{j \in H}$ $ \{ \gamma(j,c)\mid\delta(j,c) \wedge \kappa(j,c) \} $ $H = H \cup \{c\}$ $D = D \cup \{(h,c)\}$ $h=$argmin$_{j \in H}$ $ \{ \gamma(j,f)\mid\delta(j,f) \wedge \kappa(j,f) \} $ $D = D \cup \{(h,f)\}$ $D$
------- ------------------- --------------------
ADJ $\longrightarrow$ ADV
NOUN $\longrightarrow$ ADJ, NOUN, PROPN
NOUN $\longrightarrow$ ADP, DET, NUM
PROPN $\longrightarrow$ ADJ, NOUN, PROPN
PROPN $\longrightarrow$ ADP, DET, NUM
VERB $\longrightarrow$ ADV, AUX, NOUN
VERB $\longrightarrow$ PROPN, PRON, SCONJ
------- ------------------- --------------------
\[tab:rules\]
Decoding {#sec:decoding}
--------
Fig. \[fig:alg\] shows the tree-decoding algorithm. It has two blocks, namely a first block (3-11) where we assign the head of content words according to their PageRank and the constraints of the dependency rules, and a second block (12-15) where we assign the head of function words according to their proximity, direction of attachment, and dependency rules. The algorithm requires:
1. The PR-sorted list of content words $C$.
2. The set of function words $F$, sorting is irrelevant because function-head assignations are inter-independent.
3. A set $H$ for the current possible heads, and a set $D$ for the dependencies assigned at each iteration, which we represent as head-dependent tuples $(h,d)$.
4. A symbol $root$ for the root node.
5. A function $\gamma(n,m)$ that gives the linear distance between two nodes.
6. A function $\kappa(h,d)$ that returns whether the dependency $(h,d)$ has a valid attachment direction given the POS of the $d$ (cf. Sec. \[sec:directionality\]).
7. A function $\delta(h,d)$ that determines whether $(h,d)$ is licensed by the rules in Table \[tab:rules\].
The head assignations in lines 7 and 13 read as follow: the head $h$ of a word (either $c$ or $f$) is the closest element of the current list of heads ($H$) that has the right direction ($\kappa$) and respects the POS-dependency rules ($\delta$). These assignations have a back-off option to ensure the final $D$ is a tree. If the conditions determined by $\kappa$ and $\delta$ are too strict, i.e., if the set of possible heads is empty, we drop the $\delta$ head-rule constraint and recalculate the closest possible head that respects the directionality imposed by $\kappa$. If the set is empty again, we drop both constraints and assign the closest head.
Lines 4 and 5 enforce the single-root constraint. To enforce the leaf status of function nodes, the algorithm first attaches all content words ($C$), and then all function words ($F$) in the second block where H is not updated, thereby ensuring leafness for all $f \in F$. The order of head attachment is not monotonic wrt. PR between the first and second block, and can yield non-projectivities. Nevertheless, it still is a one-pass algorithm. Decoding runs in less than $O(n^2)$, namely $O(n\times\left\vert{C}\right\vert)$. However, running PR incurs the main computation cost.
Parser run example
==================
This section exemplifies a full run of UDP for the example sentence from the English test data: “They also had a special connection to some extremists”.
PageRank
--------
Given an input sentence and its POS tags, we obtain rank of each word by building a graph using head rules and running PR on it. Table \[tbl:sentence\] provides the sentence, the POS of each word, the number of incoming edges for each word after building the graph with the head rules from Sec. \[sec:pagerank\], and the personalization vector for PR on this sentence. Note that all nodes have the same personalization weight, except the estimated main predicate, the verb “had”.
\[tbl:sentence\]
Table \[tbl:graphmatrix\] shows the directed multigraph used for PR in detail. We can see, e.g., that the four incoming edges for the verb “had” from the two nouns, plus from the adverb “also” and the pronoun “They”.
After running PR, we obtain the following ranking for content words:\
$C = \langle$had,connection,extremists,special$\rangle$\
Even though the verb has four incoming edges and the nouns have five each, the personalization makes the verb the highest-ranked word.
Decoding {#decoding}
--------
Once $C$ is calculated, we can follow the algorithm in Fig. \[fig:alg\] to obtain a dependency parse. Table \[tab:trace\] shows a trace of the algorithm, with $C = \langle$had,connection,extremists,special$\rangle$ and $F = \{$They,also,a,to,some}.
The first four iterations calculate the head of content words following their PR, and the following iterations attach the function words in $F$. Finally, Fig. \[fig:tree\] shows the resulting dependency tree. Full lines are assigned in the first block (content dependents), dotted lines are assigned in the second block (function dependents). The edge labels indicate in which iteration the algorithm has assigned each dependency. Note that the algorithm is deterministic for a certain input POS sequence. Any 10-token sentence with the POS labels shown in Table \[tbl:sentence\] would yield the same dependency tree.[^4]
\[tbl:graphmatrix\]
Experiments
===========
This section describes the data, metrics and comparison systems used to assess the performance of UDP. We evaluate on the test sections of the UD1.2 treebanks [@UD12] that contain word forms. If there is more than one treebank per language, we use the treebank that has the canonical language name (e.g., *Finnish* instead of *Finnish-FTB*). We use standard unlabeled attachment score (UAS) and evaluate on all sentences of the canonical UD test sets.
Baseline {#sec:baseline}
--------
We compare our UDP system with the performance of a rule-based baseline that uses the head rules in Table \[tbl:results\]. The baseline identifies the first verb (or first content word if there are no verbs) as the main predicate, and assigns heads to all words according to the rules in Table \[tab:rules\]. We have selected the set of head rules to maximize precision on the development set, and they do not provide full coverage. The system makes any word not covered by the rules (e.g., a word with a POS such as <span style="font-variant:small-caps;">x</span> or <span style="font-variant:small-caps;">sym</span>) either dependent of their left or right neighbor, according to the estimated runtime parameter.
We report the best head direction and its score for each language in Table \[tbl:results\]. This baseline finds the head of each token based on its closest possible head, or on its immediate left or right neighbor if there is no head rule for the POS at hand, which means that this system does not necessarily yield well-formed tress. Each token receives a head, and while the structures are single-rooted, they are not necessarily connected. Note that we do not include results for the DMV model by , as it has been outperformed by a system similar to ours [@sogaard2012unsupervised]. The usual adjacency baseline for unsupervised dependency parsing, where all words depend on their left or right neighbor, fares much worse than our baseline (20% UAS below on average) even with an oracle pick for the best per-language direction, and we do not report those scores.
Evaluation setup
----------------
Our system relies solely on POS tags. To estimate the quality degradation of our system under non-gold POS scenarios, we evaluate UDP on two alternative scenarios. The first is predicted POS (UDP$_P$), where we tag the respective test set with TnT [@brants2000tnt] trained on each language’s training set. The second is a naive type-constrained two-POS tag scenario (UDP$_N$), and approximates a lower bound. We give each word either [content]{} or [function]{} tag, depending on the word’s frequency. The 100 most frequent words of the input test section receive the [function]{} tag.
Finally, we compare our parser UDP to a supervised cross-lingual system (MSD). It is a multi-source delexicalized transfer parser, referred to as [*multi-dir*]{} in the original paper by . For this baseline we train TurboParser [@martins2013turning] on a delexicalized training set of 20k sentences, sampled uniformly from the UD training data excluding the target language. MSD is a competitive and realistic baseline in cross-lingual transfer parsing work. This gives us an indication how our system compares to standard cross-lingual parsers.
Results
-------
Table \[tbl:results\] shows that UDP is a competitive system; because UDP$_G$ is remarkably close to the supervised MSD$_G$ system, with an average difference of 6.4%. Notably, UDP even outperforms MSD on one language (Hindi).
\[tbl:results\]
More interestingly, on the evaluation scenario with predicted POS we observe that our system drops only marginally (2.2%) compared to MSD (2.7%). In the least robust rule-based setup, the error propagation rate from POS to dependency would be doubled, as either a wrongly tagged head or dependent would break the dependency rules. However, with an average POS accuracy by TnT of 94.1%, the error propagation is 0.37, i.e, each POS error causes 0.37 additional dependency errors. In contrast, for MSD this error propagation is 0.46, thus higher. [^5]
For the extreme POS scenario, content vs. function POS (CF), the drop in performance for UDP is very large, but this might be too crude an evaluation setup. Nevertheless, UDP, the simple unsupervised system with PageRank, outperforms the adjacency baselines (BL) by $\sim$4% on average on the two type-based naive POS tag scenario. This difference indicates that even with very deficient POS tags, UDP can provide better structures.
Discussion
==========
In this section we provide a further error analysis of the UDP parser. We examine the contribution to the overal results of using PageRank to score content words, the behavior of the system across different parts of speech, and we assess the robustness of UDP on text from different domains.
PageRank contribution
---------------------
UDP depends on PageRank to score content words, and on two-step decoding to ensure the leaf status of function words. In this section we isolate the constribution of both parts. We do so by comparing the performance of BL, UDP, and UDP$_{NoPR}$, a version of UDP where we disable PR and rank content words according to their reading order, i.e., the first word in the ranking is the first word to be read, regardless of the specific language’s script direction. The baseline BL described in \[sec:baseline\] already ensures function words are leaf nodes, because they have no listed dependent POS in the head rules. The task of the decoding steps is mainly to ensure the resulting structures are well-formed dependency trees.
If we measure the difference between UDP$_{NoPR}$ and BL, we see that UDP$_{NoPR}$ contributes with 4 UAS points on average over the baseline. Nevertheless, the baseline is oracle-informed about the language’s best branching direction, a property that UDP does not have. Instead, the decoding step determines head direction as described in Section \[sec:directionality\]. Complementarily, we can measure the contribution of PR by observing the difference between regular UDP and UDP$_{NoPR}$. The latter scores on average 9 UAS points lower than UDP. These 9 points are caused by the difference attachment of content words in the first decoding step.
Breakdown by POS {#sec:analysis}
----------------
UD is a constantly improving effort, and not all v1.2 treebanks have the same level of formalism compliance. Thus, the interpretation of, e.g., the <span style="font-variant:small-caps;">aux</span>–<span style="font-variant:small-caps;">verb</span> or <span style="font-variant:small-caps;">det</span>–<span style="font-variant:small-caps;">pron</span> distinctions might differ across treebanks. However, we ignore these differences in our analysis and consider all treebanks to be equally compliant.
The root accuracy scores oscillate around an average of 69%, with Arabic and Tamil (26%) and Estonian (93%) as outliers. Given the PR personalization (Sec. \[sec:pagerank\]), UDP has a strong bias for choosing the first verb as main predicate. Without personalization, performance drops 2% on average. This difference is consistent even for verb-final languages like Hindi, given that the main verb of a simple clause will be its only verb, regardless of where it appears. Moreover, using PR personalization makes the ranking calculations converge a whole order of magnitude faster.
The bigram heuristic to determine adposition direction succeeds at identifying the predominant pre- or postposition preference for all languages (average <span style="font-variant:small-caps;">adp</span> UAS of 75%). The fixed direction for the other functional POS is largely effective, with few exceptions, e.g., <span style="font-variant:small-caps;">det</span> is consistently right-attaching on all treebanks except Basque (average overall <span style="font-variant:small-caps;">det</span> UAS of 84%, 32% for Basque). These alternations could also be estimated from the data in a manner similar to <span style="font-variant:small-caps;">adp</span>. Our rules do not make nouns eligible heads for verbs. As a result, the system cannot infer relative clauses. We have excluded the <span style="font-variant:small-caps;">noun</span> $\rightarrow$ <span style="font-variant:small-caps;">verb</span> rule during development because it makes the hierarchy between verbs and nouns less conclusive.
We have not excluded punctuation from the evaluation. Indeed, the UAS for the <span style="font-variant:small-caps;">punct</span> is low (an average of 21%, standard deviation of 9.6), even lower than the otherwise problematic <span style="font-variant:small-caps;">conj</span>. Even though conjunctions are pervasive and identifying their scope is one of the usual challenges for parsers, the average UAS for <span style="font-variant:small-caps;">conj</span> is much larger (an average of 38%, standard deviation of 13.5) than for <span style="font-variant:small-caps;">punct</span>. Both POS show large standard deviations, which indicates great variability. This variability can be caused by linguistic properties of the languages or evaluation datasets, but also by differences in annotation convention.
Cross-domain consistency
------------------------
Models with fewer parameters are less likely to overfit for a certain dataset. In our case, a system with few, general rules is less likely to make attachment decisions that are very particular of a certain language or dataset. have shown that rule-based parsers can be more stable to domain shift. We explore if their finding holds for UDP as well, by testing on i) the UD development data as a readily available proxy for domain shift, and ii) manually curated domain splits of select UD test sets.
\[tbl:domains\]
**Development sets** We have used the English development data to choose which relations would be included as head rules in the final system (Table \[tab:rules\]). It would be possible that some of the rules are indeed more befitting for the English data or for that particular section.
However, if we regard the results for UDP$_G$ in Table \[tbl:results\], we can see that there are 24 languages (out of 32) for which the parser performs better than for English. This result indicates that the head rules are general enough to provide reasonable parses for languages other than the one chosen for development. If we run UDP$_G$ on the development sections for the other languages, we find the results are very consistent. Any language scores on average $\pm 1$ UAS with regards to the test section. There is no clear tendency for either section being easier to parse with UDP.
**Cross-domain test sets** To further assess the cross-domain robustness, we retrieved the domain (genre) splits from the test sections of the UD treebanks where the domain information is available as sentence metadata: from Bulgarian, Croatian, and Italian. We also include a UD-compliant Serbian dataset which is not included in the UD release but which is based on the same parallel corpus as Croatian and has the same domain splits [@agic2015universal]. When averaging we pool Croatian and Serbian together as they come from the same dataset.
For English, we have obtained the test data splits matching the sentences from the original distribution of the English Web Treebank. In addition to these already available datasets, we have annotated three different datasets to assess domain variation more extensively, namely the first 50 verses of the King James Bible, 50 sentences from a magazine, and 75 sentences from the test split in QuestionBank [@judge2006questionbank]. We include the third dataset to evaluate strictly on questions, which we could do already in Italian. While the `answers` domain in English is made up of text from the Yahoo! Answers forum, only one fourth of the sentences are questions. Note these three small datasets are not included in the results on the canonical test sections in Table \[tbl:results\].
\[tbl:results-mean-stdev\]
Table \[tbl:results-mean-stdev\] summarizes the per-language average score and standard deviation, as well as the macro-averaged standard deviation across languages. UDP has a much lower standard deviation across domains compared to MSD. This holds across languages. We attribute this higher stability to UDP being developed to satisfy a set of general properties of the UD syntactic formalism, instead of being a data-driven method more sensitive to sampling bias. This holds for both the gold-POS and predicted-POS setup. The differences in standard deviation are unsurprisingly smaller in the predicted POS setup. In general, the rule-based UPD is less sensitive to domain shifts than the data-driven MSD counterpart, confirming earlier findings [@plank-vannoord:2010].
Table \[tbl:domains\] gives the detailed scores per language and domain. From the scores we can see that presidential `bulletin`, `legal` and `weblogs` are amongst the hardest domains to parse. However, the systems often do not agree on which domain is hardest, with the exception of Bulgarian `bulletin`. Interestingly, for the Italian data and some of the hardest domains UDP outperforms MSD, confirming that it is a robust baseline.
Comparison to full supervision
------------------------------
In order to assess how much information the simple principles in UDP provide, we measure how many gold-annotated sentences are necessary to reach its performance, that is, after which size the treebank provides enough information for training that goes beyond the simple linguistic principles outlined in Section \[sec:method\].
For this comparison we use a first-order non-projective TurboParser [@martins2013turning] following the setup of . The supervised parsers require around 100 sentences to reach UDP-comparable performance, namely a mean of 300 sentences and a median of 100 sentences, with Bulgarian (3k), Czech (1k), and German (1.5k) as outliers. The difference between mean and median shows there is great variance, while UDP provides very constant results, also in terms of POS and domain variation.
Conclusion
==========
We have presented UDP, an unsupervised dependency parser for Universal Dependencies (UD) that makes use of personalized PageRank and a small set of head-dependent rules. The parser requires no training data and estimates adposition direction directly from the input.
We achieve competitive performance on all but two UD languages, and even beat a multi-source delexicalized parser (MSD) on Hindi. We evaluated the parser on three POS setups and across domains. Our results show that UDP is less affected by deteriorating POS tags than MSD, and is more resilient to domain changes. Given how much of the overall dependency structure can be explained by this fairly system, we propose UDP as an additional UD parsing baseline. The parser, the in-house annotated test sets, and the domain data splits are made freely available.[^6]
UD is a running project, and the guidelines are bound to evolve overtime. Indeed, the UD 2.0 guidelines have been recently released. UDP can be augmented with edge labeling for some deterministic labels like `case` or `det`. Some further constrains can be incorporated in UDP. Moreover, the parser makes no special treatment of multiword expression that would require a lexicon, coordinations or proper names. All these three kinds of structures have a flat tree where all words depend on the leftmost one. While coordination attachment is a classical problem in parsing and out of the scope of our work, a proper name sequence can be straightforwardly identified from the part-of-speech tags, and it falls thus in the area of structures predictable using simple heuristics. Moreover, our use of PageRank could be expanded to directly score the potential dependency edges instead of words, e.g., by means of edge reification.
Acknowledgments {#acknowledgments .unnumbered}
===============
We thank the anonymous reviewers for their valuable feedback. Héctor Martínez Alonso is funded by the French DGA project VerDi. Barbara Plank thanks the Center for Information Technology of the University of Groningen for the HPC cluster. Željko Agić and Barbara Plank thank the Nvidia Corporation for supporting their research. Anders Søgaard is funded by the ERC Starting Grant LOWLANDS No. 313695.
[^1]: Out of 33 languages in UD v1.2. We exclude Japanese because the treebank is distributed without word forms and hence we can not provide results on predicted POS.
[^2]: <span style="font-variant:small-caps;">adj</span>, <span style="font-variant:small-caps;">noun</span>, <span style="font-variant:small-caps;">propn</span>, and <span style="font-variant:small-caps;">verb</span> mark content words.
[^3]: <span style="font-variant:small-caps;">nominal</span>$=\{$<span style="font-variant:small-caps;">noun, propn, pron</span>$\}$
[^4]: The resulting trees always pass the validation script in [github.com/UniversalDependencies/tools](github.com/UniversalDependencies/tools).
[^5]: Err. prop. $=(E(Parse_P)-E(Parse_G))/E(POS_P)$, where $E(x) = 1 - Accuracy(x)$.
[^6]: <https://github.com/hectormartinez/ud_unsup_parser>
|
---
abstract: 'Iterative Rational Krylov Algorithm () of [@H2] is an interpolatory model reduction approach to optimal ${\ensuremath{\mathcal{H}_{2}}}$ approximation problem. Even though the method has been illustrated to show rapid convergence in various examples, a proof of convergence has not been provided yet. In this note, we show that in the case of state-space symmetric systems, is a locally convergent fixed point iteration to a local minimum of the underlying ${\ensuremath{\mathcal{H}_{2}}}$ approximation problem.'
author:
- |
Garret Flagg, Christopher Beattie, Serkan Gugercin\
[Department of Mathematics, Virginia Tech.]{}\
[Blacksburg, VA, 24061-0123]{}\
[e-mail: {flagg,beattie,gugercin}@math.vt.edu]{}
bibliography:
- 'references.bib'
title: Convergence of the Iterative Rational Krylov Algorithm
---
Introduction
============
Consider a single-input-single-output (SISO) linear dynamical system in state-space form: $$\begin{aligned}
\label{ltisystemintro}
\dot{{\ensuremath{\boldsymbol x}}}(t) = {\ensuremath{\boldsymbol A}}\,{\ensuremath{\boldsymbol x}}(t) + {\ensuremath{\boldsymbol b}}\,u(t),\qquad
y(t) = {\ensuremath{\boldsymbol c}}^T {\ensuremath{\boldsymbol x}}(t),\end{aligned}$$ where ${\ensuremath{\boldsymbol A}}\in \IR^{n \times n}$, and ${\ensuremath{\boldsymbol b}},{\ensuremath{\boldsymbol c}}\in \IR^{n}$. In (\[ltisystemintro\]), ${\ensuremath{\boldsymbol x}}(t)\in \IR^n$, $u(t)\in \IR$, $y(t)\in \IR$, are, respectively, the *states*, *input*, and *output* of the dynamical system. The transfer function of the underlying system is $H(s) = {\ensuremath{\boldsymbol c}}^T(s{\ensuremath{\boldsymbol{I}}}-{\ensuremath{\boldsymbol A}})^{-1}{\ensuremath{\boldsymbol b}}$. $H(s)$ will be used to denote both the system and its transfer function.
Dynamical systems of the form (\[ltisystemintro\]) with large state-space dimension $n$ appear in many applications; see, e.g., [@antB] and [@KorR05]. Simulations in such large-scale settings make enormous demands on computational resources. The goal of model reduction is to construct a surrogate system $$\begin{aligned}
\label{redsysintro}
\dot{{\ensuremath{\boldsymbol x}}}_r (t) = {\ensuremath{\boldsymbol A}}_r\, {\ensuremath{\boldsymbol x}}_r (t) + {\ensuremath{\boldsymbol b}}_r u(t),\qquad
y_r(t) = {\ensuremath{\boldsymbol c}}_r^T {\ensuremath{\boldsymbol x}}_r (t),\end{aligned}$$ of much smaller dimension $r \ll n$, with ${\ensuremath{\boldsymbol A}}_r \in \IR^{r \times r}$ and ${\ensuremath{\boldsymbol b}}_r,\,{\ensuremath{\boldsymbol c}}_r \in \IR^{r}$ such that $y_r(t)$ approximates $y(t)$ well in a certain norm. Similar to $H(s)$, the transfer function $H_r(s)$ of the reduced-model (\[redsysintro\]) is given by $H_r(s) = {\ensuremath{\boldsymbol c}}_r^T(s{\ensuremath{\boldsymbol{I}}}_r-{\ensuremath{\boldsymbol A}}_r)^{-1}{\ensuremath{\boldsymbol b}}_r$. We consider reduced-order models, $H_r(s)$, that are obtained via projection. That is, we choose full rank matrices ${\ensuremath{\boldsymbol{V}_r}},{\ensuremath{\boldsymbol{W}_r}}\in \IR^{n \times r}$ such that ${\ensuremath{\boldsymbol{W}_r}}^T{\ensuremath{\boldsymbol{V}_r}}$ is invertible and define the reduced-order state-space realization with (\[redsysintro\]) and [$$\label{red_projection}
{\ensuremath{\boldsymbol{A}_r}}= (\boldsymbol{W}_r^T\boldsymbol{V}_r)^{-1}{\ensuremath{\boldsymbol{W}_r}}^T {\ensuremath{\boldsymbol A}}{\ensuremath{\boldsymbol{V}_r}},~{\ensuremath{\boldsymbol {b}_r}}= (\boldsymbol{W}_r^T\boldsymbol{V}_r)^{-1}{\ensuremath{\boldsymbol{W}_r}}^T{\ensuremath{\boldsymbol b}},~
{\ensuremath{\boldsymbol{c}_r}}= {\ensuremath{\boldsymbol{V}_r}}^T{\ensuremath{\boldsymbol c}}.$$ ]{} Within this “projection framework," selection of ${\ensuremath{\boldsymbol{W}_r}}$ and ${\ensuremath{\boldsymbol{V}_r}}$ completely determines the reduced system – indeed, it is sufficient to specify only the *ranges* of ${\ensuremath{\boldsymbol{W}_r}}$ and ${\ensuremath{\boldsymbol{V}_r}}$ in order to determine $H_r(s)$. Of particular utility for us is a result by Grimme [@Grim], that gives conditions on ${\ensuremath{\boldsymbol{W}_r}}$ and ${\ensuremath{\boldsymbol{V}_r}}$ so that the associated reduced-order system, $H_r(s)$, is a *rational Hermite interpolant* to the original system, $H(s)$.
\[thm:rationalkrylov\] Given $\Hs={\ensuremath{\boldsymbol c}}^T(s{\ensuremath{\boldsymbol{I}}}-{\ensuremath{\boldsymbol A}})^{-1}{\ensuremath{\boldsymbol b}}$, and $r$ distinct points $s_1, \dots, s_{r} \in {\ensuremath{\mathbb{C}}}$, let $$\label{eqn:VrWr}
\boldsymbol{V}_r=\lbrack (s_1\boldsymbol{I} -\boldsymbol{A})^{-1}\boldsymbol{b} \dots (s_r\boldsymbol{I} -\boldsymbol{A})^{-1}\boldsymbol{b}\rbrack
\qquad
\boldsymbol{W}_r^T=
\begin{bmatrix}
\boldsymbol{c}^T(s_{1}\boldsymbol{I}-\boldsymbol{A})^{-1}\\
\vdots \\
\boldsymbol{c}^T(s_{r}\boldsymbol{I}-\boldsymbol{A})^{-1}
\end{bmatrix}.$$
Define the reduced-order model $H_r(s) = {\ensuremath{\boldsymbol{c}_r}}^T(s {\ensuremath{\boldsymbol{I}}}_r - {\ensuremath{\boldsymbol{A}_r}})^{-1}{\ensuremath{\boldsymbol {b}_r}}$ as in (\[red\_projection\]) Then $H_r$ is a rational Hermite interpolant to $H$ at $s_1, \dots, s_{r}$: $$\label{eqn:hermite}
H(s_i) = H_r(s_i) \qquad {\rm and} \qquad H'(s_i) = H'_r(s_i)~~~{\rm for}~~~i=1,\ldots,r.$$
Rational interpolation within this “projection framework" was first proposed by Skelton *et al*. [@Skeltonlate],[@Skelt1],[@Skelt2]. Later in [@Grim], Grimme established the connection with the rational Krylov method of Ruhe [@ruhe1998rational].
Significantly, Theorem \[thm:rationalkrylov\] gives an explicit method for computing a reduced-order system that is a Hermite interpolant of the orginal system for nearly *any* set of distinct points, $\{s_1, \dots, s_{r} \}$, yet it is not apparent how one should choose these interpolation points in order to assure a high-fidelity reduced-order model in the end. Indeed, the lack of such a strategy had been a major drawback for interpolatory model reduction until recently, when an effective strategy for selecting interpolation points was proposed in [@H2] yielding reduced-order models that solve $$\label{optimalHtwo}
\| H - H_r \|_{{\ensuremath{\mathcal{H}_{2}}}} =
\min \limits_{dim(\hat{H}_r) = r} \left\| H-\hat{H}_r \right\|_{{\ensuremath{\mathcal{H}_{2}}}}.$$ where the [$\mathcal{H}_2 \text{ }$]{} system norm is defined in the usual way: $$\left \| H \right \|_{{\ensuremath{\mathcal{H}_{2}}}} = \left(\frac{1}{2\pi} \int_{-\infty}^\infty \mid H(\jmath \omega) \mid^2 d \omega \right)^{1/2}.$$
The optimization problem (\[optimalHtwo\]) has been studied extensively, see, for example, [@meieriii1967approximation; @wilson1970optimum; @H2; @spanos1992anewalgorithm; @fulcheri1998mrh; @vandooren2008hom; @gugercin2005irk; @beattie2007kbm; @beattie2009trm; @zigic1993contragredient] and references therein. (\[optimalHtwo\]) is a nonconvex optimization problem and finding global minimizers will be infeasible, typically. Hence, the usual interpretation of (\[optimalHtwo\]) involves finding *local* minimizers and a common approach to accomplish this is to construct reduced-order models satisfying first-order necessary optimality conditions. This may be posed either in terms of solutions to Lyapunov equations (e.g., [@wilson1970optimum; @spanos1992anewalgorithm; @zigic1993contragredient]) or in terms of interpolation (e.g., [@wilson1970optimum; @H2; @vandooren2008hom; @beattie2009trm]):
([@meieriii1967approximation; @H2]) \[thm:h2optimal\] Given , let $H_r(s)$ be a solution to (\[optimalHtwo\]) with simple poles $\hat{\lambda}_1,\dots,\hat{\lambda}_r$. Then $$\label{eqn:h2cond}
H(-\hat{\lambda}_i)=H_r(-\hat{\lambda}_i)~~~{\rm and}~~~
H'(-\hat{\lambda}_i)=H_r'(-\hat{\lambda}_i)~~~{\rm for}~~~
i=1,\dots,r.$$ That is, any $\mathcal{H}_2$-optimal reduced order model of order $r$ with simple poles will be a Hermite interpolant to $H(s)$ at the reflected image of the reduced poles through the origin.
Although this result might appear to reduce the problem of $\mathcal{H}_2$-optimal model approximation to a straightforward application of Theorem \[thm:rationalkrylov\] to calculate a Hermite interpolant on the set of reflected poles, $\{-\hat{\lambda}_1,\dots,-\hat{\lambda}_r\}$, these pole locations will not be known *a priori*. Nonetheless, these pole locations can be determined efficiently with the Iterative Rational Krylov Algorithm () of Gugercin *et al.* [@H2]. Starting from an arbitrary initial selection of interpolation points, iteratively corrects the interpolation points until (\[eqn:h2cond\]) is satisfied. A brief sketch of is given below.
has been remarkably successful in producing high fidelity reduced-order approximations and has been successfully applied to finding ${\ensuremath{\mathcal{H}_{2}}}$-optimal reduced models for systems of high order, $n>160,000$, see [@KRXC08]. For details on , see [@H2].
Notwithstanding typically observed rapid convergence of the iteration to interpolation points that generally yield high quality reduced models, no convergence theory for has yet been established. Evidently from the description above, may be viewed as a fixed point iteration with fixed points coinciding with the stationary points of the ${\ensuremath{\mathcal{H}_{2}}}$ minimization problem. Saddle points and local maxima of the ${\ensuremath{\mathcal{H}_{2}}}$ minimization problem are known to be repellent [@krajewski]. However, despite effective performance in practice, it has not yet been established that local minima are attractive fixed points.
In this paper, we give a proof of this for the special case of state-space-symmetric systems and establish the convergence of for this class of systems.
State-Space-Symmetric Systems
=============================
\[dfn:SSS\] =${\ensuremath{\boldsymbol c}}^T(s{\ensuremath{\boldsymbol{I}}}-{\ensuremath{\boldsymbol A}})^{-1}{\ensuremath{\boldsymbol b}}$ is *state-space-symmetric* () if [$\boldsymbol A$]{}=${\ensuremath{\boldsymbol A}}^T$ and ${\ensuremath{\boldsymbol c}}={\ensuremath{\boldsymbol b}}$.
systems appear in many important applications such as in the analysis of RC circuits and in inverse problems involving 3D Maxwell’s equations [@druskin2009solution].
A closely related class of systems is the class of zero-interlacing-pole () systems.
\[dfn:ZIP\] A system ${\displaystyle \Hs=K\frac{\prod\limits_{i=1}^{n-1}(s-z_i)}{\prod\limits_{j=1}^{n}(s-\lambda_j)} }$ is a *strictly proper system* provided that $$0>\lambda_1 >z_1 >\lambda_2 > z_2 > \lambda_3 > \dots > z_{n-1} > \lambda_n.$$
The following relation serves to characterize systems.
[@Zipchar] \[prop:zipchar\] is a strictly proper system if and only if can be written as ${\displaystyle \Hs=\sum_{i=1}^{n}\frac{b_i}{s-\lambda_i}}$ with $\lambda_i < 0$, $b_i>0$, and $ \lambda_i\ne \lambda_j$ for all $ i\ne j.$
The next result clarifies the relationship between and systems.
\[lemma:reducessslemma\] [@zippreserve] Let be . Then is minimal if and only if the poles of are distinct. Moreover, every system has a minimal realization with distinct poles, and is therefore a strictly proper system.
It can easily be verified from the implementation of given above, that for systems, the relationship [$\boldsymbol{V}_r$]{}=[$\boldsymbol{W}_r$]{} is maintained throughout the iteration, and the final reduced-order model at Step $4$ of can be obtained by $$\label{symmetry_preserve}
\begin{array}{cc}
{\ensuremath{\boldsymbol{A}_r}}={\ensuremath{\boldsymbol{Q}_r}}^T{\ensuremath{\boldsymbol A}}{\ensuremath{\boldsymbol{Q}_r}}&{\ensuremath{\boldsymbol {b}_r}}={\ensuremath{\boldsymbol{c}_r}}={\ensuremath{\boldsymbol{Q}_r}}^T{\ensuremath{\boldsymbol b}},
\end{array}$$ where [$\boldsymbol{Q}_r$]{} is an orthonormal basis for [$\boldsymbol{V}_r$]{}; the reduced system resulting from is also .
The Main Result {#sec:proof_thm:localattractor}
===============
\[thm:localattractor\] Let be applied to a minimal system $H(s)$. Then every fixed point of which is a local minimizer is locally attractive. In other words, is a locally convergent fixed point iteration to a local minimizer of the ${\ensuremath{\mathcal{H}_{2}}}$ optimization problem.
To proceed with the proof of Theorem \[thm:localattractor\], we need four intermediate lemmas. The first lemma provides insight into the structure of the zeros of the error system resulting from reducing a system.
\[lemma:zero\_structure\] Let be a system of order $n$. If $H_r(s)$ is a ZIP system that interpolates at $2r$ points $s_1,s_2,\ldots,s_{2r}$, not necessarily distinct, in $(0, \infty),$ then all the remaining zeros of the error system lie in $(-\infty,0)$.
By Lemma \[lemma:reducessslemma\], we may assume that is a strictly proper systems. Since is a strictly proper system, its poles are simple and all its residues are positive. Let $\lambda_i <0, \phi_i>0,$ for $i=1,\dots,n$ be the poles and residues of $H(s)$, respectively. Now let $$R(s)=\prod \limits_{i=1}^{2r}(s-s_i),~~P(s)=\prod\limits_{i=1}^{n-r-1}(s+z_i),~~Q(s)=\prod\limits_{i=1}^{n}(s-\lambda_i),~~\tilde{Q}(s)=\prod \limits_{i=1}^{r}(s-\tilde{\lambda}_i),$$ where $\tilde{\lambda}_i$, $s_i$, and $z_i$ are, respectively, the poles of [$H_r(s)$]{}, the interpolation points, and the remaining zeros of the error system. Then for some constant $K$, $\Hs-H_r(s)=K\frac{P(s)R(s)}{Q(s)\tilde{Q}(s)}$. First suppose that $\{\lambda_i\}_{i=1}^n \cap \{\tilde{\lambda}_k\}_{k=1}^r=\emptyset$. Then for each $\lambda_j$, $j=1,\dots,n$, $$\text{Res}(\Hs-H_r(s);\lambda_j)=K\frac{P(\lambda_j)R(\lambda_j)}{\prod\limits_{\substack{i=1\\ \lambda_i\ne\lambda_j}}^{n}(\lambda_j-\lambda_i)\tilde{Q}(\lambda_j)}=\phi_i>0.$$ Thus, ${\mathop{\mathrm{sgn}}}(KP(\lambda_j))=(-1)^{j-1}{\mathop{\mathrm{sgn}}}(\tilde{Q}(\lambda_j))$ where ${\mathop{\mathrm{sgn}}}(\alpha)$ denotes the sign of $\alpha$. Now if $(-1)^{j-1}{\mathop{\mathrm{sgn}}}(\tilde{Q}(\lambda_j))=(-1)^{j}({\mathop{\mathrm{sgn}}}(\tilde{Q}(\lambda_{j+1}))$, then $-{\mathop{\mathrm{sgn}}}(\tilde{Q}(\lambda_j))= {\mathop{\mathrm{sgn}}}(\tilde{Q}(\lambda_{j+1}))$, so $\tilde{Q}(s)$ must change sign on the interval $[\lambda_{j+1}, \lambda_j]$. Since $\tilde{Q}(s)$ is a polynomial of degree $r$, and $r<n$, $\tilde{Q}(s)$ can switch signs at most $r$ times, else $\tilde{Q}(s)\equiv 0$. But this means there are at least $n-r-1$ intervals $[\lambda_{j_k+1}, \lambda_{j_k}]$, for $k=1, \dots, n-r-1$, for which ${\mathop{\mathrm{sgn}}}(\tilde{Q}(\lambda_{j_k}))= {\mathop{\mathrm{sgn}}}(\tilde{Q}(\lambda_{j_k+1}))$, and therefore ${\mathop{\mathrm{sgn}}}(KP(\lambda_{j_k}))=-{\mathop{\mathrm{sgn}}}(KP(\lambda_{j_k+1}))$. So $KP(s)$ must change sign over at least $n-r-1$ intervals, and therefore has at least $n-r-1$ zeros on $[\lambda_n, \lambda_1]$. Again, since the error is not identically zero when $r<n$, and the degree of $KP(s)$ is $n-r-1$, this implies that all the zeros of $KP(s)$ lie in $(-\infty, 0)$.
Suppose with some $p \le r$, $\lambda_{i_j}=\tilde{\lambda}_{k_j}$ for $j=1, \dots, p$. Observe from partial fraction expansions of $\Hs$ and $H_r(s)$ that the error can be written as a rational function of degree $n+r-p-1$ over degree $n+r-p$ with distinct poles. $n$ of these poles belong to $H(s)$ and the remaining $r-p$ come from the poles of $H_r(s)$ that are distinct from the poles of $H(s)$. Now let $$R(s)=\prod \limits_{i=1}^{2r}(s-s_i),~~P(s)=\prod\limits_{i=1}^{n-r-p-1}(s+z_i),~~Q(s)=\prod\limits_{i=1}^{n}(s-\lambda_i),~~\tilde{Q}(s)=\prod \limits_{l=1}^{r-p}(s-\tilde{\lambda}_{k_l}),$$ where $\{\tilde{\lambda}_{k_l}\}_{l=1}^{r-p}=\{\tilde{\lambda}_k\}_{k=1}^r\setminus \{\lambda_i\}_{i=1}^n$. Hence, $\Hs-H_r(s)=K\frac{P(s)R(s)}{Q(s)\tilde{Q}(s)}$. Observe that there are at most $2p$ subintervals of the form $[\lambda_{i^*}, \lambda_{i^*+1}]$ or $[\lambda_{i^*-1}, \lambda_{i^*}]$, where $\lambda_{i^*} \in \{\lambda_i\}_{i=1}^n \cap \{\tilde{\lambda}_k\}_{k=1}^r$. It follows that there are at least $n-2p-1$ subintervals of the form $[\lambda_i, \lambda_{i+1}]$, where $\lambda_i, \lambda_{i+1} \not \in \{\lambda_i\}_{i=1}^n \cap \{\tilde{\lambda}_k\}_{k=1}^r$. On each such subinterval for which this is the case, we have
$$\text{Res}(\Hs-H_r(s);\lambda_i)=K\frac{P(\lambda_i)R(\lambda_i)}{\prod\limits_{\substack{j=1\\ \lambda_j\ne\lambda_i}}^{n}(\lambda_i-\lambda_j)\tilde{Q}(\lambda_i)}=\phi_i>0.$$
So ${\mathop{\mathrm{sgn}}}(KP(\lambda_i))=(-1)^{i-1}{\mathop{\mathrm{sgn}}}(\tilde{Q}(\lambda_i))$. By the same argument as above where the poles of $H(s)$ and $H_r(s)$ are distinct, either $\tilde{Q}(s)$ or $P(s)$ has a zero on the interval $[\lambda_i, \lambda_{i+1}]$. Since $\tilde{Q}(s)$ has at most $r-p$ zeros, this means that there are at least $n-2p-1-(r-p)=n-p-r-1$ subintervals between poles of $H(s)$ where $P(s)$ has zeros. Hence, the lemma is proved.
\[lemma:positiveerror\] Let $H(s)={\ensuremath{\boldsymbol b}}^T(s{\ensuremath{\boldsymbol{I}}}-{\ensuremath{\boldsymbol A}})^{-1}{\ensuremath{\boldsymbol b}}$ be , and $H_r(s)={\ensuremath{\boldsymbol {b}_r}}^T(s{\ensuremath{\boldsymbol{I}}}_r-{\ensuremath{\boldsymbol{A}_r}})^{-1}{\ensuremath{\boldsymbol {b}_r}}$ be any reduced order model of constructed by a compression of , i.e., [$\boldsymbol{A}_r$]{}=${\ensuremath{\boldsymbol{Q}_r}}^T{\ensuremath{\boldsymbol{A}_r}}{\ensuremath{\boldsymbol{Q}_r}}$, [$\boldsymbol {b}_r$]{}$={\ensuremath{\boldsymbol{Q}_r}}^T{\ensuremath{\boldsymbol b}}$. Then for any $s \ge 0$, $\Hs-H_r(s) \ge 0$.
Pick any $s \ge 0$. Then $(s{\ensuremath{\boldsymbol{I}}}_n-{\ensuremath{\boldsymbol A}})$ is symmetric, positive definite and has a Cholesky decomposition, $(s{\ensuremath{\boldsymbol{I}}}_n-{\ensuremath{\boldsymbol A}})= \boldsymbol{LL}^T$. Define ${\ensuremath{\boldsymbol Z}_r}=\boldsymbol{L}^T{\ensuremath{\boldsymbol{Q}_r}}$. Then $$\begin{aligned}
H(s)-H_r(s) =&\ {\ensuremath{\boldsymbol b}}^T\left[ (s{\ensuremath{\boldsymbol{I}}}_n-{\ensuremath{\boldsymbol A}})^{-1} -{\ensuremath{\boldsymbol{Q}_r}}\left({\ensuremath{\boldsymbol{Q}_r}}^T(s{\ensuremath{\boldsymbol{I}}}_n-{\ensuremath{\boldsymbol A}}){\ensuremath{\boldsymbol{Q}_r}}\right)^{-1}{\ensuremath{\boldsymbol{Q}_r}}^T \right]{\ensuremath{\boldsymbol b}}\\
=&\ (\boldsymbol{L}^{-1}{\ensuremath{\boldsymbol b}})^T\left[ {\ensuremath{\boldsymbol{I}}}-{\ensuremath{\boldsymbol Z}_r}\left({\ensuremath{\boldsymbol Z}_r}^T{\ensuremath{\boldsymbol Z}_r}\right)^{-1}{\ensuremath{\boldsymbol Z}_r}^T \right](\boldsymbol{L}^{-1}{\ensuremath{\boldsymbol b}}).\end{aligned}$$ Note the last bracketed expression is an orthogonal projector onto $\mathsf{Ran}({\ensuremath{\boldsymbol Z}_r})^\perp$, hence is positive semidefinite and the conclusion follows.
Our convergence analysis of will use its formulation as a fixed-point iteration. The analysis will build on the framework of [@krajewski]. Let $$\Hs=\sum \limits_{i=1}^n\frac{\phi_i}{s-\lambda_i} {\rm~~~and~~~} H_r(s)=\sum\limits_{j=1}^{r}\frac{\tilde{\phi}_j}{s-\tilde{\lambda}_j}$$ be the partial fraction decompositions of , and $H_r(s)$, respectively. Given a set of $r$ interpolation points $\{s_i\}_{i=1}^r$, identify the set with a vector ${\ensuremath{\boldsymbol{s}}}=[s_1, \dots, s_r]^T$. Construct an interpolatory reduced order model [$H_r(s)$]{} from [$\boldsymbol{s}$]{} as in Theorem \[thm:rationalkrylov\] and identify $\{\tilde{\lambda}_i\}_{i=1}^r$ with a vector $\tilde{\boldsymbol{\lambda}}=[\tilde{\lambda}_1, \dots, \tilde{\lambda}_r]^T$. Then define the function $\lambda:{\ensuremath{\mathbb{C}}}^{r} \rightarrow {\ensuremath{\mathbb{C}}}^r$ by $\lambda({\ensuremath{\boldsymbol{s}}})=-\tilde{\boldsymbol{\lambda}}$. Aside from ordering issues, this function is well defined, and the iteration converges when $\lambda({\ensuremath{\boldsymbol{s}}})={\ensuremath{\boldsymbol{s}}}$. Thus convergence of is equivalent to convergence of a fixed point iteration on the function $\lambda({\ensuremath{\boldsymbol{s}}})$. Similar to ${\ensuremath{\boldsymbol{s}}}$ and $\tilde{\boldsymbol{\lambda}}$, let $\tilde{\boldsymbol{\phi}} = [\tilde{\phi}_1, \dots, \tilde{\phi}_r]^T$. Having identified [$H_r(s)$]{} with its poles and residues, the optimal ${\ensuremath{\mathcal{H}_{2}}}$ model reduction problem may be formulated in terms of minimizing the cost function ${\ensuremath{\boldsymbol{\mathcal{J}}}}(\tilde{\boldsymbol{\phi}}, \lambda({\ensuremath{\boldsymbol{s}}})) = \|H-H_r \|_{{\ensuremath{\mathcal{H}_2 \text{ }}}}^2$, where $$\begin{aligned}
\label{h2error}
{\ensuremath{\boldsymbol{\mathcal{J}}}}(\tilde{\boldsymbol{\phi}}, \lambda({\ensuremath{\boldsymbol{s}}}))&=\sum \limits^{n}_{i=1} \phi_i(H(\lambda_i)-H_r(\lambda_i)) + \sum \limits^{r}_{j=1} \tilde{\phi}_j(H(\tilde{\lambda}_j)-H_r(\tilde{\lambda}_j))\end{aligned}$$ See [@H2] for a derivation of (\[h2error\]). Define the matrices $\boldsymbol{S}_{11},\boldsymbol{S}_{12},\boldsymbol{S}_{22} \in \IR^{r \times r}$ as $$\left[\boldsymbol{S}_{11}\right]_{i,j} = -(\tilde{\lambda}_i+\tilde{\lambda}_j)^{-1},~~\left[\boldsymbol{S}_{12}\right]_{i,j} = -(\tilde{\lambda}_i+\tilde{\lambda}_j)^{-2}~~{\rm and}~~
\left[\boldsymbol{S}_{22}\right]_{i,j} = -2(\tilde{\lambda}_i +\tilde{\lambda}_j)^{-3}$$ for $i,j=1,\ldots,r$. Also, define $\boldsymbol{R},\boldsymbol{E} \in \IR^{r\times r}$: $$\boldsymbol{R}={\rm diag}(\{\tilde{\phi}_1, \dots, \tilde{\phi}_r\}),~~~{\rm and}~~~\boldsymbol{E}={\rm diag}(\{H''(-\tilde{\lambda}_1)-H_r^{\prime\prime}(-\tilde{\lambda}_1), \dots, H''(-\tilde{\lambda}_r)-H_r^{\prime\prime}(-\tilde{\lambda}_r)\}.$$
\[lemma:E\_pos\_def\] Let be and let $H_r(s)$ be an interpolant. Then $\boldsymbol{E}$ is positive definite at any fixed point of $\lambda({\ensuremath{\boldsymbol{s}}})$.
By Lemma \[lemma:positiveerror\], $\Hs-H_r(s) \ge 0$ for all $s\in [0, \infty)$. Thus the points $H(-\tilde{\lambda}_i)-H_r(-\tilde{\lambda}_i)$ are local minima of $\Hs-H_r(s)$ on $[0, \infty)$ for $i=1,\dots,r$. It then follows that $H''(-\tilde{\lambda}_i)-H_r^{\prime\prime}(-\tilde{\lambda}_i)\ge 0$. But by Lemma \[lemma:zero\_structure\], $\Hs-H_r(s)$ has exactly $2r$ zeros in ${\ensuremath{\mathbb{C}}}_{+}$, so $H''(-\tilde{\lambda}_i)-H_r^{\prime\prime}(-\tilde{\lambda}_i) > 0$ for $i=1, \dots, r$.
\[lemma:Spos\] The matrix $
\tilde{\boldsymbol{S}}=
\begin{bmatrix}
\boldsymbol{S}_{11} &\boldsymbol{S}_{12}\\
\boldsymbol{S}_{12}& \boldsymbol{S}_{22}
\end{bmatrix}
$ is positive definite.
We will show that for any non-zero vector $\boldsymbol{z}=[z_1, z_2,\dots, z_{2r}]^T \in \IR^{2r}$ $$\boldsymbol{z}^T\boldsymbol{S}\boldsymbol{z} = \int_{0}^{\infty}\bigg[\sum\limits_{i=1}^r z_ie^{\tilde{\lambda}_i t}-t\Big(\sum \limits_{i=1}^rz_{r+i}e^{\tilde{\lambda}_i t}\Big) \bigg]^2 {\,\mathrm{d}}t >0.$$ Define $\boldsymbol{z}_r=[z_1, z_2,\dots, z_{r}]^T \in \IR^{r}$ and $\boldsymbol{z}_{2r}=[z_{r+1}, z_{r+2},\dots, z_{2r}]^T \in \IR^{r}$. Then
$$\boldsymbol{z}^T\boldsymbol{S}\boldsymbol{z}= \boldsymbol{z}_r^T\boldsymbol{S}_{11}\boldsymbol{z}_r +2\boldsymbol{z}_r^T\boldsymbol{S}_{12}\boldsymbol{z}_{2r}+\boldsymbol{z}_{2r}^T\boldsymbol{S}_{22}\boldsymbol{z}_{2r} \label{Seqn}
$$
Let $\boldsymbol{\Lambda}= \text{diag}(\tilde{\lambda}_1, \dots, \tilde{\lambda}_r)$ and $\boldsymbol{u}$ be a vector of $r$ ones. Note that $\boldsymbol{S}_{11}$ solves the Lyapunov equation $\boldsymbol{\Lambda} \boldsymbol{S}_{11} + \boldsymbol{S}_{11} \boldsymbol{\Lambda} +
\boldsymbol{uu}^T = \boldsymbol{0}$. Thus, $$\begin{aligned}
\boldsymbol{z}_r^T\boldsymbol{S}_{11}\boldsymbol{z}_r&=\int_0^{\infty}\boldsymbol{z}_r^Te^{\boldsymbol{\Lambda} t}\boldsymbol{uu}^Te^{\boldsymbol{\Lambda} t} \boldsymbol{z}_r {\,\mathrm{d}}t
= \int_{0}^{\infty} \Big(\sum \limits_{i=1}^r z_i e^{\tilde{\lambda}_i t} \Big)^2 {\,\mathrm{d}}t \label{S11eqn}\end{aligned}$$
Similarly, $\boldsymbol{S}_{12}$ solves $\boldsymbol{\Lambda} \boldsymbol{S}_{12} + \boldsymbol{S}_{12} \boldsymbol{\Lambda} - \boldsymbol{S}_{11} = \boldsymbol{0}$. An application of integration by parts gives: $$\begin{aligned}
\int_{0}^{\infty}t \Big(\sum \limits_{i=1}^{r} z_i e^{\tilde{\lambda}_i t}\Big)\Big(\sum \limits_{i=1}^{r} z_{r+i} e^{\tilde{\lambda}_i t}\Big){\,\mathrm{d}}t&=\int_{0}^{\infty}t(\boldsymbol{z}_r^T(e^{\boldsymbol{\Lambda} t}\boldsymbol{uu}^Te^{\boldsymbol{\Lambda} t})\boldsymbol{z}_{2r}){\,\mathrm{d}}t \nonumber \\
&=\boldsymbol{z}_r^T\Big[-t (e^{\boldsymbol{\Lambda} t}\boldsymbol{S}_{11}e^{\boldsymbol{\Lambda} t})\Big]_{0}^{\infty}\boldsymbol{z}_{2r}+\boldsymbol{z}_r^T\Big(\int_{0}^{\infty} e^{\boldsymbol{\Lambda} t} \boldsymbol{S}_{11}e^{\boldsymbol{\Lambda} t} {\,\mathrm{d}}t\Big)\boldsymbol{z}_{2r} \nonumber\\
&=-\boldsymbol{z}_r^T\boldsymbol{S}_{12}\boldsymbol{z}_{2r} \label{S12eqn}\end{aligned}$$ Finally, note that $\boldsymbol{S}_{22}$ solves $\boldsymbol{\Lambda} \boldsymbol{S}_{22} + \boldsymbol{S}_{22} \boldsymbol{\Lambda} - 2\boldsymbol{S}_{12} = \boldsymbol{0}$. Repeated applications of integration by parts then yields the equality: $$\boldsymbol{z}_{2r}^T\boldsymbol{S}_{22}\boldsymbol{z}_{2r}=\int_{0}^{\infty} t^2\Big(\sum \limits_{i=1}^{r} z_{r+1} e^{\tilde{\lambda}_i t} \Big)^2 {\,\mathrm{d}}t \label{S22eqn}$$
Combining equations (\[Seqn\]), (\[S11eqn\]), (\[S12eqn\]), and (\[S22eqn\]) gives the desired results since $$\mbox{$
\boldsymbol{z}^T\boldsymbol{S}\boldsymbol{z}= \int_{0}^{\infty}\bigg[\sum\limits_{i=1}^r z_ie^{\tilde{\lambda}_i t}-t\Big(\sum \limits_{i=1}^rz_{r+i}e^{\tilde{\lambda}_i t}\Big) \bigg]^2 {\,\mathrm{d}}t.
$}$$
Then it follows that the Schur complement $\boldsymbol{S}_{22}-\boldsymbol{S}_{12}\boldsymbol{S}_{11}^{-1}\boldsymbol{S}_{12}$ of $\tilde{\boldsymbol{S}}$ is also positive definite. With the setup above, we may now commence with the proof of Theorem \[thm:localattractor\].
[**Proof of Theorem \[thm:localattractor\]:**]{} It suffices to show that for any fixed point which is a local minimizer of ${\ensuremath{\boldsymbol{\mathcal{J}}}}(\tilde{\boldsymbol{\phi}}, \lambda({\ensuremath{\boldsymbol{s}}}))$, the eigenvalues of the Jacobian of $\lambda({\ensuremath{\boldsymbol{s}}})$ are bounded in magnitude by 1. As shown in [@krajewski], the Jacobian of $\lambda({\ensuremath{\boldsymbol{s}}})$ can be written as $-\boldsymbol{S}_c^{-1}\boldsymbol{K}$ where $$\boldsymbol{S}_c=\boldsymbol{S}_{22}-\boldsymbol{S}_{12}\boldsymbol{S}_{11}^{-1}\boldsymbol{S}_{12}\qquad \text{ and } \qquad
\boldsymbol{K}=\boldsymbol{E}\boldsymbol{R}^{-1}.$$ First off, note that from Lemma \[lemma:E\_pos\_def\], and the fact that is a ZIP system by Lemma \[lemma:reducessslemma\], $\boldsymbol{K}$ is positive definite. Evaluating the pencil $\boldsymbol{K}-\lambda\boldsymbol{S}_c$ at $\lambda=1$ gives $$\boldsymbol{\Phi}=-\boldsymbol{S}_{22}+\boldsymbol{E}\boldsymbol{R}^{-1}+\boldsymbol{S}_{12}\boldsymbol{S}_{11}^{-1}\boldsymbol{S}_{12},$$ This pencil is regular since $\boldsymbol{S}_c$ is positive definite by Lemma \[lemma:Spos\], and therefore $\det(\boldsymbol{K}-\lambda \boldsymbol{S}_c)$ is zero if and only if $\det(\boldsymbol{S}_c^{-1}\boldsymbol{K}-\lambda\boldsymbol{I})=0$.
Let $\nabla^2{\ensuremath{\boldsymbol{\mathcal{J}}}}$ denote the Hessian of the cost function ${\ensuremath{\boldsymbol{\mathcal{J}}}}(\tilde{\boldsymbol{\phi}}, \lambda({\ensuremath{\boldsymbol{s}}}))$. As shown in [@krajewski], $\nabla^2{\ensuremath{\boldsymbol{\mathcal{J}}}}$ can be written as $$\nabla^2{\ensuremath{\boldsymbol{\mathcal{J}}}}=
\begin{bmatrix}
\boldsymbol{I} &\boldsymbol{0}\\
\boldsymbol{0}&\boldsymbol{R}
\end{bmatrix}
\boldsymbol{M}
\begin{bmatrix}
\boldsymbol{I} &\boldsymbol{0}\\
\boldsymbol{0}&\boldsymbol{R}
\end{bmatrix},
$$ [where]{} =
\_[11]{}&\_[12]{}\
\_[12]{}&\_[22]{}-\^[-1]{}
. $$
Note that $-\boldsymbol{\Phi}$ is the Schur complement of $\boldsymbol{M}$. Hence, if the fixed point is a local minimimum, then $-\boldsymbol{\Phi}$ must be positive definite and so for $\lambda=1$ the pencil is negative definite. Since both $\boldsymbol{K}$ and $\boldsymbol{S}_c$ are positive definite, there exists a nonsingular transformation $\boldsymbol{Z}$ by which the quadratic form $\displaystyle
\boldsymbol{y}^T(\boldsymbol{K}-\lambda\boldsymbol{S}_c)\boldsymbol{y}
$ is transformed into $\displaystyle
\boldsymbol{z}^T(\boldsymbol{\Lambda}-\lambda\boldsymbol{I})\boldsymbol{z},
$ where $\boldsymbol{\Lambda}$ is a diagonal matrix formed from the solutions of $$\label{solutions}
\det(\boldsymbol{K}-\lambda\boldsymbol{S}_c)=0.$$ Thus, the solutions of (\[solutions\]) correspond to the eigenvalues of $\boldsymbol{S}_c^{-1}\boldsymbol{K}$. $\boldsymbol{\Lambda}-\boldsymbol{I}$ must be negative definite since $\boldsymbol{\Phi}$ is, and therefore the eigenvalues of the $\boldsymbol{S}_c^{-1}\boldsymbol{K}$ must be real-valued and less than one. Furthermore, note that ${\ensuremath{\boldsymbol P}}=\boldsymbol{S}_c^{-1}\boldsymbol{K}$ solves the Lyapunov equation $${\ensuremath{\boldsymbol P}}\boldsymbol{S}^{-1}_c+\boldsymbol{S}^{-1}_c{\ensuremath{\boldsymbol P}}^T=2\boldsymbol{S}^{-1}_c\boldsymbol{K}\boldsymbol{S}^{-1}_c,$$ so by the standard inertia result, all the eigenvalues of $\boldsymbol{S}_c^{-1}\boldsymbol{K}$ are positive, and the desired result follows. $\Box$
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=2.5 cm =16.2 cm
[ ]{}
[ ]{}
[*Nuclear Problems Research Institute, Bobruiskaya Str. 11,* ]{}
[*Minsk 220080 Belarus.* ]{}
[*Electronic address: bar@@inp.minsk.by* ]{}
[*Tel: 00375-172-208481* ]{}
[*Fax: 00375-172-265124*]{}[ ]{}
P- and T-odd interactions cause mixing of opposite parity levels of atom (molecule) that yields to the appearance of P- and T-odd terms of the atom (molecule) polarizability [@PLA93]. This makes possible to observe various optical phenomena, for example, photon polarization plane rotation and circular dichroism in an optically homogeneous medium placed to an electric field, polarization plane rotation (circular dichroism) phenomena for photons moving in an electric (gravitational) field in vacuum [@PLA99].
The energy of atom (molecule) in external electromagnetic field includes the term caused by the time reversal violating interactions [@PLA93]:
$$\Delta U=-\frac{1}{2}\beta _{S}^{T}\overrightarrow{E}\overrightarrow{H},
\label{1}$$
where $\beta _{S}^{T}$ is the scalar T-noninvariant polarizability of atom (molecule), $\overrightarrow{E}$ is the external electric field, $%
\overrightarrow{H}$ is the external magnetic field.
It’s well known [@Landau] that when the external field frequency $\omega
\rightarrow 0$ the polarizabilities describe the processes of magnetization of medium by a static magnetic field and electric polarization of a medium by a static electric field
The energy of interaction of magnetic moment $\overrightarrow{\mu }$ with magnetic field $\overrightarrow{H}$
$$W_{H}=-\overrightarrow{\mu }\overrightarrow{H} \label{2}$$
Comparison of (\[1\]) and (\[2\]) let one to conclude that the action of stationary electric field on an atom (molecule) induces the magnetic moment of atom
$$\overrightarrow{\mu }(\overrightarrow{E})=\frac{1}{2}\beta _{S}^{T}
\overrightarrow{E}$$
On the other hand, the energy of interaction of electric dipole moment $%
\overrightarrow{d}$ with electric field $\overrightarrow{E}$
$$W_{E}=-\overrightarrow{d}\overrightarrow{E}. \label{4}$$
As it follows from (\[1\]) and (\[4\]), magnetic field induces the electric dipole moment of atom
$$\overrightarrow{d}(\overrightarrow{H})=\frac{1}{2}\beta _{S}^{T}
\overrightarrow{H}$$
As appears from the above, atom (molecule) being placed to static electric field gets the induced magnetic moment which in its part produces magnetic field. And similarly, if atom (molecule) is placed in the area of magnetic field the induced electric dipole moment yields to the appearance of its associated electric field.
Let us consider the simplest possible experiment. Suppose that homogeneous isotropic matter (liquid or gas) is placed to the area occupied by an electric field $\overrightarrow{E}$. From the above it follows that the time reversal violation yields to the appearance of magnetic field $%
\overrightarrow{H}_{T}=4\pi \rho \overrightarrow{\mu }(\overrightarrow{E})$ parallel to $\overrightarrow{E}$ in this area ($\rho $ is the number of atoms (molecules) of matter per $cm^{3}$ ). And vice versa, the electric field $\overrightarrow{E}_{T}=4\pi \rho \overrightarrow{d}(\overrightarrow{H}
)$ appears under matter placement to the area occupied by a magnetic field $%
\overrightarrow{H}$. Let us estimate the effect value. It is easy to do by $%
\beta _{S}^{T}$ evaluation. The general case explicit expression for polarizabilities for time dependent fields were derived in [@PLA93] (see eqs. (12)-(20) therein).
Briefly the calculation technique is as follows. Let us suppose that atom is placed to the arbitrary periodic in time electric and magnetic fields. The energy of interaction of an atom (molecule) with these fields has the routine form
$$W=-\widehat{\overrightarrow{d}}\overrightarrow{E}-\widehat{\overrightarrow{
\mu }}\overrightarrow{H}+..... \label{interaction energy}$$
where $\widehat{\overrightarrow{d}}$ is the operator of atom electric dipole moment and $\widehat{\overrightarrow{\mu }}$ is the operator of atom magnetic dipole moment $$\overrightarrow{E}=\frac{1}{2}\left\{ \overrightarrow{E}_{0}\;e^{-i\omega
t}+ \overrightarrow{E}_{0}^{\ast }\;e^{i\omega t}\right\} ,\;\overrightarrow{
H}= \frac{1}{2}\left\{ \overrightarrow{H}_{0}\;e^{-i\omega t}+
\overrightarrow{H} _{0}^{\ast }\;e^{i\omega t}\right\}$$ The Shrödinger equation describing atom interaction with electromagnetic field is as follows:
$$i\hbar \frac{\partial \psi (\xi ,t)}{\partial t}=[H_{A}(\xi )+W(\xi ,t)]\psi
(\xi ,t),$$
where $H_{A}(\xi )$ is the atom Hamiltonian taking into account the weak interaction of electrons with nucleus in the center of mass of the system, $%
\xi $ is the space and spin variable of electron and nucleus, $W$ is the energy of interaction of atom with electromagnetic field of frequency $%
\omega $
$$\begin{aligned}
W &=&Ve^{-i\omega t}+V^{+}e^{i\omega t}, \label{W} \\
V &=&-\frac{1}{2}(\overrightarrow{d}\overrightarrow{E_{0}}+\overrightarrow{%
\mu }\overrightarrow{H_{0}}),V^{+}=-\frac{1}{2}(\overrightarrow{d}%
\overrightarrow{E_{0}}^{\ast }+\overrightarrow{\mu }\overrightarrow{H_{0}}%
^{\ast }) \nonumber\end{aligned}$$
Let us perform the transformation $\psi =\exp (-{i}\frac{{H_{A}}}{{\hbar }}{t%
})\varphi $. Suppose $H_{A}\psi _{n}=E_{n}\psi _{n}$ ($E_{n}=E_{n}^{(0)}-%
\frac{1}{2}i\Gamma _{n}$, $E_{n}^{(0)}$ is the atom level energy, $\Gamma
_{n}$ is the atom level width), then $\varphi =\sum_{n}b_{n}(t)\psi _{n}$. Therefore it follows from (\[W\]) $$\begin{gathered}
i\hbar \frac{\partial b_{n}(t)}{\partial t}=\sum_{f}\left\{ \langle
n|V|f\rangle \exp [i(E_{n}-E_{f}-\hbar \omega )t/\hbar ]\right. + \\
+\left. \langle n|V^{+}|f\rangle \exp [i(E_{n}-E_{f}+\hbar \omega )t/\hbar
]\right\} b_{f}(t),\;\langle \psi _{n}|\psi _{m}\rangle \ll 1. \nonumber\end{gathered}$$ Suppose $b_{n0}$ be the ground state amplitude. Let us substitute the amplitude $b_{f}$ describing the excited atom state into the equation for $%
b_{n0}$ and study this equation at time $t\gg \tau _{f}=\hbar /\Gamma
_{f}\;($or $\tau _{f}=\hbar /\Delta E);\;\Delta E=E_{f}^{(0)}-E_{n0}-\hbar
\omega $; $\Gamma _{f}$ $\gg $ $|\langle n|V|f\rangle |$ (or $\Delta E\gg $ $%
|\langle n|V|f\rangle |$). Therefore $b_{n0}$ is defined by equation $$i\hbar \frac{\partial b_{n0}(t)}{\partial t}=\widehat{U}_{eff}\;b_{n0},{\
where}$$ $$\widehat{U}_{eff}=-\sum_{f}\left( \frac{\langle n_{0}|V|f\rangle \;\langle
f|V^{+}|n_{0}\rangle }{E_{f}-E_{n0}+\hbar \omega }+\frac{\langle
n_{0}|V^{+}|f\rangle \;\langle f|V|n_{0}\rangle }{E_{f}-E_{n0}-\hbar \omega }%
\right) \label{U_eff}$$ Substituting $V$ and $V^{+}$ into (\[U\_eff\]) one can obtain $$\widehat{U}_{eff}=-\frac{1}{2}\widehat{g}_{ik}^{E}E_{0i}E_{0k}^{\ast }-\frac{%
1}{2}\widehat{g}_{ik}^{H}H_{0i}H_{0k}^{\ast }-\frac{1}{2}\widehat{g}%
_{ik}^{EH}E_{0i}H_{0k}^{\ast }-\frac{1}{2}\widehat{g}%
_{ik}^{HE}H_{0i}E_{0k}^{\ast },$$ where the polarizability of atom (molecule) is: $$\begin{aligned}
\widehat{g}_{ik}^{E} &=&-\frac{1}{2}\left( \sum_{f}\frac{\langle
n_{0}|d_{i}|f\rangle \;\langle f|d_{k}|n_{0}\rangle }{E_{f}-E_{n0}+\hbar
\omega }+\frac{\langle n_{0}|d_{k}|f\rangle \;\langle f|d_{i}|n_{0}\rangle }{%
E_{f}-E_{n0}-\hbar \omega }\right) \nonumber \\
\widehat{g}_{ik}^{H} &=&-\frac{1}{2}\left( \sum_{f}\frac{\langle n_{0}|\mu
_{i}|f\rangle \;\langle f|\mu _{k}|n_{0}\rangle }{E_{f}-E_{n0}+\hbar \omega }%
+\frac{\langle n_{0}|\mu _{k}|f\rangle \;\langle f|\mu _{i}|n_{0}\rangle }{%
E_{f}-E_{n0}-\hbar \omega }\right) \nonumber \\
\widehat{g}_{ik}^{EH} &=&-\frac{1}{2}\left( \sum_{f}\frac{\langle
n_{0}|d_{i}|f\rangle \;\langle f|\mu _{k}|n_{0}\rangle }{E_{f}-E_{n0}+\hbar
\omega }+\frac{\langle n_{0}|\mu _{k}|f\rangle \;\langle
f|d_{i}|n_{0}\rangle }{E_{f}-E_{n0}-\hbar \omega }\right) \nonumber \\
\widehat{g}_{ik}^{HE} &=&-\frac{1}{2}\left( \sum_{f}\frac{\langle n_{0}|\mu
_{i}|f\rangle \;\langle f|d_{k}|n_{0}\rangle }{E_{f}-E_{n0}+\hbar \omega }+%
\frac{\langle n_{0}|d_{k}|f\rangle \;\langle f|\mu _{i}|n_{0}\rangle }{%
E_{f}-E_{n0}-\hbar \omega }\right) \nonumber\end{aligned}$$ It should be noted that $\widehat{g}_{ik}^{E}$ and $\widehat{g}_{ik}^{H}$ are the P- and T-invariant electric and magnetic polarizability tensors and $%
\widehat{g}_{ik}^{EH}$ and $\widehat{g}_{ik}^{HE}$ are the P- and T-noninvariant polarizability tensors
Let an atom be placed at the static ($\omega \rightarrow 0$) magnetic and electric fields $\overrightarrow{E}$ and $\overrightarrow{H}$ of the same direction. Then it’s perfectly easy to obtain the effective energy of P- and T-odd interaction of an atom with these fields.
$$\widehat{U}_{eff}^{T,P}=-\frac{1}{2}\left( \sum_{f}\frac{\langle
n_{0}|d_{z}|f\rangle \;\langle f|\mu _{z}|n_{0}\rangle +\langle n_{0}|\mu
_{z}|f\rangle \;\langle f|d_{z}|n_{0}\rangle }{E_{f}-E_{n_{0}}}\right) EH$$
Axis $z$ is supposed to be parallel to $\overrightarrow{E}$. Thus from (\[1\]) $$\beta _{S}^{T}=\sum_{f}\frac{\langle n_{0}|d_{z}|f\rangle \;\langle f|\mu
_{z}|n_{0}\rangle +\langle n_{0}|\mu _{z}|f\rangle \;\langle
f|d_{z}|n_{0}\rangle }{E_{f}-E_{n_{0}}}$$ Let us estimate the $\beta _{S}^{T}$ order of magnitude. The atom state $%
|f\rangle $ does not possess the certain parity because of weak T-odd interactions. And over the weakness of $V_{T}$ the state $|f\rangle $ is mixed with the state of opposite parity of value $\eta _{T}=\frac{{V_{W}^{T}}%
}{{E_{f}-E_{n}}}$. According to $$\beta _{S}^{T}\sim \frac{\langle d\rangle \;\langle \mu \rangle }{%
E_{f}-E_{n_{0}}}\eta _{T}$$
For the heavy atoms the mixing coefficient can attain the value $\eta
_{T}\approx 10^{-14}$. Taking into account that matrix element $\langle \mu
\rangle \sim \alpha \langle d\rangle $ (where $\alpha =\frac{1}{137}$ is the fine structure constant) one can obtain $\beta _{S}^{T}\sim \eta
_{T}\;\alpha \frac{{\langle d\rangle ^{2}}}{{\Delta }}{\;}\approx
10^{-16}\cdot \frac{{8\cdot 10^{-36}}}{{10^{-12}}}\approx 10^{-40}$. Therefore, the electric field $E=10^{2}\;CGSE$ induces magnetic moment $\mu
_{T}\approx 10^{-38}$. Then, the magnetic field in the liquid target can be estimated as follows
$$H=4\pi \rho \mu _{T}\approx 10^{23}\cdot 10^{-38}=10^{-15}\;gauss$$
The magnitude of magnetic field strength can be increased, for example, by tightening of the magnetic field with superconductive shield. In this way the measured field strength can be increased by four orders when one collect the field from the area 1 $m^{2}$ to the area 1 $cm^{2}$ (Fig.1).
= 10 cm
The induced magnetic moment produces magnetic field at the electron (nucleus) of the atom. This field $H^{T}(E)\sim \mu \;\langle \frac{{1}}{{%
r^{3}}}\rangle \sim 10^{-38}\cdot 10^{26}=10^{-12}$ $gauss$. Therefore, the frequency of precession of atom magnetic moment $\mu _{A}$ in the magnetic field induced by an external electric field
$$\Omega _{E}\sim \frac{\mu _{A}\;\beta \;E\;\langle \frac{{1}}{{r^{3}}}%
\rangle }{\hbar }=\frac{10^{-20}\cdot 10^{-12}}{10^{-27}}=10^{-5}\;\sec ^{-1}$$
It should be reminded that to measure the electric dipole moment the shift of precession frequency of atom spin in the presence of both magnetic and electric fields is investigated. Then, the T-odd shift of precession frequency of atom spin includes two terms: frequency shift conditioned by interaction of atom electric dipole moment with electric field $\omega _{E}=%
\frac{{2d_{A}E}}{{\hbar }}$ and frequency shift $\Omega =\frac{{2\mu H^{T}(E)%
}}{{\ \hbar }}$ defined above. This aspect should be considered when interpreting the similar experiments. One should take note of the mixing coefficient $\eta _{T}$ essential increase when the opposite parity levels are close to each other or even degenerate. Then the effect can grow up as much as several orders $10^{5}\div 10^{6}$ (this occurs, for example, for Dy, TlF, BiS, HgF).
The similar phenomenon of magnetic field induction by electric field can occur in vacuum too.
Due to quantum electrodynamic effect of electron-positron pair creation in strong electric, magnetic or gravitational field, the vacuum is described by the dielectric $\varepsilon _{ik}$ and magnetic $\mu _{ik}$ permittivity tensors depending on these fields. The theory of $\varepsilon _{ik}$ [@Ahiezer] does not take into account the weak interaction of electron and positron with each other. Considering the T- and P-odd weak interaction between electron and positron in the process of pair creation in an electric (magnetic, gravitational) field one can obtain the density of electromagnetic energy of vacuum contains term $\beta _{v}^{T}(
\overrightarrow{E}\overrightarrow{H)}$ similar (\[1\]) (in the case of vacuum polarization by a stationary gravitational field $\beta _{g}^{T}(
\overrightarrow{H}\overrightarrow{n_{g}})$, $\overrightarrow{n_{g}}=\frac{%
\overrightarrow{g}}{g}$, $\overrightarrow{g}-$ gravitational acceleration).
As a result both electric and magnetic fields (directed along the electric field) could exist around an electric charge. But in this case $\oint
\overrightarrow{B}d\overrightarrow{S}\neq 0$ ($\overrightarrow{B}$ is the magnetic induction) that is impossible in the framework of classic electrodynamics. The existence of such field would means the existence of induced magnetic monopole. If the condition $\oint \overrightarrow{B}d%
\overrightarrow{S}=0$ is fulfilled then for the spherically symmetrical case the field appears equal to zero. Surely, the value of this magnetic field is extremely small, but the possibility of its existence is remarkable itself.
The above result can be obtained in the framework of general Lagrangian formalism. Lagrangian density can depend only on the field invariants. Two invariants are known for the quasistatic electromagnetic field: $(
\overrightarrow{E}\overrightarrow{H})$ and $(E^{2}-H^{2})$. In conventional T-invariant theory these invariants are included in the Lagrangian $L$ only as $(E^{2}-H^{2})$ and $(\overrightarrow{E}\overrightarrow{H})^{2}$, i.e. $%
L=L(E^{2}-H^{2},(\overrightarrow{E}\overrightarrow{H})^{2})$ [@Ahiezer]. But while taking into account the T-odd interactions the Lagrangian can include invariant $(\overrightarrow{E}\overrightarrow{H})$ raising to the odd power, i.e.
$$L_{T}=L_{T}(E^{2}-H^{2},(\overrightarrow{E}\overrightarrow{H})^{2},(
\overrightarrow{E}\overrightarrow{H})) \label{lagrangian}$$
Expanding (\[lagrangian\]) by weak interaction one can obtain
$$L_{T}=L(E^{2}-H^{2},(\overrightarrow{E}\overrightarrow{H})^{2})+\beta _{T}(
\overrightarrow{E}\overrightarrow{H}),$$
where $L$ is the density of Lagrangian in P- and T-invariant electrodynamics, $\beta _{T}=\beta _{T}(E^{2}-H^{2},(\overrightarrow{E}
\overrightarrow{H})^{2})$ is the constant can be found in certain theory. The explicit form of $L$ is cited in [@Ahiezer].
The additions caused by the vacuum polarization can be described by the field dependent dielectric and magnetic permittivity of vacuum. According to [@Ahiezer] the electric induction vector $\overrightarrow{D}$ and magnetic induction vector $\overrightarrow{B}$ are defined as:
$$\overrightarrow{D}=\frac{\partial L}{\partial \overrightarrow{E}},\;
\overrightarrow{B}=-\frac{\partial L}{\partial \overrightarrow{H}}$$
Similarly the electric polarization $\overrightarrow{P}$ and magnetization $%
\overrightarrow{M}$ of vacuum can be found [@Ahiezer]:
$$\begin{aligned}
\overrightarrow{P}=\frac{\partial (L_{T}-L_{0})}{\partial \overrightarrow{E}}
,\;\overrightarrow{M}=-\frac{\partial (L_{T}-L_{0})}{\partial
\overrightarrow{H}}, \\
\overrightarrow{D}=\overrightarrow{E}+4\pi \overrightarrow{P},\;
\overrightarrow{B}=\overrightarrow{H}+4\pi \overrightarrow{M.}\end{aligned}$$
In accordance with the above, the T-noninvariance yields to the appearance of additional P- and T-odd terms to the electric polarization $%
\overrightarrow{P}$ and magnetization $\overrightarrow{M}$ . There are the addition to the vector of electric polarization $\overrightarrow{P}$ proportional to the magnetic field strength $\overrightarrow{H}$ and the addition to the vector of magnetization $\overrightarrow{M}$ proportional to the electric field strength $\overrightarrow{E}$
[9]{} V.G.Baryshevsky Phys.Letters A 177 (1993) p.38-42
V.G.Baryshevsky Phys.Letters A 260 (1999) p.24-30
Landau L., Lifshitz E. Quantum mechanics, 1989, Moscow Science.
Berestetskii V., Lifshitz E., Pitaevskii L. Quantum electrodynamics, 1989, Moscow Science.
|
---
abstract: 'In this letter we investigate graviton-photon oscillation in the presence of an external magnetic field in alternative theories of gravity. Whereas the effect of an effective refractive index for the electromagnetic radiation was already considered in the literature, we develop the first approach to take into account the effect of the modification of the propagation of gravitational waves in alternative theories of gravity in the phenomenon of graviton-photon mixing.'
author:
- 'José A. R. Cembranos'
- Mario Coma Díaz
- 'Prado Martín-Moruno'
date: 'Received: date / Accepted: date'
title: 'Graviton-photon oscillation in alternative theories of gravity'
---
Introduction
============
We are witnessing the dawn of gravitational wave (GW) astronomy. Most researchers in our field are probably wondering about how this new window to our Universe will affect our theoretical knowledge. It is hight time to dust off seminal studies on GWs and get them in shape taking into account recent developments in gravitation. An optimal example is the phenomenon of graviton-photon oscillation, which has long been known by the general relativistic community. In 1960 Gertsenshtein investigated the excitation of GWs during the propagation of electromagnetic waves (EMWs) in an electric or magnetic field [@Gertsenshtein]. Later on, Lupanov considered the inverse process, that is EMWs creation by GWs crossing an electric field, as an indirect way of measuring GWs [@Lupanov]. Moreover, he argued that the effectiveness of a hypothetical detector could be improved by introducing a dielectric material with high permittivity [@Lupanov]. As it was later investigated in more detail by Zel’dovich, the change in the speed of light due to an effective refractive index induces loss of coherence and, therefore, affects the conversion process. The formalism of photon mixing with gravitons (and with low-mass particles in general) was completely developed by Raffelt and Stodolsky [@Raffelt:1987im] taking into account the effective refractive index for EMWs of quantum-electrodynamic origin. It should be emphasized that graviton-photon mixing could provide us with an indirect way to measure GWs. Although direct detection of GWs is already a reality, our ability to detect EMWs is still much better by far. Whereas the effect produced by conversion of EMWs in GWs and back again is undetectable by laboratory experiments [@Boccaletti], it has been pointed out that graviton-photon mixing of primordial GWs in the presence of cosmic magnetic fields could lead to observable footprints in the X-ray cosmic background [@Dolgov:2012be].
On the other hand, as general relativity (GR), alternative theories of gravity (ATGs) predict the emission and propagation of gravitational radiation. The nature of this gravitational radiation, however, can be very different and could have more than two independent polarizations [@Will:2014kxa]. Focusing attention on the two tensor modes, that is the GW, their propagation is typically modified with respect to the general relativistic prediction [@deRham:2014zqa; @Saltas:2014dha; @Bettoni:2016mij; @Caprini:2018mtu; @Akrami:2018yjz]. In fact, as it has been recently shown explicitly, one could consider that GWs propagate in the framework of ATGs as if they were in a diagravitational medium characterized by three different constitutive tensors, reducing just to an effective refractive index when considering perturbations around highly symmetric background spacetimes [@Cembranos:2018lcs]. Hence the possibility of using GWs observations to constrain ATGs has been taken into account seriously, as they could provide us with an *experimentum crucis* to discern what is the theory of gravity describing our Universe. Indeed, the recent measurement of GWs and their electromagnetic counterpart [@TheLIGOScientific:2017qsa] has allowed to discard a large number of ATGs as dark energy mimickers [@Lombriser:2015sxa; @Lombriser:2016yzn; @Ezquiaga:2017ekz; @Creminelli:2017sry; @Sakstein:2017xjx; @Baker:2017hug; @Akrami:2018yjz], although they could still play a relevant role during the early universe.
Following this spirit, in this letter we focus our attention on the phenomenon of graviton-photon mixing, which is also known as graviton-photon oscillation, and investigate how it is modified by a diagravitational refractive index. Up to the best of our knowledge, this is the first attempt to address this phenomenon in the framework of ATGs. Nevertheless, we would like to mention some interesting works in related topics. GWs-gauge field oscillations have been recently investigated [@Caldwell:2016sut; @Caldwell:2017sto] and its potential observational consequences analysed [@Caldwell:2018feo]. In addition, there has been developments in understanding how the background can affect the propagation of GWs in GR [@Flauger:2017ged] and also in $f(R)$-gravity [@Bamba:2018cup].
Graviton-photon mixing
======================
Let us start by briefly summarizing the description of graviton-photon mixing in GR. Raffelt and Stodolsky argued that the quantum corrections to the electrodynamic system could be relevant for graviton-photon oscillation in regions of strong magnetic fields [@Raffelt:1987im] (see also reference [@Dolgov:2012be]). Those corrections, which can be taken into account adding an Euler–Heisenberg term into the Lagrangian, lead to an effective refractive index that the photon experience even in vacuum. Apart form this term, the electromagnetic Lagrangian takes the usual form $$\begin{aligned}
\mathcal{L}_{\rm EM}=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}%\nonumber\\&+&\frac{\alpha^2}{90\,m_{\rm e}^4}\left[(F_{\mu\nu}F^{\mu\nu})^2+\frac{7}{4}(F_{\mu\nu}\tilde F^{\mu\nu})^2\right],\end{aligned}$$ where $F^{\mu\nu}$ is the electromagnetic field tensor. On the other hand, it is useful to consider that the metric perturbations, defined through $$\label{gh}
g_{\mu\nu}=\eta_{\mu\nu}+\kappa^2h_{\mu\nu},$$ are expressed in the transverse-traceless gauge. Therefore, the quadratic gravitational Lagrangian in tensor perturbations and the interaction term with the electromagnetic field can be written as $$\mathcal{L}_{\rm GR}= -\frac{1}{4}\partial_\mu h_{ij} \partial^\mu h^{ij}, \quad {\rm and} \quad \mathcal{L}_{\rm int}=\frac{\kappa}{2}\,h_{ij}T^{ij},$$ with $T^\mu{}_\nu=F^{\mu\rho}F_{\nu\rho}-\delta^\mu_\nu F_{\alpha\beta}F^{\alpha\beta}/4$. Note that the total electromagnetic field is the sum of the external magnetic field, that is $B^i_{\rm e}$, which is considered to be static and homogeneous, and the EMW, $B^j$, which field strenght is much weaker. Therefore, focusing attention on the wave-form contributions, we have $h_{ij}T^{ij}=-h_{ij}B^i_{\rm e}B^j$ and the graviton-photon dynamical system takes the form $$\begin{aligned}
\Box h_{ij}&=&\kappa\,B_i^{\rm (e)}B_j,\label{pmetric}\\
\left(\Box-m^2_{\gamma}\right)A^j&=&\kappa\,\partial_ih^{jk}F_k^{{\rm (e)}i}\label{pvector},\end{aligned}$$ where $A^j$ is the vector potential of the electromagnetic wave and $m_{\gamma}$ is the effective photon mass implied by the effective refractive index [@Dolgov:2012be]. We fix the $z$ axis along the direction of propagation of the waves without loss of generality. Then, the gravitational wave tensor and the electromagnetic vector potential can be expressed as $$\begin{aligned}
h_{ij}(t,\,z)&=&\left[h_+(z)\Epsilon^+_{ij}+h_\times(z)\Epsilon^\times_{ij}\right]e^{-i\omega t},\label{hp}\\
A_j(t,\,z)&=&i\left[A_x(z)\Epsilon^x_j+A_y(z)\Epsilon^y_j\right]e^{-i\omega t},\end{aligned}$$ where the polarization tensors have non-vanishing componentes $\Epsilon^+_{xx}=-\Epsilon^+_{yy}=1$ and $\Epsilon^\times_{xy}=\Epsilon^\times_{yx}=1$, and we have introduced a phase in the vector for convenience, as it was explicitly done in reference [@Dolgov:2012be]. Taking into account the form of the expansion (\[hp\]) in the interaction term $h_{ij}B^i_{\rm e}B^j$, one can conclude that only the external magnetic field transverse to the direction of the wave propagation contribute to the phenomenon, that is $B_{\rm T}=B_{\rm e}\sin\phi$ with $\cos\phi=\hat B_{\rm e}\cdot\hat k$. Assuming for simplicity that the transverse magnetic field lies along the $y$-direction, taking $k= i\partial_z$, so that $k_{\gamma}A=i\partial_zA$ and $k_{\rm g}h=i\partial_zh$, and noting that $|n_{\gamma}-1|\ll1$ with $k_{\gamma}=n\,\omega$, it is easy to recover the result of Raffelt and Stodolsky. This is [@Raffelt:1987im] $$\begin{aligned}
\left[\left(\omega-i\partial_z\right)+
\left({\begin{array}{cccc}
\Delta^\bot_{\gamma} & \Delta_M &0&0 \\
\Delta_M & 0 &0&0\\
0&0&\Delta^\parallel_{\gamma} & \Delta_M \\
0&0&\Delta_M & 0 \\
\end{array} }
\right)\right]
\left(\begin{array}{c}
A_\bot\\h_+\\A_\parallel\\h_\times
\end{array}\right)=0,\end{aligned}$$ with $A_\bot$ and $A_\parallel$ denoting the components of the wave vector field perpendicular and parallel to the transverse magnetic field. The mixing term depends on the gravitational coupling and on the transverse magnetic field $\Delta_M=\kappa\, B_{\rm T}/2$. On the other hand, the effective photon mass term has led to a modification in the propagation of EMWs that can be different for both polarizations (as it has been argued to be the case [@Raffelt:1987im]); that is $\Delta^{\bot, \parallel}_{\gamma}=\omega\, (n_{\gamma}^{\bot, \parallel}-1)$.
Mixing in a diagravitational medium
===================================
Now, let us firstly focus in a particular ATG and suggest later how to generalize our results. We consider Horndeski theory, which deals with the most general scalar-tensor Lagrangian leading to second order equations of motion [@Horndeski:1974wa; @Deffayet:2011gz]. This is [@Kobayashi:2011nu] $$\mathcal{L}_{\rm H}=\sum_{i=2}^5\mathcal{L}_i$$ with $$\begin{aligned}
\mathcal{L}_2 &=& K(\phi, X), \quad
\mathcal{L}_3 = -G_3(\phi, X)\Box\phi, \label{eq:G3}\\
\mathcal{L}_4&=& G_{4}(\phi, X)R+G_{4X}\left[\left(\Box\phi\right)^2-\left(\nabla_\mu\nabla_\nu\phi\right)^2\right], \label{eq:G4}\\
\label{eq:G5}
\mathcal{L}_5 &=& G_5(\phi, X) G_{\mu\nu}\nabla^\mu\nabla^\nu\phi-\frac{G_{5X}}{6}\Bigl[\left(\Box\phi\right)^3 \\
&-& 3\left(\Box\phi\right)\left(\nabla_\mu\nabla_\nu\phi\right)^2+ 2\left(\nabla_\mu\nabla_\nu\phi\right)^3\Bigr],\end{aligned}$$ where $X= - \partial_{\mu} \phi \partial^{\mu} \phi / 2$ is the scalar-field kinetic term, $G_{i X}=\partial G_{i} / \partial X $, $(\nabla_\mu\nabla_\nu\phi)^2=\nabla_\mu\nabla_\nu\phi\nabla^\mu\nabla^\nu\phi$, and $(\nabla_\mu\nabla_\nu\phi)^3=\nabla_\mu\nabla_\nu\phi\nabla^\nu\nabla^\lambda\phi\nabla_\lambda\nabla^\mu\phi$. We have in mind the potential relevance of graviton-photon mixing in cosmological scenarios, although when dealing with sub-Hubble modes the background geometry can be approximated by Minkowski space. Therefore, we assume that the scalar field is homogenous, that is $\phi=\phi(t)$. Then, the quadratic Horndeski Lagrangian for the metric perturbation can be written as [@Kobayashi:2011nu] $$\label{Hp}
\mathcal{L}_{\rm H}=\frac{\kappa^2}{4}\left[G(\phi,\,X)\,\dot h_{ij}^2-F(\phi,\,X)(\nabla h_{ij})^2\right],$$ where $$\begin{aligned}
G&=&G_4-X\left(\ddot\phi\, G_{5X}+G_{5\phi}\right)\\
F&=&G_4-2XG_{4X}+XG_{5\phi}.\end{aligned}$$ Therefore, as it is already well known, only $\mathcal{L}_4$ and $\mathcal{L}_5$ affect the propagation of GWs. It should be emphasized that we want to focus our attention on the mixing of gravitons with photons. Therefore, we are neglecting the backreaction of the scalar field into the background geometry when using expansion (\[gh\]). In addition, we will also neglect the effect of the coupling of the scalar field perturbations to the GWs, which can appear in generic situations, see for example [@Bettoni:2016mij]. Under the mentioned approximations, the first equation for the graviton-photon system in the presence of an external magnetic field, that is equation (\[pmetric\]), is modified. Now it reads $$\begin{aligned}
\left\{\Box h_{ij}-\left[\frac{\dot G}{G}\,\partial_0-\left(1-\frac{F}{G}\right)\partial_z^2\right]\right\}=\kappa_{\rm eff}\,B_i^{\rm (e)}B_j,\label{pmetricH}\end{aligned}$$ where we have defined $\kappa_{\rm eff}=1/(\kappa\, G)$; so, we recover GR for $G=1/\kappa^2$, as can be noted also from Lagrangian (\[Hp\]). We apply similar arguments to those used in the general relativistic case [@Raffelt:1987im; @Dolgov:2012be] reviewed in the previous section. It simplifies the treatment to consider that the modification on the propagation of GWs can be encapsulated in an effective diagravitational refractive index [@Cembranos:2018lcs] $$\begin{aligned}
n_{\rm g}=\frac{1}{c_T}\sqrt{1+i\frac{\nu}{\omega}}\quad{\rm with}\quad
\nu=\frac{\dot G}{G},\quad {\rm and}\quad c_T^2=\frac{F}{G},\end{aligned}$$ where, for simplicity, we will consider that these quantities are approximately constant during the interval of interest. Note that, even if the diagravitational refractive index could significantly separate from unity at early cosmic epochs, we will assume that both $|n_{\gamma}-1|\ll1$ and $|n_{\rm g}-1|\ll1$. Therefore, we have [@Cembranos:2018lcs] $$\begin{aligned}
\label{neff}
n_{\rm g}=1+i\frac{\nu }{2\,\omega}+(1-c_{T}),\end{aligned}$$ up to first order. Following the procedure outlined in the previous section, we obtain $$\begin{aligned}
\label{Matrix}
\left[\left(\omega-i\partial_z\right)+
\left({\begin{array}{cccc}
\Delta^\bot_{\gamma} & \Delta_M &0&0 \\
\Delta_M & \Delta_{\rm g}^\bot &0&0\\
0&0&\Delta^\parallel_{\gamma} & \Delta_M \\
0&0&\Delta_M & \Delta_{\rm g}^\parallel \\
\end{array} }
\right)\right]
\left(\begin{array}{c}
A_\bot\\ \tilde h_+\\A_\parallel\\ \tilde h_\times
\end{array}\right)=0,\end{aligned}$$ where now $\Delta_M=\sqrt{\kappa_{\rm eff}\kappa}\,B_{\rm T}/2$, $\Delta_{\rm g}^\bot=\Delta_{\rm g}^\parallel=\omega\,(n_{\rm g}-1)$, and $\Delta^{\bot,\parallel}_{\gamma}$ are defined as in the previous section. In addition, we have more properly defined the amplitude of the GWs polarizations states as $\tilde h_{+,\times}=\sqrt{\kappa/\kappa_{\rm eff}}\,h_{+,\times}$, which has allowed us to obtain a symmetric mixing matrix. It is worth emphasizing that, although we have assumed a Horndeski Lagranigan, other ATGs will produce similar mixing matrices as that of the system (\[Matrix\]), but with $n_{\rm g}$ and $\kappa_{\rm eff}$ given by the particular theory, and may be even $n_{\rm g}^\bot\neq n_{\rm g}^\parallel$ [@Alexander:2004wk; @Yunes:2010yf]. For example, as argued in reference [@Cembranos:2018lcs], the effective refractive index of GWs propagating in a Minkowski background could take the more general form $$\label{neff}
n_{\rm g}=1+i\frac{\nu }{2\,\omega}+(1-c_{T})-\frac{m_{\rm g}^2}{2\,\omega^2}-\frac{A}{2}\omega^{\alpha-2},$$ up to first order in $\nu/\omega$, $1-c_T$, $m_g^2/\omega^2$, and $A\,\omega^{\alpha-2}$, for theories implying a non-vanish graviton mass or Lorentz violations.
Conversion probability
======================
In view of the system (\[Matrix\]) one can note that there is no mixing between both polarizations. So, as it has been done in reference [@Dolgov:2012be] for the general relativistic case, we can split the system into two similar subsystems defining $$\begin{aligned}
\label{Matrix2}
M_\lambda=
\left({\begin{array}{cc}
\Delta^\lambda_ {\gamma} & \Delta_M \\
\Delta_M & \Delta_{\rm g}^\lambda \\
\end{array} } \right),\quad
\Psi_\lambda=\left(\begin{array}{c}
A_\lambda\\ \tilde h_\lambda
\end{array}\right),\end{aligned}$$ where the subindex $\lambda$ denotes the two possible polatizations; however, in what follows, we do not include the sub-index $\lambda$ for simplicity. As $M$ is a symmetric matrix, it can be diagonalized by a rotation. Therefore, we have $$\left[(\omega-i\partial_z)+M\right]\Psi=0=\left[(\omega-i\partial_z)+M'\right]\Psi',$$ with $M'=U^TM U$, $\Psi'=U^T\Psi$. The eigenvalues of $M$ and the angle of rotation are given by $$\begin{aligned}
m_{1,2}&=&\frac{1}{2}\left[\Delta_{\rm g}+\Delta_{\gamma}\mp\sqrt{4\Delta_{\rm M}^2+\left(\Delta_{\rm g}-\Delta_{\gamma}\right)^2}\right],\\
\tan(2\theta)&=&\frac{2\Delta_{\rm M}}{\Delta_{\rm g}-\Delta_{\gamma}}.\end{aligned}$$ So, we obtain a solution for the system with a similar expression to that presented in reference [@Dolgov:2012be], taking into account our conventions and the different definition of the quantities due to the diagravitational medium. This is $$\begin{aligned}
A(z)&=&\left[\cos^2\theta \,e^{-im_1z}+\sin^2\theta \,e^{-im_2z}\right] A(0)\nonumber\\
&-&\sin\theta\cos\theta\left[e^{-im_1z}-e^{-im_2z}\right]\tilde h(0),\\
\tilde h(z)&=&-\sin\theta\cos\theta\left[e^{-im_1z}-e^{-im_2z}\right]A(0)\nonumber\\
&+&\left[\sin^2\theta \,e^{-im_1z}+\cos^2\theta \,e^{-im_2z}\right]\tilde h(0),\end{aligned}$$ where we have absorbed a global phase $e^{i\omega z}$ in $A(z)$ and $\tilde h(z)$. Now, if we assume that initially there are no photons, the probability of graviton-photon conversion is $$\label{P}
P_{{\rm g}\rightarrow{\gamma}}=\sin^2(2\theta)\sin^2\left(\sqrt{\Delta_{\rm M}^2+\left(\Delta_{\rm g}-\Delta_{\gamma}\right)^2/4}\,\cdot z\right),$$ where $$\sin^2(2\theta)=\frac{\Delta_{\rm M}^2}{\Delta_{\rm M}^2+\left(\Delta_{\rm g}-\Delta_{\gamma}\right)^2/4}.$$ The results of the general relativistic mixing are recovered in equation (\[P\]) for $\Delta_{\rm g}=0$, which degenerate to graviton-photon conversion or resonance if $\Delta_{\gamma}=0$.
#### Maximum mixing or resonance.
In GR graviton-photon mixing is maximized for $n_{\gamma}=1$. This process corresponds to a maximum mixing angle $\theta=\pi/4$, leading to a probability $$\label{PGR}
P_{{\rm g}\rightarrow{\gamma}}=\sin^2\left(\Delta_{\rm M}\,z\right)=\sin^2\left(\kappa\,B_{\rm T}\,z/2\right).$$ Therefore, it is possible to have complete conversion of GWs into EMWs when these propagate through the magnetic field. In ATGs it is possible to have resonance even if $\Delta_{\gamma}=0$; one just needs $\Delta_{\rm g}=\Delta_{\gamma}$, that is $n_{\rm g}=n_{\gamma}$, to have $$P_{{\rm g}\rightarrow{\gamma}}=\sin^2\left(\Delta_{\rm M}z\right)=\sin^2\left(\sqrt{\kappa_{\rm eff}\kappa}\,B_{\rm T}\,z/2\right),$$ where we can see that only ATGs inducing an effective gravitational coupling modify the conversion probability.
#### Weak mixing.
We consider now the case of weak mixing, that is $\theta\ll1$, which corresponds to $(\Delta_{\rm g}-\Delta_{\gamma})^2\gg\Delta_{\rm M}^2$. Then, we have $$\begin{aligned}
P_{{\rm g}\rightarrow{\gamma}}&=&\frac{4\Delta_{\rm M}^2}{\left(\Delta_{\rm g}-\Delta_{\gamma}\right)^2}\sin^2\left(|\Delta_{\rm g}-\Delta_{\gamma}|z/2\right).\end{aligned}$$ This probability reduces to $$\begin{aligned}
P_{{\rm g}\rightarrow{\gamma}}&=&\Delta_{\rm M}^2z^2=\kappa_{\rm eff}\kappa \,B_{\rm T}^2\,z^2/4,\end{aligned}$$ for a path even shorter than the oscillation length $l_{\rm osc}=2\pi/|\Delta_{\rm g}-\Delta_{\gamma}|$. Hence, again, only ATGs inducing an effective gravitational coupling modify the mixing probability in this regime, amplifying it for $\kappa_{\rm eff}>\kappa$.
Discusion
=========
We have investigated the phenomenon of graviton-photon oscillation in ATGs and show that it reduces to resonance when both effective refractive indexes, corresponding to gravitational and electromagnetic radiation, are equal. We have obtained the probability of graviton to photon conversion in the presence of a magnetic field, which can describe the phenomenon of graviton-photon mixing due to primordial GWs crossing cosmic magnetic fields. This phenomenon may produce observable effects in the electromagnetic cosmic background.
It should be emphasized that our results are the first attempt to describe the process of graviton-photon oscillation in ATGs. Some possible extensions of our treatment could take into account the backreaction of the additional degrees of freedom used to formulate the ATG and the potential coupling of their perturbations to the tensor perturbations. In addition, we have assumed that the effective refractive index and coupling can be treated as constant parameters along the intervals of interest. Nevertheless, some non-trivial effect may emerge in regions of large variation of the effective refractive index.
This work is supported by the project FIS2016-78859-P (AEI/FEDER, UE). PMM gratefully acknowledges financial support provided through the Research Award L’Oréal-UNESCO FWIS (XII Spanish edition).
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---
abstract: 'The Visible Light Photon Counter (VLPC) features high quantum efficiency and low pulse height dispersion. These properties make it ideal for efficient photon number state detection. The ability to perform efficient photon number state detection is important in many quantum information processing applications, including recent proposals for performing quantum computation with linear optical elements. In this paper we investigate the unique capabilities of the VLPC. The efficiency of the detector and cryogenic system is measured at 543nm wavelengths to be 85$\%$. A picosecond pulsed laser is then used to excite the detector with pulses having average photon numbers ranging from 3-5. The output of the VLPC is used to discriminate photon numbers in a pulse. The error probability for number state discrimination is an increasing function of the number of photons, due to buildup of multiplication noise. This puts an ultimate limit on the ability of the VLPC to do number state detection. For many applications, it is sufficient to discriminate between 1 and more than one detected photon. The VLPC can do this with 99$\%$ accuracy.'
author:
- Edo Waks
- Kyo Inoue
- 'William D. Oliver'
- Eleni Diamanti
- Yoshihisa Yamamoto
bibliography:
- 'VLPC.bib'
title: High Efficiency Photon Number Detection for Quantum Information Processing
---
Introduction
============
Optical quantum information processing is one of the most rapidly developing segments of quantum information to date. The photon offers many distinct advantages over other implementations of a quantum bit (qubit). It is very robust to environmental noise, and can be transmitted over very long distances using optical fibers. For this reason the photonic qubit is the exclusive information carrier for quantum cryptography applications. Recent theoretical developments have also shown that single photons, combined with only linear optical components and photon counters, can be used to implement scalable quantum computers.
At the heart of any optical quantum information processing application is the ability to detect photons. Photon counters are an essential tool for virtually all quantum optics experiments. A photon counter absorbs a single photon, and outputs a macroscopic current that can be processed by subsequent digital circuits. To date, photomultiplier tubes (PMTs) and avalanche photodiodes (APDs) are the most common photon counters. In a PMT, a photon scatters a single electron from a photocathode. The electron is multiplied by successive scattering off of dynodes in order to generate a macroscopic current. PMTs are known to have superb time resolution and low pulse height dispersion, yet they typically suffer from low detection efficiencies. Optimal quantum efficiencies for a PMT typically do not exceed 40$\%$. Avalanche photodiodes feature higher quantum efficiencies. In an APD, a photon creates a single electron hole pair in a semiconductor pn junction. An avalanche breakdown mechanism multiplies this electron-hole pair into a large current. APDs can have quantum efficiencies as high as 75$\%$. The main limitations of APDs is that they have a relatively long dead time ( 35ns), and large pulse height dispersion. If two photons are simultaneously absorbed by the APD, the output pulse will not differ from the case when only one is absorbed. Thus, APDs cannot distinguish between one and more than one photon if all of the photons land within the dead time of the detector. We refer to such detectors as threshold detectors.
Recently, a new type of photon detector, the Visible Light Photon Counter (VLPC), has been shown to have some unique capabilities that conventional PMTs and APDs don’t have. The VLPC features high quantum efficiencies ( 94$\%$), and low pulse height dispersion [@KimYamamoto97; @TakeuchiKim99]. This latter property makes the VLPC useful for photon number detection [@KimTakeuchi99]. Unlike an APD, if two photons are simultaneously absorbed by the VLPC, the detector outputs a voltage pulse which is twice as high. This behavior continues for higher photon numbers. Thus, the voltage pulse of the VLPC carries information about photon number. We refer to this type of detector as a photon number detector, in contrast to threshold detectors discussed previously.
Photon number detection is very useful for quantum information processing. It has applications in quantum cryptography, particularly in conjunction with parametric down-conversion. It is also an important element for linear optical quantum computation (LOQC) as proposed by Knill, Laflamme, and Milburn [@KnillLaflamme01]. Many of the basic building blocks for this proposal fundamentally rely on the ability to distinguish between one and more than one photon with high quantum efficiency [@BartlettDiamanti02].
There are several unique aspects of the VLPC which allow it to do photon number detection. First, the VLPC is a large area detector, whose active area is about 1mm in diameter. When a photon is detected, a dead spot of several microns in diameter is formed on the detector surface, leaving the rest of the detector available for subsequent detection events. If more than one photon is incident on the detector, it will be able to detect all the photons as long as the probability that multiple photons land on the same location is small. This is a good approximation if the light is not too tightly focussed on the detector surface. In this respect the VLPC is similar to a large array of beamsplitters and threshold detectors.
There is, however, one major distinction between the VLPC and a large detector array. In an array, we can address the signal from each counter individually. In contrast, we cannot individually access each spot on the VLPC surface. Instead, the current from the entire detector is summed and accessed through a single output. We must use the height or area of the output pulse to infer the photon number. This makes the noise properties of the detector critical for photon number detection. When independent noisy voltages are summed the noise builds up. This degrades the number resolution capability of the detector, and ultimately puts a limit on the number of incident photons that can be resolved. It is important to measure this limitation in order to assess the capability of the detector to perform quantum information processing tasks. For this we need to consider the noise properties of the VLPC.
The noise properties of a photon counting system are ultimately limited by the internal multiplication noise of the detector. Photon counters typically rely on an internal multiplication gain to create a large current spike from a single photoionization event. These gain mechanisms create internal multiplication noise, meaning that the current spikes generated by a photodetection event will fluctuate in height and area. In order to do accurate photon number detection, this multiplication noise must be low. Fortunately, the VLPC has been measured to have nearly noise-free multiplication [@KimYamamoto97].
VLPC operation principle
========================
Figure \[fig:VLPCschematic\] shows the structure of the VLPC detector. Photons are presumed to come in from the left. The VLPC has two main layers, an intrinsic silicon layer and a lightly doped arsenic gain layer. The top of the intrinsic silicon layer is covered by a transparent electrical contact and an anti-reflection coating. The bottom of the detector is a heavily doped arsenic contact layer, which is used as a second electrical contact.
A single photon in the visible wavelengths can be absorbed either in the intrinsic silicon region or in the doped gain region. This absorption event creates a single electron-hole pair. Due to a small bias voltage (6-7.5V) applied across the device, the electron is accelerated towards the transparent contact while the hole is accelerated towards the gain region. The gain region is moderately doped with As impurities, which are shallow impurities lying only 54meV below the conduction band. The device is cooled to an operation temperature of 6-7K, so there is not enough thermal energy to excite donor electrons into the conduction band. These electrons are effectively frozen out in the impurity states. However, when a hole is accelerated into the gain region it easily impact ionizes these impurities, kicking the donor electrons into the conduction band. Scattered electrons can create subsequent impact ionization events resulting in avalanche multiplication.
One of the nice properties of the VLPC is that, when an electron is impact ionized from an As impurity, it leaves behind a hole in the impurity state, rather than in the valence band as in the case of APDs. The As doping density in the gain region is carefully selected such that there is partial overlap between the energy states of adjacent impurities. Thus, a hole trapped in an impurity state can travel through conduction hopping, a mechanism based on quantum mechanical tunnelling. This conduction hopping mechanism is slow, the hole never acquires sufficiency kinetic energy to impact ionize subsequent As sites. The only carrier that can create additional impact ionization events is the electron kicked into the conduction band. Thus, the VLPC has a natural mechanism for creating single carrier multiplication, which is known to significantly reduce multiplication noise [@McIntyre66]. We will return to this point in the upcoming sections.
One of the disadvantages of using shallow As impurities for avalanche gain is that these impurities can easily be excited by room temperature thermal photons. IR photons with wavelengths of up to 30$\mu m$ can directly optically excite an impurity. These excitations can create extremely high dark count levels. The bi-layer structure of the VLPC helps to suppress this. A visible photon can be absorbed both in the intrinsic and doped silicon regions. An IR photon, on the other hand, can only be absorbed in the doped region, as its energy is smaller than the bandgap of intrinsic silicon. Thus, the absorption length of IR photons is much smaller than visible photons. This suppresses the sensitivity of the device to IR photons to about 2$\%$. Despite this suppression, the background thermal radiation is very bright, requiring orders of magnitude of additional suppression. In the next section we will discuss how this is achieved.
Cryogenic system for operating the VLPC
=======================================
In order to operate the VLPC we must cool it down to cryogenic temperatures to achieve carrier freezeout of the As impurities. We must also shield it from the bright room temperature thermal radiation which it is partially sensitive to. This is achieved by the cryogenic setup shown in Figure \[fig:VLPCcryo\].
The VLPC is held in a helium bath cryostat. As small helium flow is produced from the helium bath to the cryostat cold finger by a needle valve. The helium bath is surrounded by a nitrogen jacket for radiation shielding. This improves the helium hold time. A thermal shroud, cooled to 77K by direct connection to the nitrogen jacket, covers the VLPC and low temperature shielding. This shroud is intended to improve the temperature stability of the detector by reducing the thermal radiation load. A hole at the front of the shroud allows photons to pass through. The detector itself is encased in a 6K shield made of copper. The shield is cooled by direct connection to the cold plate of the cryostat. The front windows of the 6K radiation shield, which are also cooled down to this temperature, are made of acrylic plastic. This material is highly transparent at optical frequencies, but is almost completely opaque from 2-30$\mu m$. The acrylic windows provide us with the required filtering of room temperature IR photons for operating the detector. We achieve sufficient extinction of the thermal background using 1.5-2 cm of acrylic material. In order to eliminate reflection losses from the window surfaces, the windows are coated with a broadband anti-reflection coating centered at 532nm. Room temperature transmission measurements indicate a 97.5$\%$ transmission efficiency through the acrylic windows. However, the performance of the anti-reflection coating degrades when the windows are cooled down to cryogenic temperatures. Low temperature reflection measurements indicate a 7$\%$ reflection loss. This increased loss is attributed to changes in the dielectric constant of the material, resulting in a worse impedance match for the anti-reflection coating. Better engineering of the anti-reflection coating could help eliminate these losses.
The surface of the VLPC has a broadband anti-reflection coating centered around 550nm. Nevertheless, due to the large index mismatch between silicon and air, there is still substantial reflection losses on the order of $10\%$, even at the correct wavelength. In order to eliminate this reflection loss, the detector is rotated 45 degrees to the direction of the incoming light. A spherical refocussing mirror, with reflectance exceeding 99$\%$, is used to redirect reflected light back onto the detector surface. A photon must reflect twice off of the surface in order to be lost, reducing the reflections losses to less than 1$\%$.
The VLPC features high multiplication gains of about 30,000 electrons per photo-ionization event. Nevertheless, this current must be amplified significantly in order to achieve sufficiently large signal for subsequent electronics. The current is amplified by a series of broadband RF amplifiers. In order to minimize the thermal noise contribution from the amplifiers, the first amplification stage consists of a cryogenic pre-amplifier, which is cooled to 4K by direct thermalization to the helium bath of the cryostat. The amplifier features a noise figure of 0.1 at the operating frequencies of $30-500MHz$, with a gain of roughly 20dB. The cryogenic amplifier is followed by additional commercial room temperature RF amplifiers. The noise properties of these subsequent amplifiers is not as important since the signal to noise ratio is dominated by the first cryogenic amplification stage. Using such a configuration, we achieve a 120mV pulse with a 3ns duration when using 62dB of amplifier gain.
Quantum efficiency and dark counts of the VLPC
==============================================
The quantum efficiency of the VLPC has already been studied at 650nm [@TakeuchiKim99]. Quantum efficiencies (QE) as high as 88$\%$ have been reported. The dark counts at this peak QE were 20,000 1/s. Here we present measurements using a different operating wavelength of 543nm, and a different cryogenic setup. As a quick summary of what we will discuss, we observe raw quantum efficiencies as high as 85$\%$ at this operating wavelength, with dark count rates of roughly 20,000 1/s. When correcting for reflection losses from the windows and detector surface, we estimate an intrinsic quantum efficiency of 93$\%$. These numbers are consistent with previous measurements.
The setup for measuring the quantum efficiency of the VLPC is shown in Figure \[fig:QEsetup\]. We use a helium neon laser with an output wavelength of 543nm as a light source for the measurement. An intensity stabilizer is used to stabilize the output of the laser to within about 0.1$\%$. A 50-50 beamsplitter is then used to send part of the laser to a calibrated PIN diode to measure the power. The power reading from the diode is accurate to within a 2$\%$ calibration error. Using this power reading we can calculate the photon flux $N$, in units of photons per second. This is given by the relation $$N = \frac{\lambda P}{hc},$$ where $\lambda$ is the wavelength of the laser, $P$ is the power measured by the PIN diode, $h$ is Planke’s constant, and $c$ is the velocity of light in vacuum.
The laser is attenuated by a series of carefully calibrated neutral density (ND) filters down to a flux of approximately 20,000 cps. The attenuation required for this is on the order of $10^{-9}$. This flux is sufficiently small to ensure that we are well within the linear regime of the VLPC. At count rates exceeding $10^{5}$ cps, the efficiency of the VLPC will begin to drop due to dead time effects. To measure the efficiency of the VLPC we record the count rates of the detector, which we label $N_c$, as well as the background counts $N_d$, which are measured by blocking the laser. The measured efficiency $\eta$ is given by $$\eta = \frac{N_c-N_d}{\alpha N},$$ where alpha is the transmission efficiency of the ND filters.
In Figure \[fig:QEvBias\], we show the measured quantum efficiency of the VLPC as a function of applied bias voltage across the device. Efficiencies are given for several different operating temperatures. At 7.4V bias the VLPC attains its highest quantum efficiency of 85$\%$. As the bias voltage is decreased the quantum efficiency also decreases. The reason for this is that, at lower bias voltages, electrons created by impact ionization of the initial hole are less likely to accumulate sufficient kinetic energy in the gain region to trigger an avalanche. The bias voltage cannot be increased beyond 7.4V. Beyond this bias the VLPC breaks down, resulting in large current flow through the device. This breakdown is attributed to direct tunnelling of electrons from impurity sites into the conduction band.
One will notice that as the temperature is decreased, more bias voltage is required to achieve the same quantum efficiency. This effect is attributed to a temperature dependance of the dielectric constant of the device, which results in a change in the electric field intensity in the gain region of the VLPC. As the temperature is decreased, it is speculated that the dielectric constant increases, requiring higher bias voltage to achieve the same electric field intensity. This conjecture is supported by the measurements shown in Figure \[fig:QEvDC\]. In this figure we plot the quantum efficiency as a function of dark counts, instead of bias voltage. Data is shown for the different temperatures. Increasing the bias voltage results not only increased quantum efficiency, but also in increased dark counts. Increasing the temperature also increases both quantum efficiency and dark counts. But if we plot the quantum efficiency as a function of dark counts, as is done in Figure \[fig:QEvDC\], the data for different temperatures all lie along the same curve. This suggests that the quantum efficiency and dark counts both depend on a single parameter, the electric field intensity in the gain region. The temperature and bias voltage dependance of this parameter result in the behavior shown in Figure \[fig:QEvBias\]. From Figure \[fig:QEvDC\] we see that the maximum quantum efficiency of 85$\%$ is achieved at a dark count rate of roughly 20,000 cps.
In order to infer the actual efficiency of the VLPC alone, we must correct for all other losses in our detection system. The acrylic windows are a big source of loss. As mentioned previously, the windows add a 7$\%$ reflection loss to our measurement. In addition to this loss we have a reflection loss of 1$\%$ due to the VLPC surface, despite the retro-reflector. Other effects such as detector dead time and beam focussing should contribute only negligibly small corrections to the device efficiency. Thus, the efficiency of the VLPC detector itself is estimated to be $93\%$ at 543nm wavlengths.
Noise properties of the VLPC
============================
When a photon is absorbed in a semi-conductor material, it creates a single electron hole pair. The current produced by this single pair of carriers is, in almost all cases, too weak to observe due to thermal noise in subsequent electronic components. Single photon counters get around this problem by using an internal gain mechanism to multiply the initial pair into a much greater number of carriers. Avalanche photodiodes achieve this by an avalanche breakdown mechanism in the depletion region of the diode. Photomultipliers instead rely on successive scattering off of dynodes. The VLPC achieves this gain by impact ionization of shallow arsenic impurities in silicon.
All of the above gain mechanism have an intrinsic noise process associated with them. That is, a single ionization event does not produce a deterministic number of electrons. The number of electrons the device emits fluctuate from pulse to pulse. This internal noise is referred to as excess noise, or gain noise. The amount of excess noise that a device features strongly depends on the mechanism in which gain is achieved. The excess noise is typically quantified by a parameter $F$, referred to as the excess noise factor (ENF). The ENF is mathematically defined as $$F = \frac{\langle M^2 \rangle}{\langle M \rangle^2},$$ where $M$ is the number of electrons produced by a photo-ionization event, and the brackets notation represents a statistical ensemble averages. Noise free multiplication is represented by $F=1$. In this limit, a single photo-ionization event creates a deterministic number of additional carriers. Fluctuations in the gain process will result in an ENF exceeding 1.
The noise properties of an avalanche photo-diode are well characterized. The first theoretical study of such devices was presented by McIntyre in 1966 [@McIntyre66]. McIntyre studied avalanche gain in the “Markov” limit. In this limit, the impact ionization probability for a carrier in the depletion region is a function of the local electric field intensity at the location of the carrier. In this sense, each impact ionization event is independent of past history. Under this assumption the ENF of an APD was calculated. The ENF depends on the number of carriers that can participate in the avalanche process. If both electrons and holes are equally likely to impact ionize, then $F\approx\langle M
\rangle$. In the large gain limit the ENF is very big. Restricting the impact ionization process to only electrons or holes significantly reduces the gain noise. In this ideal limit, we have $F=2$. This limit represents the best noise performance achievable within the Markov approximation.
PMTs are known to have better noise characteristics than APDs. The ENF of a typical PMT is around 1.2. This suppressed noise is because, in a PMT, a carrier is scattered off of a fixed number of dynodes. The only noise in the process is the number of electrons emitted by each dynode per electron.
The multiplication noise properties of the VLPC have been previously studied. Theoretical studies of the multiplication noise have predicted that the VLPC should feature supressed avalanche multiplication noise. This is due to two dominant effects. First, because only electrons can cause impact ionization, the VLPC features a natural single carrier multiplication process. Second, the VLPC does not require high electric field intensities to operate. This is because impact ionization events occur off of shallow arsenic impurities which are only 54meV from the conduction band. Thus, carriers do not have to acquire a lot of kinetic energy in order to scatter the impurity electrons. Because of the lower electric field intensities, a carrier requires a fixed amount of time before it can generate a second impact ionization. This delay time represents a deviation from the Markov approximation, and is predicted to suppress the multiplication noise [@LaVioletteStapelbroek89]. The ENF of the VLPC has been experimentally measured to be less than 1.03 in [@KimYamamoto97]. Thus, the VLPC features nearly noise free multiplication, as predicted by theory. This low noise property will play an important role in multi-photon detection, which we discuss next.
Multi-photon detection with the VLPC
====================================
The nearly noise-free avalanche gain process of the VLPC opens up the door to perform multi-photon detection. When more than one photon is detected by the VLPC, we expect the number of electrons emitted by the detector to be twice that of a single photon detection. If the photons arrive within a time interval which is much shorter than the electronic output pulse duration of the detection system, then we expect to see a detection pulse which is twice as high.
In the limit of noise free multiplication, this would certainly be the case. A single detection event would create $M$ electrons, while a two photon event would create $2M$ electrons. Higher order photon number detections would follow the same pattern. After amplification, the area or height of the detector pulse would allow us to perfectly discriminate the number of detected photons, even if they arrive on extremely short time scales.
In the presence of multiplication noise, the situation becomes more complicated. The pulse height of a one photon pulse will fluctuate, as will that of a two photon pulse. There becomes a finite probability that we only detect one photon, but due to multiplication noise the height of the pulse appears to be more consistent with a two photon event, and vice versa. Our ability to discriminate the number of detected photons becomes a question of signal to noise ratio.
There are ultimately two effects which will limit multi-photon detection. One is the quantum efficiency of the detector. If we label the quantum efficiency as $\eta$, then the probability of detecting $n$ photons is given by $\eta^n$, assuming detector saturation is negligible. Thus, the detection probability is exponentially small in $\eta$. For larger $n$ this may produce extremely low efficiencies. The second limitation is the electrical detection noise, as previously discussed. There are two contributions to the electrical noise. One is the excess noise of the detector, and the other is electrical noise originating from amplifiers and subsequent electronics. The latter can in principle be eliminated by engineering ultra-low noise circuitry. The former, however, is a fundamental property of the detector which cannot be circumvented, short of engineering a different detector with better noise properties.
In the absence of detection inefficiency and amplifier noise, the multiplication noise will ultimately put a limit on how many simultaneous photons we can detect. Defining $\sigma_m$ as the standard deviation of the multiplication gain, the fluctuations of an $n$ photon peak will be given by $\sqrt{n}\sigma_m$. This is because the $n$ photon pulse is simply the sum of $n$ independent single photon pulses from different locations of the VLPC active area. Summing the pulses also causes the variance to sum, resulting in the buildup of multiplication noise. The mean pulse height separation between the $n$ photon peak and the $n-1$ photon peak, however, is constant. It is simply proportional to $\langle
M \rangle$, the average multiplication gain. At some sufficiently high photon number, the fluctuations in emitted electrons will be so large that there is little distinction between an $n$ and $n-1$ photon event. We can arbitrarily establish a cutoff number at the point where the fluctuations in emitted electrons is equal to the average difference between an $n$ and $n-1$ photon detection event. In this limit, the maximum photon number we can detect is $$N_{max} = \frac{1}{F-1}.$$ Using the above condition as a cutoff, we see that even an ideal APD with $F=2$ cannot discriminate between 1 and 2 photon events. A PMT with $F=1.2$ could potentially be useful for up to 5 photon detection, but due to low quantum efficiencies of PMTS, this is typically impractical. The VLPC, with $F<1.03$ could potentially discriminate more that 30 photons. Furthermore it could potentially do this with $93\%$ quantum efficiency. However, this limit is difficult to approach due to electronic noise contribution from subsequent amplifiers.
Characterizing multi-photon detection capability
================================================
The multi-photon detection capability of the VLPC has been previously studied. Early studies used long light pulse excitations, with poor electronic time resolution so that multiple photons appeared as a single electronic pulse [@AtacPark92]. Later studies used twin photons generated from parametric down-conversion, which arrive nearly simultaneous, to investigate multi-photon detection [@KimTakeuchi99]. These studies restricted their attention to one and two photon detection. Higher photon numbers were not considered.
The experiment described below measures the photon number detection capability of the VLPC when excited by multiple photons. Figure \[fig:VLPCmultiSetup\] shows the experimental setup. A Ti:Sapphire laser, emitting pulses of about 3ps duration, is used. The duration of the optical pulses are much shorter than the electrical pulse of the VLPC detector, which is 2ns. A pulse picker is used to down-sample the repetition rate of the laser from 76MHz to 15KHz. This is done in order to avoid saturation of the detector. A synchronous countdown module, which is used as the pulse picking signal, is also used to trigger a boxcar integrator. The output of the VLPC is amplified by the amplifier configuration discussed earlier. The amplified signal is integrated by a boxcar integrator. The integrated value of a pulse is proportional to the number of electrons emitted by the detector, as long as amplifier saturation is negligible. The output of the boxcar integrator is digitized by an analog to digital converter, and stored on a computer.
Figure \[fig:OscPulse\] shows a sample oscilloscope pulse trace of a VLPC pulse after the room temperature amplifiers. The output features an initial sharp negative peak of about 2ns full width at the half maximum. A positive overshoot follows. This positive overshoot is the result of the 30MHz high pass of the cryogenic amplifiers. If we compare the variance of the electrical fluctuations before the pulse to the minimum pulse value, we determine the signal to noise ration (SNR) to be 27. The figure also illustrates the integration window used by the boxcar integrator, which captures only the negative lobe of the pulse.
In order to measure the multi-photon detection capability, we attenuate the laser to about 1-5 detected photons per pulse. For each laser pulse, the output of the VLPC is integrated and digitized. Figure \[fig:PAspectrum\] shows pulse area histograms for four different excitation powers. The area is expressed in arbitrary units determined by the analog to digital converter. Because the pulse area is proportional to the number of electrons in the pulse, the pulse area histogram is proportional to the probability distribution of the number of electrons emitted by the VLPC. This probability distribution features a series of peaks. The first peak is a zero photon event, followed by one photon, two photons, and so on. In the absence of electronic noise and multiplication noise, these peaks would be perfectly sharp, and we would be able to unambiguously distinguish photon number. Due to electronic noise however, the peaks become broadened and start to partially overlap. The broadening of the zero photon peak is due exclusively to electronic noise. Note that the boxcar integrator adds an arbitrary constant to the pulse area, so that the zero photon peak is centered around 450 instead of 0. The one photon peak is broadened by both electronic noise and multiplication noise. Thus, the variance of the one photon peak is bigger than the zero photon peak. As the photon number increases, the width of the pulses also increases due to buildup of multiplication noise. This eventually causes the smearing out of the probability distribution at around the seven photon peak.
In order to numerically analyze the results, we fit each peak to a gaussian distribution. Theoretical studies predict that the distribution of the one photon peak is a bi-sigmoidal distribution, rather than a gaussian [@LaVioletteStapelbroek89]. However, when the multiplication gain as large, as in the case of the VLPC, this distribution is well approximated by a gaussian. We use this approximation because higher photon number events are sums of multiple single photon events. A gaussian distribution has the nice property that a sum of gaussian distributions is also a gaussian distribution. In the limit of large photon numbers we expect this approximation to get even better due to the central limit theorem.
The most general fit would allow the area, mean, and variance of each peak to be independently adjustable. This allows too many degrees of freedom, which often results in the optimization algorithm falling into a local minimum. To help avoid this, we do not allow the average of each peak to be independently adjustable. Instead, we require the averages to be equally spaced, as would be expected from our model of the VLPC. Thus, the average of the i’th peak, denoted $x_i$, is determined by the relation $$x_i = x_0 + i \Delta - i^2 \alpha.$$ In the above equation, $x_0$ is the average of the zero photon peak, $\Delta$ is the spacing between peaks, and $\alpha$ is a small correction factor which can account for effects such as amplifier saturation. These three parameters are all independently adjustable. In all of our fits, $\alpha$ was much smaller than $\Delta$ indicating the peaks are, for the most part, equally spaced.
Figure \[fig:PAspectrum\] shows the results of the fits for each excitation intensity. The dotted lines plot the individual gaussian distributions for the different photon numbers, and the solid line plots the sum of all of the gaussians. The diamond markers represent the measured data points. Table \[table:FitResult\] shows the center value and standard deviation of the different peaks in panel c of the figure. In order to do photon number counting we must establish a decision region for each photon number state. This will depend, in general, on the a-priori photon number distribution. We consider the case of equal a-priori probability, which is the worst case scenario. For this case, the optimal decision threshold between two consecutive gaussian peaks is given by the point where they intersect. The value of this point can be easily solved, and is given by, $$\begin{gathered}
x_d = x_i - \frac{\sigma_i^2 (x_{i+1} -
x_i)}{\sigma_{i+1}^2-\sigma_i^2}\\
+ \frac{\sigma_i\sigma_{i+1}\sqrt{(x_{i+1} - x_i)^2- 2\left( \sigma_{i+1}^2-\sigma_i^2\right)
\ln\frac{\sigma_i}{\sigma_{i+1}}}}{\sigma_{i+1}^2-\sigma_i^2}.
\end{gathered}$$ The probability of error for this decision is given by the area of all other photon number peaks in the decision region. This probability is also shown in Table \[table:FitResult\].
Photon number Avg. Area Std. Dev. $\%$Error
--------------- ----------- ----------- ----------- --
0 0 10.6 0.01
1 135 24.8 1.1
2 275 31.7 3.4
3 416 35.3 6.1
4 561 39.0 8.5
5 709 42.2 10.6
6 859 44.5 11.3
: Results of fit for panel (c) of Figure \[fig:PAspectrum\].[]{data-label="table:FitResult"}
From the data we would like to infer whether the VLPC is being saturated at higher photon numbers. If too many photons are simultaneously incident on the detector, the detector surface may become depleted of active area. This would result in a reduced quantum efficiency for higher photon numbers. In order to investigate this possibility, we add an additional constraint to the fit that the pulse areas must scale according to a Poisson distribution. Since the laser is a Poisson light source, we expect this to be the case. However, if saturation becomes a factor, we would observe a number dependant loss. This would result in deviation from Poisson detection statistics. In Figure \[fig:PoissonSpec\] we plot the result of the fit when the peak areas scale as a Poisson distribution. One can see that the imposition of Poisson statistics does not change the fitting result in an appreciable way. Thus, we infer that detector saturation is not a strong effect at the excitation levels that we are using.
The effect of multiplication noise buildup on the pulse height spectrum can be investigated from the previous data. In general, we expect the pulse area variance to be a linearly increasing function of photon number. This is consistent with the independent detection model, in which an $n$ photon peak is a sum of $n$ single photon peaks coming from different areas of the detector. To investigate the validity of this model, we plot variance as a function of photon number in Figure \[fig:MultBuild\]. The electrical noise variance, given by the zero photon peak, is subtracted. The variance is fit to a linear model given by $$\sigma_i^2 = \sigma_0^2 + i \sigma_M^2.$$ In the above model, $i$ is the photon number, $\sigma_M^2$ is the variance contribution from multiplication noise, and $\sigma_0^2$ is a potential additive noise term. From the data, we obtain the values $\sigma_M^2=276$, and $\sigma_0^2=246$.
A surprising aspect of this result is the large value of $\sigma_0^2$. We expect that since electrical noise has been subtracted, the only remaining contribution to the variance is multiplication noise. If this were true, the value of $\sigma_0$ would be very small. Instead we obtain a value nearly equal to that of $\sigma_m^2$. This may indicate that the electrical noise is higher when the VLPC is firing, as opposed to when its not. A change in the resistance of the device during the avalanche process may effect the noise properties of subsequent amplification circuits. Further investigation is required in order to determine whether this additive noise is fundamental to the device, or can be eliminated in principle.
The above measurements of variance versus photon number gives us a very accurate measurement of the excess noise factor $F$ of the VLPC. Previous measurements of $F$ for the VLPC have determined that it is less than 1.03 [@KimYamamoto97], which is nearly noise free multiplication. This number was obtained by measuring the variance of the 1 photon peak, and comparing to the mean. However, it is difficult to separate the electrical noise contribution from the internal multiplication noise using this technique. Thus, the measurement ultimately determines only an upper bound of $F$. By considering how the variance scales with photon numbers, as we have done in Figure \[fig:MultBuild\], the multiplication noise can be accurately differentiated from additive electrical noise. This allows us to calculate an exact value for the excess noise factor. From our measurement of $\sigma_M^2$ and $\langle M \rangle$, we obtain an excess noise factor of $F=1.015$.
Conclusion
==========
In this paper we have discussed the interesting features of the VLPC for quantum information processing. The VLPC has the potential to detect photons with quantum efficiencies approaching 93$\%$. It also has the capability to do photon number detection, a critical feature for linear optical quantum computation. The photon number detection capability of the VLPC is fundamentally limited by internal noise processes in the device. For many applications, one does not require full photon number detection capability. It is sufficient to be able to distinguish between one and more than one detection event. The VLPC can do this with 99$\%$ accuracy. Although the requirements for fully scalable linear optical quantum computation are extremely demanding, the VLPC may find use in areas where limited quantum computational tasks are required. Such fields as quantum cryptography and quantum networking, where fully scalable computation is not always required, may be able to incorporate the VLPC to perform novel tasks.
|
---
author:
- |
\
Mathematical Physics Laboratory, RIKEN Nishina Center\
E-mail:
- |
Tatsuhiro Misumi\
Brookhaven National Laboratory\
E-mail:
- |
Akira Ohnishi\
Yukawa Institute for Theoretical Physics, Kyoto University\
E-mail:
title: ' QCD Phase Diagram with 2-flavor Lattice Fermion Formulations[^1]'
---
Introduction {#sec:introduction}
============
Lattice QCD has played an important role in study of the non-perturbative aspects of QCD. However, its application to the finite density system has not been established due to serious difficulty of the sign problem. In this report we propose a new framework of investigating the 2-flavor QCD with finite temperature and density by using the Karsten-Wilczek (KW) lattice fermion [@Karsten:1981gd], which possesses only two species doublers, i.e. minimally doubled fermion. This lattice formulation lifts degeneracy of 16 species without breaking its chiral symmetry by introducing a species-dependent imaginary chemical potential, instead of a species-dependent mass term introduced in the Wilson fermion formalism. Because of the chemical potential term, its discrete symmetry is not sufficient to be applied to fully Lorentz symmetric system, i.e. zero temperature and density, but enough to study the in-medium QCD. To show the usefulness of the KW fermion, we study strong-coupling lattice QCD with temperature and density.
Symmetry of KW-type minimally doubled fermion
=============================================
The KW fermion is a kind of minimally doubled fermions, involving only two species doublers by introducing a species-dependent imaginary chemical potential, which we call “flavored chemical potential". This term preserves its chiral symmetry and ultra-locality [@Misumi:2012uu], but breaks some of discrete spacetime symmetries [@Bedaque:2008xs; @Kimura:2009qe]. We then need three counter terms to take a correct Lorentz symmetric continuum limit: dimension-3, $\bar\psi i \gamma_4
\psi=i\psi^\dag \psi$, and dimension-4 terms, $\bar\psi\gamma_4\partial_4\psi$, $F_{j4}F_{j4}$ [@Capitani:2009yn]. The fermionic part of KW fermion action with the counter terms is given by $$\begin{aligned}
S_{\rm KW} & = & \sum_x
\Bigg[
\frac{1}{2} \sum_{\mu=1}^4 \bar\psi_x \gamma_\mu
\left(
U_{x,x+\hat{\mu}} \psi_{x+\hat{\mu}}
- U_{x,x-\hat{\mu}} \psi_{x-\hat{\mu}}
\right)
+ i \frac{r}{2} \sum_{j=1}^3 \bar\psi_x \gamma_4
\left(
2 \psi_x
- U_{x,x+\hat{j}} \psi_{x+\hat{j}}
- U_{x,x-\hat{j}} \psi_{x-\hat{j}}
\right)
\nonumber \\
& & \qquad
+ i \mu_3 \bar \psi_x \gamma_4 \psi_x
+ \frac{d_4}{2} \bar\psi_x \gamma_4
\left(
U_{x,x+\hat{4}} \psi_{x+\hat{4}}
- U_{x,x-\hat{4}} \psi_{x-\hat{4}}
\right)
\Bigg] .
\label{KW_action}\end{aligned}$$ The second term in the first line including $i\gamma_4$ is the flavored chemical potential term, which we also call Karsten-Wilczek(KW) term. We here introduce a parameter $r$ in analogy to Wilson fermion. $\mu_3$ and $d_4$ are parameters for the dimension-3 and dimension-4 counter terms, respectively. The corresponding Dirac operator in the momentum space yields $$a D_{\rm KW}(p) =
i \sum_{\mu=1}^4 \gamma_\mu \sin a p_\mu
+ i r \gamma_4 \sum_{j=1}^3 (1 - \cos a p_\mu)
+ i \mu_3 \gamma_4 + i d_4 \gamma_4 \sin a p_4 ,$$ which has only two zeros at $\bar{p}=(0,0,0,\frac{1}{a}\mathrm{arcsin}\left(-\frac{\mu_3}{1+d_4}\right))$ when $-1-d_4<\mu_3<1+d_4$ with $r=1$. When we expand the Dirac operator around the zeros, its dispersion relation is not Lorentz symmetric. As shown in Ref.[@Misumi:2012uu], the tuning condition for the correct dispersion relation is given by $(1+d_4)^2=1+\mu_3^2$ at the tree level. Moreover, it is shown that $\mu_3$ has to be tuned to control imaginary chemical potential in $\mathcal{O}(1/a)$.
Symmetries of the lattice action (\[KW\_action\]) are chiral symmetry, cubic symmetry corresponding to permutation of spatial three axes, CT and P [@Bedaque:2008xs]:
1. ${\mathrm{U}\,}(1)$ chiral symmetry ($\gamma_5\otimes\tau_3$ [@Creutz:2010bm; @Tiburzi:2010bm; @Drissi:2011da])
2. Cubic symmetry
3. CT
4. P
It is notable that these symmetries are the same as those of the finite-density lattice QCD: As an example, we look into the naive lattice action with chemical potential, which is given by $$S_{\rm naive} = \frac{1}{2} \sum_x
\left[
\sum_{j=1}^3
\bar\psi_x \gamma_j
\left( U_{x,x+\hat{j}} \psi_{x+\hat{j}}
- U_{x,x-\hat{j}} \psi_{x-\hat{j}}
\right)
+ \bar\psi_x \gamma_4
\left( e^{\mu} U_{x,x+\hat{4}} \psi_{x+\hat{4}}
- e^{-\mu} U_{x,x-\hat{4}} \psi_{x-\hat{4}}
\right)
\right] .
\label{naive_action}$$ The 4th direction hopping term, involving chemical potential, breaks the hypercubic symmetry into the spatial cubic symmetry, and also C, P, and T into CT and P, which are the same symmetries of (\[KW\_action\]). It means that, even if we introduce chemical potential as (\[naive\_action\]) to KW fermion, the symmetries are unchanged. The KW fermion with the exponential form chemical potential is given by, $$\begin{aligned}
S_{\rm KW} & = & \sum_x
\Bigg[
\frac{1}{2} \sum_{j=1}^3 \bar\psi_x \gamma_j
\left(
U_{x,x+\hat{j}} \psi_{x+\hat{j}}
- U_{x,x-\hat{j}} \psi_{x-\hat{j}}
\right)
+ i \frac{r}{2} \sum_{j=1}^3 \bar\psi_x \gamma_4
\left(
2 \psi_x
- U_{x,x+\hat{j}} \psi_{x+\hat{j}}
- U_{x,x-\hat{j}} \psi_{x-\hat{j}}
\right)
\nonumber \\
& & \qquad
+ \frac{1+d_4}{2} \bar\psi_x \gamma_4
\left(
e^\mu U_{x,x+\hat{4}} \psi_{x+\hat{4}}
- e^{-\mu} U_{x,x-\hat{4}} \psi_{x-\hat{4}}
\right)
+ i \mu_3 \bar \psi_x \gamma_4 \psi_x
\Bigg] .
\label{KW_action_chem}\end{aligned}$$ From the viewpoint of the universality class, these two theories, (\[naive\_action\]) and (\[KW\_action\_chem\]), should belong to the same class.
Here we remark the way of introducing chemical potential. It was pointed out in [@Hasenfratz:1983ba] that a naive form of the chemical potential, $\mu\psi^\dag\psi=\mu\bar\psi\gamma_4\psi$, violates the Abelian gauge invariance and requires a counter term to make thermodynamical quantities finite. On the other hand, in the KW fermion, the flavored chemical potential term is introduced in this naive form. It leads to necessity of tuning $\mu_3$ to deal with $\mathcal{O}(1/a)$ additive renormalization of chemical potential, as with the mass renormalization in the Wilson fermion. This renormalization effect is relevant to the phase diagram in the $(\mu_3$-$g^2)$ parameter plane, as discussed in [@Misumi:2012uu].
Strong-coupling lattice QCD
===========================
We study QCD phase diagram in the framework of the strong-coupling lattice QCD with KW-type minimally doubled fermion. We extend the strong-coupling analysis with this lattice fermion [@Kimura:2011ik] to the finite temperature and density system [@Nishida:2003uj; @Fukushima:2003vi; @Nishida:2003fb]. The effective potential in terms of the meson field is obtained by performing the 1-link integral in the strong coupling limit $(g^2\to\infty)$, and then introducing auxiliary fields to eliminate the 4-point interactions. In the case with KW fermion, we have to consider both of the scalar $\sigma = \langle \bar\psi \psi \rangle$ and vector $\pi_4 = \langle
\bar \psi i \gamma_4 \psi \rangle$ condensates. Identifying $\bar\psi\gamma_4=\psi^\dag$, the latter corresponds to the imaginary density $i\langle\psi^\dag\psi\rangle$. For the case with ${\mathrm{SU}\,}(N_c)$ gauge group and $d=D+1$ dimensions in the finite temperature and density, we obtain the the following effective potential, $$\mathcal{F}_{\rm eff}(\sigma,\pi_4;m,T,\mu,\mu_3,d_4)
= \frac{N_c D}{4}
\left(
(1+r^2) \sigma^2 + (1-r^2) \pi_4^2
\right)
- N_c \log A - \frac{T}{4} \log
\left(
\sum_{n\in{\mathbb{Z}}} \det (Q_{n+i-j})_{1\le i,j\le N_c}
\right).
\label{eff_pot}$$ In particular, the determinant part for $N_c=3$ is given by $$\begin{aligned}
&&
\sum_{n \in \mathbb{Z}} \det\left(Q_{n+i-j}\right)_{1\le i,j\le N_c}
\nonumber \\
& = &
8 \left(
1 + 12 \cosh^2 \frac{E}{T} + 8 \cosh^4 \frac{E}{T}
\right)
\left(
15 - 60 \cosh^2 \frac{E}{T} + 160 \cosh^4 \frac{E}{T}
-32 \cosh^6 \frac{E}{T} + 64 \cosh^8 \frac{E}{T}
\right)
\nonumber \\
&&
+ 64 \cosh \frac{\mu_B}{T} \cosh \frac{E}{T}
\left(
-15 + 40 \cosh^2 \frac{E}{T} + 96 \cosh^4 \frac{E}{T}
+ 320 \cosh^8 \frac{E}{T}
\right)
\nonumber \\
&&
+ 80 \cosh \frac{2\mu_B}{T}
\left(
1 + 6 \cosh^2 \frac{E}{T} + 24 \cosh^4 \frac{E}{T}
+ 80 \cosh^6 \frac{E}{T}
\right)
\nonumber \\
&&
+ 80 \cosh \frac{3\mu_B}{T}
\cosh \frac{E}{T}
\left(
- 1 + \cosh^2 \frac{E}{T}
\right)
+ 2 \cosh \frac{4\mu_B}{T},\end{aligned}$$ with $$E = \mathrm{arcsinh} (B/A) , \quad
A^2 = (1+d_4)^2 +
\left(
\mu_3 + D r - \frac{D}{2} (1-r^2) \pi_4
\right)^2, \quad
B = m + \frac{D}{2} (1+r^2) \sigma.$$ Here the baryon chemical potential is defined as $\mu_B = 3 \mu$. Remark that the next-leading order terms in $\mathcal{O}(1/\sqrt{D})$ are omitted in the derivation. See [@Misumi:2012ky] for the detailed calculation.
In the zero temperature case, we can solve the equilibrium condition analytically. For $D=3$ ($d=4$) with $m=0$ and $r=1$ the potential is given by $$\mathcal{F}_{\rm eff}(\sigma) =
\frac{9}{2} \sigma^2 - \frac{3}{2}
\log \left( (1+d_4)^2 + (\mu_3+3)^2 \right)
- \max
\left\{
3 \ \mathrm{arcsinh}
\left(
\frac{3 \sigma}{\sqrt{(1+d_4)^2+(\mu_3+3)^2}}
\right), \mu_B
\right\} .$$ In this case there are two local minima of the free energy as a function of $\sigma$ at $\sigma=0$ and $\sigma=\sigma_0$. This $\sigma_0$ can be determined by the gap equation, $\partial{\mathcal{F}_{\rm eff}}/ \partial
\sigma\Big|_{\sigma=\sigma_0}=0$, $$\sigma_0^2 = \frac{(1+d_4)^2+(\mu_3+3)^2}{18}
\left[
\sqrt{1+\frac{36}{((1+d_4)^2+(\mu_3+3)^2)^2}} - 1
\right] .$$ Comparing these two local minima, we can show that the global minimum changes from $\sigma = \sigma_0$ to $\sigma = 0$ at the critical chemical potential as $$\mu_B^{\rm critical}(T=0) =
3 \, \mathrm{arcsinh}
\left(
\frac{3 \sigma_0}{\sqrt{(1+d_4)^2+(\mu_3+3)^2}}
\right)
- \frac{9}{2} \sigma_0^2 .
\label{zero_temp_crit_chem_pot}$$ This chiral phase transition is of 1st order because the order parameter $\sigma$ changes discontinuously at this critical chemical potential. We can also evaluate the baryon density $\rho_B = - \partial {\mathcal{F}_{\rm eff}}/
\partial \mu_B$ at $T = 0$. It turns out to be empty $\rho_B =0$ when $\mu_B < \mu_B^{\rm critical}$. On the other hand, when $\mu_B > \mu_B^{\rm critical}$, it is saturated as $\rho_B = 1$.
![(Left) Phase diagram for the chiral transition with $r=1$, $\mu_3=-0.9$ and $d_4=0$. Green and red lines show 2nd and 1st transition lines, respectively. The transition order is changed from 2nd to 1st at the tricritical point $(\mu_B^{\rm tri},T^{\rm
tri})=(0.804,0.234)$. (Right) Three-dimensional chiral phase diagram for $T$, $\mu_B$ and $\mu_3$ for $m=0$ where $\mu_{3}$ runs within half of the physical range $-3<\mu_3<\sqrt{32/7}-3$. Green, red and purple lines show 2nd, 1st order transitions and tricritical point, respectively.[]{data-label="pb_r1"}](d_0.9.eps "fig:"){width="16em"} ![(Left) Phase diagram for the chiral transition with $r=1$, $\mu_3=-0.9$ and $d_4=0$. Green and red lines show 2nd and 1st transition lines, respectively. The transition order is changed from 2nd to 1st at the tricritical point $(\mu_B^{\rm tri},T^{\rm
tri})=(0.804,0.234)$. (Right) Three-dimensional chiral phase diagram for $T$, $\mu_B$ and $\mu_3$ for $m=0$ where $\mu_{3}$ runs within half of the physical range $-3<\mu_3<\sqrt{32/7}-3$. Green, red and purple lines show 2nd, 1st order transitions and tricritical point, respectively.[]{data-label="pb_r1"}](3d_plot.eps "fig:"){width="19em"}
We then discuss the phase diagram with respect to chiral symmetry. We now concentrate on the case with $r=1$ for simplicity because the effective potential (\[eff\_pot\]) is independent of $\pi_4$ in such a case. The 2nd order chiral phase boundary is given by the condition, such that the coefficient of $\sigma^2$ in the effective potential (\[eff\_pot\]) becomes zero. When the order of the phase transition is changed from 2nd to 1st, the coefficient of $\sigma^4$ as well as $\sigma^2$ should vanish. The left panel of Fig. \[pb\_r1\] shows the phase boundary of the chiral transition with $r=1$, $\mu_{3}=-0.9$ and $m=0$ for $d_4=0$. The counter term parameter is taken from the physical region $-\sqrt{32/7}<\mu_3+3<\sqrt{32/7}$ [@Misumi:2012uu]. The order of the phase transition is changed from 2nd to 1st at the tricritical point $(\mu_B^{\rm tri},T^{\rm tri})=(0.804,0.234)$. We also depict $\sigma$ condensate and the baryon density $\rho_B=-\partial \mathcal{F}_{\rm eff}/\partial \mu_{B}$ as functions of $\mu_{B}$ with several fixed $T$ in Fig. \[condensate\_r1\]. We find that there are 1st ($T<T^\mathrm{tri}$) and 2nd ($T>T^\mathrm{tri}$) order phase transitions for $\sigma$, followed by the phase transition of the density $\rho_B$. For $m\not=0$, we can show that the crossover transition instead appears with the 2nd order critical point.
![Chiral condensate $\sigma$ and the baryon density $\rho_B$ for (left) $T=0.3$ and (right) $T=0.2$ with $d_4=0$. Top and bottom panels show the massless $m=0$ and massive $m=0.1$ cases. There are 1st and 2nd phase transitions for $\sigma$. In the case of $m\not=0$, there appears the crossover behavior instead of the 2nd order transition. []{data-label="condensate_r1"}](condensate_T0.3_m0.0.eps "fig:"){width="18em"} ![Chiral condensate $\sigma$ and the baryon density $\rho_B$ for (left) $T=0.3$ and (right) $T=0.2$ with $d_4=0$. Top and bottom panels show the massless $m=0$ and massive $m=0.1$ cases. There are 1st and 2nd phase transitions for $\sigma$. In the case of $m\not=0$, there appears the crossover behavior instead of the 2nd order transition. []{data-label="condensate_r1"}](condensate_T0.2_m0.0.eps "fig:"){width="18em"}\
![Chiral condensate $\sigma$ and the baryon density $\rho_B$ for (left) $T=0.3$ and (right) $T=0.2$ with $d_4=0$. Top and bottom panels show the massless $m=0$ and massive $m=0.1$ cases. There are 1st and 2nd phase transitions for $\sigma$. In the case of $m\not=0$, there appears the crossover behavior instead of the 2nd order transition. []{data-label="condensate_r1"}](condensate_T0.3_m0.1.eps "fig:"){width="18em"} ![Chiral condensate $\sigma$ and the baryon density $\rho_B$ for (left) $T=0.3$ and (right) $T=0.2$ with $d_4=0$. Top and bottom panels show the massless $m=0$ and massive $m=0.1$ cases. There are 1st and 2nd phase transitions for $\sigma$. In the case of $m\not=0$, there appears the crossover behavior instead of the 2nd order transition. []{data-label="condensate_r1"}](condensate_T0.2_m0.1.eps "fig:"){width="18em"}
These results are qualitatively consistent with those with strong-coupling lattice QCD with staggered fermions, while there are some quantitative differences. For example, the KW phase diagram is suppressed in $T$ direction compared to that in staggered. We here compare the ratio of the transition baryon chemical potential at $T=0$ to the critical temperature at $\mu_B=0$, $R^{0}=\mu_c(T=0)/T_c(\mu_B=0)$. In staggered fermion, this ratio is $R^0_\mathrm{st} \simeq 3 \times 0.56 / (5/3) \sim 1$ [@Fukushima:2003vi; @Nishida:2003fb], while $R^0_\mathrm{KW} \simeq 0.767 / 0.356 \sim 2.2$. In the real world, this ratio is larger, $R^0 \gtrsim M_N / 170~\mathrm{MeV} \sim 5.5$. When the finite coupling and Polyakov loop effects are taken into account for staggered fermion, $T_c(\mu_B=0)$ decreases, $\mu_c(T=0)$ stays almost constant, then $R^0$ value increases [@Miura:2009nu]. Larger $R^0$ with KW fermion in the strong coupling limit may suggest smaller finite coupling corrections in the phase boundary. Another interesting point is the location of the tricritical point. In KW fermion, the ratio is $R^\mathrm{tri}_\mathrm{KW}=0.804/0.234 \simeq
3.4$, while $R^\mathrm{tri}_\mathrm{st}=1.73/0.866 \simeq 2.0$ for unrooted staggered fermion [@Fukushima:2003vi; @Nishida:2003fb]. It would be too brave to discuss this value, but $R^\mathrm{tri}_\mathrm{KW}$ is consistent with the recent Monte-Carlo simulations (see references in [@Ohnishi:2011aa]), which implies that the critical point does not exist in the low baryon chemical potential region, $\mu_B/T \lesssim 3$. These observations reveal usefulness of KW fermion for research on QCD phase diagram.
Apart from the phase transitions, the $\mu_B$ dependence of $\sigma$ and $\rho_B$ seems to have some characteristics in Fig. \[condensate\_r1\]. At $T=0.3 > T^\mathrm{tri}$ with m=0, $\sigma$ and $\rho_B$ undergoes the 2nd-order phase transition at $\mu_B\simeq 0.5$, and at a larger $\mu_B$ ($\mu_B\simeq 1.15$), increasing rate of $\rho_B$ as a function of $\mu_B$ becomes higher again. At lower temperature, $T=0.2 < T^\mathrm{tri}$, partial restoration of the chiral symmetry is seen before the first order phase transition. Since we have not taken care of the diquark condensate, these continuous changes are not related to the color superconductor. Other types of matter, such as quarkyonic matter [@McLerran:2007qj], partial chiral restored matter [@Miura:2009nu], or nuclear matter, may be related to the above characteristics.
In this report we focus on the case with $r=1$ and $d_4=0$ for simplicity. It is shown in [@Misumi:2012ky] that effects of these parameters are just quantitative.
Summary {#sec:summary}
=======
We have proposed a new framework for investigating the two-flavor finite-($T$,$\mu$) QCD phase diagram. We have shown that the discrete symmetries of KW fermion strongly suggest its applicability to the in-medium lattice QCD. To support our idea, we study the strong-coupling lattice QCD in the medium and derive the phase diagram of chiral symmetry for finite temperature and chemical potential. We have obtained the phase diagram with 1st, 2nd-order and crossover critical lines, which is qualitatively in agreement to results from the model study.
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[^1]: RIKEN-MP-57
|
---
abstract: 'We investigate the problem of learning Bayesian networks in an agnostic model where an ${\epsilon}$-fraction of the samples are adversarially corrupted. Our agnostic learning model is similar to – in fact, stronger than – Huber’s contamination model in robust statistics. In this work, we study the fully observable Bernoulli case where the structure of the network is given. Even in this basic setting, previous learning algorithms either run in exponential time or lose dimension-dependent factors in their error guarantees. We provide the first computationally efficient agnostic learning algorithm for this problem with dimension-independent error guarantees. Our algorithm has polynomial sample complexity, runs in polynomial time, and achieves error that scales nearly-linearly with the fraction of adversarially corrupted samples.'
author:
- |
Ilias Diakonikolas[^1]\
University of Southern California\
[diakonik@usc.edu]{}\
- |
Daniel M. Kane[^2]\
University of California, San Diego\
[dakane@cs.ucsd.edu]{}\
- |
Alistair Stewart[^3]\
University of Southern California\
[alistais@usc.edu]{}
bibliography:
- 'allrefs.bib'
title: 'Robust Learning of Fixed–Structure Bayesian Networks'
---
Introduction {#sec:intro}
============
Motivation and Background {#ssec:background}
-------------------------
Probabilistic graphical models [@Koller:2009] provide an appealing and unifying formalism to succinctly represent structured high-dimensional distributions. The general problem of inference in graphical models is of fundamental importance and arises in many applications across several scientific disciplines, see [@Wainwright:2008] and references therein.
In this work, we study the problem of learning graphical models from data [@Neapolitan:2003; @RQS11]. There are several variants of this general learning problem depending on: (i) the precise family of graphical models considered (e.g., directed, undirected), (ii) whether the data is fully or partially observable, and (iii) whether the structure of the underlying graph is known a priori or not (parameter estimation versus structure learning). This learning problem has been studied extensively along these axes during the past five decades, see, e.g., [@Chow68; @Dasgupta97; @Abbeel:2006; @WainwrightRL06; @AnandkumarHHK12; @SanthanamW12; @LohW12; @BreslerMS13; @BreslerGS14a; @Bresler15] for a few references, resulting in a beautiful theory and a collection of algorithms in various settings.
The main vulnerability of all these algorithmic techniques is that they crucially rely on the assumption that the samples are precisely generated by a graphical model in the given family. This simplifying assumption is inherent for known guarantees in the following sense: if there exists even a very small fraction of arbitrary outliers in the dataset, the performance of known algorithms can be totally compromised. It is an important research direction to explore the natural setting when the aforementioned assumption holds only in an approximate sense. Specifically, we propose the following family of questions:
\[question\] Let ${\cal P}$ be a family of graphical models describing distributions over ${\mathbb{R}}^d$. Suppose we are given a set of $N$ samples drawn from some unknown distribution ${{\widetilde{P}}}$ over ${\mathbb{R}}^d$, such that there exists $P \in {\cal P}$ that is ${\epsilon}$-close to ${{\widetilde{P}}}$, in total variation distance. Can we efficiently find a distribution $Q \in {\cal P}$ that is $f({\epsilon})$-close, in total variation distance, to ${{\widetilde{P}}}$? (Here, $f: {\mathbb{R}}\to {\mathbb{R}}_+$ can be any increasing function that satisfies $\lim_{x \to 0}f(x)=0$.)
More specifically, we would like to design robust learning algorithms for Question \[question\] whose sample complexity, $N$, is close to the information-theoretic minimum, and whose computational complexity is polynomial in $N$. We emphasize that the crucial requirement is that the error guarantee of the algorithm is [*independent*]{} of the dimensionality $d$ of the problem.
Question \[question\] fits in the framework of robust statistics [@Huber09; @HampelEtalBook86]. Classical estimators from this field can be classified into two categories: either (i) they are computationally efficient but incur an error that scales [*polynomially*]{} with the dimension $d$, or (ii) they are provably robust (in the aforementioned sense) but are hard to compute. In particular, essentially all known estimators in robust statistics (e.g., the Tukey depth [@Tukey75]) have been shown [@JP:78; @Bernholt; @HardtM13] to be intractable in the high-dimensional setting. We also note that the robustness requirement does not typically pose information-theoretic impediments for the learning problem. In most cases of interest (see, e.g, [@CGR15; @CGR15b; @DiakonikolasKKL16]), the sample complexity of robust learning is comparable to its (easier) non-robust variant. The challenge is to design computationally efficient algorithms.
#### Related Work.
We start by noting that efficient robust estimators are known for various [*one-dimensional*]{} structured distributions (see, e.g., [@DDS12soda; @CDSS13; @CDSS14; @CDSS14b; @ADLS15]). However, the robust learning problem becomes surprisingly challenging in high dimensions. Very recently, there has been algorithmic progress on this front: Two recent papers [@DiakonikolasKKL16; @LaiRV16] give polynomial-time algorithms with improved error guarantees for certain “simple” high-dimensional structured distributions. Specifically, [@LaiRV16] provides error guarantees that scale logarithmically with the dimension, while [@DiakonikolasKKL16] obtains the first [*dimension-independent*]{} error guarantees. The results of [@DiakonikolasKKL16] apply to simple distributions, including Bernoulli product distributions, Gaussians, and mixtures thereof (under some natural restrictions). We remark that the algorithmic approach of this work builds on the framework of [@DiakonikolasKKL16] together with additional technical and conceptual ideas.
Formal Setting and Our Results {#ssec:results}
------------------------------
In this work, we study Question \[question\] in the context of [*Bayesian networks*]{} [@Jensen:2007]. We focus on the fully observable case when the underlying network is given. In the [*non-agnostic*]{} setting, this learning problem is straightforward: the “empirical estimator” (which coincides with the maximum likelihood estimator) is known to be sample and computationally efficient [@Dasgupta97]. In sharp contrast, even this most basic regime is surprisingly challenging in the agnostic setting. For example, the very special case of agnostically learning a Bernoulli product distribution (corresponding to a trivial network with no edges) was analyzed only recently in [@DiakonikolasKKL16]. To formally state our results, we will need some terminology:
\[def:learning\] Let ${\cal P}$ be a family of probability distributions on $\{0, 1\}^d$. A randomized algorithm $A^{\cal P}$ is an [*agnostic distribution learning algorithm for $\cal P$,*]{} if for any ${\epsilon}>0,$ and any probability distribution ${{\widetilde{P}}}: \{0, 1\}^d \to {\mathbb{R}}_+$, on input ${\epsilon}$ and sample access to ${{\widetilde{P}}}$, with probability $9/10,$ algorithm $A^{\cal P}$ outputs a hypothesis $Q \in \mathcal{P}$ such that ${d_{\mathrm TV}}({{\widetilde{P}}}, Q) \leq f({\mathrm{OPT}}) + {\epsilon}$, where ${\mathrm{OPT}}{\stackrel{{\mathrm {\footnotesize def}}}{=}}\inf_{P \in \mathcal{P}} {d_{\mathrm TV}}({{\widetilde{P}}}, P)$, and $f: {\mathbb{R}}\to {\mathbb{R}}_+$ is monotone increasing and satisfies $\lim_{x \to 0}f(x)=0$.
We note that the value of ${\mathrm{OPT}}$ in the above definition is assumed to be unknown to the learning algorithm. By standard results, see e.g., Theorem 6 in [@CDSS14b], the agnostic learning problem can be reduced to its special case that ${\mathrm{OPT}}\leq {\epsilon}.$ We will henceforth work with this simplified definition without loss of generality.
We point out that our agnostic learning model subsumes Huber’s ${\epsilon}$-contamination model [@Huber64], which prescribes that the noisy distribution ${{\widetilde{P}}}$ is of the form $(1-{\epsilon}) P + {\epsilon}R$, where $P \in {\cal P}$ and $R$ is some arbitrary distribution.
#### Bayesian Networks.
Fix a directed acyclic graph, $G$, whose vertices are labelled $[d] {\stackrel{{\mathrm {\footnotesize def}}}{=}}\{1,2,\ldots,d\}$, so that all edges point from vertices with smaller index to vertices with larger index. A probability distribution $P$ on $\{0,1\}^d$ is defined to be a *Bayesian network* (or *Bayes net*) with graph $G$ if for each $i \in [d]$, we have that $\Pr_{X\sim P}\left[X_i = 1 \mid X_1,\ldots,X_{i-1}\right]$ depends only on the values $X_j$, where $j$ is a parent of $i$ in $G$. Such a distribution $P$ can be specified by its [*conditional probability table*]{}, the vector of conditional probabilities of $X_i=1$ conditioned on every possible combination of values of the coordinates of $X$ at the parents of $i$. In order to clarify this notation, we introduce the following terminology:
Let $S$ be the set $\{(i,a):1\leq i\leq d, a\in \{0,1\}^{\mathrm{Parents}(i)}\}$. Let $m=|S|$. For $(i,a)\in S$, the *parental configuration* $\Pi_{i,a}$ is defined to be the event that $X_{\mathrm{Parents}(i)} =a$. Once $G$ is fixed, we may associate to a Bayesian network $P$ the corresponding conditional probability table $p\in [0,1]^S$ given by $p_{i,a}=\Pr_{X\sim P}\left[X_i=1 \mid \Pi_{i,a}\right].$ Furthermore, note that $P$ is determined by $p$.
We will frequently index $p$ as a vector. That is, we will use the notation $p_k$, for $1 \leq k \leq m$, and the associated events $\Pi_k$, where each $k$ stands for an $(i,a) \in S$ lexicographically ordered.
#### Our Results.
We give the first efficient agnostic learning algorithm for Bayesian networks with a known graph $G$. Our algorithm has polynomial sample complexity and running time, and provides an error guarantee that scales near-linearly with the fraction of adversarially corrupted samples, under the following natural restrictions: First, we assume that each parental configuration is reasonably likely. Intuitively, this assumption seems necessary because we will need to observe each configuration many times in order to learn the associated conditional probability to good accuracy. Second, we assume that each of the conditional probabilities are balanced, i.e., bounded away from $0$ and $1$. This assumption is needed for technical reasons. In particular, we will need this to show that having a good approximation to the conditional probability table will imply that the corresponding Bayesian network is close in variation distance.
Formally, we say that a Bayesian network is *$c$-balanced*, for some $c>0$, if all coordinates of the corresponding conditional probability table are between $c$ and $1-c$. Using this terminology, we state our main result:
\[thm:bal-bn\] Let ${\epsilon}> 0$. Let $P$ be a $c$-balanced Bayesian network on $\{0, 1\}^d$ with known graph $G$ with the property that each parental configuration of any node has probability at least $\Omega({\epsilon}^{1/2} \log^{1/4} (1/{\epsilon}))$, i.e., $\min_{(i, a) \in S} \Pr_P[\Pi_{i,a}] \geq \Omega({\epsilon}^{1/2} \log^{1/4} (1/{\epsilon}))$. Let ${{\widetilde{P}}}$ be a distribution on $\{0,1\}^d$ with ${d_{\mathrm TV}}(P,{{\widetilde{P}}}) \leq {\epsilon}$. There is an algorithm that given $G$, ${\epsilon}$, and sample access to ${{\widetilde{P}}}$, with probability $9/10$, outputs a Bayesian network $Q$ with ${d_{\mathrm TV}}(P,Q) \leq {\epsilon}\sqrt{\log1/{\epsilon}}/ \min_{(i,a) \in S} \Pr_P[\Pi_{i,a}]$. The algorithm takes at most $\tilde{O}(dm^2/{\epsilon}^2)$ samples and runs in polynomial time.
We remark that the condition on the minimum probability of any parental configuration follows from the assumption of being $c$-balanced when the fan-in of all nodes is bounded (e.g., for the case of trees). Indeed, if every node has at most $f$ parents, then $\min_{(i, a) \in S} \Pr_P[\Pi_{i,a}] \geq c^f$. So, as long as $c^f \geq \Omega({\epsilon}^{1/2} \log^{1/4} (1/{\epsilon}))$, this condition automatically holds. On the other hand, this condition is impossible if a node has $\Omega(\log(1/{\epsilon}))$ many parents, since then some parental configuration must have small probability.
Overview of Algorithmic Techniques {#sec:techniques}
----------------------------------
Our approach builds on the filtering-based framework of [@DiakonikolasKKL16] which was used to obtain agnostic learning algorithms for various simple families, including that of learning a binary product distribution. The latter can be seen as a starting point for our agnostic algorithm in this paper. At a high level, both algorithms work as follows: To each sample $X$, we associate a vector $F(X)$ so that learning the mean of $F(X)$ to good accuracy is sufficient to recover the distribution. In the case of binary products, $F(X)$ is simply $X$, while in our case it will need to take into account additional information about conditional means. From this point, our algorithm will try to do one of two things: Either we show that the sample mean of $F(X)$ is close to the mean of the correct distribution – in which case we can already approximate the distribution in question – or we are able to produce a [*filter*]{}, i.e., an efficient method which throws away some of our samples, but is guaranteed to leave us with a distribution [*closer*]{} to the one we are trying to learn than we had before. If we produce a filter, we then iterate this algorithm on those samples that pass the filter until we are left with a good approximation.
To achieve the aforementioned scheme, we create a matrix $M$ which is roughly the sample covariance matrix of $F(X)$. We show that if the errors in our distribution are sufficient to notably disrupt the sample mean of $F(X)$, there must be many erroneous samples that are all far from the mean in roughly the same direction. If this is the case, it can be detected as a large eigenvalue in the matrix $M$. Specifically, if $M$ has [*no*]{} large eigenvalue, then we show that our sample mean is sufficient. If, on the other hand, $M$ [*does*]{} have a large eigenvalue, this will correspond to some direction in which many of our sample values of $F(X)$ will be far from the mean. Then, concentration bounds on $F(X)$, will imply that almost all samples far from the mean are erroneous, and thus filtering them out will provide an improved distribution.
We next look into the major ingredients required for the application of this filter-based framework to Bayesian networks, and compare our new proof with that given for balanced binary products in [@DiakonikolasKKL16] on a somewhat more technical level.
To begin with, in each case, we need a function $F$ so that learning the mean of $F(X)$ is sufficient to learn the distribution. In the case of binary products, $F(X)=X$ is sufficient, since the coordinate means determine the distribution. For Bayesian networks, the situation is somewhat more complicated. In particular, we need to learn all of the relevant *conditional* means. Now each sample, $X$, will give us information on some of the relevant conditional means, but not all of them. In order to “fill out” our vector, we will need to produce entries corresponding to conditional means for which the condition failed to happen. We do this by filling such entries with an approximation to the relevant conditional mean.
Next, we need to consider the eigenvalues of $M$. It is easy to show that sampling error can be ignored if we take sufficiently many samples, so we can instead consider the matrix given by the actual covariances of the distribution in question. We break this matrix into three parts. One coming from the distribution we are trying to learn, one coming from the subtractive error (i.e., the points that have *smaller* probability under the noisy distribution), and one coming from the additive error. In both cases, the noise-free distribution has a diagonal covariance matrix. For binary products this is trivial, but for Bayesian networks the analogous result takes some additional technical work. Furthermore, the term coming from the subtractive error will have no large eigenvalues. This is because of *concentration* bounds. These will imply that the tails of $F(X)$ are not wide enough so that taking a small amount of mass away from them can introduce much error. For the binary product case, this is by standard Chernoff bounds, but for Bayesian networks, we must instead rely on martingale arguments and Azuma’s inequality. Together, this says that any large eigenvalues are due to the additive error, which can be large. Finally, we will need to reuse our concentration bounds to show that if our additive errors are reasonably frequently far from the mean in a known direction, then they can be reliably distinguished from good samples.
Organization
------------
Section \[sec:prelims\] contains the results specific to Bayesian networks that we need to specialize our filtering algorithm for this setting. Section \[sec:alg\] gives the details of the algorithm and its proof of correctness. In Section \[sec:concl\], we conclude and propose directions for future work.
Technical Preliminaries {#sec:prelims}
=======================
The structure of this section is as follows: First, we need to be able to bound the total variation distance between two Bayes nets in terms of their conditional probability tables. Second, we define a function $F(x,q)$ – which takes a sample and returns an $m$-dimensional vector by filling out the coordinates corresponding to unobserved conditional means – and show some of its properties. Finally, we derive a concentration bound from Azuma’s inequality.
We note that a few technical proofs from this section have been deferred to Appendix \[app:prelims\].
\[lem:hel-bn\] Let $P$ and $Q$ be Bayesian networks with the same dependency graph $G$. In terms of the conditional probability tables $p$ and $q$ of $P$ and $Q$, we have: $${d_{\mathrm H}}(P,Q)^2 \leq 2 \sum_{k=1}^m \sqrt{\Pr_P[\Pi_k] \Pr_Q[\Pi_k]} \frac{(p_k-q_k)^2}{(p_k+q_k)(2-p_k-q_k)} \;.$$
The dependence of this bound on both $\Pr_P[\Pi_k]$ and $\Pr_Q[\Pi_k]$ is not ideal. One reason that we need to specify a minimum probability for $\Pr_P[\Pi_k]$ is to argue that $\Pr_Q[\Pi_k]$ is close to this and so get a bound containing only $\Pr_P[\Pi_k]$. We can obtain a simpler expression for $c$-balanced binary Bayesian networks and those in which the minimum probability of any of the $\Pi_k$ is larger than the final error bound:
\[cor:bal-min\] Suppose that: (i) $\min_{k \in [m]} \Pr_P[\Pi_k] \geq 2{\epsilon}$, and (ii) $P$ or $Q$ is $c$-balanced, and (iii) $\frac{3}{c}\sqrt{\sum_k \Pr_P[\Pi_k] (p_k-q_k)^2} \leq {\epsilon}$. Then we have that ${d_{\mathrm TV}}(P,Q) \leq {\epsilon}.$
Note that a given sample $x$ from $P$ only gives us information about a subset of the conditional probabilities. Specifically, $x_i$ gives us information about $p_{i,a}$ if and only if $x \in \Pi_{i,a}$. To get an $m$-dimensional vector from $x$, we need to specify all the other unobserved coordinates. We will do so by setting these coordinates to their conditional means or an approximation or guess for these means.
Let $F(x,q)$ for $\{0,1\}^d \times {\mathbb{R}}^m \rightarrow {\mathbb{R}}^m$ be defined as follows: If $x \in \Pi_{i,a}$, then $F(x,q)_{i,a}=x_i$, otherwise $F(x,q)_{i,a}=q_{i,a}$.
When $q=p$, we have that the expectation of the $(i,a)$-th coordinate of $F(X,p)$, for $X \sim P$, is the same conditioned on either $\Pi_{i,a}$ or $\neg \Pi_{i,a}$. Using the conditional independence properties of Bayesian networks, we will show that the covariance of $F(x,p)$ is diagonal. Our algorithm makes crucial use of this fact to detect whether or not the empirical conditional probability table of the noisy distribution is close to the true conditional probability table.
First, we note that $F$ is invertible in the following sense:
\[lem:inverse\] For all $x \in \{0,1\}^d$, $q \in [0,1]^m$ and $1 \leq j \leq k$, $x_1, \dots, x_j$ can be recovered from the vector of coordinates $F(x,p)_{i,a}$ for all $(i,a) \in S$ with $j \leq i$, and these coordinates of $F$ can be computed from only $x_1,\ldots,x_j$.
The coordinates of $F(X,p)$ have the following conditional independence property:
\[lem:cond-ind\] For $X \sim P$ and any $(i,a) \in S$, if we condition on either $\Pi_{i,a}$ or $\neg \Pi_{i,a}$, then $F(X,p)_{i,a}$ is independent of $F(X,p)_{j,a'}$ for all $(j,a') \in S$ with $j \leq i$ and $(j,a')\neq (i,a)$.
Now we can show our claim about the mean and covariance of $F(X,p)$:
\[lem:F-moments\] For $X \sim P$, we have ${\mathbb{E}}(F(X,p)) = p$. The covariance matrix of $F(X,p)$ satisfies ${\mathrm{Cov}}[F(X,p)]= {\mathrm{diag}}(\Pr_P[\Pi_k]p_k(1-p_k))$.
Although the coordinates of $F(X,p)$ are not independent, the first and second moments are the same as that of a product distribution of the marginal of each coordinate.
We will also need a suitable concentration inequality. Thanks to conditional independence properties, we can use Azuma’s inequality to show:
\[lem:Azuma\] For $X \sim P$ and any unit vector $v \in {\mathbb{R}}^d$, we have $$\Pr[|v \cdot (F(X,q)-q)| \geq T + \|p-q\|_2] \leq 2\exp(-T^2/2) \;.$$
It follows from Lemmas \[lem:inverse\] and \[lem:cond-ind\] that ${\mathbb{E}}_{X \sim P}[F(X,p)_k \mid F(X,p)_1, \dots, F(X,p)_{k-1}]=p_k$ for all $1 \leq k \leq m$ (see Claim \[clm:cond-mean\] in the appendix). Thus, the sequence $\sum_{k=1}^\ell v_k (F(X,p)_k - p_k)$ for $1 \leq \ell \leq m$ is a martingale, and we can apply Azuma’s inequality. Note that the support size of $v_k (F(X,p)_k - p_k)$ is $v_k$ and thus we have $$\Pr[|v \cdot (F(X,p)-p)| \geq T] \leq 2\exp(-T^2/2\|v\|_2) = 2\exp(-T^2/2) \;.$$ Consider an $x \in \{0,1\}^d$. If $x \in \Pi_{i,a}$, then we have $F(x,p)_{i,a}=F(x,q)_{i,a}=x_i$ and so $$|(F(x,p)_{i,a} - p_{i,a}) - (F(x,q)_{i,a}-q_{i,a})| = |p_{i,a}-q_{i,a}| \;.$$ If $x \notin \Pi_{i,a}$, then $F(x,p)_{i,a} = p_{i,a}$ and $F(x,q)_{i,a} = q_{i,a}$, hence $$|(F(x,p)_{i,a} - p_{i,a}) - (F(x,q)_{i,a}-q_{i,a})| = 0 \;.$$ Thus, we have $$\|(F(x,p)-p) - (F(x,q)-q)\|_2 \leq \|p-q\|_2 \;.$$ An application of the Cauchy-Schwarz inequality gives that $|v \cdot (F(X,q)-q)| \geq T + \|p-q\|_2$ implies that $|v \cdot (F(x,p)-p)| \geq T$. The probability of the former holding for $X$ must therefore be at most the probability that the latter holds for $X$.
Efficient Agnostic Learning Algorithm {#sec:alg}
=====================================
Our main result will follow by iterating the efficient procedure described in the following proposition:
\[prop:bal\] Let ${\epsilon}> 0$. Let $P$ be a Bayesian network on $\{0, 1\}^d$ with known graph $G$ with the property that each parental configuration of any node has probability at least $\Omega({\epsilon}^{1/2}\log^{1/4}(1/{\epsilon}))$, i.e., $\min_{k \in [m]} \Pr_P[\Pi_k] \geq \Omega({\epsilon}^{1/2}\log^{1/4}(1/{\epsilon}))$. Let ${{\widetilde{P}}}$ be a distribution with ${{\widetilde{P}}} = w_P P + w_E E - w_L L$ for distributions $E$ and $L$ with disjoint support and $w_E,w_L \geq 0$ and $w_E+w_L \leq 2 {\epsilon}$. There is an algorithm that given $G$, ${\epsilon}$ and sample access to ${{\widetilde{P}}}$, with probability $1-1/(100(2d+1))$, either
1. outputs a Bayesian network $Q$ with ${d_{\mathrm TV}}(P,Q) \leq {\epsilon}\sqrt{\log1/{\epsilon}}/ \min_k \Pr_P[\Pi_k]$, or
2. gives a filter $A$ such that ${{\widetilde{P}}}$ conditioned on $A$ accepting can be written as $w'_P P + w'_E E' + w'_L L'$, where $E'$ and $L'$ are distributions with disjoint support, $w'_E,w'_L \geq 0$ and $w'_E+w'_L\leq w_E+w_L - {\epsilon}/d$. Furthermore, the probability that $A$ rejects a random element of ${{\widetilde{P}}}$ is at most $O((w_E+w_L)-(w_E'+w_L'))$.
The algorithm uses $\tilde{O}(m^2/{\epsilon}^2)$ samples and runs in polynomial time.
If this algorithm produces a filter, we will iterate it using all filters generated in previous iterations, until it outputs a distribution. By the equation, ${{\widetilde{P}}} = w_P P + w_E E - w_L L$, we mean that the pmfs of these distributions satisfy ${{\widetilde{P}}}(x) = w_P P(x) + w_E E(x) - w_L L(x)$ for all $x \in \{0,1\}^d$. We note that when we initially ${d_{\mathrm TV}}({{\widetilde{P}}},P) \leq {\epsilon}$, there are distributions $E$ and $L$ that satisfy this:
\[fact:simple\] Let $P$ and $Q$ be probability distributions supported over a discrete set $D$. There exist probability distributions $E$ and $L$ over $D$ with disjoint supports, so that for all $x \in D$ it holds:
- $Q(x)= P(x) + {d_{\mathrm TV}}(P, Q) \cdot E(x) -{d_{\mathrm TV}}(P, Q) \cdot L(x),$ and
- ${d_{\mathrm TV}}(P, Q) \cdot L(x) \leq P(x).$
For $x \in D$, we define $ E(x) {\stackrel{{\mathrm {\footnotesize def}}}{=}}\frac{\max\{ Q(x) - P(x), 0 \}}{{d_{\mathrm TV}}(P, Q)} ,$ and $ L(x) {\stackrel{{\mathrm {\footnotesize def}}}{=}}\frac{\max\{ P(x) - Q(x), 0 \}}{{d_{\mathrm TV}}(P, Q)}.$ It is clear that $E$ and $L$ have disjoint supports and that they satisfy conditions (i) and (ii).
We will show that, if an iteration produces a filter, the filter will reject more samples from $E$ than from $P$. This property has the effect of decreasing $w_E$. However, any false positives result in an increase in $w_L$. Perhaps counterintuitively, this implies that the total variation distance between ${{\widetilde{P}}}$ and $P$ can increase (slightly) from iteration to iteration. However, we are able to show that $w_L+w_E$ decreases with high probability. We note that Proposition \[prop:bal\] suffices to establish Theorem \[thm:bal-bn\]. We include the simple proof here for the sake of completeness.
Consider the first iteration of the overall algorithm. By Fact \[fact:simple\], we can write ${{\widetilde{P}}} = P +{d_{\mathrm TV}}(P,{{\widetilde{P}}}) \cdot E - {d_{\mathrm TV}}(P,{{\widetilde{P}}}) \cdot L$ for distributions $E$ and $L$ with disjoint support. Since we have that ${d_{\mathrm TV}}(P,{{\widetilde{P}}}) \leq {\epsilon}$, it follows that $w_E+w_L \leq 2 {\epsilon}$.
If in each iteration before the $i$-th one we output a filter satisfying the second condition of Proposition \[prop:bal\], then at the $i$-th iteration, we have $w^{(i)}_E+w^{(i)}_L \leq 2 {\epsilon}- (i-1){\epsilon}/d$. Since $w^{(i)}_E, w^{(i)}_L \geq 0$, this can only happen for $2d$ iterations. By a union bound, the algorithm succeeds with probability at least $99/100$ for the first $2d+1$ iterations. Therefore, with at least this probability, it will output a Bayesian network satisfying the first condition.
Furthermore, after any number of these iterations the probability that a sample from our original distribution ${{\widetilde{P}}}$ is accepted by all of our filters is at least $\exp(-O(w_E+w_L)) = \exp(O({\epsilon})) \geq 9/10$. Finally, note that each iteration draws $\tilde{O}(m^2/{\epsilon}^2)$ samples from ${{\widetilde{P}}}$, and so the total sample complexity is $\tilde{O}(dm^2/{\epsilon}^2)$.
The main part of this section is devoted to the proof of Proposition \[prop:bal\], and is structured as follows: In Section \[sec:algo-description\], we describe the algorithm [Filter-Known-Topology]{} establishing Proposition \[prop:bal\] in tandem with an overview of its analysis. In Section \[ssec:setup\], we present a number of useful structural lemmas. Sections \[ssec:small-norm\] and \[ssec:big-norm\] give the proof of correctness for the two regimes of the algorithm [Filter-Known-Topology]{}.
Algorithm [Filter-Known-Topology]{} {#sec:algo-description}
-----------------------------------
The algorithm establishing Proposition \[prop:bal\] is presented below:
#### Overview of Analysis.
Recall that we write ${{\widetilde{P}}} = w_P \cdot P + w_E \cdot E - w_L \cdot L$ for disjoint distributions $E$ and $L$, where $w_P + w_E - w_L=1$ and $w_E, w_L \geq 0$ with $w_E + w_L \leq 2 {\epsilon}$. Thus, $|w_P -1| \leq 2 {\epsilon}$ and for all events $B$, $w_L \Pr_L[B] \leq w_P \Pr_P[B]$ and $w_E \Pr_E[B] \leq \Pr_{{{\widetilde{P}}}}[B]$.
The basic idea of the analysis is as follows: By taking enough samples, we can ensure that $q$ is close to the true conditional probability table of ${{\widetilde{P}}}$. This will give us a good approximation to the actual mean vector, $p$ (and thus a good approximation to the distribution $P$) as long as the conditional expectations of ${{\widetilde{P}}}$ are close to the conditional expectations of $P$. In other words, we will be in good shape so long as the error between $P$ and ${{\widetilde{P}}}$ does not introduce a large error in the conditional probability table.
Thinking more concretely about this error, we may split it into two parts: $w_L L$, the subtractive error, and $w_E E$ the additive error. Using concentration results for $P$, it can be shown that the subtractive errors cannot cause significant problems for the conditional probability table. It remains to consider additive errors. These clearly can produce notable problems in the conditional probability table, since any given sample can produce terms $\sqrt{d}$ away from the mean. If many of the additional terms line up in some direction, this can lead to a notable discrepancy.
However, if many of these errors line up in some direction (which is necessary in order to have a large impact on the mean), the effects will be reflected in the first two moments. More concretely, if for some unit vector $v$, the expectation of $v\cdot F(E,q)$ is very far from the expectation of $v\cdot F(P,q)$, this will force the variance of $v\cdot F({{\widetilde{P}}},q)$ to be large. This implies two things: First, it tells us that if $v\cdot F({{\widetilde{P}}},q)$ is small for all $v$ (a condition equivalent to $\|M\|_2$ being small), we know that $q$ is a good approximation to the true conditional probability table. On the other hand, if $\|M\|_2$ is large, we can find a large eigenvector, which corresponds to a unit vector $v$ so that $v\cdot F({{\widetilde{P}}}, q)$ has large variance. Given such a vector, we note that a reasonable fraction of this variance must be coming from samples taken from $E$ that have $v\cdot F(X,q)$ very far from the mean. On the other hand, using concentration bounds for $v\cdot F(P,q)$, we know that very few valid samples are this far from the mean. This discrepancy will allow us to create a filter in which we throw away any samples $X$ where $v\cdot F(X,q)$ is too far from $v\cdot q$. Our filter can be built in such a way so that the vast majority of the rejected samples came from $E$, thus decreasing the variation distance.
Setup and Structural Lemmas {#ssec:setup}
---------------------------
In order to understand the second moment matrix, $M$, we will need to break down this matrix in terms of several related matrices defined below:
- Let $M_{{{\widetilde{P}}}}$ be the matrix with zero diagonal and $(i, j)$ entry ${\mathbb{E}}_{X \sim {{\widetilde{P}}}}[(F(X,q)_i-q_i)(F(X,q)_j - q_j)]$ when $i \neq j$.
- Let $M_P$ be the matrix with zero diagonal and $(i, j)$ entry ${\mathbb{E}}_{X \sim P}[(F(X,q)_i-q_i)(F(X,q)_j - q_j)]$ when $i \neq j$.
- Let $M_L$ be the matrix with $(i, j)$ entry ${\mathbb{E}}_{X \sim L}[(F(X,q)_i-q_i)(F(X,q)_j - q_j)]$ for all $i, j$.
- Let $M_E$ be the matrix with $(i, j)$ entry ${\mathbb{E}}_{X \sim E}[(F(X,q)_i-q_i)(F(X,q)_j - q_j)]$ for all $i, j$.
We begin our analysis with some basic bounds, showing that with high probability that $\alpha,q,$ and $M$ are all appropriately close to the things that they are designed to estimate.
\[lem:empirical-moments-good\] With probability at least $1-\frac{1}{200(2d+1)}$, $\alpha - \min_k \Pr_{{{\widetilde{P}}}}[\Pi_k] \leq {\epsilon}$, $\|q-{\mathbb{E}}_{X \sim {{\widetilde{P}}}}[F(X,q)]\|_2 \leq {\epsilon}$ and $\|M-M_{{{\widetilde{P}}}}\|_2 \leq {\epsilon}$.
The proof of this lemma is given in Appendix \[app:setup\]. We will henceforth assume that the conditions of Lemma \[lem:empirical-moments-good\] hold.
Our next step is to analyze the spectrum of $M$, and in particular show that $M$ is close in spectral norm to $w_E M_E$. To do this, we begin by considering the spectral norm of $M_P$. In particular, we claim that it is relatively small. This can be shown by a relatively straightforward computation, as we know exactly the second moments of $F(P,q)$ (see Appendix \[app:setup\] for the simple proof):
\[lem:MP-bound\] $\|M_P\|_2 \leq \sum_k \Pr_P[\Pi_k] (p_k-q_k)^2$.
Next, we wish to bound the contribution to $M_{{{\widetilde{P}}}}$ coming from the subtractive error. We show that this is small due to concentration bounds on $P$. The idea is that for any unit vector $v$ we have that $v\cdot F(P,q)$ is tightly concentrated around $q$. Since $w_L L$ can at worst consist of a small fraction of the tail of this distribution, the expectation of $v\cdot (F(L,q)-q)$ cannot be too large.
\[lem:ML-bound\] $w_L \|M_L\|_2 \leq O({\epsilon}\log(1/{\epsilon})+{\epsilon}\|p-q\|_2^2)$.
For any event $A$, we have that $w_L \Pr_L[A] \leq w_P \Pr_P[A] \leq 2 \Pr_P[A]$. Thus, using [Lemma \[lem:Azuma\]]{}, we obtain: $$\Pr_{X \sim L}\left[|v \cdot (F(X,q)-q)| \geq T + \|p-q\|_2\right] \leq (4/w_L)\exp(-T^2/2) \;.$$ By definition, $\|M_L\|_2$ is the maximum over unit vectors $v$ of $v^T M_L v$. For any unit vector $v$, we have $$\begin{aligned}
w_L v^T M_L v & = w_L {\mathbb{E}}_{X \sim L}\left[ (v\cdot (F(X,q)-q))^2 \right] \\
& = 2 w_L \int_{0}^\infty \Pr_{X \sim L} \left[ |v\cdot (F(X,q)-q)| \geq T \right] T dT \\
& \ll \int_{0}^{\|p-q\|_2 + 4 \sqrt{\ln(1/w_L)}} w_L T dT + \int_{\|p-q\|_2 + 4 \sqrt{\ln(1/w_L)}}^\infty \exp\left(-(T-\|p-q\|_2)^2/2 \right) T dT \\
& \ll w_L \|p-q\|_2^2 + w_L(\log(1/w_L) + 1) \\
& \ll {\epsilon}\log(1/{\epsilon})+ {\epsilon}\|p-q\|_2^2 \;. \tag*{(since $w_L \leq 2{\epsilon}$)}\end{aligned}$$
Finally, combining the above results, since $M_{P}$ and $M_L$ have little contribution to the spectral norm of $M_{{{\widetilde{P}}}}$, most of it must come from $M_E$.
\[lem:MApprox\] $\|M - w_E M_E\|_2 \leq O\left({\epsilon}\log(1/{\epsilon}) + \sum_k \Pr_P[\Pi_k] (p_k-q_k)^2\right)$.
Note that $M_{{{\widetilde{P}}}}=w_P M_P + w_E M_E - w_L M_L$. By the triangle inequality $$\begin{aligned}
\|M - w_E M_E\|_2 & \leq \|M-M_{{\widetilde{P}}}\|_2 + w_P\|M_P\|_2 w_L\|M_L\|_2 \\
& \leq O({\epsilon}) + O\left(\sum_k \Pr_P[\Pi_k] (p_k-q_k)^2 + {\epsilon}\log(1/{\epsilon})+{\epsilon}\|p-q\|_2^2\right) \;,\end{aligned}$$ using Lemmas \[lem:empirical-moments-good\], \[lem:MP-bound\]and \[lem:ML-bound\]. Note that ${\epsilon}\|p-q\|_2^2 \leq \sum_k \Pr_P[\Pi_k] (p_k-q_k)^2$, since we assumed that $\min_k \Pr_P [\Pi_k] \geq {\epsilon}$.
Next we show that the contributions that $L$ and $E$ make to the expectation of $F({{\widetilde{P}}},q)$ can be bounded in terms of the spectral norms of $M_L$ and $M_E$. Combining with above results about these norms, this will imply that if $\|M\|_2$ is small that ${\mathbb{E}}_{X \sim \tilde P}[F(X,q)]$ is close to ${\mathbb{E}}_{X \sim P}[F(X,q)]$, which will then necessarily be close to $p$.
\[lem:wrong-mean-covar\] $\left|{\mathbb{E}}_{X \sim L}[F(X,q)] - q \right| \leq \sqrt{\|M_L\|_2}$ and $|{\mathbb{E}}_{X \sim E}[F(X,q)] - q| \leq \sqrt{\|M_E\|_2}$.
Let $X$ be any random variable supported on ${\mathbb{R}}^m$ and $x \in {\mathbb{R}}^m$. Then we have $${\mathbb{E}}[(X-x)(X-x)^T]={\mathbb{E}}\left[(X-{\mathbb{E}}[X])(X-{\mathbb{E}}[X])^T\right] + ({\mathbb{E}}[X]-x)({\mathbb{E}}[X]-x)^T.$$ Since this first term, the covariance of $X$, is positive semidefinite, we have $$\|{\mathbb{E}}\left[(X-x)(X-x)^T\right]\|_2 \geq \|({\mathbb{E}}[X]-x)({\mathbb{E}}[X]-x)^T\|_2 = \|{\mathbb{E}}[X]-x\|_2^2.$$ Applying this for $x=q$ and $X=F(Y,q)$, for $Y \sim L$ or $Y\sim E$, completes the proof.
\[lem:mean-transformed\] $({\mathbb{E}}_{X \sim P}[F(X,q)] - q)_k = \Pr_P[\Pi_k] (p_k-q_k)$.
We have that when $\Pi_k$ does not occur, $F(X,q)_k=q_k$. Thus, we can write: $${\mathbb{E}}_{X \sim P}[F(X,q)_k-q_k] = \Pr_P[\Pi_k] {\mathbb{E}}_{X \sim P}\left[F(X,q)_k-q_k \mid \Pi_k \right] = \Pr_P[\Pi_k] (p_k-q_k) \;,$$ as desired.
The Case of Small Spectral Norm {#ssec:small-norm}
-------------------------------
Using the lemmas from Section \[ssec:setup\], we can bound the variation distance between $P$ and $Q$, the Bayesian net with conditional probability table $q$, in terms of $\|M\|_2$:
\[lem:mean-dist-from-norm\] $\sqrt{\sum_k \Pr_P[\Pi_k]^2 (p_k-q_k)^2)} \leq 2\sqrt{{\epsilon}\|M\|_2} + O({\epsilon}\sqrt{\log(1/{\epsilon})})$.
For notational simplicity, let $D={\mathrm{diag}}(\sqrt{\Pr_P[\Pi_k]})$, so we have $$\sqrt{\sum_k \Pr_P[\Pi_k] (p_k-q_k)^2} = \|D(p-q)\|_2.$$ Let $\mu^P$, $\mu^{{{\widetilde{P}}}}$, $\mu^L$ and $\mu^E$ be ${\mathbb{E}}[F(X,q)]$ for $X$ distributed as $P$, ${{\widetilde{P}}}$, $L$ or $E$ respectively. Then we have $$\mu^{{{\widetilde{P}}}} = w_P \mu^P - w_L \mu^L + w_E \mu^E \;.$$ Note that, by Lemma \[lem:mean-transformed\], $\mu^{{{\widetilde{P}}}} - q = D^2(p-q)$. By Lemma \[lem:empirical-moments-good\], $\|q-\mu^{{{\widetilde{P}}}}\|_2 \leq {\epsilon}$. Then, by the triangle inequality, we obtain $$\begin{aligned}
\|D^2(p-q)\|_2 & = \|\mu^P - q\|_2 \leq \|\mu^P-\mu^{{{\widetilde{P}}}}\|_2 + {\epsilon}\\
& \leq w_L\|\mu^L-\mu^P\|_2 + w_E\|\mu^E-\mu^P\|_2 + {\epsilon}\\
& \leq w_L\|\mu^L-q\|_2 + w_E\|\mu^E-q\|_2 + O\left({\epsilon}\|D^2(p-q)\|_2\right) + {\epsilon}\\
& \leq w_L \sqrt{\|M_L\|_2} + w_E \sqrt{\|M_E\|_2} + O\left({\epsilon}\|D^2(p-q)\|_2\right) + {\epsilon}\\
& \leq O\left({\epsilon}\sqrt{\log(1/{\epsilon})}\right) + \sqrt{2{\epsilon}\|M\|_2} + O\left(\sqrt{{\epsilon}}\|D(p-q)\|_2\right) \\
& \leq O\left({\epsilon}\sqrt{\log(1/{\epsilon})}\right) + (3/2)\sqrt{{\epsilon}\|M\|_2} + \|D^2(p-q)\|_2/4 \;,\end{aligned}$$ where we used the assumption that ${\epsilon}$ is smaller than a sufficiently small constant. Rearranging the inequality completes the proof.
\[cor:correct\] If $\|M\|_2 \leq O({\epsilon}\log(1/{\epsilon})/\alpha)$, then ${d_{\mathrm TV}}(P,Q)=O({\epsilon}\sqrt{\log(1/{\epsilon})}/\min_k \Pr_P[\Pi_k])$.
By Lemma \[lem:empirical-moments-good\], $|\alpha-\min_k \Pr_{{{\widetilde{P}}}}[\Pi_k]| \leq {\epsilon}$. Since ${d_{\mathrm TV}}({{\widetilde{P}}},P) = \max\{w_E,w_L\} \leq 2{\epsilon}$, we have that $|\alpha-\min_k \Pr_P[\Pi_k]| \leq 3{\epsilon}$. Since by assumption $\min_k \Pr_P[\Pi_k] \geq \sqrt{{\epsilon}}$, we have that $\alpha = \Theta(\min_k \Pr_P[\Pi_k])$.
Therefore, we have $$\begin{aligned}
\sqrt{\sum_k \Pr_P[\Pi_k] (p_k-q_k)^2} \leq \sqrt{\sum_k \Pr_P[\Pi_k]^2 (p_k-q_k)^2}/\sqrt{\min_k \Pr_P[\Pi_k]} \\
\leq (2\sqrt{{\epsilon}\|M\|_2} + O({\epsilon}\sqrt{\log(1/{\epsilon})}))/\sqrt{\min_k \Pr_P[\Pi_k]} \tag*{(by Lemma \ref{lem:mean-dist-from-norm})} \\
\leq O({\epsilon}\sqrt{\log(1/{\epsilon})}/\min_k \Pr_P[\Pi_k]) \;.\end{aligned}$$ Now we can apply Corollary \[cor:bal-min\] with $O({\epsilon}\sqrt{\log(1/{\epsilon})}/\min_k \Pr_P[\Pi_k])$ in place of ${\epsilon}$. Note that condition (i) follows from our assumption that $\min_k \Pr_P[\Pi_k] = \Omega({\epsilon}^{1/2} \log^{1/4}(1/{\epsilon}))$.
The Case of Large Spectral Norm {#ssec:big-norm}
-------------------------------
Now we consider the case when $\|M\|_2 \geq C{\epsilon}\ln(1/{\epsilon})/\alpha$. We begin by showing that $p$ and $q$ are not too far apart from each other.
\[clm:delta-distance\] $\|p-q\|_2 \leq \delta := 3 \sqrt{{\epsilon}\|M\|_2}/\alpha$.
By Lemma \[lem:mean-dist-from-norm\], we have that $\sqrt{\sum_k \Pr_P[\Pi_k]^2 (p_k-q_k)^2} \leq 2\sqrt{{\epsilon}\|M\|_2} + O({\epsilon}\sqrt{\log(1/{\epsilon})})$. For sufficiently large $C$, this last term is smaller than $\frac{1}{2} \sqrt{{\epsilon}\|M\|_2} \geq {\epsilon}\sqrt{C \ln(1/{\epsilon})}$. Then we have $\sqrt{\sum_k \Pr_P[\Pi_k]^2 (p_k-q_k)^2} \leq (5/2)\sqrt{{\epsilon}\|M\|_2}$. However, $$\left(\min_k \Pr_P[\Pi_k]\right)^2 \sum_k (p_k-q_k)^2 \leq \sum_k \Pr_P[\Pi_k]^2 (p_k-q_k)^2$$ and $\min_k \Pr_P[\Pi_k] = \alpha+O({\epsilon})$.
Next we show that most of the variance of $v^*\cdot F({{\widetilde{P}}},q)$ comes from the additive error.
\[clm:norm-mostly-E\] $w_E v^{\ast T} M_E v^{\ast} \geq \frac{1}{2} v^{\ast T} M v^{\ast}$.
By Lemma \[lem:MApprox\] and Claim \[clm:delta-distance\], we deduce $$\begin{aligned}
\|M - w_E M_E\|_2
&\leq O\left({\epsilon}\log(1/{\epsilon}) + \sum_k \Pr_P[\Pi_k] (p_k-q_k)^2\right) \\
&\leq O\left({\epsilon}\log(1/{\epsilon}) + \left({\epsilon}/\min_k \Pr_P[\Pi_k]\right) \|M\|_2\right) \;.\end{aligned}$$ By assumption $\min_k \Pr_P[\Pi_k] \geq C'{\epsilon}$ for sufficiently large $C'$, so we can bound this second term by $\|M\|_2$. For large enough $C$, the first term is bounded by $\|M\|_2/6$. Thus, $\|M - w_E M_E\|_2 \leq \|M\|_2/2$. Since $v^{\ast T} M v^{\ast}=\|M\|_2$, we obtain $w_E v^{\ast T} M_E v^{\ast} \geq \frac{1}{2} v^{\ast T} M v^{\ast}$, as required.
This will imply that the tails of $w_E E$ are reasonably thick. In particular, we show that there must be a threshold $T>0$ satisfying a property similar to the one desired in Step 9 of our algorithm.
\[lem:exists-T-dist\] There exists a $T \geq 0$ such that $$\Pr_{X \sim {{\widetilde{P}}}}[|v^\ast \cdot (F(X,q) - q)| \geq T + \delta] > 6\exp(-T^2/2) + 6{\epsilon}/d \;.$$
Suppose for the sake of contradiction that this does not hold. For all events $A$, it holds that $w_E \Pr_E[A] \leq \Pr_{{{\widetilde{P}}}}[A]$. Thus, we have $$w_E \Pr_{X \sim E}[|v^\ast \cdot (F(X,q) - q)| \geq T + \delta] \leq 6\exp(-T^2/2) + 6{\epsilon}/d \;.$$ Note that for any $x \in \{0,1\}^d$, we have that $$|v^\ast \cdot (F(x,q)-q)| \leq \|F(x,q)-q\|_2 \leq \sqrt{d} \;,$$ since $F(x,q)$ and $q$ differ on at most $d$ coordinates. We have the following sequence of inequalities: $$\begin{aligned}
\|M\|_2 & \ll w_E v^{\ast T} M v^{\ast} \\
& = 2 w_E \int_0^{\sqrt{d}} \Pr_{x \sim E}\left[|v^\ast \cdot (F(X,q)-q)| \geq T\right] T dT \\
& \leq 2 w_E \int_0^{\delta + 4 \sqrt{\ln(1/w_E)}} T dT + \int_{\delta + 4 \sqrt{\ln(1/w_E)}}^{\sqrt{d}} \left( \exp(-(T-\delta)^2) T + ({\epsilon}T/d) \right) dT \\
& \ll w_E \delta^2 + w_E \ln(1/w_E) + {\epsilon}\\
& \ll {\epsilon}\delta^2 + {\epsilon}\log(1/{\epsilon}) \\
& \ll ({\epsilon}^2/\alpha^2)\|M\|_2 + \alpha \|M\|_2/C \;.\end{aligned}$$ Since ${\epsilon}^2/\alpha^2 =O({\epsilon})$ and $\alpha \leq 1$, for sufficiently large $C$, this gives the desired contradiction.
We will also need that the samples $t^i$ actually give us a good approximation to the tails of $v^{\ast}\cdot F({{\widetilde{P}}},q)$. Fortunately, this happens with high probability using the DKW inequality [@DKW56; @Massart90]:
\[lem:dkw\] With probability at least $1-1/(200(2d+1))$, for all $T > 0$, we have $$|\frac{\#\{i:|v^\ast \cdot (F(t^i,q)-q)|>T+\delta\}}{N} - \Pr_{X \sim {{\widetilde{P}}}}[|F(X,q) - q| \geq T + \delta]| \leq {\epsilon}/d \;.$$
We assume that this is the case. We then have that Step \[step:findT\] will find a $T>0$:
\[cor:exists-T-empirical\] There exists a $T>0$ such that $$\frac{\#\{i:|v^\ast \cdot (F(t^i,q)-q)|>T+\delta\}}{N} > 6\exp(-T^2/2)+5{\epsilon}/d \;.$$
Finally, we consider the distribution ${{\widetilde{P}}}$ conditioned on $A$ accepting. We claim that the significant majority of rejected samples came from $E$, leading to a cleaner distribution. In particular, this will complete the proof of Proposition \[prop:bal\]:
\[clm:calc\] We can write ${{\widetilde{P}}}'$ which is ${{\widetilde{P}}}$ conditioned on $A$ accepting as ${{\widetilde{P}}}' = w'_P P + w'_E E' - w'_L L'$, where $L'$ and $E'$ have distinct supports, $w'_L,w'_E > 0$ and $w'_E +w'_L \leq w_E + w_L - {\epsilon}/d$. Furthermore, the probability that $A$ rejects a random element of ${{\widetilde{P}}}$ is at most $O((w_E+w_L)-(w_E'+w_L'))$.
Let $A_R$ be the event that $A$ rejects.
By Corollary \[cor:exists-T-empirical\], Step \[step:findT\] finds a $T>0$ such that $$\frac{\#\{i:|v^\ast \cdot (F(t^i,q)-q)|>T+\delta\}}{m} > 6\exp(-T^2/2)+5{\epsilon}/d \;.$$ By Lemma \[lem:dkw\], we have that $$\Pr_{X \sim {{\widetilde{P}}}}\left[|F(X,q) - q| \geq T + \delta\right] > 6\exp(-T^2/2)+4{\epsilon}/d \;.$$ On the other hand, by Lemma \[lem:Azuma\], $$\Pr_{X \sim P}[|F(X,q) - q| \geq T + \delta] \leq 2\exp(-T^2/2).$$ Thus, we have $\Pr_{{{\widetilde{P}}}}[A_R] \geq 3 \Pr_P[A_R]$. Since $$\Pr_{{{\widetilde{P}}}}[A_R] \leq w_P \Pr_P[A_R] + w_E \Pr_E[A_R]$$ and $w_P \leq 1+O({\epsilon})$, we must have $$\Pr_{{{\widetilde{P}}}}[A_R] \leq (2+O({\epsilon})) w_E \Pr_E[A_R]$$ and $$\Pr_P[A_R] \leq (1/3 + O({\epsilon})) w_E \Pr_E[A_R].$$ Then, we get that $$\begin{aligned}
\left(1-\Pr_{{{\widetilde{P}}}}[A_R]\right) {{\widetilde{P}}}'(x) & = \left(1-\Pr_{{{\widetilde{P}}}}[A_R]\right)({{\widetilde{P}}} \mid A_R)(x) \\
& = w_P P(x) + w_E\left(1-\Pr_E[A_R]\right) E(x) + w_L\left(1-\Pr_L[A_R]\right) L(x) - w_P \Pr_P[A_R] P(x) \;.\end{aligned}$$ Thus, we have $$\begin{aligned}
w'_L & = \frac{w_L\left(1-\Pr_L[A_R]\right) - w_P \Pr_P[A_R]}{1-\Pr_{{{\widetilde{P}}}}[A_R]} \\
& \leq w_L + (1 + O({\epsilon})) \Pr_P[A_R] + O\left({\epsilon}\Pr_{{{\widetilde{P}}}}[A_R]\right) \\
& \leq w_L + \left(1/3 + O({\epsilon})\right)w_E \Pr_E[A_R] + O\left({\epsilon}\Pr_{{{\widetilde{P}}}}[A_R]\right) \;.\end{aligned}$$ Also we have $$\begin{aligned}
w'_E & = \frac{w_E\left(1-\Pr_E[A_R]\right)}{1-\Pr_{{{\widetilde{P}}}}[A_R]} \\
& \leq w_E\left(1-\Pr_E[A_R]\right) + O\left({\epsilon}\Pr_{{{\widetilde{P}}}}[A_R]\right) \;.\end{aligned}$$ Thus, $$\begin{aligned}
w_L + w_E - w'_L - w'_E
& \geq \left(2/3 - O({\epsilon})\right)w_E \Pr_E[A_R] - O\left({\epsilon}\Pr_{{{\widetilde{P}}}}[A_R]\right) \\
& \geq \left(1/3 - O({\epsilon})\right) \Pr_{{{\widetilde{P}}}}[A_R] \geq {\epsilon}/d \;.\end{aligned}$$ Note that the penultimate inequality also gives that $$\Pr_{{{\widetilde{P}}}}[A_R] \leq \left(3+O({\epsilon})\right)(w_L + w_E - w'_L - w'_E).$$ This completes the proof.
Conclusions and Future Directions {#sec:concl}
=================================
In this paper we initiated the study of the efficient robust learning for graphical models. We described a computationally efficient algorithm for robustly learning Bayesian networks with a known topology, under some natural conditions on the conditional probability table. We believe that these conditions can be eliminated by a more careful application of our techniques. A challenging open problem that requires additional ideas is to generalize our results to the setting of structure learning, i.e., the case when the underlying directed graph is unknown.
This work is part of a broader agenda of systematically investigating the robust learnability of high-dimensional structured probability distributions. There is a wealth of natural probabilistic models that merit investigation in the robust setting, including undirected graphical models (e.g., Ising models), and graphical models with hidden variables (i.e., incorporating latent structure).
#### Acknowledgements.
We are grateful to Daniel Hsu for suggesting the model of Bayes nets, and for pointing us to [@Dasgupta97].
Omitted Proofs from Section \[sec:prelims\] {#app:prelims}
===========================================
[**Lemma \[lem:hel-bn\]**]{}
*Let $P$ and $Q$ be Bayesian networks with the same dependency graph $G$. In terms of the conditional probability tables $p$ and $q$ of $P$ and $Q$, we have: $${d_{\mathrm H}}(P,Q)^2 \leq 2 \sum_{k=1}^m \sqrt{\Pr_P[\Pi_k] \Pr_Q[\Pi_k]} \frac{(p_k-q_k)^2}{(p_k+q_k)(2-p_k-q_k)} \;.$$*
Let $A$ and $B$ be two distributions on $\{0,1\}^d$. Then we have: $$\label{squared-hellinger}
1-{d_{\mathrm H}}^2(A,B) = \sum_{x \in \{0,1\}^d} \sqrt{\Pr_A[x]\Pr_B[x]} \;.$$ For a fixed $i \in [d]$, the events $\Pi_{i,a}$ form a partition of $\{0,1\}^d$. Dividing the sum above into this partition, we obtain $$\begin{aligned}
1-{d_{\mathrm H}}^2(A,B) & = \sum_a \sum_{x \in \Pi_{i,a}} \sqrt{\Pr_A[x]\Pr_B[x]} \\
& = \sum_a \sqrt{\Pr_A[\Pi_{i,a}]\Pr_B[\Pi_{i,a}]} \sum_{x \in \{0,1\}^d} \sqrt{\Pr_{A \mid \Pi_{i,a}}[x]\Pr_{B \mid \Pi_{i,a}} [x]} \\
& = \sum_a \sqrt{\Pr_A[\Pi_{i,a}]\Pr_B[\Pi_{i,a}]} \left(1-{d_{\mathrm H}}^2\left(A\mid\Pi_{i,a}, B\mid\Pi_{i,a} \right) \right) \;.\end{aligned}$$
Let $P_{\leq i}$ be the distribution over the first $i$ coordinates of $P$ and define $Q_{\leq i}$ similarly for $Q$. Let $P_i$ and $Q_i$ be the distribution of the $i$-th coordinate of $P$ and $Q$ respectively. We will apply the above to $P_{\leq i-1}$ and $P_{\leq i}$. First note that the $i$-th coordinate of $P_{\leq i} \mid \Pi_{i,a}$ and $Q_{\leq i}\mid\Pi_{i,a}$ is independent of the others, thus we may factorize the RHS of Equation $(\ref{squared-hellinger})$ for these distributions, obtaining: $$1-{d_{\mathrm H}}^2\left(P_{\leq i}\mid\Pi_{i,a}, Q_{\leq i}\mid\Pi_i,a\right)=
\left(1-{d_{\mathrm H}}^2\left(P_{\leq i-1}\mid\Pi_{i,a}, Q_{\leq i-1}\mid\Pi_{i,a}\right)\right) \cdot \left(1-{d_{\mathrm H}}^2\left(P_{i}\mid\Pi_{i,a}, Q_{i}\mid\Pi_{i,a}\right)\right) \;.$$
Thus, we have: $$\begin{aligned}
& 1-{d_{\mathrm H}}^2\left(P_{\leq i}, Q_{\leq i}\right) = \sum_a \sqrt{\Pr_P[\Pi_{i,a}] \Pr_Q[\Pi_{i,a}]} \left(1-{d_{\mathrm H}}^2\left(P_{\leq i}\mid\Pi_{i,a}, Q_{\leq i}\mid\Pi_{i,a}\right)\right) \\
& = \sum_a \sqrt{\Pr_P[\Pi_{i,a}] \Pr_Q[\Pi_{i,a}]} \left(1-{d_{\mathrm H}}^2\left(P_{\leq i-1}\mid \Pi_{i,a}, Q_{\leq i-1} \mid\Pi_{i,a}\right)\right)
\left(1-{d_{\mathrm H}}^2\left(P_{i}\mid\Pi_{i,a}, Q_{i}\mid\Pi_{i,a}\right)\right) \\
& = 1-{d_{\mathrm H}}^2\left(P_{\leq i-1}, Q_{\leq i -1} \right) - \sum_a \sqrt{\Pr_P[\Pi_{i,a}] \Pr_Q[\Pi_{i,a}]} \left(1-{d_{\mathrm H}}^2(P_{\leq i-1}\mid\Pi_{i,a},Q_{\leq i-1}\mid\Pi_{i,a})\right)
{d_{\mathrm H}}^2\left(P_{i}\mid\Pi_{i,a}, Q_{i}\mid\Pi_{i,a}\right) \\
& \geq 1-{d_{\mathrm H}}^2\left(P_{\leq i-1},Q_{\leq i-1}\right) - \sum_a \sqrt{\Pr_P[\Pi_{i,a}] \Pr_Q[\Pi_{i,a}]} {d_{\mathrm H}}^2\left(P_{i}\mid\Pi_{i,a}, Q_{i}\mid\Pi_{i,a}\right) \;.\\
\end{aligned}$$ By induction on $i$, we have $${d_{\mathrm H}}^2(P,Q) \leq \sum_{(i,a) \in S} \sqrt{\Pr_P[\Pi_{i,a}] \Pr_Q[\Pi_{i,a}]} {d_{\mathrm H}}^2\left(P_{i}\mid\Pi_{i,a}, Q_{i}\mid\Pi_{i,a}\right) \;.$$ Now observe that the $P_{i}\mid\Pi_{i,a}$ and $Q_{i}\mid\Pi_{i,a}$ are Bernoulli distributions with means $p_{i,a}$ and $q_{i,a}$. For $p,q \in [0,1]$, we have: $$\begin{aligned}
{d_{\mathrm H}}^2(\textrm{Bernoulli}(p),\textrm{Bernoulli}(q)) & = (\sqrt{p}-\sqrt{q})^2 + (\sqrt{1-p}-\sqrt{1-q})^2 \\
& = \frac{(p-q)^2}{(\sqrt{p}+\sqrt{q})^2} + \frac{((1-p)-(1-q))^2}{(\sqrt{1-p}+\sqrt{1-q})^2} \\
&= (p-q)^2 \cdot (\frac{1}{(\sqrt{p}+\sqrt{q})^2}+\frac{1}{(\sqrt{1-p}+\sqrt{1-q})^2}) \\
& \leq (p-q)^2 \cdot (\frac{1}{p+q}+\frac{1}{2-p-q}) \\
& = (p-q)^2 \cdot \frac{2}{(p+q)(2-p-q)} \;,\\\end{aligned}$$ and thus $${d_{\mathrm H}}(P,Q)^2 \leq 2 \sum_{k} \sqrt{\Pr_P[\Pi_k]\Pr_Q[\Pi_k]} \frac{(p_k-q_k)^2}{(p_k+q_k)(2-p_k-q_k)} \;,$$ as required.
[**Corollary \[cor:bal-min\]**]{} [*Suppose that: (i) $\min_{k \in [m]} \Pr_P[\Pi_k] \geq 2{\epsilon}$, and (ii) $P$ or $Q$ is $c$-balanced, and (iii) $\frac{3}{c}\sqrt{\sum_k \Pr_P[\Pi_k] (p_k-q_k)^2} \leq {\epsilon}$. Then we have that ${d_{\mathrm TV}}(P,Q) \leq {\epsilon}.$* ]{}
When either $P$ or $Q$ is $c$-balanced the denominators in Lemma \[lem:hel-bn\] satisfies $(p_k+q_k)(2-p_k-q_k) \geq c$ and so we have ${d_{\mathrm TV}}(p,q) \leq \frac{2}{c}\sqrt{\sum_k \sqrt{\Pr_P[\Pi_k]\Pr_Q[\Pi_k]} (p_k-q_k)^2}.$ Thus, it suffices to show that $\Pr_Q[\Pi_k] \leq 2\Pr_P[\Pi_k]$ or indeed that $\Pr_Q[\Pi_k] \leq \Pr_P[\Pi_k]+{\epsilon}$. This would follow from our conclusion that ${d_{\mathrm TV}}(P,Q) \leq {\epsilon}$.
We can show by induction on $i$ that for all $i$ and $a$, $\Pr_Q[\Pi_{i,a}] \leq \Pr_P[\Pi_{i,a}]+{\epsilon}$. Suppose we have that, for $1 \leq i \leq i'$, $\Pr_Q[\Pi_{i,a}] \leq \Pr_P[\Pi_{i,a}]+{\epsilon}$. Then we have that ${d_{\mathrm TV}}(P_{\leq i'-1},Q_{\leq i'-1}) \leq {\epsilon}$. But for any $a$, $\Pi_{i',a}$ depends only on $i < i'$. Thus, $|\Pr_P[\Pi_{i',a}] - \Pr_Q[\Pi_{i',a}]| \leq {d_{\mathrm TV}}(P_{\leq i'-1},Q_{\leq i'-1}) \leq {\epsilon}$, and therefore $\Pr_Q[\Pi_{i',a}] \leq \Pr_P[\Pi_{i,a}]+{\epsilon}$ for all $a$.
[**Lemma \[lem:inverse\]**]{} [*For all $x \in \{0,1\}^d$, $q \in [0,1]^m$ and $1 \leq j \leq k$, $x_1, \dots, x_j$ can be recovered from the vector of coordinates $F(x,p)_{i,a}$ for all $(i,a) \in S$ with $j \leq i$, and these coordinates of $F$ can be computed from only $x_1,\ldots,x_j$.* ]{}
Note that whether or not $x \in \Pi_{i,a}$ depends only on $x_1,\dots, x_{i-1}$. Thus by the definition of $F(x,p)$ to find $F(x,p)_{i,a}$, we need to know $x_i$ and whether $x \in \Pi_{i,a}$ and so $F(x,p)_{i,a}$ depends only on $x_1,\dots, x_{i-1}$. Thus $F(x,p)_{i,a}$ for all $(i,a) \in S$ with $j \leq i$ is a function of $x_1, \dots, x_j$.
We need to show that $x_1, \dots, x_j$ is a function of $F(x,p)_{i,a}$ for all $(i,a) \in S$ with $j \leq i$. We have that $x_i = F(x,p)_{i,a}$ for the unique $a$ with $x \in \Pi_{i,a}$ which depends only on $x_1,\dots, x_{i-1}$. Since $x_1$ has no parents, we have $x_i = F(x,p)_{1,a'}$ for the empty bitstring $a'$. By an easy induction, we can determine $x_1, \dots, x_j$.
[**Lemma \[lem:cond-ind\]**]{} [*For $X \sim P$ and any $(i,a) \in S$, if we condition on either $\Pi_{i,a}$ or $\neg \Pi_{i,a}$, then $F(X,p)_{i,a}$ is independent of $F(X,p)_{j,a'}$ for all $(j,a') \in S$ with $j \leq i$ and $(j,a')\neq (i,a)$.* ]{}
Conditioned on $\neg \Pi_{i,a}$, $F(X,p)_{i,a}$ is constantly $p_{i,a}$ so is independent of all random variables. We consider the case when we condition on $\Pi_{i,a}$, when $F(X,p_{i,a})= X_i$. Since $\Pi_{i,a}$ implies $\neg \Pi_{i,a'}$ for all $a' \neq a$, in this case $X_i$ is independent of the constant $F(X,p)_{i,a'}$. So, we only need to consider the case when $j \leq i$. However, fixing $F(X,p)_{j,a'}$ for all $(j,a') \in S$ with $j<i$ is equivalent to fixing $X_1,\ldots X_{i-1}$ and so $$\Pr[X_i=1 \mid F(X,p)_{j,a'} \text{ for all $(j,a') \in S$ with $j<i$}]= \Pr[X_i=1|X_1,\dots,X_{i-1}]=\Pr[X_i=1|\Pi_{i,a}]$$
for all combinations of $F(X,p)_{j,a'}$ for all $(j,a') \in S$ with $j<i$ which imply $\Pi_{i,a}$.
[**Lemma \[lem:F-moments\]**]{} [*For $X \sim P$, we have ${\mathbb{E}}\left[ F(X,p) \right] = p$. The covariance matrix of $F(X,p)$ satisfies ${\mathrm{Cov}}[F(X,p)]= {\mathrm{diag}}(\Pr_P[\Pi_k]p_k(1-p_k))$.* ]{}
Note that ${\mathbb{E}}[F(X,p)]_{k} = \Pr_P[\Pi_k]p_k+(1-\Pr_P[\Pi_k])p_k=p_k$ for all $1 \leq k \leq m$.
Next we show that for any $(i,a) \neq (j,a')$, that ${\mathbb{E}}[(F(X,p)_{i,a}-p_{i,a})(F(X,p)_{j,a'}-p_{j,a'})]=0$. We assume without loss of generality that $i\geq j$. On the one hand, if $i=j$ then $\Pi_{i,a}$ and $\Pi_{j,a'}$ cannot simultaneously hold. Therefore, with probability $1$ either $F(X,p)_{i,a}=p_{i,a}$ or $F(X,p)_{j,a'}=p_{j,a'}$. Hence, $(F(X,p)_{i,a}-p_{i,a})(F(X,p)_{j,a'}-p_{j,a'})$ is $0$ with probability $1$, and thus has expectation $0$. Otherwise, if $i>j$, we claim that even after conditioning on the value of $F(X,p)_{j,a'}$, that the expected value of $F(X,p)_{i,a}$ is $p_{i,a}$. In fact, we claim that even after conditioning on all of the values of $X_1,\ldots,X_{i-1}$ that the expectation of $F(X,p)_{i,a}$ is $p_{i,a}$. This is because subject to this conditioning either $\Pi_{i,a}$ holds, in which case, $F(X,p)_{i,a}=X_i$, which is $1$ with probability $p_{i,a}$, or $\Pi_{i,a}$ doesn’t hold, and $F(X,p)$ is deterministically $p_{i,a}$. This completes the proof that $F(X,p)_{i,a}$ and $F(X,p)_{j,a'}$ are uncorrelated.
Finally, for any $(a,i) \in S$, $${\mathbb{E}}\left[(F(X,p)_{i,a}-p_{i,a})^2\right]=\Pr_P[\Pi_{i,a}] {\mathbb{E}}\left[(X_i-p_{i,a})^2 \mid \Pi_{i,a}\right]
=\Pr_P[\Pi_{i,a}] p_{i,a}(1-p_{i,a}).$$ This completes the proof.
\[clm:cond-mean\] We have that ${\mathbb{E}}_{X \sim P}[F(X,p)_k \mid F(X,p)_1, \dots, F(X,p)_{k-1}]=p_k$, for all $1 \leq k \leq m.$
Firstly we claim that ${\mathbb{E}}[F(X,p)_k \mid F(X,p)_1, \dots, F(X,p)_{k-1}]=p_k$ for all $1 \leq k \leq m$. Let $(i,a) \in S$ be such that $F(X,p)_k=F(X,p)_{i,a}$. Then, by definition, $F(X,p)_1, \dots, F(X,p)_{k-1}$ includes $F(X)_{j,a'}$ for all $(j,a') \in S$ with $j < i$. By Lemma \[lem:inverse\], these determine the value of the parents of $X_i$, i.e., whether or not $\Pi_{i,a}$ occurs. If $F(X,p)_1, \dots, F(X,p)_{k-1}$ imply $\Pi_{i,a}$, then by Lemma \[lem:cond-ind\], $${\mathbb{E}}[F(X,p)_k \mid F(X,p)_1, \dots, F(X,p)_{k-1}]={\mathbb{E}}[F(X,p)_k \mid \Pi_k]=p_k \;,$$ and if they imply $\neg \Pi_k$, then we similarly have: $${\mathbb{E}}[F(X,p)_k \mid F(X,p)_1, \dots, F(X,p)_{k-1}]={\mathbb{E}}[F(X,p)_k \mid \neg \Pi_k]=p_k \; .$$ We have that ${\mathbb{E}}[F(X,p)_k \mid F(X,p)_1, \ldots, F(X,p)_{k-1}]=p_k$ for all combinations of $F(X,p)_1, \ldots, F(X,p)_{k-1}$.
Omitted Proofs from Section \[sec:alg\] {#app:alg}
=======================================
Omitted Proofs from Section \[ssec:setup\] {#app:setup}
------------------------------------------
[**Lemma \[lem:empirical-moments-good\]**]{} [*With probability at least $1-\frac{1}{200(2d+1)}$, $\alpha - \min_k \Pr_{{{\widetilde{P}}}}[\Pi_k] \leq {\epsilon}$, $\|q-{\mathbb{E}}_{X \sim {{\widetilde{P}}}}[F(X,q)]\|_2 \leq {\epsilon}$ and $\|M-M_{{{\widetilde{P}}}}\|_2 \leq {\epsilon}$.* ]{}
By the Chernoff and union bounds, with probability at least $1-\frac{1}{600(2d+1)}$, we have that our empirical estimates for $\Pr_{{{\widetilde{P}}}}[\Pi_k]$ are correct to within ${\epsilon}$ using $O(\log m/{\epsilon}^2)$ samples. This implies that $\alpha - \min_k \Pr_{{{\widetilde{P}}}}[\Pi_k] \leq {\epsilon}$.
Note that for fixed $i, a$ we have that $$q_{i,a}-{\mathbb{E}}_{X \sim {{\widetilde{P}}}}[F(X,q)]_{i,a} = \Pr_{{{\widetilde{P}}}}[\Pi_{i, a}] \left(q_{i,a}- \Pr_{X \sim {{\widetilde{P}}}}[X_i=1 \mid \Pi_{i,a}]\right)$$ and that the latter conditional probability does not depend on $q$. For $\tau > 0$, suppose that we have $\Omega (m^2 \Pr_{{{\widetilde{P}}}}[\Pi_{i, a}]^2 \log(1/\tau)/{\epsilon}^2)$ samples for which $\Pi_{i, a}$ holds. Then, by the Chernoff bound, it follows that $$\left|q_{i,a}- \Pr_{X \sim {{\widetilde{P}}}}[X_i=1 \mid \Pi_{i,a}] \right|\leq \frac{{\epsilon}}{m\Pr_{{{\widetilde{P}}}}[\Pi_{i, a}]}$$ with probability at least $1-\tau$. Again by the Chernoff bound, this happens with probability at least $1-\tau$ when we have $O(m^2 \log(1/\tau)/{\epsilon}^2)$ samples from ${{\widetilde{P}}}$. By a union bound, we have $\|q-{\mathbb{E}}_{X \sim {{\widetilde{P}}}}[F(X,q)]\|_\infty \leq {\epsilon}/m$ with probability at least $1-\frac{1}{600(2d+1)}$ with $O(m^2 \log(m)/{\epsilon}^2)$ samples. In this case, we have $\|q-{\mathbb{E}}_{X \sim {{\widetilde{P}}}}[F(X,q)]\|_2 \leq {\epsilon}$ as required.
Now we assume that this holds and need to show that $\|M-M_{{{\widetilde{P}}}}\|_2 \leq {\epsilon}$. Let $\mu^{{{\widetilde{P}}}} = {\mathbb{E}}_{X \sim {{\widetilde{P}}}}[F(X,q)]$. When $i=j$, $M_{i,j}=(M_{{{\widetilde{P}}}})_{i,j}=0$. Consider some unequal $i, j$. Since $\|\mu^{{{\widetilde{P}}}} - q\|_\infty \leq {\epsilon}/m$, we have $$\begin{aligned}
(M_{{{\widetilde{P}}}})_{i,j} &= {\mathbb{E}}_{X \sim {{\widetilde{P}}}}[(F(X,q)_i-q_i)(F(X,q)_j - q_j)] \\
&= {\mathbb{E}}_{X \sim {{\widetilde{P}}}}[(F(X,q)_i-\mu^{{{\widetilde{P}}}}_i)(F(X,q)_j - \mu^{{{\widetilde{P}}}}_j)]+O({\epsilon}/m) \;.\end{aligned}$$ Similarly, if $S$ is the empirical distribution over the samples, we have $$\begin{aligned}
M_{i,j} &= {\mathbb{E}}_{X \sim S}[(F(X,q)_i-q_i)(F(X,q)_j - q_j)] \\
&= {\mathbb{E}}_{X \sim S}[(F(X,q)_i-\mu^{{{\widetilde{P}}}}_i)(F(X,q)_j - \mu^{{{\widetilde{P}}}}_j)]+O({\epsilon}/m) \;.\end{aligned}$$ Now note that this is a sum of independent centered random variables each with absolute value at most one. By Hoeffding’s inequality, it follows that with probability $2\exp(-N{\epsilon}^2/2m^2)$ we have that $|M_{i,j} - (M_{{{\widetilde{P}}}})_{i,j}| \leq {\epsilon}/m$. Since $N=\Omega(m^2\log(m)/{\epsilon}^2)$, by a union bound, this holds for all $i,j$ with probability at least $1-\frac{1}{600(2d+1)}$. Then, we have that $\|M-M_{{{\widetilde{P}}}}\|_2 \leq \|M-M_{{{\widetilde{P}}}}\|_F \leq {\epsilon}$.
[**Lemma \[lem:MP-bound\]**]{} [*$\|M_P\|_2 \leq \sum_k \Pr_P[\Pi_k] (p_k-q_k)^2$.* ]{}
Note that when $x \in \{0,1\}^d$ is such that $x \in \neg \Pi_k$, we have that $F(x,q)_k = q_k$ and that $\Pr_P[\Pi_i \wedge \Pi_j] \leq \min \{\Pr_P[\Pi_i], \Pr_P[\Pi_j] \} \leq \sqrt{\Pr_P[\Pi_i]\Pr_P[\Pi_j]}$. By definition we have that $(M_P)_{i,i}=0$ and, for $i \neq j$, we can write $$(M_P)_{i,j} = {\mathbb{E}}_{X \sim P}\left[(F(X,q)_i-q_i)(F(X,q)_j - q_j)\right] = \Pr_P[\Pi_i \wedge \Pi_j] (p_i-q_i)(p_j-q_j) \;.$$ Either way, it holds that $$(M_P)_{i,j}^2 \leq \Pr_P[\Pi_i]\Pr_P[\Pi_j] (p_i-q_i)^2(p_j-q_j)^2 \;.$$ Thus, we obtain $$\begin{aligned}
\|M_P\|_2^2 & \leq \|M_P\|_F^2 = \sum_{i,j} (M_P)_{i,j}^2 \\
& \leq \sum_{i,j} \Pr_P[\Pi_i]\Pr_P[\Pi_j] (p_i-q_i)^2(p_j-q_j)^2 \\
& = \left(\sum_k \Pr_P[\Pi_k] (p_k-q_k)^2\right)^2 \;,\end{aligned}$$ as desired.
[^1]: Supported in part by a Marie Curie Career Integration grant.
[^2]: Supported in part by NSF Award CCF-1553288 (CAREER). Some of this work was performed while visiting the University of Edinburgh.
[^3]: Supported in part by a Marie Curie Career Integration grant.
|
---
abstract: 'Gamma-ray induced air showers are notable for their lack of muons, compared to hadronic showers. Hence, air shower arrays with large underground muon detectors can select a sample greatly enriched in photon showers by rejecting showers containing muons. IceCube is sensitive to muons with energies above $\sim$500 GeV at the surface, which provides an efficient veto system for hadronic air showers with energies above 1 PeV. One year of data from the 40-string IceCube configuration was used to perform a search for point sources and a Galactic diffuse signal. No sources were found, resulting in a 90% C.L. upper limit on the ratio of gamma rays to cosmic rays of $1.2\times10^{-3}$ for the flux coming from the Galactic Plane region ( $-80^\circ \lesssim l \lesssim -30^\circ$; $-10^\circ \lesssim b \lesssim 5^\circ$) in the energy range 1.2 – 6.0 PeV. In the same energy range, point source fluxes with $E^{-2}$ spectra have been excluded at a level of $(E/\mathrm{TeV})^2 \mathrm{d}\Phi/\mathrm{d}E \sim 10^{-12} - 10^{-11}$ cm$^{-2}$s$^{-1}$TeV$^{-1}$ depending on source declination. The complete IceCube detector will have a better sensitivity, due to the larger detector size, improved reconstruction and vetoing techniques. Preliminary data from the nearly-final IceCube detector configuration has been used to estimate the 5 year sensitivity of the full detector. It is found to be more than an order of magnitude better, allowing the search for PeV extensions of known TeV gamma-ray emitters.'
author:
- 'M. G. Aartsen'
- 'R. Abbasi'
- 'Y. Abdou'
- 'M. Ackermann'
- 'J. Adams'
- 'J. A. Aguilar'
- 'M. Ahlers'
- 'D. Altmann'
- 'K. Andeen'
- 'J. Auffenberg'
- 'X. Bai'
- 'M. Baker'
- 'S. W. Barwick'
- 'V. Baum'
- 'R. Bay'
- 'K. Beattie'
- 'J. J. Beatty'
- 'S. Bechet'
- 'J. Becker Tjus'
- 'K.-H. Becker'
- 'M. Bell'
- 'M. L. Benabderrahmane'
- 'S. BenZvi'
- 'J. Berdermann'
- 'P. Berghaus'
- 'D. Berley'
- 'E. Bernardini'
- 'D. Bertrand'
- 'D. Z. Besson'
- 'D. Bindig'
- 'M. Bissok'
- 'E. Blaufuss'
- 'J. Blumenthal'
- 'D. J. Boersma'
- 'S. Bohaichuk'
- 'C. Bohm'
- 'D. Bose'
- 'S. Böser'
- 'O. Botner'
- 'L. Brayeur'
- 'A. M. Brown'
- 'R. Bruijn'
- 'J. Brunner'
- 'S. Buitink'
- 'M. Carson'
- 'J. Casey'
- 'M. Casier'
- 'D. Chirkin'
- 'B. Christy'
- 'K. Clark'
- 'F. Clevermann'
- 'S. Cohen'
- 'D. F. Cowen'
- 'A. H. Cruz Silva'
- 'M. Danninger'
- 'J. Daughhetee'
- 'J. C. Davis'
- 'C. De Clercq'
- 'S. De Ridder'
- 'F. Descamps'
- 'P. Desiati'
- 'G. de Vries-Uiterweerd'
- 'T. DeYoung'
- 'J. C. D[í]{}az-Vélez'
- 'J. Dreyer'
- 'J. P. Dumm'
- 'M. Dunkman'
- 'R. Eagan'
- 'J. Eisch'
- 'R. W. Ellsworth'
- 'O. Engdeg[å]{}rd'
- 'S. Euler'
- 'P. A. Evenson'
- 'O. Fadiran'
- 'A. R. Fazely'
- 'A. Fedynitch'
- 'J. Feintzeig'
- 'T. Feusels'
- 'K. Filimonov'
- 'C. Finley'
- 'T. Fischer-Wasels'
- 'S. Flis'
- 'A. Franckowiak'
- 'R. Franke'
- 'K. Frantzen'
- 'T. Fuchs'
- 'T. K. Gaisser'
- 'J. Gallagher'
- 'L. Gerhardt'
- 'L. Gladstone'
- 'T. Glüsenkamp'
- 'A. Goldschmidt'
- 'G. Golup'
- 'J. A. Goodman'
- 'D. Góra'
- 'D. Grant'
- 'A. Gro[ß]{}'
- 'S. Grullon'
- 'M. Gurtner'
- 'C. Ha'
- 'A. Haj Ismail'
- 'A. Hallgren'
- 'F. Halzen'
- 'K. Hanson'
- 'D. Heereman'
- 'P. Heimann'
- 'D. Heinen'
- 'K. Helbing'
- 'R. Hellauer'
- 'S. Hickford'
- 'G. C. Hill'
- 'K. D. Hoffman'
- 'R. Hoffmann'
- 'A. Homeier'
- 'K. Hoshina'
- 'W. Huelsnitz'
- 'P. O. Hulth'
- 'K. Hultqvist'
- 'S. Hussain'
- 'A. Ishihara'
- 'E. Jacobi'
- 'J. Jacobsen'
- 'G. S. Japaridze'
- 'O. Jlelati'
- 'A. Kappes'
- 'T. Karg'
- 'A. Karle'
- 'J. Kiryluk'
- 'F. Kislat'
- 'J. Kläs'
- 'S. R. Klein'
- 'J.-H. Köhne'
- 'G. Kohnen'
- 'H. Kolanoski'
- 'L. Köpke'
- 'C. Kopper'
- 'S. Kopper'
- 'D. J. Koskinen'
- 'M. Kowalski'
- 'M. Krasberg'
- 'G. Kroll'
- 'J. Kunnen'
- 'N. Kurahashi'
- 'T. Kuwabara'
- 'M. Labare'
- 'H. Landsman'
- 'M. J. Larson'
- 'R. Lauer'
- 'M. Lesiak-Bzdak'
- 'J. Lünemann'
- 'J. Madsen'
- 'R. Maruyama'
- 'K. Mase'
- 'H. S. Matis'
- 'F. McNally'
- 'K. Meagher'
- 'M. Merck'
- 'P. Mészáros'
- 'T. Meures'
- 'S. Miarecki'
- 'E. Middell'
- 'N. Milke'
- 'J. Miller'
- 'L. Mohrmann'
- 'T. Montaruli'
- 'R. Morse'
- 'R. Nahnhauer'
- 'U. Naumann'
- 'S. C. Nowicki'
- 'D. R. Nygren'
- 'A. Obertacke'
- 'S. Odrowski'
- 'A. Olivas'
- 'M. Olivo'
- 'A. O’Murchadha'
- 'S. Panknin'
- 'L. Paul'
- 'J. A. Pepper'
- 'C. Pérez de los Heros'
- 'D. Pieloth'
- 'N. Pirk'
- 'J. Posselt'
- 'P. B. Price'
- 'G. T. Przybylski'
- 'L. Rädel'
- 'K. Rawlins'
- 'P. Redl'
- 'E. Resconi'
- 'W. Rhode'
- 'M. Ribordy'
- 'M. Richman'
- 'B. Riedel'
- 'J. P. Rodrigues'
- 'F. Rothmaier'
- 'C. Rott'
- 'T. Ruhe'
- 'B. Ruzybayev'
- 'D. Ryckbosch'
- 'S. M. Saba'
- 'T. Salameh'
- 'H.-G. Sander'
- 'M. Santander'
- 'S. Sarkar'
- 'K. Schatto'
- 'M. Scheel'
- 'F. Scheriau'
- 'T. Schmidt'
- 'M. Schmitz'
- 'S. Schoenen'
- 'S. Schöneberg'
- 'L. Schönherr'
- 'A. Schönwald'
- 'A. Schukraft'
- 'L. Schulte'
- 'O. Schulz'
- 'D. Seckel'
- 'S. H. Seo'
- 'Y. Sestayo'
- 'S. Seunarine'
- 'C. Sheremata'
- 'M. W. E. Smith'
- 'M. Soiron'
- 'D. Soldin'
- 'G. M. Spiczak'
- 'C. Spiering'
- 'M. Stamatikos'
- 'T. Stanev'
- 'A. Stasik'
- 'T. Stezelberger'
- 'R. G. Stokstad'
- 'A. Stö[ß]{}l'
- 'E. A. Strahler'
- 'R. Ström'
- 'G. W. Sullivan'
- 'H. Taavola'
- 'I. Taboada'
- 'A. Tamburro'
- 'S. Ter-Antonyan'
- 'S. Tilav'
- 'P. A. Toale'
- 'S. Toscano'
- 'M. Usner'
- 'D. van der Drift'
- 'N. van Eijndhoven'
- 'A. Van Overloop'
- 'J. van Santen'
- 'M. Vehring'
- 'M. Voge'
- 'M. Vraeghe'
- 'C. Walck'
- 'T. Waldenmaier'
- 'M. Wallraff'
- 'M. Walter'
- 'R. Wasserman'
- 'Ch. Weaver'
- 'C. Wendt'
- 'S. Westerhoff'
- 'N. Whitehorn'
- 'K. Wiebe'
- 'C. H. Wiebusch'
- 'D. R. Williams'
- 'H. Wissing'
- 'M. Wolf'
- 'T. R. Wood'
- 'K. Woschnagg'
- 'C. Xu'
- 'D. L. Xu'
- 'X. W. Xu'
- 'J. P. Yanez'
- 'G. Yodh'
- 'S. Yoshida'
- 'P. Zarzhitsky'
- 'J. Ziemann'
- 'S. Zierke'
- 'A. Zilles'
- 'M. Zoll'
title: Search for Galactic PeV Gamma Rays with the IceCube Neutrino Observatory
---
[^1]
[^2]
[^3]
[^4]
[^5]
Introduction
============
Gamma-rays are an important tool for studying the cosmos; unlike cosmic rays (CRs), they point back to their sources and can identify remote acceleration regions. Air Cherenkov telescopes have identified numerous sources of high-energy ($E > 1$ TeV) gamma-rays (see e.g. [@Aharonian:2008]): within our galaxy, gamma-rays have been observed coming from supernova remnants (SNRs), pulsar wind nebulae (PWNe), binary systems, and the Galactic Center. Extra-galactic sources include starburst galaxies and Active Galactic Nuclei (AGNs). Surface air-shower arrays like Milagro have performed all-sky searches for TeV gamma-rays. Although these detectors are less sensitive to point sources than Air Cherenkov telescopes, they have identified several Galactic pointlike and extended sources [@Abdo:2009]. Interactions of CRs with interstellar matter and radiation in the Galaxy produce a diffuse flux. Hadrons interacting with matter produce neutral pions, which decay into gamma rays, while CR electrons produce gamma-rays via inverse Compton scattering on the radiation field. Milagro has measured this diffuse Galactic flux in the TeV energy range with a median energy of 15 TeV and reported an excess in the Cygnus region, which might originate from CRs from local sources interacting with interstellar dust clouds [@Abdo:2008]. IceCube’s predecessor AMANDA-II has also looked for TeV photons from a giant flare from SGR 1806-20, using 100 GeV muons. AMANDA’s large muon collection area compensated for the small cross-section for photons to produce muons [@Achterberg:2006az].
At higher energies, extra-galactic sources are unlikely to be visible, because more energetic photons are predicted to interact with cosmic microwave background radiation (CMBR), and with infrared starlight from early galaxies, producing $e^+e^-$ pairs [@Gould66]. At 1 PeV, for example, photon propagation is limited to a range of about 10 kpc. It is unknown whether Galactic accelerators exist that can produce gamma rays of such high energy, but an expected flux results from interaction of (extragalactic) CRs with the interstellar medium (ISM) and dense molecular clouds.
To date, the best statistics on photons with energies in the range from $\sim$300 TeV to several PeVs come from the Chicago Air Shower Array - Michigan Muon Array (CASA-MIA), built at the Dugway Proving Ground in Utah. CASA consisted of 1089 scintillation detectors placed on a square array with 15 m spacing. MIA consisted of 1024 scintillation counters buried under about 3 m of earth, covering an area of 2500 m$^2$. It served as a muon veto, with a threshold of about 0.8 GeV.
CASA-MIA set a limit on the fraction of photons in the cosmic-ray flux of $10^{-4}$ at energies above 600 TeV [@Chantell:1997gs]. The experiment also sets a limit of $2.4\times10^{-5}$ on the fraction of photons in the CR flux coming from within 5$^\circ$ of the galactic disk [@Borione:1997fy] at 310 TeV. This is near the theoretical expectation due to cosmic-ray interactions with the interstellar medium. For a Northern hemisphere site like CASA-MIA, Ref. [@Aharonian1991] predicts a gamma-ray fraction of $2\times10^{-5}$ for the average gas column density.
In this work, we present a new approach for detecting astrophysical PeV gamma rays, based on data of the surface component, IceTop, and the in-ice array of IceCube. IceTop measures the electromagnetic component of air showers, while the in-ice array is sensitive to muons that penetrate the ice with energies above 500 GeV. While most CR showers above 1 PeV contain many muons above this threshold, only a small fraction of PeV gamma-ray showers carry muons that are energetic enough to reach the in-ice array. Therefore, gamma-ray candidates are selected among muon-poor air showers detected with IceTop and whose axis is reconstructed as passing through the in-ice array.
This approach of selecting muon-poor showers as gamma-ray candidates is fundamentally different from the earlier AMANDA-II gamma-ray search described above, which was only sensitive to gamma-ray showers that *do* contain high energy muons ($> 100$ GeV).
We present a limit on the gamma-ray flux coming from the Galactic Plane, based on one year of data with half of the IceCube strings and surface stations installed. We also discuss the sensitivity of the completed detector.
The IceCube detector {#sec:detector}
====================
![\[fig:detector\] IceCube consists of a $\sim$km$^2$ surface air shower array and 86 strings holding 60 optical modules each, filling a physical volume of a km$^{3}$. The region in the center of the buried detector is more densely instrumented. See text for details.](fig1.pdf){width="\linewidth"}
IceCube (see Fig. \[fig:detector\]) is a particle detector located at the geographic South Pole. The in-ice portion consists of 86 strings that reach 2450 m deep into the ice. Most of the strings are arranged in a hexagonal grid, separated by $\sim$125 m. Each of these strings holds 60 digital optical modules (DOMs) separated by $\sim$ 17 m covering the range from 1,450 m to 2,450 m depth. Eight strings form a denser instrumented area called DeepCore. The DOMs detect Cherenkov light produced by downward-going muons in cosmic-ray air showers and from charged particles produced in neutrino interactions. The data used in this analysis was collected in 2008/9, when the 40 strings shown in Fig. \[fig:layout\] were operational.
Each DOM is a complete detector system, comprising a 25 cm diameter Hamamatsu R7081-02 phototube [@Abbasi:2010vc], shaping and digitizing electronics [@:2008ym], calibration hardware, plus control electronics and power supply. Most of the buried PMTs are run at a gain of $10^7$. Digitization is initiated by a discriminator, with a threshold set to 0.25 times the typical peak amplitude of a single photoelectron waveform. Each DOM contains two separate digitizing systems; the Analog Transient Waveform Digitizer (ATWD) records 400 ns of data at 300 Megasamples/s, with a 14 bit dynamic range (divided among 3 parallel channels), while the fast Analog-to-Digital Converter (fADC) records 6.4 $\mu$s of data at 40 Megasamples/s, with 10 bits of dynamic range. A system transmits timing signals between the surface and each DOM, providing timing calibrations across the entire array of about 2 ns [@Achterberg:2006md; @Halzen:2010yj].
The IceTop surface array [@icetopdp] is located on the surface directly above the in-ice detectors. It consists of 81 stations, each consisting of two ice-filled tanks, about 5 m apart. For the 2008 data used here, 40 stations were operational (IC40, see Fig. \[fig:layout\]). Each tank is 1.8 m in diameter, filled with ice to a depth of about 90 cm. The tanks are initially filled with water, and the freezing of the water is controlled to minimize air bubbles and preserve the optical clarity of the ice. Each tank is instrumented with two DOMs, a high-gain DOM run at a PMT gain of $5\times10^6$, and a low-gain DOM, with a PMT gain of $5\times 10^5$. The two different gains were chosen to maximize dynamic range; the system is quite linear over a range from 1 to $10^5$ photoelectrons. A station is considered hit when a low-gain DOM in one tank fires in coincidence with a high-gain DOM in the other; the thresholds are set to about 20 photoelectrons.
When an in-ice DOM is triggered it sends a Local Coincidence (LC) message to its nearest two neighbors above and nearest two neighbors below. If the DOM also receives an LC message from one of its neighbors within 1 $\mu$s it is in Hard Local Coincidence (HLC). In that case the full waveform information of both the ATWD and fADC chip is stored. For IC40 and earlier configurations, isolated or Soft Local Coincidence (SLC) hits were discarded. In newer configurations, the SLC hits are stored albeit with limited information. Keeping the full waveforms would require too much bandwidth, since the rate of isolated hits per DOM due to noise is $\sim$500 Hz. Instead, only the three fADC bins with the highest values and their hit times are stored.
In Sec. \[sec:ic40ana\] we present a gamma-ray analysis of the IC40 dataset. In Sec. \[sec:fullsensitivity\] we study the expected sensitivity of the full IceCube detector, and discuss how the inclusion of SLC hits increases the background rejection.
![\[fig:layout\] Map of location of all 86 strings of the completed IceCube detector. The blue dots represent the 40 string configuration that is used for this analysis. At surface level each of these 40 strings is complemented by an IceTop station consisting of two tanks. The large (red) circles indicate the ‘inner strings’ of the IC40 configuration.](fig2.pdf){width="\linewidth"}
Detection principle {#sec:principle}
===================
We create a sample of gamma-ray candidates by selecting air showers that have been successfully reconstructed by IceTop and have a shower axis that passes through the IceCube instrumented volume. The geometry limits this sample to showers having a maximum zenith angle of $\sim$30 degrees. Since IceCube is located at the geographic South Pole, the Field of View (FOV) is roughly constrained to the declination range of $-60$ to $-90$ degrees, as shown in Fig. \[fig:FOV\]. This FOV includes the Magellanic Clouds and part of the Galactic Plane. Gamma-rays at PeV energies are strongly attenuated over extra-galactic distances, thus limiting the observable sources to those localized in our galaxy. At distances of $\sim$ 50 kpc and $\sim$60 kpc, the PeV gamma-ray flux from the Large and Small Magellanic Cloud is suppressed by several orders of magnitude.
The contours in the background of Fig. \[fig:FOV\] are the integrated neutral atomic hydrogen (HI) column densities under the assumption of optical transparency based on data from the Leiden/Argentine/Bonn survey [@LABsurvey]. These densities are not incorporated into the analysis and are only plotted to indicate the Galactic Plane. We do expect, however, that gamma-ray sources are correlated with the HI column density. Firstly, Galactic CR accelerators are more abundant in the high density regions of the Galaxy. Secondly, the gamma-ray flux of (extra-)galactic CRs interacting with the ISM naturally correlates with the column density. However, it has to be noticed that this correlation is not linear, because of the attenuation of gamma-rays over a 10 kpc distance scale. Furthermore, the column densities do not include molecular hydrogen which can also act as a target for CRs.
![\[fig:FOV\] Contours of integrated neutral atomic hydrogen (HI) column densities [@LABsurvey], in Galactic coordinates (flat projection). The blue circle indicates the gamma-ray FOV for IceCube in the present IC40 analysis. The red rectangle indicate the regions for which CASA-MIA [@Chantell:1997gs] has set an upper limit on the Galactic diffuse photon flux in the 100 TeV $-$ 1 PeV energy range. IceCube’s FOV is smaller but covers a different part of the Galactic Plane, close to the Galactic Center.](fig3.pdf){width="\linewidth"}
The gamma-ray candidate events are searched for in a background of CR showers that have exceptionally few muons or are directionally misreconstructed. In the latter case the muon bundle reaches kilometers deep into the ice but misses the instrumented volume. This background is hard to predict with Monte Carlo (MC) simulations. Cosmic-ray showers at PeV energies and with a low number of energetic muons are rare. For example, at 1 PeV less than 0.1% of the simulated showers contain no muons with an energy above 500 GeV, approximately what is needed to reach the detector in the deep ice when traveling vertically. Determining how many hadronic showers produce a signal in a buried DOM would require an enormous amount of MC data to reach sufficient statistics, plus very strong control of the systematic uncertainties due to muon production, propagation of muons and Cherenkov photons through the ice, and the absolute detector efficiencies. It would also have to be able to accurately predict the errors in air shower reconstruction parameters. For example, this analysis is very sensitive to the tails of the distribution of the error on the angular reconstruction of IceTop. Even MC sets that are large enough to populate these tails are not expected to properly describe them.
To avoid these issues, we determine the background directly from data. As a result, we are not able to measure a possible isotropic contribution to the gamma-ray flux, because these gamma-rays would be regarded as background. Instead, we search for localized excesses in the gamma-ray flux.
We search for a correlation of the arrival directions of the candidate events with the Galactic Plane, and scan for point sources. The acceptance of IceTop-40 is a complex function of azimuth and zenith due to its elongated shape and the requirement that the axis of the detected shower passes through IC40 (with the same elongated shape). However, since the arrival time is random (there are no systematic gaps in detector uptime w.r.t. sidereal time) the reconstructed right ascension (RA) of an isotropic flux of CR showers is uniform. The correct declination distribution of the background is very sensitive to the number of background showers introduced by errors in the IceTop angular reconstruction of the air shower as a function of the zenith angle, and is taken to be unknown. However, the flat distribution of background over RA is enough to allow for a search for gamma-ray sources.
Recently, IceCube found an anisotropy in the arrival direction of CRs on the southern hemisphere [@ic:anisotropy]. These deviations with RA have been established for samples of CRs with median energies of 20 TeV and 400 TeV. The two energy ranges show a very different shape of the anisotropy, but the level of the fractional variations in flux is at a part-per-mil level for both [@ic:anisotropy2]. An anisotropy with comparable magnitude in the PeV energy range is too small to affect this analysis (the IC40 final sample contains 268 events).
IC40 analysis {#sec:ic40ana}
=============
Event selection
---------------
Between April 5 2008 and May 20 2009, IceCube took data with a configuration of 40 strings and 40 surface stations (IC40), using several trigger conditions based on different signal topologies. This analysis uses the 8 station surface trigger, which requires a signal above threshold in both tanks of at least 8 IceTop stations. An additional signal in IceCube is not required for this trigger, but all HLC hits in the deep detector within a time window of 10 $\mu s$ before and after the surface trigger are recorded.
The air shower parameters are reconstructed from the IceTop hits with a series of likelihood maximization methods. The core position is found by fitting the lateral distribution of the signal, using $$S(r) = S_{\mathrm{ref}} \left(\frac{r}{R_{\mathrm{ref}}}\right)^{-\beta-\kappa \log_{10}\left(\frac{r}{R_{\mathrm{ref}}}\right)}
\label{eq:latfit}$$ where $S$ is the signal strength, $r$ is the distance to the shower axis, $S_{\mathrm{ref}}$ is the fitted signal strength at the reference distance $R_{\mathrm{ref}}=125$ m, $\beta$ is the slope parameter reflecting the shower age, and $\kappa=0.303$ is a constant determined from simulation [@icetopdp; @IceCube:2012wn]. Signal times are used to find the arrival direction of the air shower. The time delay due to the shape of the shower plane is described by the sum of a Gaussian function and a parabola, both centered at the shower core, which yields a resolution of $1.5^{\circ}$ for IC40. The relationship between the reconstructed energy $E_{\mathrm{reco}}$ and $S_{\mathrm{ref}}$ is based on MC simulations for proton showers and depends on the zenith angle.
IceCube data is processed in different stages: in the first two levels the raw data is calibrated and filtered, and various fitting algorithms are applied, of which only the IceTop reconstruction described above is used in this analysis. In the selection of photon shower candidates from the data sample we distinguish two more steps: level three (L3) and level four (L4). Level three includes all the conditions on reconstruction quality, geometry and energy that make no distinction between gamma-ray showers and CR showers. The L4 cut is designed to separate gamma rays from CRs.
Two parameters are used to constrain the geometry and ensure the shower axis passes through the instrumented volume of IceCube. The IceTop containment parameter $C_{\mathrm{IT}}$ is a measure for how centralized the core location is in IceTop. When the core is exactly in the center of the array $C_{\mathrm{IT}}=0$, while $C_{\mathrm{IT}}=1$ means that it is exactly on the edge of the array. More precisely, $C_{\mathrm{IT}}=x$ means that the core would have been on the edge of the array if the array would be $x$ times its actual size. The string distance parameter $d_{\mathrm{str}}$ is the distance between the point where the shower reaches the depth of the first level of DOMs and the closest inner string. Inner string, in this sense, means a string which is not on the border of the detector configuration. IC40 has 17 inner strings (see Fig. \[fig:layout\]). The L3 cuts are:
- Quality cut on lateral distribution fit: $1.55 < \beta < 4.95$ (cf. Eq. \[eq:latfit\])
- Geometry cut: $C_{\mathrm{IT}} < C_{\mathrm{max}}$
- Geometry cut: $d_{\mathrm{str}} < d_{\mathrm{max}}$
- Energy cut: $E_{\mathrm{reco}} > E_{\mathrm{min}} $
The energy and geometry cuts are optimized in a later stage (Sec. \[sec:gptest\]).
The L4 stage imposes only one extra criterion to the event: there should be no HLC hits in IceCube. This removes most of the CR showers, if $E_{\mathrm{min}}$ is chosen sufficiently high. The remaining background consists of CR showers with low muon content and mis-reconstructed showers, the high energy muons from which do not actually pass through IceCube.
The event sample after the L4 cut might be dominated by remaining background but it can be used to set an upper limit to the number of gamma-rays in the sample. Since the event sample after L3 cuts is certainly dominated by CRs, the ratio between the number of events after L4 and L3 cuts can be used to calculate an upper limit to the ratio of gamma-ray-to-CR showers.
The remaining set of candidate gamma ray events is tested for a correlation with the Galactic Plane (Sec. \[sec:gptest\]) and the presence of point like sources (Sec. \[sec:pssearch\]). First, the results of simulations are presented, which provide several quantities needed for sensitivity calculations.
Simulation {#sec:mc}
----------
Although we determine the background from data only, simulations are needed to investigate systematic differences in the detector response to gamma-ray showers and cosmic-ray showers. More specifically, we are interested in the energy reconstruction of gamma-ray showers, the fraction of gamma-ray showers that is rejected by the muon veto system, and a possible difference in effective detector area between both types of showers.
Gamma-ray and proton showers are simulated with CORSIKA v6.900, using the interaction models FLUKA 2008 and SYBILL 2.1 for low and high energy hadronic interactions, respectively. For both primaries 10,000 showers are generated within an energy range of 800 TeV to 3 PeV with a $E^{-1}$ spectrum. Because the shower axes are required to pass through IceCube, the zenith angle is restricted to a maximum of 35 degrees. The observation altitude for IceTop is 2835 m. Atmospheric model MSIS-90-E is used, which is South Pole atmosphere for July 01, 1997. Seasonal variations in the event rate are of the order of 10% [@tilav09].
The detector simulation is done with the IceTray software package. Each simulated shower is fed into the detector simulation ten times with different core positions and azimuthal arrival direction, for a total of 100,000 events for both gamma rays and protons. This resampling of showers is a useful technique for increasing the statistics when examining quantities like the resolution of the energy reconstruction of IceTop. However, it cannot be used for quantities with large shower-to-shower variations, such as the number of high energy muons.
![\[fig:Ereco\] Ratio between reconstructed and true energy of simulated gamma-ray showers as a function of their true energy. At low energies the overestimation of the gamma-ray energy is largely due to a bias effect of the eight-station filter. At higher energies, this overestimation decreases. ](fig4.pdf){width="\linewidth"}
![\[fig:Ethresh\] Distribution of true energy of gamma-ray (red, solid) and proton (blue, dotted) showers for an energy cut at $E_{\mathrm{reco}}>1.4$ PeV, weighted to a $E^{-2.7}$ spectrum.](fig5.pdf){width="\linewidth"}
The energy of gamma-ray showers is overestimated by the reconstruction algorithm. Fig. \[fig:Ereco\] shows the distribution of the logarithm of the ratio between the reconstructed and true primary energy as function of true energy, weighted to a $E^{-2.7}$ spectrum. There are two reasons for the energy offset. First, there is a selection effect of the eight-station filter, which has a bias (below a few PeV) towards showers that produce relatively large signals in the IceTop tanks. This also affects the reconstructed energy of CR showers. At higher energies, the offset decreases but the reconstructed energy of gamma-ray showers is still slightly overestimated because the energy calibration of IceTop is performed with respect to proton showers.
Figure \[fig:Ethresh\] shows the distribution of true energies of gamma-ray and proton showers for the energy cut $E_{\mathrm{reco}}>1.4$ PeV (which will be adopted in Sec. \[sec:gptest\]). After this cut, 95% of the gamma-ray showers have a true energy above 1.2 PeV, while 95% of the proton showers have an energy above 1.3 PeV.
The fraction of gamma-ray showers that is falsely rejected because the showers contain muons that produce a signal in IceCube is found by applying the cuts to the MC simulation.
After applying the L3 cuts (defined in Sec. \[sec:gptest\]) to the simulated gamma-ray sample there are 737 events left in the sample, of which 121 produce a signal in IceCube. Taking into account an energy weighting of $E^{-2.7}$, this corresponds to 16%.
Showers that have no energetic muons can still be rejected if an unrelated signal is detected by IceCube. This could be caused by noise hits or unrelated muon tracks that fall inside the time window. This noise rate is determined directly from the data and leads to 14% signal loss. The total fraction of gamma-ray showers that is falsely rejected is therefore 28%.
Finally, because the composition, shower size and evolution of gamma-ray and CR showers are different, one might expect a difference in the number of triggered stations and the quality of the reconstruction, which could lead to different effective areas. Such an effect would be of importance when calculating the relative contribution of gamma-rays to the total received flux. We compare the effective area for gamma rays and protons by counting the number of events that are present at L3. We compensate for the energy reconstruction offset by reducing the reconstructed gamma-ray energy by a factor 1.16. The ratio of the effective area for gamma-rays to that for protons is then found to be 0.99.
It should be emphasized that we do not use the simulation to determine the number of muons and their energy distribution from CR showers. This would require a simulation set that includes heavier nuclei instead of protons only. Moreover, various hadronic interaction models generate significantly different muon fluxes [@pierog06]. Instead, this analysis estimates the rate of CR showers that do not trigger IceCube using the data itself.
Galactic Plane test {#sec:gptest}
-------------------
The IC40 data set consists of 368 days of combined IceCube and IceTop measurements. The data from August 2009 is used as a burn sample, which means that it is used to tune the parameters of the analysis. After this tuning the burn sample is discarded and the remaining data is used for the analysis.
IceCube is sensitive to gamma rays above 1 PeV. Earlier searches by CASA-MIA in a slightly lower energy range (100 TeV – 1 PeV) with better sensitivity have not established a correlation of gamma-rays with the Galactic Plane (see Fig. \[fig:sensgalp\]). For a Galactic diffuse flux below the CASA-MIA limit [@Chantell:1997gs] no gamma rays are expected in the IC40 burn sample. However, IceCube observes a different part of the Galactic Plane (see Fig. \[fig:FOV\]), close to the Galactic Center, so the possibility exists that previously undetected sources or local enhancements in CR and dust densities create an increase in the flux from that part of the sky.
In order to find a possible correlation of candidate events in our burn sample to the Galactic Plane, different sets of L3 cut parameters are applied to find a set that produces the most significant correlation. Afterwards, the cut parameters are fixed and the burn sample is discarded. The fixed cut parameters are then used in the analysis of the rest of the IC40 data set to test whether the correlation is still present. Note that these cuts are applied at L3, so they affect both the event samples after L3 and L4 cuts. This is important because the ratio between the number of events after L4 and L3 cuts is used to calculate the ratio between gamma ray and CR showers.
There are three cut parameters that are optimized by using the burn sample: $E_{\mathrm{min}}$, $C_{\mathrm{max}}$ and $d_{\mathrm{max}}$. This is done by scanning through all combination of parameters within the following range: 600 TeV $ \leq E_{\mathrm{min}} \leq $ 2 PeV with steps of 100 TeV, $0.5 \leq C_{\mathrm{max}} \leq 1.0$ with steps of 0.1, and 50 m $\leq d_{\mathrm{max}} \leq 90$ m with steps of 10 m. For each combination, the number of events $N_S$ in the burn sample after the L4 cut, located in the source region, is counted. The source region is defined as within 10 degrees of the Galactic Plane. Then, the data set is scrambled multiple times by randomizing the RA of each data point. For each scrambled data set the number of events in the source region is again counted. The best combination of cut parameters is the set which has the lowest fraction of scrambled data sets for which this number is equal to or higher than $N_S$.
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The result of the scan is given in the three panels of Fig. \[fig:optim\]. For each cut parameter the fraction of scrambled data sets that has a number of events in the source region equal to or exceeding the amount in the original data set is plotted for different cut values. For each plot the values of the other two cut parameters are kept constant at their optimal value. The actual search is done in three dimensions. The ratio is lowest for $E>1.4$ PeV, $C_{\mathrm{IT}}<0.8$ and $d_{\mathrm{str}}<60$ m. With this combination of cuts only 0.011% of the scrambled data sets produce an equal or higher number of events in the source region.
Note that while this procedure of optimizing cuts should be effective in the presence of sufficient signal, the small fraction obtained here and its erratic behavior with changing cut values are consistent with fluctuations of the background CR distribution alone, given the large number of possible cut combinations which were scanned. Nonetheless, the cut parameter values found with this procedure seem very reasonable (similar values are found with an alternative method, see Sec. \[sec:fullsensitivity\]).
{width="\linewidth"}
The optimized cuts are applied to the complete IC40 data set minus the burn sample. There are 268 candidate events of which 28 are located in the source region. Figure \[fig:skymap\] is a map of the sky showing all 268 events. The colors in the background indicate the integrated HI column densities, cf. Fig. \[fig:FOV\] (see discussion Sec. \[sec:principle\]). These are meant to guide the eye and are not part of the analysis.
The significance of the correlation with the Galactic Plane is tested by producing data sets with scrambled RA. An equal or higher number of source region events is found in 21% of the scrambled data sets, corresponding to a non-significant excess of +0.9$\sigma$.
We follow the procedure of Feldman & Cousins [@FC98] to construct an upper limit for the ratio of gamma rays to CRs. The background is determined by selecting a range of RA that does not contain the source region. Within this range the data points are scrambled multiple times and for each scrambled set the number of events in a pre-defined region of the same shape and size as the source region is counted. This yields a mean background of 24.13 events for the source region. Using a 90% confidence interval, the upper limit on the number of excess gamma rays from this region is 14.
Since 28% of gamma-ray showers are rejected by the veto from the buried detector, the maximum number of excess gamma-rays from the Galactic Plane is $14/0.72=19.4$. From Fig. \[fig:Ethresh\], it is known that the energy cut corresponds to a threshold of 1.2 PeV for gamma-rays, and 1.3 PeV for protons. Given that at L3 the sample is dominated by CR showers, and assuming a CR and gamma-ray power law of $\gamma=-2.7$, a 90% C.L. upper limit of $1.2\times 10^{-3}$ on the ratio of gamma-ray showers to CR showers in the source region can be derived in the energy range 1.2 – 6.0 PeV. The upper bound of 6.0 PeV is the value for which 90% of the events are inside the energy range. This value falls outside the range for which gamma-ray showers were simulated. However, there is no indication that the energy relation plotted in Fig. \[fig:Ereco\] behaves erratically above 3 PeV.
This is a limit on the average excess of the ratio of gamma-rays to CRs in the source region with respect to the rest of the sky, i.e. a limit on the Galactic component of the total gamma-ray flux. A possible isotropic component is not included. Systematic uncertainties lead to a 18% variation of the upper limit, as determined in Sec. \[sec:systerror\].
Unbinned point source search {#sec:pssearch}
----------------------------
An additional search for point-like sources tests the possibility that a single source dominates the PeV gamma-ray sky. This source does not necessarily lie close to the Galactic Plane. An unbinned point source search is performed on the sky within the declination range of $-85^{\circ}$ to $-60^{\circ}$, using a method that follows [@Braun08]. The region within 5 degrees around the zenith is omitted, because the method relies on scrambled data sets that are produced by randomizing the RA of the events. Close to the zenith this randomization scheme fails due to the small number of events.
The data is described by an unknown amount of signal events on top of a flat distribution of background events. In an unbinned search, a dense grid of points in the sky is scanned. For each point a maximum likelihood fit is performed for the relative contribution of source events over background events.
{width="\linewidth"}
For a particular event $i$ the probability density function (PDF) is given by $$P_i(n_S)=\frac{n_S}{N} S_i + \left( 1- \frac{n_S}{N} \right) B_i$$ where $n_S$ is the number of events that is associated to the source, $B_i$ is the background PDF, and $$S_i= \frac{1}{2 \pi \sigma^2} \exp\left(-\frac{\Delta \Psi^2}{2 \sigma^2}\right)$$ is the two-dimensional Gaussian source PDF, in which $\Delta\Psi$ is the space angle between the event and the source test location, and $\sigma=1.5^{\circ}$ is the angular resolution of IceTop. The background PDF $B_i$ is only dependent on the zenith angle, and is derived from the zenith distribution of the data. For each point in the sky there is a likelihood function $$L(n_S)=\Pi_i P_i(n_S),$$ and associated test statistic $$\lambda= -2 \left( \log(L(0)) - \log(L(n_S) \right),$$ which is maximized for $n_S$. In the optimization procedure, $n_S$ is allowed to have a negative value, which mathematically corresponds to a local flux deficit.
The procedure is similar to the search method for the neutrino point sources with IceCube [@icecube:ps], except that the source and background PDF do not contain an energy term. Because the range of energies in the event sample is relatively small (90% of the events have an energy between 1 and 6 PeV), an energy PDF is unlikely to improve the sensitivity.
![\[fig:hotspots\] Distribution of the largest value of $\lambda$ observed in each scrambled data set. The red dotted line indicated the value for $\lambda$ that corresponds to the hottest spot in the data.](fig9.pdf){width="\linewidth"}
Figure \[fig:pointsources\] displays a map of the sky with declination between $-85^{\circ}$ and $-60^{\circ}$ showing the events in this region and contours of the test statistic $\lambda$. The maximum value is $\lambda = 2.1$ at $\delta=-65.4^{\circ}$ and RA$=28.7^{\circ}$, corresponding to a fit of $n_s=3.5$ signal events. The overall significance of this value for $\lambda$ is found by producing $10,000$ scrambled data sets by randomizing the RA of each event. Figure \[fig:hotspots\] shows the distribution of $\lambda$ associated to the hottest spot in each scrambled data set. The median test statistic value for the hottest spot in the scrambled data sets is $\lambda=2.7$, so the actual data set is consistent with a flat background.
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Upper limits on the gamma-ray flux can be derived for each point in the sky by assuming that all events are gamma-rays. Since many events are in fact muon-poor or misreconstructed showers, this leads to a conservative upper limit. Because the acceptance of IC40 decreases as a function of declination (see Fig. \[fig:acceptance\]), the limit is more constraining at lower declination. Figure \[fig:pointsourceslimits\] is a sky map of the 90% C.L. upper limit in the energy range $E=1.2 - 6.0$ PeV for $E^{-2}$ source spectra. Point source fluxes are excluded at a level of $(E/\mathrm{TeV})^2 \mathrm{d}\Phi/\mathrm{d}E \sim 10^{-12} - 10^{-11}$ cm$^{-2}$s$^{-1}$TeV$^{-1}$ depending on source declination. Corrections for signal efficiency and detector noise are taken into account. Systematic uncertainties lead to a 18% variation of the upper limit.
Systematic Errors {#sec:systerror}
=================
Since this analysis derives the background from data, the systematic uncertainties due to the background estimation are small. The previously discussed cosmic-ray anisotropy measurement (see Sec. \[sec:principle\] and [@ic:anisotropy2]) is too small to have an impact on this analysis. Since there are no systematic gaps in detector uptime with respect to sidereal time-of-day in our sample, the coverage of RA is homogeneous.
Therefore, we focus on the systematic uncertainties in the signal efficiency, due to uncertainties in the surface detector sensitivity, and in the muon production rate for photon showers.
The uncertainty in the surface detector sensitivity is studied in Ref. [@IceCube:2012wn]; Table 2 there gives the uncertainties for hadronic showers as a function of shower energy and zenith angle. Although there are differences between hadronic and electromagnetic showers, most factors that contribute to this figure apply to both types of showers. Strongly contributing factors include atmospheric fluctuations, calibration stability and uncertainties in response of detector electronics (PMT saturation and droop). The contribution from the uncertainty in modeling the hadronic interaction is clearly different for electromagnetic showers, and is discussed below. For $E<10$ PeV, and zenith angle less than 30$^{\circ}$, there is a 6.0% systematic uncertainty in energy, and a 3.5% systematic uncertainty in flux. For an $E^{-2.7}$ spectrum, a 6.0% uncertainty in energy translates into a 17.0% uncertainty in flux, or, adding in quadrature, 17.4% flux uncertainty.
The uncertainty in the muon production from hadronic showers emerges from theoretical uncertainties. It depends on the hadronic photoproduction and electroproduction cross-sections for energies between 10 TeV and 6 PeV. Figure 1 of [@Couderc:2009tq] compares two cross-sections from two different photoproduction models, and finds (for water with a similar atomic number and mass number as air), a difference that rises from about 20% at 10 TeV to 60% at 1 PeV. The bulk of the particles in the shower are at lower energies, so we adopt a 20% uncertainty on the muon production rate via photoproduction. In addition, there is also a contribution of muon pair creation. To reach the in-ice DOMs, muons need at least 500 GeV. At 1 TeV, the fractional contribution of muon pair creation is $\sim 10$% [@Stanev85]. Since muon pair production is not included in SYBILL 2.1, we arrive at 30% uncertainty in total muon production rate. This uncertainty is applied to the 16% of photon showers that are lost because they contain muons for a final 4.8% uncertainty in sensitivity due to the unknown muon production cross-section.
We add the uncertainties due to detector response and muon production in quadrature, and arrive at an overall 18 % uncertainty in sensitivity.
IceCube 5-year sensitivity {#sec:fullsensitivity}
==========================
The sensitivity of the full IceCube detector to a gamma-ray flux from the Galactic Plane benefits from multiple improvements that can be made with respect to the analysis presented above. In this section we use preliminary data from the IC79 configuration (79 strings, 73 surface stations, 2010/2011) to estimate the sensitivity that the full IceCube detector can reach in 5 years. Since the full detector (IC86: 86 strings and 81 surface stations) is slightly larger than the IC79 configuration, the predicted sensitivity will be slightly underestimated. Also, the new cuts proposed below are not yet optimized, as this would require the actual IC86 data set.
Air shower reconstruction
-------------------------
This analysis is very sensitive to the quality of the core reconstruction. If the shower core is not reconstructed accurately, a muon bundle that passes outside the in-ice array might be incorrectly assumed to be aimed at the detector. Because of the absence of a signal in the in-ice DOMs, the event is then misinterpreted as a gamma-ray candidate. A more accurate core reconstruction algorithm has been developed for IceTop and will improve the CR rejection in post-IC40 analyses. In addition, the angular resolution of the larger array is improved, increasing the sensitivity to point sources.
Isolated hits
-------------
The SLC mode (which is available since IC59, see Sec. \[sec:detector\]) increases the sensitivity to CR showers with low muon content. A muon with just enough energy to reach IceCube, might not emit enough Cherenkov light to trigger multiple neighboring DOMs. By tightening the L4 cut so that no SLC hits are allowed to be present in the data, the efficiency with which CR showers can be rejected increases. At the same time, actual gamma-ray showers may be rejected in case of a noise hit in a single DOM. To keep this chance low, SLC hits only count as veto hits if they can be associated to the shower muon bundle both spatially and temporally.
{width="0.48\linewidth"} {width="0.48\linewidth"}
Figure \[fig:slchits\] shows the distribution of isolated hits in the complete detector as a function of time relative to the arrival time of the air shower as measured by IceTop. The plots show data at L3 level, applying the same cut values as in the IC40 analysis. The left plot shows the distribution of SLC hits for all events, while the right plot shows the same distribution but restricted to the subset of events which contain [*only*]{} SLC hits, i.e. events with no HLC hits. Hits associated with the muon bundle are seen throughout the detector, although the number of hits varies with depth because of variations in the optical properties of the ice due to naturally varying levels of contaminants such as dust, which attenuate Cherenkov photons The large number of isolated hits in the two bottom rows is an edge effect: the DOMs have fewer neighbors, so the chance for a hit to be isolated increases. In principle, the same effect could occur at the top two rows. However, the muon bundle deposits more energy in this region and the probability for any hit to have neighbor hits is larger here.
The muon-poor showers that produce no HLC hits (right-hand plot) can still cause some isolated hits in the top of the detector. These events can be removed with an additional cut on SLC hits. Because isolated hits can also be produced by noise, only a small area is selected in which SLC hits are used as a veto. A simple additional L4 cut is that all events are removed that have an SLC hit meeting the following three criteria:
- it is within 200 m from the reconstructed shower axis,
- it is within a time window of 4.8–7.5 $\mu$s after the shower arrival time, and
- it is in one of the six top layers of DOMs (spanning a vertical extend of 85 m).
Note that the lower bound of the time window (4.8 $\mu$s) corresponds to the time it takes for a muon traveling vertically to reach the top layer of in-ice DOMs starting from the surface. Muons from an inclined shower will arrive even later. The number of background events that are discarded in the L4 cut is increased by $\sim$30%, while the SLC noise rate in the data implies a decrease in signal efficiency of $\sim$5%.
With the completed detector it will be possible to optimize the SLC cuts further by making the time window dependent on zenith angle and DOM depth. The effect of this optimization was not yet studied here.
Re-optimization of cuts
-----------------------
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For the IC40 analysis the cut parameters were optimized to increase the detection probability of a possible correlation of gamma rays with the Galactic Plane. To increase the sensitivity of future searches with the completed IC86 configuration, the cut values were re-evaluated to increase the number of candidate events without losing background rejection power. This was achieved by evaluating the ratio between the number of events after L3 and L4 cuts.
While the L3 event sample is completely dominated by CRs, the L4 sample is a combination of possible gamma-ray showers, muon-poor CR showers and misreconstructed CR showers. The fraction of gamma-rays and muon-poor CRs in the detected events is independent from the cuts on geometry parameters $d_{\mathrm{str}}$ and $C_{\mathrm{IT}}$. The number of misreconstructed CR showers, on the other hand, will increase if the geometry cut values are chosen too loosely. Therefore, the ratio between the number of L4 and L3 events as a function of the cut parameter should be flat up to some maximum value after which it starts to increase. This maximum value is the preferred cut value since it maximizes the number of candidate events without lowering the background rejection power. It also maximizes the FOV, as looser geometry cuts imply a larger maximum zenith angle.
Figure \[fig:ratios\] shows the number of L3 (red) and L4 (blue) events together with their ratio (black dotted line; right-hand axis) as a function of the three main cut parameters (with the other cut parameters kept constant at their final value).
The rejection efficiency for $d_{\mathrm{str}}$ is fairly stable up to 60 m. The number of events rapidly decreases above this value, while the rejection becomes worse. In this case, the alternative method of optimization yields the same result as the method used in the IC40 analysis. For the containment size $C_{\mathrm{IT}}$ the ratio remains stable up to the edge of the array ($C_{\mathrm{IT}}=1$) after which it starts to rise. It appears the cut can be relaxed with respect to the IC40 analysis. In the following we will use $d_{\mathrm{str}}<60$ m and $C_{\mathrm{IT}}<1.0$.
The efficiency of the energy cut increases, as expected, with increasing energy, leveling off around $\sim$ 2.0 PeV. Since the total number of events falls off rapidly for increasing energy, the most sensitive region will be $\sim 2-3$ PeV. However, since the spectra of possible sources in this energy regime are unknown, it is not clear what energy cut would produce the optimal sensitivity. Instead, the sensitivity is calculated for ten energy bins in the range 1–10 PeV (see Fig. \[fig:sensgalp\]).
Increased acceptance
--------------------
![\[fig:acceptance\] Acceptance (effective area integrated over solid angle) for showers with an axis through both IceTop and IceCube for IC40, $C_{\mathrm{IT}}$=0.8 (black), and IC86, $C_{\mathrm{IT}}$=1.0 (blue).](fig13.pdf){width="\linewidth"}
With a larger array the acceptance, defined here as the effective area integrated over the solid angle of each 1$^{\circ}$ bin in zenith angle, increases considerably. Because of the condition that the shower axis has to be inside the instrumented area of both IceCube and IceTop, the increase is especially dramatic at larger zenith angles. Fig. \[fig:acceptance\] shows the acceptance for IC40 with $C_{\mathrm{IT}} < 0.8$ and the complete IC86 array with $C_{\mathrm{IT}} < 1.0$. Not only does the acceptance increase at large zenith angles, the range of possible zenith angles is also extended (to $\approx 45^{\circ}$). This extends the FOV to cover a larger part of the Galactic Plane and probe an area closer to the Galactic Center. The Galactic Center itself is still outside the FOV at $\delta \approx -29^{\circ}$, corresponding to a zenith angle of $61^{\circ}$.
Sensitivity
-----------
The sensitivity that can be reached with 5 years of data from the completed IceCube configuration can be estimated with preliminary data from IC79. It is assumed that the fraction of gamma-rays that are missed due to noise hits is the same as in the IC40 analysis. The full detector obviously has more noise hits, but this can be compensated by refining the in-ice cut by only allowing vetoes from DOMs that can be associated to the shower muon bundle in space and time (cf. the SLC cut described above). The sensitivity is calculated by producing scrambled data sets with randomized RA. Figure \[fig:sensgalp\] shows the 90% C.L. sensitivity to a diffuse flux from within 10 degrees of the Galactic Plane that can be achieved with 5 years of full detector data. The blue dashed line indicates the integrated limit between 1 and 10 PeV, while the blue dots indicate the sensitivity in six smaller energy bins. The upper limits found by CASA-MIA and IC40 (present work) are also included in the plot. The KASCADE [@KASCADE] results are not included since they set a limit on the all-sky gamma-ray flux.
Figure \[fig:sensps\] shows the sensitivity to point sources that is possible with 5 years of IceCube data. The sensitivity is a strong function of declination because the acceptance decreases at larger zenith angles. Point sources are expected to lie close to the Galactic Plane which reaches its lowest declination at $-63^{\circ}$. Within the IceCube field of view there are several PWNe and other gamma-ray sources detected by H.E.S.S. [@HESS], listed in Table \[table:sources\]. For these sources no significant cut-off was observed up to the maximum energy of 10 TeV, where statistics gets low. The blue dots indicate the flux that these sources would have at 1 PeV if their spectrum remains unchanged up to that energy. No correction for gamma-ray attenuation between the source and observer has been applied in this calculation. The extrapolation over two order of magnitude causes large uncertainties in the gamma-ray flux due to propagation of the errors on the spectral indices.
![\[fig:sensgalp\] Existing limits (red triangles for CASA-MIA and purple line for present IC40 analysis) and IceCube sensitivity to a diffuse gamma-ray flux from a region within 10 degrees from the Galactic Plane. The blue dashed line indicates the five year sensitivity of the completed detector, while the blue dots represent the sensitivity in smaller energy bins.](fig14.pdf){width="\linewidth"}
![\[fig:sensps\] IceCube 5 year sensitivity to point sources as a function of declination. The solid (dashed) black line indicates the sensitivity to an $E^{-2} (E^{-2.5}) $ flux. The dashed red line indicates the lowest declination reached by the Galactic Plane. The blue points indicate the flux at 1 PeV with extrapolated uncertainties of the sources listed in Table \[table:sources\] in the absence of a cut-off.](fig15.pdf){width="\linewidth"}
Source RA decl. Flux at 1 TeV (cm$^{-2}$s$^{-1}$TeV)$^{-1})$ $\Gamma$ Classification
--------------------- ---------------------- ------------------------------------------- ---------------------------------------------- ------------------------- ------------------------------
HESS J1356-645 $13^{h}56^{m}00^{s}$ $-64^{\circ}30^{\prime}00^{\prime\prime}$ $(2.7\pm0.9\pm0.4) \times 10^{-12}$ $ 2.2\pm 0.2 \pm 0.2$ PWN [@src:J1356-645]
HESS J1303-631 $13^{h}03^{m}00^{s}$ $-63^{\circ}11^{\prime}55^{\prime\prime}$ $(4.3\pm0.3) \times 10^{-12}$ $ 2.44\pm 0.05 \pm 0.2$ PWN [@src:J1303-631]
RCW 86 $14^{h}42^{m}43^{s}$ $-62^{\circ}28^{\prime}48^{\prime\prime}$ $(3.72\pm0.5\pm0.8) \times 10^{-12}$ $ 2.54\pm 0.12 \pm 0.2$ shell-type SNR [@src:RCW86]
HESS J1507-622 $15^{h}06^{m}53^{s}$ $-62^{\circ}21^{\prime}00^{\prime\prime}$ $(1.5\pm0.4\pm0.3) \times 10^{-12}$ $ 2.24\pm 0.16 \pm 0.2$ no ID [@src:J1507-622]
Kookaburra (Rabbit) $14^{h}18^{m}04^{s}$ $-60^{\circ}58^{\prime}31^{\prime\prime}$ $(2.64\pm0.2\pm0.53) \times 10^{-12}$ $ 2.22\pm 0.08\pm 0.1$ PWN [@src:kookaburra]
HESS J1427-608 $14^{h}27^{m}52^{s}$ $-60^{\circ}51^{\prime}00^{\prime\prime}$ $(1.3\pm0.4) \times 10^{-12}$ $ 2.2\pm 0.1 \pm 0.2$ no ID [@src:J1427-608]
Kookaburra (PWN) $14^{h}20^{m}09^{s}$ $-60^{\circ}45^{\prime}36^{\prime\prime}$ $(3.48\pm0.2\pm0.7) \times 10^{-12}$ $ 2.17\pm 0.06\pm 0.1$ PWN [@src:kookaburra]
MSH 15-52 $15^{h}14^{m}07^{s}$ $-59^{\circ}09^{\prime}27^{\prime\prime}$ $(5.7\pm0.2\pm1.4) \times 10^{-12}$ $ 2.27\pm 0.03 \pm 0.2$ PWN [@src:MSH15-52]
HESS J1503-582 $15^{h}03^{m}38^{s}$ $-58^{\circ}13^{\prime}45^{\prime\prime}$ $(1.6\pm0.6) \times 10^{-12}$ $ 2.4\pm 0.4\pm 0.2$ dark (FWV?) [@src:J1503-582]
HESS J1026-582 $10^{h}26^{m}38^{s}$ $-58^{\circ}12^{\prime}00^{\prime\prime}$ $(0.99\pm0.34) \times 10^{-12}$ $ 1.94\pm 0.2\pm 0.2$ PWN [@src:westerlund2]
Westerlund 2 $10^{h}23^{m}24^{s}$ $-57^{\circ}47^{\prime}24^{\prime\prime}$ $(3.25\pm0.5) \times 10^{-12}$ $ 2.58\pm 0.19 \pm 0.2$ MSC [@src:westerlund2]
Conclusions
===========
We have presented a new method of searching for high energy gamma-rays using the IceCube detector and its surface array IceTop. One year of data from IC40 was used to perform a search for point sources and a Galactic diffuse signal. No sources were found, resulting in a 90% C.L. upper limit on the ratio of gamma rays to cosmic rays of $1.2\times 10^{-3}$ for the flux coming from the Galactic Plane region ( $-80^\circ \lesssim l \lesssim -30^\circ; -10^\circ \lesssim b \lesssim 5^\circ$) in the energy range 1.2 – 6.0 PeV. Point source fluxes with $E^{-2}$ spectra have been excluded at a level of $(E/\mathrm{TeV})^2 \mathrm{d}\Phi/\mathrm{d}E \sim 10^{-12} - 10^{-11}$ cm$^{-2}$s$^{-1}$TeV$^{-1}$ depending on source declination. The full detector was shown to be much more sensitive, because of its larger size, improved reconstruction techniques and the possibility to record isolated hits.
This analysis offers interesting observation possibilities. IceCube can search for a diffuse Galactic gamma-ray flux with a sensitivity comparable to CASA-MIA, but at higher energies. This sensitivity is reached, however, by studying a much smaller part of the Galactic Plane than CASA-MIA. IceCube is therefore especially sensitive to localized sources, which might be Galactic accelerators or dense targets for extragalactic CRs.
The H.E.S.S. and CANGAROO-III [@CANGAROO] telescopes have found several high energy gamma-ray sources in IceCube’s FOV. Most of these sources are identified as or correlated with PWNe. Their energy spectrum has been measured up to a couple of tens of TeV. At this energy, statistics become low and for most sources no cut-off has been established. If these spectra extend to PeV energies without a break, IceCube will be able to detect them. It is also possible that an additional spectral component in the PeV energy range is present if a nearby dense molecular cloud acts as a target for the PWN beam [@Bednarek]. IceCube will be able to study these systems and place constraints on their behavior at very high energies, or possibly detect PeV gamma-rays for the first time.
We acknowledge the support from the following agencies: U.S. National Science Foundation-Office of Polar Programs, U.S. National Science Foundation-Physics Division, University of Wisconsin Alumni Research Foundation, the Grid Laboratory Of Wisconsin (GLOW) grid infrastructure at the University of Wisconsin - Madison, the Open Science Grid (OSG) grid infrastructure; U.S. Department of Energy, and National Energy Research Scientific Computing Center, the Louisiana Optical Network Initiative (LONI) grid computing resources; National Science and Engineering Research Council of Canada; Swedish Research Council, Swedish Polar Research Secretariat, Swedish National Infrastructure for Computing (SNIC), and Knut and Alice Wallenberg Foundation, Sweden; German Ministry for Education and Research (BMBF), Deutsche Forschungsgemeinschaft (DFG), Research Department of Plasmas with Complex Interactions (Bochum), Germany; Fund for Scientific Research (FNRS-FWO), FWO Odysseus programme, Flanders Institute to encourage scientific and technological research in industry (IWT), Belgian Federal Science Policy Office (Belspo); University of Oxford, United Kingdom; Marsden Fund, New Zealand; Australian Research Council; Japan Society for Promotion of Science (JSPS); the Swiss National Science Foundation (SNSF), Switzerland.
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[^1]: Physics Department, South Dakota School of Mines and Technology, Rapid City, SD 57701, USA
[^2]: corresponding author: `s.j.buitink@rug.nl`\
KVI, University of Groningen, Zernikelaan 25, 9747 AA Groningen, Netherlands
[^3]: Los Alamos National Laboratory, Los Alamos, NM 87545, USA
[^4]: also Sezione INFN, Dipartimento di Fisica, I-70126, Bari, Italy
[^5]: NASA Goddard Space Flight Center, Greenbelt, MD 20771, USA
|
---
abstract: 'In i-theory a typical layer of a hierarchical architecture consists of HW modules pooling the dot products of the inputs to the layer with the transformations of a few templates under a group. Such layers include as special cases the convolutional layers of Deep Convolutional Networks (DCNs) as well as the non-convolutional layers (when the group contains only the identity). Rectifying nonlinearities – which are used by present-day DCNs – are one of the several nonlinearities admitted by i-theory for the HW module. We discuss here the equivalence between group averages of linear combinations of rectifying nonlinearities and an associated kernel. This property implies that present-day DCNs can be exactly equivalent to a hierarchy of kernel machines with pooling and non-pooling layers. Finally, we describe a conjecture for theoretically understanding hierarchies of such modules. A main consequence of the conjecture is that hierarchies of trained HW modules minimize memory requirements while computing a selective and invariant representation.'
author:
- Fabio Anselmi
- Lorenzo Rosasco
- Cheston Tan
- Tomaso Poggio
bibliography:
- 'whydeepbib.bib'
title: Deep Convolutional Networks are Hierarchical Kernel Machines
---
[^1]
Introduction
============
The architectures now called Deep Learning Convolutional networks appeared with the name of convolutional neural networks in the 1990s – though the supervised optimization techniques used for training them have changed somewhat in the meantime. Such architectures have a history that goes back to the original Hubel and Wiesel proposal of a hierarchical architecture for the visual ventral cortex iterating in different layers the motif of simple and complex cells in V1. This idea led to a series of quantitative, convolutional cortical models from Fukushima ([@Fukushima1980]) to HMAX (Riesenhuber and Poggio, [@riesenhuber2000]). In later versions (Serre et al., [@serre2007]) such models of primate visual cortex have achieved object recognition performance at the level of rapid human categorization. More recently, deep learning convolutional networks trained with very large labeled datasets (Russakovsky et al. [@russakovsky2014], Google [@LastConvnets2014], Zeiler and Fergus [@Zeiler2013visualizing]) have achieved impressive performance in vision and speech classification tasks. The performance of these systems is ironically matched by our present ignorance of why they work as well as they do. Models are not enough. A theory is required for a satisfactory explanation and for showing the way towards further progress. This brief note outlines a framework towards the goal of a full theory.
Its organization is as follows. We first discuss how i-theory applies to existing DLCNs. We then show that linear combinations of rectification stages can be equivalent to kernels. Deep Learning Convolutional Networks can be similar to hierarchies of HBFs ([@Poggio89atheory]).
DCNs are hierarchies of kernel machines
=======================================
In this section, we review the basic computational units composing deep learning architectures of the convolution type. Then we establish some of their mathematical properties by using i-theory as described in [@anselmi2013unsupervised; @Anselmi2015; @Rosascokernels2015].
DCNs and i-theory
-----------------
The class of learning algorithms called deep learning, and in particular convolutional networks, are based on a basic operation in multiple layers. We describe it using the notation of i-theory.
The operation is the inner product of an input with another point called a template (or a filter, or a kernel), followed by a non linearity, followed by a group average. The output of the first two steps can be seen to roughly correspond to the neural response of a so called simple cell [@Hubel1962]. The collection of inner products of a given input with a template and its [*transformations*]{} in i-theory corresponds to a so called convolutional layer in DCNs. More precisely, given a template $t$ and its transformations $gt$, here $g\in \G$ is a finite set of transformations (in DCNs the only transformations presently used are translations), we have that each input $x$ is mapped to $\scal{x}{gt},
\quad g\in \G$. The values are hence processed via a non linear [*activation*]{} function, e.g. a sigmoid $(1+e^{-s})^{-1}$, or a rectifier $|s+b|_+=\max\{-b,s\}$ for $s,b\in
\R$ (the rectifier nonlinearity was called ramp by Breiman[@Breiman1991]). In summary, the first operation unit can be described, for example by $$x\mapsto |\scal{x}{gt}+b|_+.$$
The last step, often called [*pooling*]{}, aggregates in a single output the values of the different inner products previously computed that correspond to transformations of the same template, for example via a sum $$\label{convlayer}
\sum_g |\scal{x}{gt}+b|_+, \quad t\in \TT, b\in \R$$ or a max operation
$$\label{convlayer2}
\max_g |\scal{x}{gt}+b|_+, \quad t\in \TT, b\in \R$$
This corresponds to the neural response of a so called complex cell in [@Hubel1962].
A HW module is a kernel machine
-------------------------------
It is trivial that almost any (positive or negative defined) nonlinearity after the dot product in a network yields a kernel. Consider a 3-layers network with the first layer being the input layer ${x}$. Unit $i$ in the second layer (comprising $N$ units) computes $|\scal{t_i}{x} + b_i|_+ = \phi_i(x),
i=1,\cdots,N$. Thus each unit in the third layer performing a dot product of the vector of activities from the second layer with weights $\phi_i(y)$ (from layer two to three) computes $K(x,y) = \sum_j
\phi_j(x)\phi_j(y)$ which is guaranteed to be well defined because the sum is finite. Key steps of the formal proof, which holds also in the general case, can be found in Rosasco and Poggio, [@Rosascokernels2015] and references there; see also Appendix \[summaryISK\] and Appendix \[ExampleKernel\] for an example.\
A different argument shows that it is “easy” to obtain kernels in layered networks. Assume that inputs $x \in \R^n$ as well as the weights $t$ are normalized, that is $x,t \in S^n$ where $S^n$ is the unit sphere.\
In this case dot products are radial functions (since $\scal{x}{t} =
\frac{1}{2} (2 -(|x-t|^2))$) of $r^2$. The kernel $r^2$ can be shaped by linear combinations (with bias) of rectifier units (linear combinations of ramps can generate sigmoidal units, linear combinations of which are equivalent to quasi-Gaussian, “triangular” functions) [^2]
Thus for normalized inputs dot products with nonlinearities can easily be equivalent to radial kernels. Pooling before a similarity operation (thus pooling at layer $n$ in a DCN before dot products in layer $n+1$) maintains the kernel structure (since $\widetilde K
(x,x')=\int dg \int dg' K(gx,g'x')$ is a kernel if $K$ is a kernel). Thus [*DCNs with normalized inputs are hierarchies of radial kernel machines, also called Radial Basis Functions (RBFs)*]{}.
Kernels induced by linear combinations of features such as
$$\label{convlayer}
\sum_g |\scal{x}{gt}+b|_+, \quad t\in \TT, b\in \R$$
are selective and invariant (if $G$ is compact, see Appendix \[summaryISK\]). The max operation $\max_g |\scal{x}{gt}+b|_+$ has a different form and does not satisfy the condition of the theorems in the Appendix. On the other hand the pooling defined in terms of the following soft-max operation (which approximates the max for “large” $n$)
$$\label{softmax2}
\sum_g \frac{(\scal{x}{gt})^n}{\sum_{g'} (1+\scal{x}{g't})^{n-1}}, \quad t\in \TT\,$$
induces a kernel that satisfies the conditions of the theorems summarized in Appendix \[summaryISK\]. It is well known that such an operation can be implemented by simple circuits of the lateral inhibition type (see [@kouh2008canonical]). On the other hand the pooling in Appendix \[Mex\] (see [@CohenS14]) does not correspond in general to a kernel, does not satisfy the conditions of Appendix \[summaryISK\] and is not guaranteed to be selective.
Since weights and “centers” of the RBFs are learned in a supervised way, the kernel machines should be more properly called HyperBF, see Appendix \[HBF\].
![[*A “Gaussian” of one variable can be written as the linear combination of ramps (e.g. rectifiers): a) a sigmoid-like function can be written as linear combinations of ramps b) linear combinations of sigmoids give gaussian-like triangular functions.*]{}\[equivalence\]](equivalence.pdf){width="60.00000%"}
Summary: every layer of a DCN is a kernel machine
-------------------------------------------------
Layers of a Deep Convolutional Network using linear rectifiers (ramps) can be described as
$$\label{rectlayer}
\sum_g |\scal{x}{gt}+b|_+, \quad t\in \TT, b\in \R$$
where the range of pooling ($\sum_g$) may be degenerate (no pooling) in which case the layer is not a convolutional layer.
Such a layer corresponds to the kernel
$$\label{kernel}
\tilde{K}(x,x')=\int\;dg\;dg'\;K_{0}(x,g,g').$$
with
$$\label{kernel}
K_{0}(x,g,g')=\int\;dt\;db\;|\scal{gt}{x}+b|_{+}|\scal{g't}{x'}+b|_{+}.$$
An alternative pooling to \[rectlayer\] is provided by softmax pooling
$$\label{softlayer}
\sum_g \frac{(\scal{x}{gt})^n}{\sum_{g'} (1+\scal{x}{g't})^{n-1}}, \quad t\in \TT\,$$
Present-day DCNs seem to use equation \[softlayer\] for the pooling layers and equation \[rectlayer\] for the non-convolutional layers. The latter is the degenerate case of equation \[softlayer\] (when $G$ contains only the identity element).
A conjecture on what a hierarchy does
=====================================
The previous sections describe an extension of i-theory that can be applied exactly to any layer of a DLCN and any of the nonlinearities that have been used: pooling, linear rectifier, sigmoidal units. In this section we suggest a framework for understanding hierarchies of such modules, which we hope may lead to new formal or empirical results.
![[*A hierarchical, supervised architecture built from eHW-modules. Each red circle represents the signature vector computed by the associated module (the outputs of complex cells) and double arrows represent its receptive fields – the part of the (neural) image visible to the module (for translations this is also the pooling range). The “image” is at level $0$, at the bottom.*]{}\[Figure1\]](fig1.pdf){width="70.00000%"}
DLCNs and HVQ {#DLCNsHVQ}
-------------
We consider here an HW module with pooling (no pooling corresponds to 1 pixel stride convolutional layers in DLN). Under the assumption of normalized inputs, this is equivalent to a HBF module with Gaussian-like radial kernel and “movable” centers (we assume standard Euclidean distance, see Poggio, Girosi,[@Poggio1990b]; Jones, Girosi and Poggio,[@GirJonPog95]).
Notice that one-hidden-layer HBF can be much more efficient in storage (e.g. bits used for all the centers) than classical RBF because of the smaller number of centers (HBFs are similar to a multidimensional free-knots spline whereas RBFs correspond to classical spline).
The next step in the argument is the observation that a network of radial Gaussian-like units become in the limit of $\sigma \to 0$ a look-up table with entries corresponding to the centers. The network can be described in terms of [*soft Vector Quantization*]{} (VQ) (see section 6.3 in Poggio and Girosi, [@Poggio89atheory]). Notice that hierarchical VQ (dubbed HVQ) can be even more efficient than VQ in terms of storage requirements (see e.g. [@HVQ]). This suggests that a hierarchy of HBF layers may be similar (depending on which weights are determined by learning) to HVQ. Note that [*compression is achieved when parts can be reused in higher level layers as in convolutional networks*]{}. Notice that the center of one unit at level $n$ of the “convolutional” hierarchy of Figure \[Figure1\] is a combinations of parts provided by each of the lower units feeding in it. This may even happen without convolution and pooling as shown in the following extreme example.\
\
[**Example**]{} Consider the case of kernels that are in the limit delta-like functions (such as Gaussian with very small variance). Suppose as in Figure \[Example\] that there are four possible quantizations of the input $x$: $x_1, x_2, x_3, x_4$. One hidden layer would consist of four units $\delta(x-x_i), i=1,\cdots,4$. But suppose that the vectors $x_1, x_2, x_3,x_4$ can be decomposed in terms of two smaller parts or features $x'$ and $x"$, e.g. $x_1 =x'\oplus x"$, $x_2=x'\oplus x'$, $x_3=x"\oplus x"$ and $x_4=x"\oplus x'$. Then a two layer network could have two types of units in the first layer $\delta(x-x')$ and $\delta(x-x")$; in the second layer four units will detect the conjunctions of $x'$ and $x"$ corresponding to $x_1, x_2, x_3,x_4$. The memory requirements will go from $4N$ to $2N/2+8$ where $N$ is the length of the quantized vectors; the latter is much smaller for large $N$. Memory compression for HVQ vs VQ – that is for multilayer networks vs one-layer networks – increases with the number of (reusable) parts. Thus for problems that are [ *compositional*]{}, such as text and images, hierarchical architectures of HBF modules minimize memory requirements.
![[*See text, Example in section \[DLCNsHVQ\]*]{}.\[Example\]](compositionality.pdf){width="110.00000%"}
Classical theorems (see refrences in [@GirPog-Kol89; @GirPog-best90] show that one hidden layer networks can approximate arbitrarily well rather general classes of functions. A possible advantage of multilayer vs one-layer networks that emerges from the analysis of this paper is memory efficiency which can be critical for large data sets and is related to generalization rates.
Remarks
=======
- Throughout this note, we discuss the potential properties of multilayer networks, that is the properties they have with the “appropriate” sets of weights when supervised training is involved. The assumption is therefore that greedy SGD using very large sets of labeled data, can find the “appropriate sets of sets of weights".
- Recently, several authors have expressed surprise when observing that the last hidden unit layer contains information about tasks different from the training one (e.g. [@Yamins2014]). This is in fact to be expected. The last layer of HBF is rather independent of the training target and mostly depends on the input part of the training set (see theory and gradient descent equations in Poggio and Girosi, [@Poggio89atheory] for the one-hidden layer case). This is exactly true for one-hidden layer RBF networks and holds approximatively for HBFs. The weights from the last hidden layer to the output are instead task/target dependent.
- The result that linear combinations of rectifiers can be equivalent to kernels is [*robust*]{} in the sense that it is true for several different nonlinearities such as rectifiers, sigmoids etc. Ramps (e.g. rectifiers) are the most basic ones. [*Such robustness is especially attractive for neuroscience.*]{}
Appendices
==========
HW modules are equivalent to kernel machines (a summary of the results in [@Rosascokernels2015]) {#summaryISK}
------------------------------------------------------------------------------------------------
In the following we summarize the key passages of [@Rosascokernels2015] in proving that HW modules are kernel machines:
1. The feature map $$\phi(x,t,b) = |\scal{t}{x}+b|_{+}$$ (that can be associated to the output of a simple cell, or the basic computational unit of a deep learning architecture) can also be seen as a kernel in itself. The kernel can be a universal kernel. In fact, under the hypothesis of normalization of the vectors $x,t$ we have that $2(|1-\scal{t}{x}|_{+}+|\scal{t}{x}-1|_{+})=2|1-\scal{t}{x}|=\nor{x-t}_{2}^{2}$ which is a universal kernel (see also th 17 of [@micchelli2006]).\
The feature $\phi$ leads to a kernel $$K_{0}(x,x') = \phi^{T}(x)\phi(x') = \int\;db\;dt\;|\scal{t}{x}+b|_{+}|\scal{t}{x'}+b|_{+}$$ which is a universal kernel being a kernel mean embedding (w.r.t. $t,b$, see [@SrGrFu10]) of the a product of universal kernels.
2. If we explicitly introduce a group of transformations acting on the feature map input i.e. $\phi(x,g,t,b) = |\scal{gt}{x}+b|_{+}$ the associated kernel can be written as $$\tilde{K}(x,x')=\int\;dg\;dg'\int\;dt\;db\;|\scal{gt}{x}+b|_{+}|\scal{g't}{x'}+b|_{+}=\int\;dg\;dg'\;K_{0}(x,g,g').$$ $\tilde{K}(x,x')$ is the group average of $K_{0}$ (see [@burkhardt2007]) and can be seen as the mean kernel embedding of $K_{0}$ (w.r.t. $g,g'$, see [@SrGrFu10]).
3. The kernel $\tilde{K}$ is invariant and, if $G$ is compact, selective i.e. $$\tilde{K}(x,x')=1\;\Leftrightarrow\;x\sim x'.$$ The invariance follows from the fact that any $G-$group average function is invariant to $G$ transformations. Selectivity follows from the fact that $\tilde{K}$ a universal kernel being a kernel mean embedding of $K_{0}$ which is a universal kernel (see [@SrGrFu10]).
If the distribution of the templates $t$ follows a gaussian law the kernel $K_{0}$, with an opportune change of variable, can be seen as a particular case of the $n$th order arc-cosine kernel in [@cho2009] for $n=1$.
An example of an explicit calculation for an inner product kernel {#ExampleKernel}
-----------------------------------------------------------------
Here we note how a simple similarity measure between functions of the form in eq. correspond to a kernel in the case when the inner product between two HW module outputs at the first layer, say $\mu(I),\mu(I')$, is calculated using a step function nonlinearity. Note first that a heaviside step function can be approximated by a sigmoid like function derived by a linear combination of rectifiers of the form: $$H(x)\sim \alpha(|x|_{+}-|x-\frac{1}{\alpha}|_{+})$$ for very large values of $\alpha$. With this specific choice of the nonlinearity we have that the inner product of the HW modules outputs (for fixed $t$) at the first layer is given by: $$\scal{\mu(I)}{\mu(I')} = \int \int dg\;dg'( \int db H(b-\scal{I}{gt})H(b-\scal{I'}{g't})).$$ with $$\mu^{t}_{b}(I)=\int\;dg\;H(b-\scal{I}{gt}).$$ Assuming that the scalar products, $\scal{I}{gt},\scal{I'}{g't}$, range in the interval $[-p,p]$ a direct computation of the integral above by parts shows that, $x=\scal{I}{gt}, x'=\scal{I'}{gt}$: $$\begin{aligned}
& \int_{-p}^{p} db H(b-x)H(b-x')\\
&= H(b-x)\big((b-x')H(b-x')-(x-x')H(x-x')\big)\rvert_{-p}^{p}\\
&= p-\frac{1}{2}(x+x'+|x-x'|)\end{aligned}$$ and being $$\begin{aligned}
\max\{x, x'\} &= \max\left\{x - \frac{1}{2}(x+x'), x' - \frac{1}{2}(x+x')\right\} + \frac{1}{2}(x+x')\\
&= \max\left\{\frac{1}{2}(x'-x), \frac{1}{2}(x - x')\right\} + \frac{1}{2}(x+x')\\
&= \max\left\{-\frac{1}{2}(x-x'), \frac{1}{2}(x-x')\right\} + \frac{1}{2}(x+x')\\
&= +\left|\frac{1}{2}(x-x')\right| + \frac{1}{2}(x+x')\\
&= \frac{1}{2}(x+x'+|x-x'|).\end{aligned}$$ we showed that $$K(\scal{I'}{t}, \scal{I}{t}) = C-\max(\scal{I'}{t},\scal{I}{t})$$ which defines a kernel. If we include the pooling over the transformations and templates we have $$\label{kernelembedding}
\tilde{K}(I,I') = p-\int\;d\lambda(t)\;dg\;dg'\;\max(\scal{I'}{gt},\scal{I}{g't}).$$ where $\lambda(t)$ is a probability measure on the templates $t$. $\tilde{K}$ is again a kernel.\
A similar calculation can be repeated at the successive layer leading to kernels of kernels structure.
Mex {#Mex}
---
Mex is a generalization of the pooling function. From [@CohenS14] eq. 1 it is defined as: $$\label{Mexeq}
Mex_{(\{c_{i}\},\xi)}=\frac{1}{\xi}\log\Big(\frac{1}{n}\sum_{i=1}^{n}\exp(\xi c_{i})\Big)$$ We have $$\begin{aligned}
&& Mex_{(\{c_{i}\},\xi)}\xrightarrow[\xi \to \infty]{} Max_{i}{(c_{i})}\\
&& Mex_{(\{c_{i}\},\xi)}\xrightarrow[\xi \to 0]{} Mean_{i}{(c_{i})}\\
&& Mex_{(\{c_{i}\},\xi)}\xrightarrow[\xi \to -\infty]{} Min_{i}{(c_{i})}.\end{aligned}$$ We can also choose values of $\xi$ in between the ones above, the interpretation is less obvious. The Mex pooling does not define a kernel since is not positive definite in general (see also Th 1 in [@CohenS14])
Hyper Basis Functions: minimizing memory in Radial Basis Function networks {#HBF}
--------------------------------------------------------------------------
We summarize here an old extension by Poggio and Girosi [@PoGir94extension] of the classical kernel networks called Radial Basis Functions (RBF). In summary (but see the paper) they extended the theory by defining a general form of these networks which they call Hyper Basis Functions. They have two sets of modifiable parameters: [*moving centers*]{} and [*adjustable norm-weights*]{}. Moving the centers is equivalent to task-dependent clustering and changing the norm weights is equivalent to task-dependent dimensionality reduction.\
A classical RBF has the form $$\label{eq:green-rad}
f({\bf x}) = \sum_{i = 1}^N c_i K(\| {\bf x} -
{\bf x}_i \|^2),$$ which is a sum of radial functions, each with its [*center*]{} ${\bf x}_i$ on a distinct data point. Thus the number of radial functions, and corresponding centers, is the same as the number of examples. Eq. is a minimizer solution of $$\label{H}
H[f] = \sum_{i=1}^{N} (y_{i}-f(x_{i}))^{2} + \lambda\nor{Pf}^{2}\;\;\lambda\in \R^{+}$$ where $P$ is a constrain operator (usually a differential operator).\
HBF extend RBF in two directions:
1. The computation of a solution of the form has a complexity (number of radial functions) that is independent of the dimensionality of the input space but is on the order of the dimensionality of the training set (number of examples), which can be very high. Poggio and Girosi showed how to justify an approximation of equation in which the number of centers is much smaller than the number of examples and the positions of the centers are modified during learning. The key idea is to consider a specific form of an approximation to the solution of the standard regularization problem.
2. Moving centers are equivalent to the free knots of nonlinear splines. In the context of networks they were first suggested as a potentially useful heuristics by Broomhead and Lowe [@BroLow88] and used by Moody and Darken [@MooDar89].
Poggio and Girosi called [*Hyper Basis Functions*]{}, in short [*HyperBFs*]{}, the most general form of regularization networks based on these extensions plus the use of a weighted norm.
### Moving Centers
The solution given by standard regularization theory to the approximation problem can be very expensive in computational terms when the number of examples is very high. The computation of the coefficients of the expansion can become then a very time consuming operation: its complexity grows polynomially with $N$, (roughly as $N^3$) since an $N\times N$ matrix has to be inverted. In addition, the probability of ill-conditioning is higher for larger and larger matrices (it grows like $N^3$ for a $N \times N$ uniformly distributed random matrix) [@Demmel87]. The way suggested by Poggio and Girosi to reduce the complexity of the problem is as follows. While the exact regularization solution is equivalent to generalized splines with [ *fixed*]{} knots, the approximated solution is equivalent to generalized splines with [*free*]{} knots.
A standard technique, sometimes known as Galerkin’s method, that has been used to find approximate solutions of variational problems, is to expand the solution on a finite basis. The approximated solution $f^*({\bf x})$ has then the following form: $$f^*({\bf x}) = \sum_{i = 1}^n c_{i} \phi_{i}({\bf x})
\label{eq:galerkin}$$ where $\{ \phi_{i} \}_{i = 1}^n$ is a set of linearly independent functions [@Mikhlin65]. The coefficients $c_{i}$ are usually found according to some rule that guarantees a minimum deviation from the true solution. A natural approximation to the exact solution will be then of the form:
$$f^*({\bf x}) = \sum_{\alpha = 1}^n c_{\alpha} G({\bf x} ; {\bf
t}_{\alpha})
\label{eq:GRBF}$$
where the parameters ${\bf t}_{\alpha}$, that we call “centers”, and the coefficients $c_{\alpha}$ are unknown, and are in general fewer than the data points ($n \leq N$). This form of solution has the desirable property of being an universal approximator for continuous functions [@Poggio89atheory] and to be the only choice that guarantees that in the case of $n = N$ and $\{ {\bf t}_{\alpha}\}_{\alpha = 1}^n = \{
{\bf x}_i\}_{i = 1}^n$ the correct solution (of equation ) is consistently recovered. We will see later how to find the unknown parameters of this expansion.
### How to learn centers’ positions
Suppose that we look for an approximated solution of the regularization problem of the form $$\label{eq:GRBF-W}
f^*({\bf x}) = \sum_{\alpha = 1}^n c_{\alpha} G(\nor{{\bf x}- {\bf
t}_{\alpha}}^{2})$$ We now have the problem of finding the $n$ coefficients $c_{\alpha}$, the $d
\times ~n$ coordinates of the centers ${\bf t}_{\alpha}$. We can use the natural definition of optimality given by the functional $H$. We then impose the condition that the set $\{ c_{\alpha}, {\bf
t}_{\alpha} | \alpha = 1, ..., n \}$ must be such that they minimizes $H[f^*]$, and the following equations must be satisfied: $${\partial H[f^*] \over \partial c_{\alpha}} = 0~,~~~~~
{\partial H[f^*] \over \partial {\bf t}_{\alpha}} = 0, ~~~
\alpha = 1, ..., n~.$$ Gradient-descent is probably the simplest approach for attempting to find the solution to this problem, though, of course, it is not guaranteed to converge. Several other iterative methods, such as versions of conjugate gradient and simulated annealing [@KirGelVec83] may be more efficient than gradient descent and should be used in practice. Since the function $H[f^*]$ to minimize is in general non-convex, a stochastic term in the gradient descent equations may be advisable to avoid local minima. In the stochastic gradient descent method the values of $c_{\alpha}$, ${\bf
t}_{\alpha}$ and $\bf M$ that minimize $H[f^*]$ are regarded as the coordinates of the stable fixed point of the following stochastic dynamical system: $$\dot c_\alpha = - \omega
{\partial H[f^*] \over \partial c_\alpha} + \eta_\alpha (t), ~~\alpha = 1, \dots , n$$ $$\dot {\bf t}_\alpha = - \omega
{\partial H[f^*] \over \partial {\bf t}_\alpha} + \mbox{\boldmath $\mu$}_\alpha (t), ~~\alpha = 1, \dots , n$$ where $\eta_\alpha (t)$, $\mbox{\boldmath $\mu$}_\alpha (t)$ are white noise of zero mean and $\omega$ is a parameter determining the microscopic timescale of the problem and is related to the rate of convergence to the fixed point. Defining $$\Delta_i \equiv y_i - f^*({\bf x}) =
y_i - \sum_{\alpha = 1}^n c_\alpha G(\| {\bf x}_i -
{\bf t}_\alpha \|^2)$$ we obtain $$H[f^*] = H_{{\bf c}, {\bf t}}= \sum_{i=1}^{N} (\Delta_i)^2.$$ The important quantities – that can be used in more efficient schemes than gradient descent – are
- for the $c_\alpha$ $$\label{eq:grad-c}
{{\partial H[f^*]} \over {\partial c_\alpha}} = - 2 \sum_{i = 1}^N
\Delta_i G(\| {\bf x}_i - {\bf t}_\alpha \|^2)~~;$$
- for the centers $t_\alpha$ $$\label{eq:grad-t}
{{\partial H[f^*]} \over {\partial {\bf t}_\alpha}} = 4 c_\alpha
\sum_{i = 1}^N \Delta_i G'(\| {\bf x}_i - {\bf
t}_\alpha \|^2)({\bf x}_i - {\bf
t}_\alpha)$$
[**Remarks**]{}
1. Equation has a simple interpretation: the correction is equal to the sum over the examples of the products between the error on that example and the “activity” of the “unit” that represents with its center that example. Notice that $H[f^*]$ is quadratic in the coefficients $c_\alpha$, and if the centers are kept fixed, it can be shown [@Poggio89atheory] that the optimal coefficients are given by $$\label{eq:pseudo}
{\bf c} = (G^T~G + \lambda g)^{-1} G^T {\bf y}$$ where we have defined $({\bf y})_i = y_i$, $({\bf
c})_\alpha = c_\alpha$, $(G)_{i \alpha} = G( {\bf x}_i ; {\bf
t}_\alpha)$ and $(g)_{\alpha \beta} = G({\bf t}_\alpha ; {\bf
t}_{\beta})$. If $\lambda$ is let go to zero, the matrix on the right side of equation converges to the pseudo-inverse of $G$ [@Albert72] and if the Green’s function is radial the approximation method of [@BroLow88] is recovered.
2. Equation is similar to task-dependent clustering [@Poggio89atheory]. This can be best seen by assuming that $\Delta_i$ are constant: then the gradient descent updating rule makes the centers move as a function of the majority of the data, that is of the position of the clusters. In this case a technique similar to the k-means algorithm is recovered, [@MacQueen67; @MooDar89]. Equating ${\partial H[f^*]} \over
{\partial {\bf t}_\alpha}$ to zero we notice that the optimal centers ${\bf t}_\alpha$ satisfy the following set of nonlinear equations: $${\bf t}_\alpha = {{\sum_i P_i^\alpha {\bf x}_i} \over
{\sum_i P_i^\alpha}} ~~~\alpha = 1, \dots , n$$ where $P_i^\alpha = \Delta_i G'(\| {\bf x}_i -
{\bf t}_\alpha \|^2)$. The optimal centers are then a weighted sum of the data points. The weight $P_i^\alpha$ of the data point $i$ for a given center ${\bf t}_\alpha$ is high if the interpolation error $\Delta_i$ is high there [*and*]{} the radial basis function centered on that knot changes quickly in a neighborhood of the data point. This observation suggests faster update schemes, in which a suboptimal position of the centers is first found and then the $c_\alpha$ are determined, similarly to the algorithm developed and tested successfully by Moody and Darken [@MooDar89].
### An algorithm
It seems natural to try to find a reasonable initial value for the parameters ${\bf c}, {\bf t}_\alpha$, to start the stochastic minimization process. In the absence of more specific prior information the following heuristics seems reasonable.
- Set the number of centers and set the centers’ positions to positions suggested by cluster analysis of the data (or more simply to a subset of the examples’ positions).
- Use matrix pseudo-inversion to find the $c_\alpha$.
- Use the ${\bf t}_\alpha$, and $c_\alpha$ found so far as initial values for the stochastic gradient descent equations.
Experiments with movable centers and movable weights have been performed in the context of object recognition (Poggio and Edelman, [@PogEde90]; Edelman and Poggio, [@EdePog90]) and approximation of multivariate functions.
### Remarks
1. Equation is similar to a clustering process.
2. In the case of $N$ examples, $n=N$ fixed centers, there are enough data to constrain the $N$ coefficients $c_\alpha$ to be found. Moving centers add another $n d$ parameters ($d$ is the number of input components). Thus the number of examples $N$ must be sufficiently large to constrain adequately the free parameters – $n$ d-dimensional centers, $n$ coefficients $c_\alpha$. Thus $$N>>n + nd.$$
Acknowledgment {#acknowledgment .unnumbered}
==============
This work was supported by the Center for Brains, Minds and Machines (CBMM), funded by NSF STC award CCF – 1231216. This work was also supported by A\*STAR JCO VIP grant $\#$1335h00098. Part of the work was done in Singapore at the Institute for Infocomm under REVIVE funding. TP thanks A\*Star for its hospitality.
[^1]: Email addresses: anselmi@mit.edu; lrosasco@mit.edu; cheston-tan@i2r.a-star.edu.sg; corresponding author: tp@ai.mit.edu. The main part of this work was done at the Institute for Infocomm Research with funding from REVIVE
[^2]: Notice that in one dimension the kernel $|x-y|$ can be written in terms of ramp functions as $|x-y|
=|x-y|_+ + |-(x-y)|_+$. See Figure \[equivalence\] and [@GirJonPog95].
|
---
author:
- Francesco Gianoli
- Thomas Risler
- 'Andrei S. Kozlov'
title: 'Lipid bilayer mediates ion-channel cooperativity in a model of hair-cell mechanotransduction'
---
echanoelectrical transduction (MET) in the inner ear occurs when mechanical forces deflect the stereocilia of hair cells, changing the open probability of mechanosensitive ion channels located in the stereociliary membrane [@hudspeth_how_1989; @peng_integrating_2011; @fettiplace_physiology_2014]. Channel gating (opening and closing) and stereocilia motion are directly coupled by tip links, extracellular filaments that connect the tip of each stereocilium to the side of its taller neighbor [@pickles_cross-links_1984]. Tip links act as molecular springs, whose tension determines the channels’ open probability. Reciprocally, channel gating affects tension in the tip links and consequently the stiffness of the whole hair bundle. This phenomenon, known as gating compliance [@howard_compliance_1988], is a key feature of hair-cell mechanics and contributes to the auditory system’s high sensitivity and sharp frequency tuning [@fettiplace_physiology_2014; @hudspeth_putting_2000]. Its mechanism, however, remains unclear. The classical model of mechanotransduction ascribes gating compliance to the gating swing, a change in the extension of the tip link due to the conformational rearrangement of a single MET channel upon gating [@howard_compliance_1988; @markin_gating-spring_1995]. To reproduce experimental data theoretically, however, the amplitude of the gating swing must be comparable to, or even greater than, the size of a typical ion channel [@martin_negative_2000; @van_netten_channel_2003; @tinevez_unifying_2007; @sul_gating_2010]. This requirement constitutes an issue that is often acknowledged [@peng_integrating_2011; @fettiplace_physiology_2014; @albert_comparative_2016] but still unresolved.
{width="\textwidth"}
The classical model posits a single MET channel connected to the tip link’s upper end, near myosin motors that regulate tip-link tension [@hudspeth_making_2008]. Electrophysiological recordings, however, point to two channels per tip link [@denk_calcium_1995; @ricci_tonotopic_2003; @beurg_large-conductance_2006; @beurg_localization_2009], which is in accord with its dimeric structure [@kazmierczak_cadherin_2007; @kachar_high-resolution_2000]. Furthermore, high-speed Ca^2+^ imaging shows that the channels are located at the tip link’s lower end [@beurg_localization_2009]. This result has been corroborated by the expression patterns of key mechanotransduction proteins within the hair bundle, which interact with protocadherin-15, the protein constituting the lower end of the tip link (reviewed in ref. [@zhao_elusive_2015]). Together, these findings turned textbook views of molecular mechanotransduction in the inner ear literally upside-down [@kandel_principles_2013; @spinelli_bottoms_2009]. Moreover, experimental data suggest that the lipid bilayer surrounding the channels modulates their open probability as well as the rates of slow and fast adaptation [@hirono_hair_2004; @peng_adaptation_2016], although it is not clear how.
In this work, we propose and explore a quantitative model of hair-cell mechanotransduction that incorporates the main pieces of evidence accumulated since the publication of the classical gating-spring model some 30 y ago [@howard_compliance_1988]. Our proposal relies on the cooperative gating of two MET channels per tip link, which are mobile within the membrane and coupled by elastic forces mediated by the lipid bilayer. The model accounts for the number and location of MET channels and reproduces the observed hair-cell mechanics quantitatively, using only realistic parameters. Furthermore, it provides a framework that can help understand some as-yet-unexplained features of hair-cell mechanotransduction.
Results {#results .unnumbered}
=======
Model Description {#model-description .unnumbered}
-----------------
{width="90.00000%"}
We describe here the basic principles of our model, illustrated in Fig. \[FigTwoChannelModel\]. Structural data indicate that the tip link is a dimeric, string-like protein that branches at its lower end into two single strands, which anchor to the top of the shorter stereocilium [@kachar_high-resolution_2000; @kazmierczak_cadherin_2007]. The model relies on three main hypotheses. First, each strand of the tip link connects to one MET channel, mobile within the membrane. Second, an intracellular spring—referred to as the adaptation spring—anchors each channel to the cytoskeleton, in agreement with the published literature [@fettiplace_physiology_2014; @howard_hypothesis:_2004; @zhang_ankyrin_2015; @powers_stereocilia_2012]. Third, and most importantly, the two MET channels interact via membrane-mediated elastic forces, which are generated by the mismatch between the thickness of the hydrophobic core of the bare bilayer and that of each channel [@nielsen_energetics_1998]. Such interactions have been observed in a variety of transmembrane proteins, including the bacterial mechanosensitive channels of large conductance (MscL) [@wiggins_analytic_2004; @ursell_cooperative_2007; @phillips_emerging_2009; @grage_bilayer-mediated_2011; @haselwandter_connection_2013]. Since the thickness of the channel’s hydrophobic region changes during gating, this hydrophobic mismatch induces a local deformation of the membrane that depends on the channel’s state [@wiggins_analytic_2004; @ursell_cooperative_2007; @phillips_emerging_2009; @haselwandter_connection_2013]. For a closed channel, the hydrophobic mismatch is small, and the membrane is barely deformed. An open channel’s hydrophobic core, however, is substantially thinner, and the bilayer deforms accordingly [@ursell_cooperative_2007; @phillips_emerging_2009]. When the two channels are sufficiently near each other, the respective bilayer deformations overlap, and the overall membrane shape depends both on the states of the channels as well as on the distance between them. As a result, the pair of MET channels is subjected to one of three different energy landscapes: open–open (OO), open–closed (OC), or closed–closed (CC) [@ursell_cooperative_2007]. The effects of this membrane-mediated interaction are most apparent at short distances: The potentials strongly disfavor the OC state, favor the OO state, and generate an attractive force between the two channels when they are both open.
Channel motion as a function of the imposed external force can be pictured as follows (Fig. \[FigTwoChannelModel\] and Movie S1). When tip-link tension is low, the two channels are most likely to be closed, and they are kept apart by the adaptation springs; at this large inter-channel distance, the membrane-mediated interaction between them is negligible (Fig. \[FigTwoChannelModel\]*A*). When a positive deflection is applied to the hair bundle, tension in the tip link rises. Consequently, the channels move toward one another and their open probabilities increase (Fig. \[FigTwoChannelModel\]*B*). When the inter-channel distance is sufficiently small, the membrane’s elastic energy favors the OO state, and both channels open cooperatively. As a result, the attractive membrane interaction in the OO state enhances their motion toward one another (red horizontal arrows, Fig. \[FigTwoChannelModel\]*B* and Movie S1), which provides an effective gating swing that is larger than the conformational change of a single channel (red vertical arrow, Fig. \[FigTwoChannelModel\]*B*). Eventually, the channels close—for example due to Ca^2+^ binding [@choe_model_1998; @cheung_ca2+_2006]—and the membrane-mediated interactions become negligible (Fig. \[FigTwoChannelModel\]*C*). Now the adaptation springs can pull the channels apart. Their lateral movement away from each other increases tip-link tension and produces the twitch, a hair-bundle movement associated with fast adaptation [@cheung_ca2+_2006; @benser_rapid_1996; @ricci_active_2000].
{width="95.00000%"} \[TableParameters\]
Mathematical Formulation {#mathematical-formulation .unnumbered}
------------------------
We represent schematically our model in Fig. \[FigSchematicModel\]. Fig. \[FigSchematicModel\]*A* illustrates the geometrical arrangement of a pair of adjacent stereocilia. They have individual pivoting stiffness $k_{\rm SP}$ at their basal insertion points. The displacement coordinate $X$ of the hair bundle’s tip along the axis of mechanosensitivity and the coordinate $x$ along the tip link’s axis are related by a geometrical factor $\gamma$. With $H$ the height of the tallest stereocilium in the hair bundle and $D$ the distance between its rootlet and that of its neighbor, $\gamma$ is approximately equal to $D/H$ [@howard_compliance_1988]. The transduction unit schematized in Fig. \[FigSchematicModel\]*A* is represented in more detail in Fig. \[FigSchematicModel\] *B* and *C*. In Fig. \[FigSchematicModel\]*B*, the stereociliary membrane is orthogonal to the tip link’s central axis. Depending on tip-link tension, the channels are likely to be closed (small tip-link tension, Fig. \[FigSchematicModel\]*B*, *Left*) or open (large tip-link tension, Fig. \[FigSchematicModel\]*B*, *Right*), and positioned at different locations. The tip link is modeled as a spring of constant stiffness $k_{\rm t}$ and resting length $l_{\rm t}$. It has a current length $x_{\rm t}$ and branches into two rigid strands of length $l$, a distance $d$ away from the membrane. Each strand connects to one MET channel. Due to the global geometry of the hair bundle (Fig. \[FigSchematicModel\] *A* and *B*), $x_{\rm t} + d=\gamma(X-X_0)$, where $X_0$ is a reference position of the hair-bundle tip related to the position of the adaptation motors, to which the upper part of the tip link is anchored (see also Hair-Bundle Force and Stiffness and Fig. S2). The channels’ positions are symmetric relative to the tip-link axis, with their attachments to the tip link a distance $2a$ from each other. The channels have cylindrical shapes with axes perpendicular to the membrane plane. They have a diameter $2\rho$ when closed and $2\rho + \delta$ when open, where $\delta$ corresponds to the conformational change of each channel along the membrane plane; we refer to it as the single-channel gating swing. Each tip-link branch inserts a distance $\rho/2$ from the inner edge of each channel. The adaptation springs are parallel to the direction of channel motion. They have stiffness $k_{\rm a}$ and resting length $l_{\rm a}$ and are anchored to two fixed reference positions a distance $L$ away from the tip-link axis.
Under tension, the stereociliary membrane can present different degrees of tenting [@kachar_high-resolution_2000; @assad_tip-link_1991]. To account for this geometry, and more generally for the non-zero curvature of the membrane at the tips of stereocilia, we introduce in Fig. \[FigSchematicModel\]*C* an angle $\alpha$ between the perpendicular to the tip link’s axis and each of the half membrane planes, along which the channels move. The simpler, flat geometry of Fig. \[FigSchematicModel\]*B* is recovered in the case where $\alpha = 0$.
The inter-channel forces mediated by the membrane are described by three elastic potentials $V_{{\rm b},n}(a)$, one for each state $n$ of the channel pair (OO, OC, and CC), and are functions of the distance $a$ (Fig. \[FigSchematicModel\]D). The index $n$ can be 0, 1, or 2, corresponding to the number of open channels in the transduction unit. We choose analytic expressions and parameters that mimic the shapes of the potentials used to model similar interactions between bacterial MscL channels [@ursell_cooperative_2007; @haselwandter_connection_2013] (*Materials and Methods*). In addition to the membrane-mediated elastic force $f_{{\rm b},n} = - {\rm d}V_{{\rm b},n}/{\rm d}a$, force balance on the channels depends on the force $f_{\rm t} = k_{\rm t}(x_{\rm t} - l_{\rm t})$ exerted by the tip link on its two branches and on the force $f_{\rm a} = k_{\rm a}(a_{\rm adapt} - a - n\delta/2)$ exerted by the adaptation springs, where $a_{\rm adapt} = L - l_{\rm a} - 3\rho/2$ is the value of $a$ for which the adaptation springs are relaxed when both channels are closed. Taking into account the geometry and the connection between $x_{\rm t}$ and $X$ given previously, force balance on either of the two channels reads: $$\label{EqForceBalance}
\begin{aligned}
k_{\rm t}[\gamma(X - X_0) - d - l_{\rm t}] = 2\frac{d + a \sin{\alpha}}{a + d \sin{\alpha}} \times \\ \times \left [k_{\rm a} \left ( a_{\rm adapt} - a -\frac{n}{2}\delta \right ) - \frac{{\rm d}V_{{\rm b},n}(a)}{{\rm d}a} \right ]\, .
\end{aligned}$$ In addition, the geometry implies: $$\label{EqGeometry}
d = \sqrt{l^2 - (a \cos{\alpha})^2} - a \sin{\alpha}\, .$$ Putting the expressions of $d$ and $V_{{\rm b},n}$ as functions of $a$ into Eq. \[EqForceBalance\] allows us to solve for $X$ as a function of $a$, for each state $n$. Inverting these three functions numerically gives three relations $a_n(X)$, which are then used to express all the relevant quantities as functions of the displacement coordinate $X$ of the hair bundle, taking into account the probabilities of the different states. Further details about this procedure are presented in *Materials and Methods*.
Finally, global force balance is imposed at the level of the whole hair bundle, taking into account the pivoting stiffness of the stereocilia at their insertion points into the cuticular plate of the cell (Fig. \[FigTwoChannelModel\]*B*, *Inset*, and Fig. \[FigSchematicModel\]*A*): $$F_{\rm ext} = K_{\rm sp}(X - X_{\rm sp}) + F_{\rm t}\, ,
\label{EqBundleForceBalance}$$ where $F_{\rm ext}$ is the total external force exerted at the tip of the hair bundle along the $X$ axis, $K_{\rm sp}$ is the combined stiffness of the stereociliary pivots along the same axis, $X_{\rm sp}$ is the position of the hair-bundle tip for which the pivots are at rest, and $F_{\rm t} = N\gamma f_{\rm t}$ is the combined force of the tip links projected onto the $X$ axis, with $N$ being the number of tip links.
In these equations, two related reference positions appear: $X_0$ and $X_{\rm sp}$. As the origin of the $X$ axis is arbitrary, only their difference is relevant. The interpretation of $X_{\rm sp}$ is given just above. As for $X_0$, it sets the amount of tension exerted by the tip links, since the force exerted by the tip link on its two branches reads $f_{\rm t} = k_{\rm t}[\gamma(X - X_0) - d - l_{\rm t}]$. To fix $X_0$—or equivalently the combination $X_0+l_{\rm t}/\gamma$, which appears in this expression—we rely on the experimentally observed hair-bundle movement that occurs when tip links are cut, and which is typically on the order of 100 nm [@assad_tip-link_1991; @jaramillo_displacement-clamp_1993]. Therefore, imposing $X = 0$ as the resting position of the hair-bundle tip with intact tip links, Eq. \[EqBundleForceBalance\] must be satisfied with $F_{\rm ext} = 0$, $X = 0$, and $X_{\rm sp} = 100$ nm, which formally sets the value of $X_0$ for any predefined $l_{\rm t}$. Solving for $X_0$, however, requires a numerical procedure, the details of which are presented in [*Materials and Methods*]{}.
All parameters characterizing the system together with their default values are listed in Table \[TableParameters\]. The geometrical projection factor $\gamma$ and number of stereocilia $N$ are set, respectively, to 0.14 and 50 [@howard_compliance_1988]. The combined stiffness of the stereociliary pivots $K_{\rm sp}$ is set to 0.65 mN$\cdot$m$^{-1}$ [@jaramillo_displacement-clamp_1993]. We use a tip-link stiffness $k_{\rm t}$ and an adaptation-spring stiffness $k_{\rm a}$ of 1 mN$\cdot$m$^{-1}$ to obtain a total hair-bundle stiffness in agreement with experimental observations [@howard_compliance_1988]. The length $l$ of the tip-link branch can be estimated by analyzing the structure of protocadherin-15, a protein constituting the tip link’s lower end. Three extracellular cadherin (EC) repeats are present after the kink at the EC8–EC9 interface, which suggests that $l$ is $\sim$12–14 nm [@araya-secchi_elastic_2016]. This estimate agrees with studies based on high-resolution electron microscopy of the tip link [@kachar_high-resolution_2000]. We allow for the branch to fully relax the adaptation springs by choosing $a_{\rm adapt}=2\cdot l$. The parameters $\delta$ and $\rho$ correspond respectively to the amplitude of the conformational change of a single channel in the membrane plane upon gating and to the radius of the closed channel (see Fig. \[FigSchematicModel\]*B*). Since the hair-cell MET channel has not yet been crystallized, we rely on the crystal structures of another mechanosensitive protein, the bacterial MscL channel, and choose $\delta=2$ nm and $\rho=2.5$ nm [@ursell_cooperative_2007]. Finally, the channel gating energy $E_{\rm g}$ is estimated in the literature to be on the order of 5–20 $k_{\rm B}T$ [@corey_kinetics_1983; @hudspeth_hair-bundle_1992; @ricci_mechano-electrical_2006]. We use 9 $k_{\rm B}T$ as a default value.
We now focus on the predictions of this model regarding the main biophysical characteristics of hair-bundle mechanics: open probability, force and stiffness as functions of displacement, the twitch during fast adaptation, and effects of Ca^2+^ concentration on hair-bundle mechanics.
Open Probability {#open-probability .unnumbered}
----------------
To determine the accuracy of the model and to investigate the effect of its parameters, we first focus on the predicted open probability ($P_{\rm open}$) as a function of the hair-bundle displacement $X$, for four sets of parameters (Fig. \[FigOpenprobability\]).
![Open probability curves as functions of hair-bundle displacement. All curves share a common set of parameters, whose values are specified in Table \[TableParameters\]. Parameter values that are not common to all curves are specified below. In addition, for each curve, the value of $X_0$ is set such that the external force $F_{\rm ext}$ applied to the hair bundle vanishes at $X = 0$. (Orange) ($k_{\rm t} = 1$ mN$\cdot$m$^{-1}$, $\alpha = 0 \degree$, $k_{\rm a} = 1$ mN$\cdot$m$^{-1}$, $E_{\rm g} = 9$ $k_{\rm B}T$). The curve is roughly sigmoidal and typical of experimental measurements. (Orange dashed) Fit to a two-state Boltzmann distribution as resulting from the classical gating-spring model, with expression $1/(1 + \exp[z(X_0 - X)/{k_{\rm B}T}])$, where $z \simeq 0.36$ pN, $X_0 \simeq 35$ nm, and $k_{\rm B}T \simeq 4.1$ zJ. Here, $z$ corresponds to the gating force in the framework of the classical gating-spring model. (Blue) ($k_{\rm t} = 2$ mN$\cdot$m$^{-1}$, $\alpha = 15 \degree$, $k_{\rm a} = 1$ mN$\cdot$m$^{-1}$, $E_{\rm g} = 8.8$ $k_{\rm B}T$). The values of $X_0$ and $E_{\rm g}$ have been chosen so that the force is zero at $X = 0$ within the region of negative stiffness, which is required for a spontaneously oscillating hair bundle [@martin_negative_2000]. Channel gating occurs here over a narrower range of hair-bundle displacements. (Blue dashed) Fit to a two-state Boltzmann distribution, with $z \simeq 1.0$ pN, $X_0 \simeq 1.4$ nm, and $k_{\rm B}T \simeq 4.1$ zJ. (Red) ($k_{\rm t} = 1$ mN$\cdot$m$^{-1}$, $\alpha = 0 \degree$, $k_{\rm a} = 1$ mN$\cdot$m$^{-1}$, $E_{\rm g} = 9$ $k_{\rm B}T$, no membrane potentials). The channels remain closed over the whole range of displacements shown in the figure. (Green) ($k_{\rm t} = 1$ mN$\cdot$m$^{-1}$, $\alpha = 0 \degree$, $k_{\rm a} = 200$ mN$\cdot$m$^{-1}$, $E_{\rm g} = 9$ $k_{\rm B}T$, no membrane potentials). The curve presents a plateau around $P_{\rm open} = 0.5$. []{data-label="FigOpenprobability"}](fig_Popen.pdf){width="\linewidth"}
For our default parameter set defined above (see also Table \[TableParameters\]), the open probability as a function of hair-bundle displacement is a sigmoid that matches the typical curves measured experimentally (orange, continuous curve). It is well fit by a two-state Boltzmann distribution (orange, dashed curve). In this case, the range of displacements over which the channels gate is $\sim$100 nm, in line with experimental measurements [@ricci_mechanisms_2002; @he_mechanoelectrical_2004; @jia_mechanoelectric_2007; @hudspeth_integrating_2014]. Recorded ranges vary, however, from several tens to hundreds of nanometers, depending on whether the hair bundle moves spontaneously or is stimulated, and depending on the method of stimulation (ref. [@meenderink_voltage-mediated_2015] and reviewed in ref. [@fettiplace_physiology_2014]). Increasing the tip-link stiffness $k_{\rm t}$ and the angle $\alpha$ compresses this range to a few tens of nanometers (blue curve), matching that measured for spontaneously oscillating hair bundles [@meenderink_voltage-mediated_2015].
Decreasing the amplitude of the single-channel gating swing $\delta$ instead broadens the range and shifts it to larger hair-bundle displacements (Fig. S1). In contrast to the classical model of mechanotransduction, where channel gating is intimately linked to the existence of the single-channel gating swing, here, gating still takes place when $\delta = 0$ due to the membrane elastic potentials. To demonstrate the crucial role played by these potentials, we compare in Fig. \[FigOpenprobability\] the open probability curves obtained using the default set of parameters with (orange) and without (red) the bilayer-mediated interaction. Without the membrane contribution, the channels remain closed over the whole range of hair-bundle displacements. The associated curve (red) is barely visible close to the horizontal axis of Fig. \[FigOpenprobability\]. It is possible, however, to have the channels gate over this range of displacements without the membrane contribution by choosing a value of $k_{\rm a}$ sufficiently large for the lateral channel motion to be negligible. This configuration mimics the case of immobile channels, as in the classical gating-spring model on timescales that are smaller than the characteristic time of slow adaptation. The resulting curve (green) does not match any experimentally measured open-probability relations: It displays a plateau at $P_{\rm open} = 0.5$ corresponding to the OC state. This state is prevented in the complete model with mobile channels by the membrane-mediated forces. Reintroducing these forces while keeping the same large value of $k_{\rm a}$ hardly changes the open-probability relation, because the channels are maintained too far from each other by the adaptation springs to interact via the membrane. Therefore, we only display one of the two curves here. We illustrate further the influence of the value of $k_{\rm a}$ as well as of the amplitude of the elastic membrane potentials in Fig. S5.
We conclude from these results that our model can reproduce the experimentally observed open-probability relations using only realistic parameters, and that the membrane-mediated interactions as well as the ability of the MET channels to move within the membrane are essential features of the model.
Hair-Bundle Force and Stiffness {#hair-bundle-force-and-stiffness .unnumbered}
-------------------------------
Two other classical characteristics of hair-cell mechanics are the force- and stiffness- displacement relations. In Fig. \[FigForceStiffness\], we display them using the same sets of parameters and color coding as in Fig. \[FigOpenprobability\].
![Hair-bundle force (*A*) and stiffness (*B*) as functions of hair-bundle displacement. The different sets of parameters are the same as the ones used in Fig. \[FigOpenprobability\], following the same color code. (Orange) The force–displacement curve shows a region of gating compliance, characterized by a decrease in its slope over the gating range of the channels, recovered as a decrease in stiffness over the same range. (Blue) The force–displacement curve shows a region of negative slope, characteristic of a region of mechanical instability. The corresponding stiffness curve shows associated negative values. (Red) Without the membrane elastic potentials, the channels are unable to open and the hair-bundle mechanical properties are roughly linear, except for geometrical nonlinearities. (Green) The curves display two regions of gating compliance, better visible on the stiffness curve. []{data-label="FigForceStiffness"}](fig_Force_Stiffness.pdf){width="\linewidth"}
The predicted forces necessary to move the hair bundle by tens of nanometers are on the order of tens of piconewtons, in line with the literature [@howard_compliance_1988; @van_netten_channel_2003]. With our reference set of parameters, the force is weakly nonlinear, associated with a small drop in stiffness (orange curves). When the range of displacements over which the channels gate is sufficiently narrow, a nonmonotonic trend appears in the force, corresponding to a region of negative stiffness (blue curves). In the absence of membrane-mediated interactions—in which case the channels do not gate—the force-displacement curve is nearly linear and the stiffness nearly constant (red curves). The relatively small stiffness variation along the curve is due to the geometry, which imposes a nonlinear relation between hair-bundle displacement and channel motion (Eqs. \[EqForceBalance\] and \[EqGeometry\]). When channel motion is prevented by a large value of $k_{\rm a}$, two separate regions of gating compliance appear, corresponding to the two transitions between the three states (CC, OC, and OO) (green curves). The red curves of this figure demonstrate no contribution from the channels whereas the green curves are again unlike any experimentally measured ones. These results confirm the importance in our model of both the lateral mobility of the channels and the membrane-mediated elastic forces.
We next investigate whether we can reproduce the effects on the force–displacement relation of the slow and fast adaptation (reviewed in refs. [@eatock_adaptation_2000] and [@holt_two_2000]). Slow adaptation is attributed to a change in the position of myosin motors that are connected to the tip link’s upper end and regulate its tension [@howard_compliance_1988; @corey_analysis_1983; @eatock_adaptation_1987; @howard_mechanical_1987; @crawford_activation_1989; @hacohen_regulation_1989; @assad_active_1992; @wu_two_1999; @kros_reduced_2002]. Here, this phenomenon corresponds to a change in the value of the reference position $X_0$. This parameter affects tip-link tension via the force exerted by the tip link on its two branches: $f_{\rm t} = k_{\rm t}[\gamma(X - X_0) - d - l_{\rm t}]$ (Mathematical Formulation). Starting from the parameters associated with the blue curve and varying $X_0$, we obtain force-displacement relations that are in agreement with experimental measurements (Fig. S2) [@martin_negative_2000; @le_goff_adaptive_2005].
Fast adaptation is thought to be due to an increase in the gating energy $E_{\rm g}$ of the MET channels, for example, due to Ca^2+^ binding to the channels, which decreases their open probability [@choe_model_1998; @cheung_ca2+_2006; @ricci_active_2000; @wu_two_1999; @kennedy_fast_2003]. Starting from the same default curve and changing $E_{\rm g}$ by 1 $k_{\rm B}T$, we obtain a shift in the force–displacement relation (Fig. S3). In this case, the amplitude of displacements over which channel gating occurs remains roughly the same, but the associated values of the external force required to produce these displacements change. Such a shift has been measured in a spontaneously oscillating, weakly slow-adapting cell by triggering acquisition of force–displacement relations after rapid positive or negative steps [@le_goff_adaptive_2005]. During a rapid negative step, the channels close, which we attribute to fast adaptation with an increase in $E_{\rm g}$. In Fig. S3, increasing $E_{\rm g}$ by 1 $k_{\rm B}T$ increases the value of the force for the same imposed displacement. This mirrors the results in ref. [@le_goff_adaptive_2005], where a similar outcome is observed when comparing the curve measured after rapid negative steps with that measured after rapid positive steps. From Figs. S2 and S3, we conclude that our model is capable of reproducing the effects of both slow and fast adaptation on the force–displacement relation.
In summary, the model reproduces realistic force–displacement relations when both lateral channel mobility and membrane-mediated interactions are present. These relations exhibit a region of gating compliance and can even show a region of negative stiffness while keeping all parameters realistic.
A Mechanical Correlate of Fast Adaptation, the Twitch {#a-mechanical-correlate-of-fast-adaptation-the-twitch .unnumbered}
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Next, we investigate whether our model can reproduce the hair-bundle negative displacement induced by rapid reclosure of the MET channels, known as the twitch [@cheung_ca2+_2006; @benser_rapid_1996; @ricci_active_2000]. It is a mechanical correlate of fast adaptation, an essential biophysical property of hair cells, which is believed to allow for rapid cycle-by-cycle stimulus amplification [@choe_model_1998]. To reproduce the twitch observed experimentally [@cheung_ca2+_2006; @benser_rapid_1996; @ricci_active_2000], we compute the difference in the positions of the hair bundle before and after an increment of $E_{\rm g}$ by 1 $k_{\rm B}T$, and plot it as a function of the external force (Fig. \[FigTwitch\]).
![Twitch as a function of the external force exerted on the hair bundle (main image) and normalized twitch as a function of the open probability (*Inset*). The different sets of parameters are the same as the ones in Figs. \[FigOpenprobability\] and \[FigForceStiffness\] for the orange, blue and green curves. The additional purple curve is associated with the same parameter set as that of the orange curve, except for the number of intact tip links, set to $N=25$ rather than $N=50$. (Orange) The maximal twitch amplitude for the standard set of parameters is $\sim$5 nm. (Blue) Because of the region of mechanical instability associated with negative stiffness, the corresponding curve for the twitch is discontinuous, as shown by the two regions of near verticality in the blue curve. This corresponds to the two regions of almost straight lines in the normalized twitch. Both of these linear parts are displayed as guides for the eye. (Green) The channels gate independently, producing two distinct maxima of the twitch amplitude. (Purple) The twitch peaks at a smaller force and its amplitude is reduced compared with the orange curve. Plotted as a function of the open probability, however, the two curves are virtually identical. []{data-label="FigTwitch"}](fig_Twitch.pdf){width="\linewidth"}
With the same parameters as in Figs. \[FigOpenprobability\] and \[FigForceStiffness\], we find twitch amplitudes within the range reported in the literature [@cheung_ca2+_2006; @benser_rapid_1996; @ricci_active_2000]. They reach their maxima for intermediate, positive forces and drop to zero for large negative or positive forces, as experimentally observed. The twitch is largest and peaks at the smallest force when the hair bundle displays negative stiffness (blue curve), since the channels open then at the smallest displacements. For the green curve, the channels gate independently, producing two distinct maxima of the twitch amplitude, mirroring the biphasic open-probability relation. Note that no curve is shown with the parameter set corresponding to the red curves of Figs. \[FigOpenprobability\] and \[FigForceStiffness\], since the twitch is nearly nonexistent in that case.
Twitch amplitudes reported in the literature are variable, ranging from $\sim$4 nm in single, isolated hair cells [@cheung_ca2+_2006], to $>$30 nm in presumably more intact cells within the sensory epithelium [@benser_rapid_1996; @ricci_active_2000]. A potential source of variability is the number of intact tip links, since these can be broken during the isolation procedure. We show that decreasing the number of tip links in our model shifts the twitch to smaller forces and decreases its amplitude (Fig. \[FigTwitch\], purple vs. orange curves). Twitch amplitudes are further studied for different values of the adaptation-spring stiffness and amplitudes of the elastic membrane potentials in Fig. S5. To compare further with experimental data [@benser_rapid_1996; @ricci_active_2000], we also present the twitch amplitude normalized by its maximal value, and plot it as a function of the channels’ open probability (Fig. \[FigTwitch\], *Inset*). The twitch reaches its maximum for an intermediate level of the open probability and drops to zero for smaller or larger values, as measured experimentally [@benser_rapid_1996; @ricci_active_2000].
Another factor that strongly affects both the amplitude and force dependence of the twitch is the length $l$ of the tip-link branching fork. For a long time, the channels were suspected to be located at the tip link’s upper end, where the tip-link branches appear much longer [@kachar_high-resolution_2000]. With long branches, the twitch is tiny and peaks at forces that are too large (Fig. S4), unlike what is experimentally measured. This observation provides a potential physiological reason why the channels are located at the tip link’s lower end rather than at the upper end as previously assumed [@zhao_elusive_2015; @spinelli_bottoms_2009]. There are two more reasons why our model requires the channels to be located at the lower end of the tip link. First, as shown in Figs. \[FigOpenprobability\]–\[FigTwitch\], some degree of membrane tenting increases the sensitivity and nonlinearity of the system, as well as the amplitude of the twitch. While it is straightforward to obtain the necessary membrane curvature at the tip of a stereocilium, this is not the case on its side. Second, while pulling on the channels located at the tip compels them to move toward one another, doing so with the channels located on the side would instead make them slide down the stereocilium, impairing the efficiency of the mechanism proposed in this work.
In summary, our model reproduces correctly the hair-bundle twitch as well as its dependence on several key parameters. It therefore includes the mechanism that can mediate the cycle-by-cycle sound amplification by hair cells.
Effect of Ca^2+^ Concentration on Hair-Bundle Mechanics {#effect-of-ca2-concentration-on-hair-bundle-mechanics .unnumbered}
-------------------------------------------------------
Our model can also explain the following important results that have so far evaded explanation. First, it is established that, with increasing Ca^2+^ concentration, the receptor current vs. displacement curve shifts to more positive displacements, while its slope decreases [@corey_kinetics_1983]. Second, within the framework of the classical gating-spring model, Ca^2+^ concentration appears to affect the magnitude of the gating swing [@tinevez_unifying_2007]: When a hair bundle is exposed to a low, physiological, Ca^2+^ concentration of 0.25 mM, the force–displacement relation presents a pronounced region of negative slope, and the estimated gating swing is large, on the order of 9–10 nm. But when the same hair bundle is exposed to a high Ca^2+^ concentration of $\sim$1 mM, the region of negative stiffness disappears, and the estimated gating swing becomes only half as large.
In our model, it is the decrease of the interchannel distance following channel opening that, transmitted onto the tip link’s main axis, effectively plays the role of the classical gating swing (Figs. \[FigTwoChannelModel\] and \[FigSchematicModel\] and Movie S1). To quantify the change of tip-link extension as the channels open, we introduce a new quantity, which we call the gating-associated tip-link extension (GATE). It is defined mathematically as $d_{\rm OO}-d_{\rm CC}$, where $d_{\rm OO}$ and $d_{\rm CC}$ are the respective values of the distance $d$ in the OO and CC states. To study the influence of Ca^2+^ concentration on the GATE, we hypothesize that Ca^2+^ ions favor the closed conformation of the channels over the open one, that is, that the energy difference $E_{\rm g}$ between the two states increases with Ca^2+^ concentration [@choe_model_1998; @cheung_ca2+_2006].
![GATE and open probability as functions of hair-bundle displacement (*A*) and GATE as a function of the open probability (*B*), for different values of the channel gating energy $E_{\rm g}$. (*A*) Open-probability curves are generated by using the default parameter set of the blue curves of Figs. \[FigOpenprobability\]–\[FigTwitch\], and otherwise different values of the channel gating energy $E_{\rm g}$, as indicated directly on *B*. The GATE as a function of $X$ (red curve) depends only on the geometry of the system, such that only one curve appears here. We indicate, in addition, directly on the image the values of the single-channel gating force $z$ obtained by fitting each open-probability relation with a two-state Boltzmann distribution, as done in Fig. \[FigOpenprobability\] for the orange and blue curves. We report, together with these values, the relative magnitudes of the single-channel gating swing $g_{\rm swing}$ obtained with the classical model. (*B*) The GATE values are plotted as functions of the open probability, for each chosen value of $E_{\rm g}$. []{data-label="FigGATE"}](fig_GATE.pdf){width="\linewidth"}
Within this framework, we expect to see the following effect of Ca^2+^ on the GATE, via the change of $E_{\rm g}$: Higher Ca^2+^ concentrations correspond to higher gating energies, causing the channels to open at greater positive hair-bundle displacements. Greater displacements in turn correspond to smaller values of the inter-channel distance before channel opening. Since the final position of the open channels is always the same (at $a=a_{\rm min}$, where the OO membrane potential is minimum; Fig. \[FigSchematicModel\]*D*), the change of the interchannel distance induced by channel opening is smaller for higher Ca^2+^ concentrations. As a result, the GATE experienced by the tip link is smaller for higher Ca^2+^ concentrations, in agreement with the experimental findings cited above.
We study this effect quantitatively in Fig. \[FigGATE\]. In Fig. \[FigGATE\]*A*, we plot simultaneously the GATE and $P_{\rm open}$ as functions of the hair-bundle displacement $X$, for five values of $E_{\rm g}$. Although the function ${\rm GATE}(X)$ spans the whole range of displacements, the relevant magnitudes of the GATE are constrained by the displacements for which channel opening is likely to happen; we use as a criterion that $P_{\rm open}$ must be between 0.05 and 0.95. The corresponding range of displacements depends on the position of the $P_{\rm open}$ curve along the horizontal axis, which ultimately depends on $E_{\rm g}$. We display in Fig. \[FigGATE\]*A* the two ranges of hair-bundle displacements (dashed vertical lines) associated with the smallest and largest values of $E_{\rm g}$, together with the amplitudes of the GATE within these intervals (dashed horizontal lines). For $E_{\rm g} = 6$ $k_{\rm B}T$ (blue curve and GATE interval), the size of the GATE is on the order of $4.1\text{--}5.3$ nm, whereas it is on the order of $1.3\text{--}2.3$ nm for $E_{\rm g} = 14$ $k_{\rm B}T$ (orange curve and GATE interval). In general, larger values of the channel gating energy $E_{\rm g}$ cause smaller values of the GATE.
To compare directly with previous analyses, we next fit the open-probability relations of Fig. \[FigGATE\]*A* with the gating-spring model, obtaining the corresponding single-channel gating forces $z$. This procedure allows us to quantify the change of the magnitude of an effective gating swing $g_{\rm swing}$ with $E_{\rm g}$ by the formula $z=g_{\rm swing}k_{\rm gs}\gamma$, where $k_{\rm gs}$ is the stiffness of the gating spring. We give directly on the panel the relative values of $g_{\rm swing}$ obtained by this procedure. Taking, for example, $k_{\rm gs}=1$ mN$\cdot$m$^{-1}$, $g_{\rm swing}$ ranges from 8.3 nm for $E_{\rm g} = 6$ $k_{\rm B}T$ to 3.6 nm for $E_{\rm g} = 14$ $k_{\rm B}T$.
In Fig. \[FigGATE\]*B*, we show the GATE as a function of $P_{\rm open}$ for the different values of $E_{\rm g}$. For each curve, the amplitude of the GATE is a decreasing function of $P_{\rm open}$ that presents a broad region of relatively weak dependence for most $P_{\rm open}$ values. These results demonstrate that the GATE defined within our model decreases with increasing values of $E_{\rm g}$, corresponding to increasing Ca^2+^ concentrations. In addition, the same dependence is observed for the effective gating swing estimated from fitting the classical gating-spring model to our results, as it is when fit to experimental data [@tinevez_unifying_2007].
Finally, we can see from Fig. \[FigGATE\]*A* that the predicted open-probability vs. displacement curves shift to the right and their slopes decrease with increasing values of $E_{\rm g}$, a behavior in agreement with experimental data (see above and ref. [@corey_kinetics_1983]). Together with the decrease in the slope, the region of negative stiffness becomes narrower (Fig. S3) and even disappears for a sufficiently large value of $E_{\rm g}$ (Fig. S3, yellow curve). This weakening of the gating compliance has been measured in hair bundles exposed to a high Ca^2+^ concentration [@tinevez_unifying_2007].
In summary, our model explains the shift in the force-displacement curve as well as the changes of the effective gating swing and stiffness as functions of Ca^2+^ concentration.
Discussion {#discussion .unnumbered}
==========
We have designed and analyzed a two-channel, cooperative model of hair-cell mechanotransduction. The proposed geometry includes two MET channels connected to one tip link. The channels can move relative to each other within the stereociliary membrane and interact via its induced deformations, which depend on whether the channels are open or closed. This cross-talk produces cooperative gating between the two channels, a key feature of our model. Most importantly, because the elastic membrane potentials are affected by channel gating on length scales larger than the proteins’ conformational rearrangements, and because the channels can move in the membrane over distances greater than their own size, the model generates an appropriately large effective gating swing without invoking unrealistically large conformational changes. Moreover, even when the single-channel gating swing vanishes, the effective gating swing determined by fitting the classical model to our results does not. In this case, the conformational change of the channel is orthogonal to the membrane plane and its gating is triggered only by the difference in membrane energies between the OO and CC states. We have shown that our model reproduces the hair bundle’s characteristic current– and force–displacement relations as well as the existence and characteristics of the twitch, the mechanical correlate of fast adaptation. It also explains the puzzling effects of the extracellular Ca^2+^ concentration on the magnitude of the estimated gating swing and on the spread of the negative-stiffness region, features that are not explained by the classical gating-spring model.
In addition to reproducing these classical features of hair-cell mechanotransduction, our model may be able to account for other phenomena that have had so far no—or only unsatisfactory—explanations. One of them is the flick, a small, voltage-driven hair-bundle motion that requires intact tip links but does not rely on channel gating [@cheung_ca2+_2006; @ricci_active_2000; @meenderink_voltage-mediated_2015]. It is known that changes in membrane voltage modulate the membrane mechanical tension and potentially the membrane shape by changing the interlipid distance [@zhang_voltage-induced_2001; @breneman_hair_2009], but it is not clear how this property can produce the flick. This effect could be explained within our framework as a result of a change in the positions of the channels following the change in interlipid distance driven by voltage. This would in turn change the extension of the tip link and thus cause a hair-bundle motion corresponding to the flick.
Another puzzling observation from the experimental literature is the recordings of transduction currents that appear as single events but with conductances twofold to fourfold that of a single MET channel [@pan_tmc1_2013; @beurg_conductance_2014]. Because tip-link lower ends were occasionally observed to branch into three or four strands at the membrane insertion [@kachar_high-resolution_2000], one tip link could occasionally be connected to as many channels. According to our model, these large-conductance events could therefore reflect the cooperative openings of coupled channels.
Our model predicts that changing the membrane properties must affect the interaction between the MET channels, potentially disrupting their cooperativity and in turn impairing the ear’s sensitivity and frequency selectivity. For example, if the bare bilayer thickness were to match more closely the hydrophobic thickness of the open state of the channel rather than that of the closed state, the whole shape of the elastic membrane potentials would be different. In such a case, the open probability vs. displacement curves would be strongly affected, and gating compliance and fast adaptation would be compromised. Potentially along these lines, it was observed that chemically removing long-chain—but not short-chain—phospholipid PiP2 blocked fast adaptation [@hirono_hair_2004]. With a larger change of membrane thickness, one could even imagine reversing the roles of the OO and CC membrane-mediated interactions. This would potentially change the direction of fast adaptation, producing an “anti-twitch”, a positive hair-bundle movement due to channel reclosure. Such a movement has indeed been measured in rat outer hair cells [@kennedy_force_2005]. Whether it was produced by this or a different mechanism remains to be investigated.
Our model fundamentally relies on the hydrophobic mismatch between the MET channels and the lipid bilayer. Several studies have demonstrated that the lipids with the greatest hydrophobic mismatch with a given transmembrane protein are depleted from the protein’s surrounding. The timescale of this process is on the order of a 100 ns for the first shell of annular lipids [@beaven_gramicidin_2017]. It is much shorter than the timescales of MET-channel gating and fast adaptation. Therefore, it is possible that lipid rearrangement around a MET channel reduces the hydrophobic mismatch and thus decreases the energy cost of the elastic membrane deformations, lowering in turn the importance of the membrane-mediated interactions in hair-bundle mechanics. However, such lipid demixing in the fluid phase of a binary mixture is only partial, on the order of 5–10% [@yin_hydrophobic_2012]. Furthermore, ion channels are known to bind preferentially specific phospholipids such as PiP2 [@suh_pip2_2008], further suggesting that the lipid composition around a MET channel does not vary substantially on short timescales. We therefore expect the effect of this fast lipid mobility to be relatively minor. Slow, biochemical changes of the bilayer composition around the channels, however, could have a stronger effect. It would be interesting for future studies to investigate the role played by lipid composition around a MET channel on its gating properties and how changes in this composition affect hair-cell mechanotransduction.
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---
abstract: 'We consider two functions $\phi$ and $\psi$, defined as follows. Let $x,y \in (0,1]$ and let $A,B,C$ be disjoint nonempty subsets of a graph $G$, where every vertex in $A$ has at least $x|B|$ neighbors in $B$, and every vertex in $B$ has at least $y|C|$ neighbors in $C$. We denote by $\phi(x,y)$ the maximum $z$ such that, in all such graphs $G$, there is a vertex $v \in C$ that is joined to at least $z|A|$ vertices in $A$ by two-edge paths. If in addition we require that every vertex in $B$ has at least $x|A|$ neighbors in $A$, and every vertex in $C$ has at least $y|B|$ neighbors in $C$, we denote by $\psi(x,y)$ the maximum $z$ such that, in all such graphs $G$, there is a vertex $v \in C$ that is joined to at least $z|A|$ vertices in $A$ by two-edge paths. In their recent paper [@ourpaper], M. Chudnovsky, P. Hompe, A. Scott, P. Seymour, and S. Spirkl introduced these functions, proved some general results about them, and analyzed when they are greater than or equal to $1/2, 2/3,$ and $1/3$. Here, we extend their results by analyzing when they are greater than or equal to $3/4, 2/5,$ and $3/5$.'
author:
- Patrick Hompe
title: Some results on concatenating bipartite graphs
---
Introduction
============
The paper focuses on the functions $\phi$ and $\psi$ introduced in [@ourpaper]. We repeat the definitions and some fundamental results here for completeness, but refer the reader to [@ourpaper] for their proofs and a more complete introduction to these functions. I should note that most of what follows for the rest of the introduction is taken word-for-word from [@ourpaper].
All graphs in this paper are finite, and have no loops or multiple edges. We denote the semi-open interval $\{x:0< x\le 1\}$ of real numbers by $(0,1]$. Let $x,y\in (0,1]$; and let $A,B,C$ be disjoint nonempty subsets of a graph $G$, where every vertex in $A$ has at least $x|B|$ neighbours in $B$, and every vertex in $B$ has at least $y|C|$ neighbours in $C$. If we ask for a real number $z$ such that we can guarantee that some vertex in $A$ can reach at least $z|C|$ vertices in $C$ by two-edge paths, then $z$ must be at most $y$, since perhaps all the vertices in $B$ have the same neighbours in $C$. But in the reverse direction the question becomes much more interesting; that is, we ask for $z$ such that some vertex in $C$ can reach at least $z|A|$ vertices in $A$ by two-edge paths. Then there might well be values of $z>\max(x,y)$ with this property.
Let us say this more precisely. A [*tripartition*]{} of a graph $G$ is a partition $(A,B,C)$ of $V(G)$ where $A,B,C$ are all nonempty stable sets. For $x,y\in (0,1]$, we say a graph $G$ is [*$(x,y)$-constrained, via a tripartition $(A,B,C)$,*]{} if
- every vertex in $A$ has at least $x|B|$ neighbours in $B$;
- every vertex in $B$ has at least $y|C|$ neighbours in $C$; and
- there are no edges between $A$ and $C$.
For $v\in V(G)$, $N(v)$ denotes its set of neighbours, and $N^2(v)$ is the set of vertices with distance exactly two from $v$. We write $N^2_A(v)$ for $N^2(v)\cap A$, and so on. Here is a basic result from [@ourpaper], which we state without proof:
\[suptomax\] Let $x,y\in (0,1]$, and let $Z$ be the set of all $z\in (0,1]$ such that, for every graph $G$, if $G$ is $(x,y)$-constrained via $(A,B,C)$ then $|N^2_A(v)|\ge z|A|$ for some $v\in C$. Then $\sup\{z\in Z\}$ belongs to $Z$.
We define $\phi(x,y)$ to be $\sup\{z\in Z\}$, as defined in \[suptomax\]. There is a second related problem concerning a function $\psi$. Let us say $G$ is [*$(x,y)$-biconstrained*]{} ([*via*]{} $(A,B,C)$) if $G$ is $(x,y)$-constrained via $(A,B,C)$, and in addition
- every vertex in $B$ has at least $x|A|$ neighbours in $A$, and
- every vertex in $C$ has at least $y|B|$ neighbours in $B$.
Let $\psi(x,y)$ be the analogue of $\phi(x,y)$ for biconstrained graphs; that is, the maximum $z$ such that for all $G$, if $G$ is $(x,y)$-biconstrained via $(A,B,C)$, then $|N^2_A(v)|\ge z|A|$ for some $v\in C$. (As before, this maximum exists.)
We list the following additional basic results from [@ourpaper] without proof.
\[maxbound\] $\phi(x,y)\ge \max(x,y)$ for all $x,y>0$.
\[cayleybound\] For all $x,y\in (0,1]$, $$\phi(x,y)\le \frac{\lceil kx\rceil +\lceil ky\rceil -1}{k}$$ for every integer $k\ge 1$.
\[intk\] For every integer $k\ge 1$, if $x,y>0$ and $\max(x,y) = 1/k$ then $\phi(x,y) = 1/k$.
\[trivialbounds\] For all $x,y\in (0,1]$, $$\max(x,y)\le \phi(x,y)\le \psi(x,y)\le \frac{\lceil kx\rceil +\lceil ky\rceil -1}{k}$$ for every integer $k\ge 1$.
\[phisymmetry\] $\phi(x,y)=\phi(y,x)$ for all $x,y$.
Now, one of the approaches used in [@ourpaper] to try to understand the behavior of these functions was to analyze when they were greater than or equal to $1/2$, $2/3$, and $1/3$. Here, we continue along these lines, looking at when each of $\phi$ and $\psi$ are greater than or equal to $3/4$, $2/5$, and $3/5$.
General Results for $\phi$ and $\psi$
=====================================
In this section we prove several general results concerning the functions $\phi$ and $\psi$ that will be used several times in the remainder of the paper. For $B_1 \subseteq B$, let $N_A(B_1) \subseteq A$ be the set of vertices in $A$ with a neighbor in $B_1$. The first result is a very useful lemma which says, roughly speaking, the larger $B_1$ is, the larger $N_A(B_1)$ is.
\[1-y\] Suppose $G$ is $(x,y)$-biconstrained via $(A,B,C)$ and satisfies $|N_A^2(w)| < z|A|$ for all $w \in C$. Then for $k \ge 1$, if $B_k \subseteq B$ with $$b_k = \frac{|B_k|}{|B|} > (k-1)(1-y) + \max(1-y,1-x/(1-y))$$ then $|N_A(B_k)| > (x+k(1-z))|A|$.
[[**Proof.**]{} ]{}We proceed by induction on $k$. Let $A_k = N_A(B_k)$. If $k=1$, every vertex in $B \setminus B_1$ has at least $y|C|$ neighbors in $C$, so there exists $w \in C$ with at least $y|B \setminus B_1|$ neighbors in $B \setminus B_1$. For all $v \in A \setminus A_1$, $v$ has at least $x|B|$ neighbors in $B \setminus B_1$, and the condition $b_1 > 1 - x/(1-y)$ is equivalent to $y(1-b_1)+x>1-b_1$, so it follows that $v$ has a neighbor in $N(w)$, and thus every $v \in A \setminus A_1$ is a second-neighbor of $w$. Then since $b_1 > 1-y$, $w$ has a neighbor in $B_1$, and consequently $x|A|$ neighbors in $A_1$, so it follows that $z|A| > |N_A^2(w)| \ge (|A|-|A_1|)+x|A|$, which becomes $|A_1| > (x+1-z)|A|$, as desired.
Now, suppose the statement is true for all $k \le n$. For $k = n + 1 \ge 2$, choose $w \in C$ with at least $y|B-B_k|$ neighbors in $B_k$. Let $A_k = N_A(k)$. For all $v \in A-A_k$, $v$ has at least $x|B|$ neighbors in $B-B_k$, so we have: $$b_k > (k-1)(1-y) + 1-x/(1-y) \ge 1-x/(1-y)$$ It follows that $y(1-b_k)+x>1-b_k$, and thus every $v \in A-A_k$ is a second-neighbor of $w$. Now, if $U = N(w) \cap B_k$, we have that: $$\frac{|U|}{|B|} \ge b_k+y-1 > (k-2)(1-y)+\max(1-y,1-x/(1-y))$$ Then by induction we have that $U$ has more than $(x+(k-1)(1-z))|A|$ neighbors in $A$, which are all in $|N_A^2(w) \cap A_k|$. Thus: $$z|A| > |N_A^2(w)| > (|A|-|A_k|) + (x+(k-1)(1-z))|A|$$ and consequently $|A_k| > (x+k(1-z))|A|$ as desired. This proves \[1-y\].
The following collection of results comes from \[1\].
\[5.7\] For $x,y,z \in (0,1]$, if $y > 1/2$ and $4x^2y(1-z) \ge (z-x)^2$ then $\phi(x,y) \ge z$. If in addition $4x^2y(1-z)>(z-x)^2$, then $\phi(x,y)>z$.
\[2/3psilb\] We have:
- $\psi(x,y)<2/3$ if $1/2\le y\le 3/5$ and $2x+y\le 1$;
- $\psi(x,y)<2/3$ if $3/5 \le y$ and $x+3y \le 2$, with $x+3y < 2$ if x or y is irrational;
- $\psi(x,y)<2/3$ if $4/7 \le x \le 11/17$ and $x+3y \le 1$.
\[2/3philb\] For all $x,y \in (0,1]$ with $y \le 1/2$, if $\frac{x}{1-x}+\frac{y}{1-2y} \le 2$, then $\phi(x,y) < 2/3$.
\[1/kphilb\] Let $k \in \mathbb{Z}$ be an integer, and let $x,y \in (0,1]$ with $\frac{x}{1-kx}+\frac{y}{1-ky} \le 1$, with strict inequality if $x$ or $y$ is irrational; then $\phi(x,y) < \frac{1}{k+1}$.
\[lowerboundextension\] Suppose that $z/(1-z)=\phi(x/(1-x),y/(1-y))$; then $\phi(x,y) \le z$.
\[rotating\] The statements $\phi(x,y) \le 1-z$, $\phi(z,x) \le 1-y$, and $\phi(y,z) \le 1-x$ are equivalent.
\[rotating\] is referred to as “rotating” for the remainder of this paper.
\[2/3phiub\] If $y \le 1/2$ and $x > (1-y)^2 / (1-2y^2)$ then $\phi(x,y) \ge 2/3$.
Now, we prove some results for obtaining lower bounds, first for $\phi$:
\[philb\] Suppose $x \ge 1/2$ and $y \le 1/2$. If $\phi\left((2x-1)/x,y/(1-y)\right) < (2z-1)/z$, then $\phi(x,y) < z$.
[[**Proof.**]{} ]{}Let $x'=(2x-1)/x$, $y'=y/(1-y)$, and let $G'$ be an $(x,y)$-constrained graph, via $(A',B',C')$, such that $|N_A^2(w)| = z'|A'| < (2z-1)|A'|/z$ for all $w \in C'$. Add three vertices $a,b,c$ to the graph, with edges from $a$ to every vertex in $B'$, edges from $b$ to every vertex in $A'$, and an edge between $b$ and $c$. Let this new graph be $G = (A,B,C)$, and assign weights as follows: $$\begin{aligned}
w(a)&=& 1-z\\
w(v)&=&z/|A'| \text{ for each }v\in A'\\
w(b)&=&1-x\\
w(v)&=& x/|B'| \text{ for each }v\in B'\\
w(c)&=& y\\
w(v)&=& (1-y)/|C'| \text{ for each }v\in C'\\\end{aligned}$$ Then $G$ is $(x,y)$-constrained, since $xx'+(1-x) = x$, $(1-y)y'=y$, and for all $w \in C$ we have: $$|N_A^2(w)| \le (1-z)+zz'<(1-z)+(2z-1)=z$$ Thus $\phi(x,y) < z$, as desired. This proves \[philb\].
The next two results are similar, but for $\psi$:
\[psilb1\] Suppose $z' = \psi(x',y')$. If $x,y \in (0,1]$ such that $x \le 1/(2-x')$, $y \le y'/(1+y')$, $x \le z$, $x + (1-x')y/y' \le 1$, $z \ge 1/(2-z')$, and $$x \le \frac{z-z'+x'(1-z)}{1-z'}$$ then $\psi(x,y) \le z$. It follows that if $z'=\psi(x',y')$ and $x,y \in (0,1]$ such that $x \le 1/(2-x')$, $y \le y'/(1+y')$, $x < z$, $x + (1-x')y/y' \le 1$, $z > 1/(2-z')$, and $$x < \frac{z-z'+x'(1-z)}{1-z'}$$ then $\psi(x,y) < z$.
[[**Proof.**]{} ]{}Let $G'$ be an $(x,y)$-biconstrained graph, via $(A',B',C')$, such that $|N_A^2(w)| \le z'|A'|$ for all $w \in C'$. Add three vertices $a,b,c$ to the graph, with edges from $a$ to every vertex in $B'$, edges from $b$ to every vertex in $A'$, and an edge from $b$ and $c$. Let this new graph be $G = (A,B,C)$, and assign weights as follows: $$\begin{aligned}
w(a)&=& p\\
w(v)&=&(1-p)/|A'| \text{ for each }v\in A'\\
w(b)&=&q\\
w(v)&=& (1-q)/|B'| \text{ for each }v\in B'\\
w(c)&=& y\\
w(v)&=& (1-y)/|C'| \text{ for each }v\in C'\\\end{aligned}$$ We will choose $p,q$ such that this graph is $(x,y)$-biconstrained and has $|N_A^2(w)| \le z$ for all $w \in C$. The conditions are $1-p \ge x$, $1-q \ge x$, $q \ge y$, $(1-q)x'+q \ge x$, $(1-p)x'+p \ge x$, $(1-y)y' \ge y$, $(1-q)y' \ge y$, and finally $1-p \le z$, $(1-p)z'+p \le z$. These are equivalent to the following: $$\begin{aligned}
\max\left(1-z,\frac{x-x'}{1-x'}\right) \le &p \le \min\left(1-x, \frac{z-z'}{1-z'}\right) \\
\max\left(y,\frac{x-x'}{1-x'}\right) \le &q \le \min\left(1-x,1-\frac{y}{y'}\right)\end{aligned}$$ Thus, it suffices to show that the lower bound on $p$ is at most the upper bound on $p$, and similarly for $q$. We obtain eight conditions, which simplify to those given in the problem statement. The strict inequality formulation immediately follows, since for some $\epsilon>0$ we have $\psi(x,y) \le z-\epsilon < z$. This proves \[psilb1\].
\[psilb2\] Suppose $z' = \psi(x',y')$. If $x,y \in (0,1]$ such that $y \le 1/(2-y')$, $x \le x'/(1+x')$, $x \le x'z$, $(1-y')x/x' + y \le 1$, $z \ge 1/(2-z')$, and $x \le (z-z')/(1-z')$, then $\psi(x,y) \le z$.\
\
It follows that if $z' = \psi(x',y')$ and $x,y \in (0,1]$ such that $y \le 1/(2-y')$, $x \le x'/(1+x')$, $x < x'z$, $(1-y')x/x' + y \le 1$, $z > 1/(2-z')$, and $x < (z-z')/(1-z')$, then $\psi(x,y) < z$.
[[**Proof.**]{} ]{}Let $G'$ be an $(x,y)$-biconstrained graph, via $(A',B',C')$, such that $|N_A^2(w)| \le z'|A'|$ for all $w \in C'$. Add three vertices $a,b,c$ to the graph, with an edge from $a$ to $b$, edges from $b$ to every vertex in $C'$, and edges from $c$ to every vertex in $B'$. Let this new graph be $G = (A,B,C)$, and assign weights as follows: $$\begin{aligned}
w(a)&=& p\\
w(v)&=&(1-p)/|A'| \text{ for each }v\in A'\\
w(b)&=&q\\
w(v)&=& (1-q)/|B'| \text{ for each }v\in B'\\
w(c)&=& 1-y\\
w(v)&=& y/|C'| \text{ for each }v\in C'\\\end{aligned}$$ The conditions that this graph is $(x,y)$-biconstrained with $|N_A^2(w)| \le z|A|$ for all $w \in C$ can be written as follows: $$\begin{aligned}
\max\left(x,1-z\right) \le &p \le \min\left(1-\frac{x}{x'}, \frac{z-z'}{1-z'}\right) \\
\max\left(x,\frac{y-y'}{1-y'}\right) \le &q \le \min\left(1-y,1-\frac{x}{x'}\right)\end{aligned}$$ We just need to check that the lower bound for $p$ is at most the upper bound for $p$, and similarly for $q$. This gives eight conditions, which simplify to those given in the problem statement. The strict inequality formulation immediately follows, since for some $\epsilon>0$ we have $\psi(x,y) \le z-\epsilon < z$. This proves \[psilb2\].
\[3.3gen\] Suppose $x,y \in \mathbb{Q}$, and $a/b \le 1/2$, $bx/a+y \le 1$, $x+by/a \le 1$, and either $ay \le x$ or $ax \le y$; then $\psi(x,y) < a/b$.\
\
In addition, if at least one of $x,y$ is irrational, and $a/b \le 1/2$, $bx/a+y < 1$, $x+by/a < 1$, and either $ay < x$ or $ax < y$, then $\psi(x,y) < a/b$.
[[**Proof.**]{} ]{}It suffices to prove the case with $x,y \in \mathbb{Q}$, since in the other case we can increase $x,y$ if necessary so that they are rational. Now, suppose first that $ay \le x$. Let $k+1 = \frac{b}{a}$, so the conditions are $(k+1)x+y \le 1$ and $x+(k+1)y \le 1$. Choose an integer $N \ge 1$ such that $p = xN / (1-kx)$ and $q = yN / (1-kx)$ are integers, and thus $p + q \le (x+y)N/(x+y) = N$. Let $G_1$ be the graph with vertices $\{a_1,...,a_N, a^*, b_1,...,b_N, c_1,...,c_N\}$ where each $a_i$ is adjacent to $b_i,...,b_{i+p-1}$, each $b_i$ is adjacent to $c_i,...,c_{i+q-1}$, and $a^*$ is adjacent to all of the $b_i$.
Let $m = b-a$. Let $G_2$ be the graph with vertices $\{a_1',...,a_m', b_1',...,b_m', c_1',...,c_m'\}$, where each $a_i'$ is adjacent to $b_i',...,b_{i+a-1}'$, and each $b_i'$ is adjacent to $c_i'$. Let $G$ be the disjoint union of $G_1$ and $G_2$. Let $r$ satisfy $(k+1)rN = \frac{1}{k+1}-x$. Assign weights as follows:
$$\begin{aligned}
w(a_i)&=& kr(N+1)/N\\
w(a_i')&=&1/b-r/a\\
w(a^*)&=&1/(k+1)-Nkr\\
w(b_i)&=& (1-kx)/N\\
w(b_i')&=& x/a\\
w(c_i)&=& (1-kay)/N\\
w(c_i') &=& y\\\end{aligned}$$
Now, to verify that this graph is $(x,y)$-biconstrained via $(A,B,C)$ with $|N_A^2(v)| < (a/b)|A|$ for all $v \in C$, we need to check that the following set of conditions hold:
$$\begin{aligned}
ay &\le& x\\
x &\le& p(1-kx)/N\\
y &\le& q(1-kx)/N\\
y &\le& q(1-kay)/N\\
x &\le& 1/(k+1) - r\\
x &\le& pkr(N+1)/N + 1/(k+1) - Nkr\\
1/(k+1) &>& 1/(k+1)-Nkr+(p+q-1)kr(N+1)/N\\\end{aligned}$$
The first follows from assumption. The next two follow from the definition of $p$ and $q$. The fourth is equivalent to $ay \le x$. The fifth follows from the definition of $r$. The sixth is equivalent to $rk(N-x(N+1)/(1-kx)) \le 1/(k+1)-x$, which follows from the definition of $r$. Finally, the seventh condition is equivalent to $(p+q-1)(N+1)/N < N$, which is true since $p+q \le N$. Thus, $\psi(x,y) < a/b$, as desired.
So suppose $ay > x$, and consequently $ax \le y$. Again, let $k+1 = b/a$. Choose an integer $N \ge 1$ such that $p=xN/(1-ky)$, $q = yN/(1-ky)$ are integers, and thus $p+q \le (x+y)N/(x+y) = N$. Let $G_1$ be as above. Let $m = b-a$, and let $G_2$ be the graph with vertices $\{a_1',...,a_m', b_1',...,b_m', c_1',...,c_m'\}$, where each $a_i'$ is adjacent to $b_i'$, and each $b_i'$ is adjacent to $c_i',...,c_{i+a-1}'$ (this is the earlier graph $G_2$ flipped). Let $G$ be the disjoint union of $G_1$ and $G_2$. Let $r$ satisfy $(k+1)rN \le 1/(k+1)-x$ and $r \le 1/(k+1)-y$. Assign weights as follows:
$$\begin{aligned}
w(a_i)&=& kr(N+1)/N\\
w(a_i')&=&1/b-r/a\\
w(a^*)&=&1/(k+1)-Nkr\\
w(b_i)&=& (1-ky)/N\\
w(b_i')&=& y/a\\
w(c_i)&=& (1-ky)/N\\
w(c_i') &=& y/a\\\end{aligned}$$
Now, to verify that this graph is $(x,y)$-biconstrained via $(A,B,C)$ with $|N_A^2(v)| < (a/b)|A|$ for all $v \in C$, we need to check that the following set of conditions hold:
$$\begin{aligned}
ax &\le& y\\
x &\le& p(1-ky)/N\\
y &\le& q(1-ky)/N\\
x &\le& 1/b - r/a\\
x &\le& pkr(N+1)/N + 1/(k+1) - Nkr\\
1/(k+1) &>& 1/(k+1)-Nkr+(p+q-1)kr(N+1)/N\\\end{aligned}$$
The first follows from assumption. The next two follow from the definition of $p$ and $q$. For the fourth, we have $ax \le y \le 1/(k+1) - r = a/b-r$ by the definition of $r$. The fifth and sixth follow as before. Thus, $\psi(x,y) < a/b$. This proves \[3.3gen\].
The 3/4 Level
=============
In this section we give results for when $\psi(x,y) \ge 3/4$ and $\phi(x,y) \ge 3/4$. The results are shown in Figure 1.
(0,0) – (12.5,0) node\[anchor=north\][$x$]{}; (0,0) – (0,12.5) node\[anchor=east\] [$y$]{}; at (8.4,7.7)
$(1/2,1/2)$
;
at (15\*74/99, 15\*1/99); at (15\*1/99, 15\*74/99); at (15\*71/95, 15\*1/95); at (15\*1/95, 15\*71/95); at (15\*68/91, 15\*1/91); at (15\*1/91, 15\*68/91); at (15\*65/87, 15\*1/87); at (15\*1/87, 15\*65/87); at (15\*62/83, 15\*1/83); at (15\*1/83, 15\*62/83); at (15\*59/79, 15\*1/79); at (15\*1/79, 15\*59/79); at (15\*56/75, 15\*1/75); at (15\*1/75, 15\*56/75); at (15\*53/71, 15\*1/71); at (15\*1/71, 15\*53/71); at (15\*50/67, 15\*1/67); at (15\*1/67, 15\*50/67); at (15\*47/63, 15\*1/63); at (15\*1/63, 15\*47/63); at (15\*44/59, 15\*1/59); at (15\*1/59, 15\*44/59); at (15\*41/55, 15\*1/55); at (15\*1/55, 15\*41/55); at (15\*38/51, 15\*1/51); at (15\*1/51, 15\*38/51); at (15\*35/47, 15\*1/47); at (15\*1/47, 15\*35/47); at (15\*32/43, 15\*1/43); at (15\*1/43, 15\*32/43); at (15\*29/39, 15\*1/39); at (15\*1/39, 15\*29/39); at (15\*26/35, 15\*1/35); at (15\*1/35, 15\*26/35); at (15\*23/31, 15\*1/31); at (15\*1/31, 15\*23/31); at (15\*20/27, 15\*1/27); at (15\*1/27, 15\*20/27); at (15\*17/23, 15\*1/23); at (15\*1/23, 15\*17/23); at (15\*14/19, 15\*1/19); at (15\*1/19, 15\*14/19); at (15\*11/15, 15\*1/15); at (15\*1/15, 15\*11/15); at (15\*8/11, 15\*1/11); at (15\*1/11, 15\*8/11); at (15\*5/7, 15\*1/7); at (15\*1/7, 15\*5/7); at (15\*2/3, 15\*1/3); at (15\*1/3, 15\*2/3); at (15\*1/2, 15\*1/2); at (15\*1/2, 15\*1/2); (15\*3/4, 15\*0/1) – (15\*74/99, 15\*0/1) – (15\*74/99, 15\*1/99) – (15\*71/95, 15\*1/99) – (15\*71/95, 15\*1/95) – (15\*68/91, 15\*1/95) – (15\*68/91, 15\*1/91) – (15\*65/87, 15\*1/91) – (15\*65/87, 15\*1/87) – (15\*62/83, 15\*1/87) – (15\*62/83, 15\*1/83) – (15\*59/79, 15\*1/83) – (15\*59/79, 15\*1/79) – (15\*56/75, 15\*1/79) – (15\*56/75, 15\*1/75) – (15\*53/71, 15\*1/75) – (15\*53/71, 15\*1/71) – (15\*50/67, 15\*1/71) – (15\*50/67, 15\*1/67) – (15\*47/63, 15\*1/67) – (15\*47/63, 15\*1/63) – (15\*44/59, 15\*1/63) – (15\*44/59, 15\*1/59) – (15\*41/55, 15\*1/59) – (15\*41/55, 15\*1/55) – (15\*38/51, 15\*1/55) – (15\*38/51, 15\*1/51) – (15\*35/47, 15\*1/51) – (15\*35/47, 15\*1/47) – (15\*32/43, 15\*1/47) – (15\*32/43, 15\*1/43) – (15\*29/39, 15\*1/43) – (15\*29/39, 15\*1/39) – (15\*26/35, 15\*1/39) – (15\*26/35, 15\*1/35) – (15\*23/31, 15\*1/35) – (15\*23/31, 15\*1/31) – (15\*20/27, 15\*1/31) – (15\*20/27, 15\*1/27) – (15\*17/23, 15\*1/27) – (15\*17/23, 15\*2/23);
(15\*1/7, 15\*5/7) – (15\*1/6, 15\*2/3) – (15\*1/3, 15\*2/3) – (15\*1/3,15\*1/2) – (15\*1/2, 15\*1/2) – (15\*1/2, 15\*1/3) – (15\*2/3, 15\*1/3) – (15\*2/3, 15\*1/6); (15\*5/7,15\*2/21) – (15\*5/7, 15\*1/7);
plot ([15\*(1-2\*)]{},[15\*]{}); plot ([15\*(1-3\*)]{},[15\*]{}); plot ([15\*]{},[15\*(1-2\*)]{}); plot ([15\*(3-4\*)]{},[15\*]{});
at (9,9) [ ]{}; at (4.5,4) [ ]{}; at (5,7.2)
$(1/3,1/2)$
; at (6.25,5.1)
$(1/2,1/3)$
; at (11.15, 5)
$(2/3,1/3)$
; at (3.8, 9.5)
$(1/3,2/3)$
; at (8.75,2.5)
$(2/3,1/6)$
;
plot ([15\*((5-6\*)/(11-12\*))]{}, [15\*()]{}); plot ([15\*(2-3\*)]{}, [15\*()]{}); plot ([15\*()]{}, [15\*((4\*-3)\*(4\*-3)/(16\*)]{}); plot ([15\*(1--(2-3\*)\*(2\*-1))]{}, [15\*()]{}); plot([15\*(1/2 - (2\*-1)/(4-4\*))]{}, [15\*]{}); (15\*1/3, 15\*.625) – (15\*1/3, 15\*2/3);
(15\*1/2,15\*1/2) – (15\*1/2, 15\*.4393); plot ([15\*((4\*-3)\*(4\*-3)/(16\*))]{}, [15\*()]{});
plot ([15\*(1-2\*+2\*)]{}, [15\*()]{}); (15\*5/9, 15\*1/3) – (15\*2/3, 15\*1/3) – (15\*2/3, 15\*.2113); plot ([15\*(.25\*(3-()/(1-)))]{}, [15\*()]{}); plot ([15\*(1--(2/3-)/(1-))]{}, [15\*()]{}); plot ([15\*(1-2\*)]{}, [15\*()]{}); (15\*.75, 15\*.125) – (15\*.75, 0); (15\*.25, 15\*2/3) – (15\*1/3, 15\*2/3); plot ([15\*()]{},[15\*((1/6)\*((-1)\*-sqrt(+12\*)+6))]{}); (15\*.25, 15\*2/3) – (15\*.25, 15\*.6875); (0, 15\*.75) – (15\*.15, 15\*.75); (15\*.5, 15\*.5) – (15\*.4393, 15\*.5); plot ([15\*(3-4\*)]{},[15\*]{});
at (-1, 15\*3/4)
$y=3/4$
; at (15\*3/4, -.4)
$x=3/4$
;
(r1) at (1, 12)
$\ref{maxbound}$
; (r1) to (1,15\*3/4);
(r2) at (3.8, 15\*3/4+.5)
$\ref{3/4ub5}$
; (r2) to (2.6,11.1); (r2) – (3.5,10.4);
(r1) at (4.3, 11)
$\ref{3/4ub4}$
; (r1) to (4.3,10.2);
(r1) at (6, 10.5)
$\ref{3/4ub3}$
; (r1) to (5.1,9.65); (r1) to (6.2,8.1); (r1) to (5.8,8.4);
(r1) at (7, 9.5)
$\ref{edgecounting}$
; (r1) to (5.5,9);
(r1) at (6.3, 6.45)
$\ref{3/4from5.7}$
; (r1) to (7,7.35); (r1) to (7.35,7);
(r1) at (9, 6.5)
$\ref{3/4ub1}$
; (r1) to (8,6); (r1) to (9,5.3);
(r1) at (11.5, 3.5)
$\ref{3/4ub2}$
; (r1) to (10.15,3.7); (r1) to (10.3,3.15); (r1) to (10.6,2.65); (r1) to (11.2,2.2);
(r1) at (12, 1)
$\ref{maxbound}$
; (r1) to (15\*3/4,1);
(r1) at (1.3, 10)
$\ref{3/4psilb}$
; (r1) to (1.3,10.75); (r1) to (2.2,10.2);
(r1) at (10, 1.05)
$\ref{3/4psilb}$
; (r1) to (10.75,1.3); (r1) to (10.2,2.2);
(0,0) – (12.5,0) node\[anchor=north\][$x$]{}; (0,0) – (0,12.5) node\[anchor=east\] [$y$]{}; at (7.8,7.8)
$(1/2,1/2)$
;
(15\*1/4,15\*2/3) – (15\*1/3,15\*2/3)–(15\*1/3,15\*56/97) – (15\*8/23, 15\*56/97) – (15\*8/23, 15\*8/15) – (15\*35/88, 15\*8/15) – (15\*35/88, 15\*10/19) – (15\*39/98, 15\*10/19) – (15\*39/98, 15\*1/2) – (15\*1/2,15\*1/2) – (15\*1/2, 15\*39/98) – (15\*10/19, 15\*39/98) –(15\*10/19, 15\*35/88) –(15\*8/15, 15\*35/88) – (15\*8/15, 15\*8/23) – (15\*56/97, 15\*8/23) – (15\*56/97, 15\*1/3) – (15\*2/3,15\*1/3) – (15\*2/3,15\*1/4);
plot ([15\*((3-10\*)/(4-13\*))]{},[15\*]{});
plot ([15\*]{},[15\*((3-10\*)/(4-13\*))]{});
plot ([15\*()]{}, [15\*((4\*-3)\*(4\*-3)/(16\*))]{}); plot ([15\*]{},[15\*(1-)]{});
plot ([15\*((4\*-3)\*(4\*-3)/(16\*))]{},[15\*]{}); plot ([15\*(1-)]{},[15\*()]{});
plot([15\*((1-)/(1+-3\*))]{},[15\*()]{});
plot([15\*]{},[15\*((1-)/(1+-3\*))]{});
(15\*3/4, 15\*.1886) – (15\*3/4, 0); (0, 15\*3/4) – (15\*.1886, 15\*3/4); (15\*.4393, 15\*.5) – (15\*.5, 15\*.5) – (15\*.5, 15\*.4393);
at (9,9) [ ]{}; at (4.3,5.1) [ ]{};
at (11.15, 5)
$(2/3,1/3)$
; at (6, 10.3)
$(1/3,2/3)$
; at (3.8, 9.5)
$(1/4,2/3)$
; at (8.75,3.7)
$(2/3,1/4)$
;
at (-1, 15\*3/4)
$y=3/4$
; at (15\*3/4, -.4)
$x=3/4$
;
(r1) at (1.5, 12.1)
$\ref{maxbound}$
; (r1) to (1.5,15\*3/4+.1);
(r1) at (12.1, 1.5)
$\ref{maxbound}$
; (r1) to (15\*3/4+.1,1.5);
(r1) at (8.5, 7.25)
$\ref{3/4from5.7}$
; (r1) to (8,6.5); (r1) to (7.65,7);
(r1) at (12, 3.75)
$\ref{3/4phiub}$
; (r1) to (11,3.6);
(r1) at (9.75, 1.75)
$\ref{3/4philb}$
; (r1) to (10.7,2.7);
\[3/4ub1\] Suppose $y > 1/3$, $x \ge 1/2$, and $2y-2y^2 > 1-x$. Then $\psi(x,y) \ge 3/4$.
[[**Proof.**]{} ]{}We can assume that $x,y$ are rational, and by multiplying vertices if necessary that $y|B| \in \mathbb{Z}$. Let $v_1 \in C$, and let $B_1 \subseteq N(v_1)$ be such that $|B_1| = y|B|$. Choose $v_2 \in C$ with at least $y|B \setminus B_1|=y(1-y)|B|$ neighbors in $B \setminus B_1$. Then, letting $A_i = N_A^2(v_i)$, we have that $A_1 \cup A_2 = A$, since $y+y(1-y) > 1-x$ by assumption. Let $B_2 \subseteq N(v_2)$ be such that $|B_2| = y|B|$. If $B_1 \cap B_2 \ne \emptyset$, then it follows that $|A_1 \cap A_2| \ge x|A|$, and consequently: $$|A_1|+|A_2| \ge (1 + x)|A| \ge 3|A|/2$$ Then it follows that $|A_i| \ge 3|A|/4$ for some $A_i$, a contradiction.
Thus, $B_1$ and $B_2$ are disjoint. Choose $v_3 \in C$ with at least $y(1-2y)|B|$ neighbors in $B \setminus (B_1 \cup B_2)$. Since $y > 1/3$ and $B_1,B_2$ are disjoint, without loss of generality $B_1 \cap N(v_3) \ne \emptyset$. Then by assumption $y+y(1-2y)>1-x$, so $|B_1 \cup N(v_3)|>1-x$ and consequently if $A_3 = N_A^2(v_3)$, we have that $A_1 \cup A_3 = A$. But then since there is a vertex in $B_1 \cap N(v_3)$, we have that $|A_1 \cap A_3| \ge x|A|$, and thus: $$|A_1|+|A_3| \ge (1+x)|A| \ge 3|A|/2$$ and one of $A_1,A_3$ has size at least $3|A|/4$, a contradiction. This proves \[3/4ub1\].
\[3/4ub2\] Suppose that $x > 2/3$, $x+2y>1$, $x+y/(4(1-2y)) \ge 3/4$, and either $y(5-6y)/(3(1-y)) \ge 1-x$ or $x+y/(4(1-y)) \ge 3/4$. Then $\psi(x,y) \ge 3/4$.
[[**Proof.**]{} ]{}Let $G$ be $(x,y)$-biconstrained via $(A,B,C)$. Let $H$ be the graph with $V(H) = V(C)$ and $(uv) \in E(H)$ if and only if $u$ and $v$ have a common neighbor in $B$. Let $\alpha(H)$ denote the size of the largest stable set in $H$. I claim $\alpha(H) < 4$. Suppose on the contrary that there exists a stable set $\{v_1,v_2,v_3,v_4\}$ of size four in $H$. Then because $x+2y>1$, we know that every pair of these vertices have their second neighborhoods cover $A$, and consequently every vertex in $A$ is a second neighbor of at least three of these vertices. Then, by averaging, one of the $v_i$ is a second neighbor of at least $3|A|/4$ of the vertices in $A$, a contradiction.
Now, we consider each of the three possible values for $\alpha(H)$ individually. If $\alpha(H)=1$, take $v_1 \in C$, and let $A_1 = N_A^2(v_1)$. Then every vertex $v \in C$ has a common neighbor with $v_1$. Now, choose $w \in B$ with at least $x|A \setminus A_1|/(1-y) > x|A|/(4-4y)$ neighbors in $A \setminus A_1$, and let $v_2 \in C$ be a neighbor of $w$. Then: $$\begin{aligned}
|N_A^2(v_2)| > (x+x/(4-4y))|A| > (2/3 + 1/6)|A| = 5|A|/6 \ge 3|A|/4\end{aligned}$$ which is a contradiction.
If $\alpha(H) = 2$, let $\{v_1,v_2\}$ be a maximum stable set. Then every vertex $v \in C$ has a common neighbor with $v_1$ or $v_2$.\
\
(1) [*For all $v_1,v_2,v_3\in C$, if $|N(v_1) \cap N(v_2)| \ne 0$ and $|N(v_2) \cap N(v_3)| \ne 0$, then $|N(v_1) \cap N(v_3)| \ne 0$.*]{}\
\
Suppose $N(v_1)$ and $N(v_3)$ are disjoint, and let $w_1, w_3$ be common neighbors of $v_2$ with $v_1$ and $v_3$, respectively. Then if $A_i = N_A^2(v_i)$, since $x+2y>1$ we have $A_1 \cup A_3 = A$, and consequently $|A_1 \cap A_3| < |A|/2$. Then $N_A(w_1) \cap N_A(w_3) \subseteq A_1 \cap A_3$, so $$|N_A^2(v_2)| \ge |N_A(w_1) \cup N_A(w_3)| > (2x-1/2)|A| > 5|A|/6$$ which gives a contradiction. This proves (1).
Thus, by (1) we have that every vertex $v \in C$ has a common neighbor with exactly one of $v_1,v_2$. This implies that the bipartite graph $(B,C)$ has two components, and furthermore that these components are complete. Let the components be $P_1$ and $P_2$, and let $B_i = P_i \cap B$, $C_i = P_i \cap C$. We have that $|C_i| \ge y|C|$, which implies that $|C_i| \le (1-y)|C|$.
Now, suppose first that $x+\frac{y}{4(1-y)} \ge 3/4$. Then, without loss of generality $|B_1| \ge |B|/2$, and thus every vertex in $A \setminus A_1$ has a neighbor in $B_1$ and therefore hits at least fraction $\frac{y}{1-y}$ of the vertices in $C_1$, so there exists $w \in C_1$ with at least $y|A \setminus A_1|/(1-y) > \frac{y}{4(1-y)}|A|$ neighbors in $A \setminus A_1$. Then $w$ has a common neighbor with $v_1$ since $C_1$ is a complete component, and thus: $$\begin{aligned}
|N_A^2(w)| > \frac{y|A|}{4(1-y)}+x|A| \ge 3|A|/4\end{aligned}$$ which is a contradiction.
So we can assume $x+y/(4(1-y)) < 3/4$, which implies that $\frac{y(5-6y)}{3-3y} \ge 1-x$ by assumption. Without loss of generality $|C_1| \le |C|/2$. If $|B_1| \ge |B|/3$, then every vertex $v \in A \setminus A_1$ has a neighbor in $B_1$, and consequently $y|C|$ neighbors in $C_1$. Then every $v \in A \setminus A_1$ hits at least fraction $2y$ of the vertices in $C_1$, so it follows that there exists $w \in C_1$ with at least $2y|A \setminus A_1|>y|A|/2$ neighbors in $A \setminus A_1$. Now, we know that $w$ has a common neighbor in $B$ with $v_1$, and thus we have: $$\begin{aligned}
|N_A^2(w)| > (x+y/2)|A| > (x+(1-x)/4)|A| > 3|A|/4\end{aligned}$$ since $x > 2/3$ and $x+2y>1$.
Thus we can assume that $|B_1| < |B|/3$. Now, every vertex in $B_2$ hits at least fraction $y/(1-y)$ of the vertices of $C_2$. Choose $u \in C_2$, and let $P = N(u)$. Choose $v \in C_2$ with at least $y(|B_2|-|P|)/(1-y)$ neighbors in $B_2 - P$. I claim that every vertex in $A$ is a second-neighbor of either $u$ or $v$. Indeed, if $S = N(u) \cup N(v)$, then: $$\begin{aligned}
|S| + x|B| - |B| &\ge |P| + \frac{y(|B_2|-|P|)}{1-y} + x|B|- |B| \\
&= \frac{y}{1-y}|B_2| + \frac{1-2y}{1-y}|P| +(x-1)|B| \\
&> \frac{2y}{3(1-y)}|B| + \frac{y(1-2y)}{1-y}|B| +(x-1)|B| \\
&= \left(\frac{y(5-6y)}{3-3y} + x - 1\right)|B| \\
&\ge 0\end{aligned}$$ by assumption. Thus, $N_A^2(u) \cup N_A^2(v) = A$, and since $u$ and $v$ have a common neighbor in $B_2$ we have that: $$|N_A^2(u)| + |N_A^2(v)| \ge (x+1)|A| > 5|A|/3$$ and thus one of $u,v$ has at least $5|A|/6$ second-neighbors in $A$, a contradiction.
Finally, if $\alpha(H) = 3$, let $\{v_1,v_2,v_3\}$ be a maximum stable set of $H$, and let $A_i = N_A^2(v_i)$. Then since $x+2y>1$, the pairwise unions of the $A_i$ are equal to $A$, so it follows that the pairwise intersections of the $A_i$ all have size less than $|A|/2$. Since there is no stable set of size $4$ in $C$, it follows that every $v \in C$ has a common neighbor with at least one of the $v_i$, and by (1) we know that every vertex $v \in C$ has a common neighbor with exactly one of the $v_i$. It follows that $(B,C)$ has three connected components, and furthermore that they are all complete. Call them $P_1,P_2,P_3$ and let $B_i = P_i \cap B$, $C_i = P_i \cap C$. Now, suppose without loss of generality that $|B_1| \ge |B|/3$. Then since $x > 2/3$, every vertex in $A \setminus A_1$ has a neighbor in $B_1$ and thus sees at least fraction $y/(1-2y)$ of $C_1$, so there exists $w \in C_1$ with at least $y|A \setminus A_1|/(1-2y) > y|A|/(4-8y)$ second-neighbors in $A \setminus A_1$. Then: $$|N_A^2(w)| \ge (x+y/(4-8y))|A| \ge 3|A|/4$$ is clearly true if $x+y/(4(1-y)) \ge 3/4$. If, on the other hand, we have $y(5-6y)/(3-3y) \ge 1-x$, then we have $$x+\frac{y}{4-8y} \ge 1 - \frac{y(5-6y)}{3-3y} + \frac{y}{4-8y} \ge \frac{3}{4}$$ is equivalent to $48y^3-67y^2+26y-3 \le 0$, which is true for $y \le .223$. Since all points of the upper bound curve satisfy $y < .223$, it follows that it suffices to show the claim under the additional assumption that $y \le .223$, so it follows that $x+y/(4-8y) \ge 3/4$, as desired. This proves \[3/4ub2\].
\[3/4ub3\] Suppose $y > 1/2$, $x > 1/3$, $3y+x>2$, and either $(3x-1)/(12-12y-4) + x \ge 1/2$ or $x \ge (3-4y)/(4-4y)$, then $\psi(x,y) \ge 3/4$.
Let $G$ be $(x,y)$-biconstrained via $(A,B,C)$, and let the graph $H$ have $V(H) = V(C)$, with $(uv) \in E(H)$ if and only if $|N(u) \cup N(v)| \le (1-x)|B|$. First, we note that there is no stable set in $H$ of size at least four. For if there was a stable set $\{v_1,v_2,v_3,v_4\}$, then since every pair have their second-neighbor sets in $A$ having union equal to $A$, it follows that each vertex in $A$ is a second-neighbor of at least three of the $v_i$, so by averaging one of the $v_i$ has at least $3|A|/4$ second-neighbors in $A$, a contradiction.\
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(1) [*For all $v_1,v_2\in C$, if $|N(v_1)\cup N(v_2)|\le(1-x)|B|$ then $|N^2_A(v_1)\cap N^2_A(v_2)|>(x+1/4)|A|$.*]{}\
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We have $$|N(v_1)\cap N(v_2)| > 2y-(1-x)>1-y$$ by assumption, and also that $x>1/3$ and $y>1/2$ imply $$|N(v_1)\cap N(v_2)| > 2y+x-1>1/3>1-x/(1-y)$$ Thus, \[1-y\] gives that $|N^2_A(v_1)\cup N^2_A(v_2)|>(x+1/4)|A|$. This proves (1).
Now, we deal with the cases $\alpha(H) = 1$ and $\alpha(H) = 2$ together. If $\alpha(H)=1$, let $v_1$ and $v_2$ be any two vertices, and if $\alpha(H)=2$, let $\{v_1,v_2\}$ be a maximum stable set. Then partition $C = C_1 \cup C_2$ such that for all $v \in C_i$ we have that $|N(v) \cup N(v_i)| \le (1-x)|B|$. Then by (1) we have that for all $v \in C_i$, $v$ and $v_i$ have more than $(x+1/4)|A|$ common second-neighbors in $A$.
Now, suppose without loss of generality that $|C_1| \ge 1/2$. If $B_1 = N(v_1)$, $A_1 = N_A^2(v_1)$, choose $u \in B \setminus B_1$ with at least $x|A_1|/(1-y) > x|A|/(4-4y)$ neighbors in $A \setminus A_1$, and let $v \in C_1$ be a neighbor of $u$. Then: $$\begin{aligned}
|N_A^2(v)| > x|A|/(4-4y) + (x+1/4)|A| \ge 3|A|/4\end{aligned}$$ since $x > 1/3$, $y > 1/2$, a contradiction.
If $\alpha(H)=3$, let $\{v_1,v_2,v_3\}$ be a maximum stable set. Let $B_i = N(v_i)$ and $A_i = N_A^2(v_i)$. Suppose first that $(3x-1)/(12-12y-4) + x \ge 1/2$. Partition $C = C_1 \cup C_2 \cup C_3$ such that for all $v \in C_i$ we have that $|N(v) \cup N(v_i)| \le (1-x)|B|$. If $|C_i| > (1-y)|C|$ for some $i$, then every vertex in $B$ has a neighbor in $C_i$, so, identically to before, choose $u \in B$ with more than $x|A|/(4-4y)$ neighbors in $A-A_i$, and let $w \in C_i$ be a neighbor of $u$. Then since $x/(4-4y)+x+1/4 \ge 3/4$, we again obtain a contradiction. So, we can assume that $|C_i| \le (1-y)|C|$, which implies that every vertex in $B$ has neighbors in at least two of the $C_i$. Partition $B = S_1 \cup S_2 \cup S_3$ such that every vertex in $S_1$ has a neighbor in both $C_2$ and $C_3$, and similarly for $S_2$ and $S_3$. Then, without loss of generality, $|S_1| \le 1/3$. Every vertex in $S_2 \cup S_3$ has a neighbor in $C_1$, by definition. Then, every vertex in $A \setminus A_1$ hits at least fraction $(x-1/3)/(1-y-1/3)$ of the vertices in $B \setminus (B_1 \cup S_1)$, so there exists $u \in B \setminus (B_1 \cup S_1)$ with more than $(3x-1)/(12-12y-4)$ neighbors in $A \setminus A_1$. Let $w \in C_1$ be a neighbor of $u$ in $C_1$, which exists since $u \notin S_1$. Then: $$\begin{aligned}
|N_A^2(w)| > (3x-1)|A|/(12-12y-4) + x|A| + 1/4|A| \ge 3|A|/4\end{aligned}$$ which is a contradiction.
Finally, if $(3x-1)/(12-12y-4) + x < 1/2$ then we have that $x \ge (3-4y)/(4-4y)$ by assumption (note here we are still in the $\alpha(H)=3$ case). I claim each component of $H$ is complete. If not, then there exists an induced path of length two in $H$, namely three vertices $u_1,u_2,u_3 \in C$ such that (letting $B_i = N(u_i), A_i = N_A^2(u_i)$): $$\begin{aligned}
|B_1 \cup B_2| \le (1-x)|B| \\
|B_2 \cup B_3| \le (1-x)|B| \\
|B_1 \cup B_3| > (1-x)|B| \end{aligned}$$
Now, if $|B_1 \cap B_3| > (1-y)|B|$, since $x > y(1-y)$ for $x > 1/3$ and $y > 1/2$, we have by \[1-y\] that $|A_1 \cap A_3| > (x + 1/4)|A| \ge |A|/2$, but then since $A = A_1 \cup A_3$, we have $|A_1| + |A_3| \ge 3|A|/2$, so one of $A_1,A_3$ has size at least $3|A|/4$, a contradiction. Thus $|B_1 \cap B_3| \le (1-y)|B|$, and thus $|B_3 \setminus B_1| \ge (2y-1)||B|$. But we know $|B_2 \setminus B_1|\le (1-x-y)|B|$, since $|B_1| \ge y|B|$ and $|B_1 \cup B_2| \le (1-x)|B|$, and similarly $|B_3 \setminus B_2| \le (1-x-y)|B|$. Then, since: $$B_3 \setminus B_1 \subseteq (B_2 \setminus B_1) \cup (B_3 \setminus B_2)$$ we have that: $$\begin{aligned}
(2y-1)|B| \le |B_3 \setminus B_1| \le |B_2 \setminus B_1| + |B_3 \setminus B_2| \le 2(1-x-y)|B|\end{aligned}$$ which is equivalent to $4y+2x \le 3$. Using $x \ge (3-4y)/(4-4y)$, this becomes $2(3-4y)/(4-4y) \le 3-4y$, which is equivalent to $y \le 1/2$, contradicting our assumption that $y > 1/2$.
Thus, there is no induced path of length two in $H$, and consequently every component of $H$ is complete. Then, since there exists a stable set of size three, it follows that $H$ has three connected components (since there does not exist a stable set of size four), which are furthermore all complete. Write $C = C_1 \cup C_2 \cup C_3$ where the $C_i$ are the three components of $H$. As before, we can assume $|C_i| \le (1-y)|C|$. I claim that there does not exist a vertex $w \in B$ with a neighbor in all three of the $C_i$. Suppose there does exist such a vertex, and let $v_1,v_2,v_3$ be neighbors of $w$ in $C_1,C_2,C_3$ respectively. Let $B_i = N(v_i)$, $A_i = N_A^2(v_i)$. We know that the pairwise unions of the $B_i$ have size greater than $(1-x)|B|$, so thus the pairwise unions of the $A_i$ are equal to $A$. Furthermore, the pairwise intersections of the $A_i$ all have size less than $|A|/2$. Let $S_1 = B_2 \cap B_3$, and similarly define $S_2, S_3$. Let $T_1 = A_2 \cap A_3$, and similarly define $T_2, T_3$. Then since every vertex in $A$ is a second-neighbor of at least two of the $v_i$, it follows that every vertex in $A$ is in at least one of the $T_i$. Now, since $w \in N(v_1) \cap N(v_2) \cap N(v_3)$, we have that $w \in S_1 \cap S_2 \cap S_3$ as well. Then $|T_i|<|A|/2$, and we know $T_1 \cup T_2 \cup T_3 = A$, so $T_2$ and $T_3$ cover $A \setminus T_1$, which has size greater than $|A|/2$. Thus, without loss of generality, more than $|A|/4$ vertices of $T \setminus T_1$ belong to $T_2$. But $|T_1 \cap T_2| \ge x|A| > |A|/4$, since $N(w) \subseteq T_1 \cap T_2$, so we obtain $|T_2| > |A|/2$, which is a contradiction.
So there cannot exist a vertex $w \in B$ with a neighbor in all three of the $C_i$. In addition, if $w \in B$ has $N(w)$ contained in some $C_i$, then $|C_i| \ge y|C| > (1-y)|C|$, contrary to our earlier assumption that $|C_i| \le (1-y)|C|$ (this case is the same as the two-component case). Thus, every vertex $w \in B$ has neighbors in exactly two of the $C_i$, so it follows that we can partition $B = R_1 \cup R_2 \cup R_3$ such that every vertex in $R_1$ has neighbors in $C_2$ and $C_3$ but not $C_1$, and similarly for $R_2,R_3$. Then, without loss of generality, $|R_1| \le 1/3$. For all $w \in B \setminus R_1$, $w$ has its neighbors contained in, for example, $C_1$ and $C_2$ (the other case, $C_1$ and $C_3$, is identical). Since $|C_2|\le(1-y)|C|$, it follows that $w$ has at least $y|C|-(1-y)|C|=(2y-1)|C|$ neighbors in $C_1$. Then, since $x > 1/3$, we have that every vertex in $A \setminus A_1$ has a neighbor in $B \setminus R_1$, and consequently has at least fraction $(2y-1)/(1-y)$ second-neighbors in $C_1$, so there exists $u \in C_1$ with greater than $(2y-1)|A|/(4-4y)$ second-neighbors in $A \setminus A_1$, and consequently: $$\begin{aligned}
|N_A^2(u)| > (x+1/4+(2y-1)/(4-4y))|A| \ge 3|A|/4\end{aligned}$$ by the assumption $x \ge (3-4y)/(4-4y)$. This proves \[3/4ub3\].
\[edgecounting\] Suppose $1/2 < y < 2/3$ and $(2-3y)(2y-1)>1-x-y$; then $\psi(x,y) \ge 3/4$.
[[**Proof.**]{} ]{}We can assume that $y \in \mathbb{Q}$, and that $y|B| \in \mathbb{Z}$, by decreasing $y$ and blowing up vertices if necessary. Now, let $H$ be a graph with $V(H) = C$ and $(uv) \in E(H)$ if and only if $|N(u) \cup N(v)| \le (1-x)|B|$. We proceed by cases by the value of $\alpha(H)$, noting that $\alpha(H) < 4$ since if $\{v_1,v_2,v_3,v_4\}$ was a stable set of size four in $H$, then we would have that every vertex $v \in A$ is a second-neighbor of at least three of the $v_i$, and consequently some $v_i$ has at least $3|A|/4$ second-neighbors in $A$.
Now, suppose $\alpha(H) = 1$. Choose $v_1, v_2 \in C$ at random. The expectation of the size of $|N(v_1) \cup N(v_2)|$ is at least $(1-(1-y)^2)|B| = (2y-y^2)|B| > (1-x)|B|$ since $1-x < 2/3 < 2y-y^2$ for $1/2 < y < 2/3$. Thus, we cannot have $\alpha(H) = 1$.
Now, suppose $\alpha(H) = 2$. Let $\{v_1,v_2\}$ be a maximum stable set in $H$. Then partition $C = C_1 \cup C_2$ such that for all $v \in C_i$ we have $|N(v_i) \cup N(v)| \le (1-x)|B|$. Let $B_i \subseteq N(v_i)$ such that $|B_i|=y|B|$. We are going to count the number of edges from $B$ to $C$ which do not go from a vertex in $B_1$ to a vertex in $C_1$, or a vertex in $B_2$ to a vertex in $C_2$. Call this set of edges $E$. On the one hand, we have $|E| \le (1-x-y)|B||C|$, since every vertex in $C_i$ can have at most $(1-x-y)|B|$ edges outside of $B_i$. On the other hand, if we let $c_i = |C_i|$, looking at the vertices in $B$ we have that the vertices of $B_1 \setminus B_2$ need at least $(y-c_1)|C|$ edges to $C_2$, and similarly for $B_2 \setminus B_1$, and the vertices in $B \setminus (B_1 \cup B_2)$ each need at least $y|C|$ edges in $E$. So if we let $t|B| = |B_1 \setminus B_2|$, we get that $$1-x-y \ge \frac{|E|}{|B||C|} \ge t(y-c_1)+t(y-c_2)+(1-y-t)y$$ and the right-hand side is $$t(y-1)+(1-y)y \ge (1-y)(y-1)+(1-y)y = (1-y)(2y-1)$$ but by assumption we have $1-x-y < (2-3y)(2y-1) < (1-y)(2y-1)$ since $y > 1/2$, a contradiction.
Finally, suppose $\alpha(H) = 3$. Let $\{v_1,v_2,v_3\}$ be a maximum stable set in $H$ and partition $C = C_1 \cup C_2 \cup C_3$ such that for all $v \in C_i$ we have $|N(v_i) \cup N(v)| \le (1-x)|B|$. Choose $B_i \subseteq N(v_i)$ such that $|B_i| = y|B|$, and let $a_1 = |B_1 \setminus (B_2 \cup B_3)|$, $a_2 = |B_2 \setminus (B_1 \cup B_3)|$, $a_3 = |B_3 \setminus (B_1 \cup B_2)|$, $b_1 = |(B_2 \setminus B_1) \cap (B_3 \setminus B_1)|$, $b_2 = |(B_1 \setminus B_2) \cap (B_3 \setminus B_2)|$, $b_3 = |(B_1 \setminus B_3) \cap (B_2 \setminus B_3)|$, $b_4 = |B_1 \cap B_2 \cap B_3|$, and $a_4 = |B \setminus (B_1 \cup B_2 \cup B_3)|$. Also, let $c_i = |C_i|/|C|$ and $d_i = 1 - c_i$.
As before, let $E$ be the set of edges from $B$ to $C$ that are not between any of the pairs $(B_i,C_i)$ for $1 \le i \le 3$. Now, we have $3y+a_4-1 = \left(\sum_{i=1}^4 b_i\right) + b_4$, and so $\sum_{i=1}^4 b_i = 3y+a_4-b_4-1$, and consequently $\sum_{i=1}^4 a_i = 2-3y-a_4+b_4$. We proceed by casework on how many of the values of the $d_i$ are at least $y$. In all cases, we have $|E|/(|B||C|) \le (1-x-y)$ by the same argument as before, and in all cases we show that $|E|/(|B||C|) \ge (2-3y)(2y-1)$ for the desired contradiction.
First, suppose that $c_1 > y$. Then every vertex in $B \setminus B_1$ needs at least $(y-(1-y))|C| = (2y-1)|C|$ edges in $E$. It follows that $|E| \ge (1-y)(2y-1)|B||C| > (2-3y)(2y-1)|B||C|$, a contradiction. So, for the remainder of the proof, we can assume $c_i \le y$ for all $i$.
Now, suppose that $d_i \ge y$ for all $i$, which is equivalent to $c_i \le 1-y$. Now, in this case we have: $$\begin{aligned}
\frac{|E|}{|B||C|} &\ge \left(\sum_{i=1}^3 a_i \right)(2y-1)+a_4 y \\
&= (2-3y-2a_4+b_4)(2y-1)+a_4 y \\
&\ge (2-3y)(2y-1)+a_4(2-3y) \\
&\ge (2-3y)(2y-1)\end{aligned}$$
Now, suppose that $d_1 < y$, $d_2 \ge y$, $d_3 \ge y$ (without loss of generality, this is the case where exactly one of the $d_i$ is less than $y$). Note first that $$\begin{aligned}
b_1 &= (1-y)-(a_2+a_3+a_4) \\
&= (1-y)-(2-3y-a_4+b_4-a_1) \\
&= 2y-1+a_4-b_4+a_1\end{aligned}$$ and thus we have $$\begin{aligned}
\frac{|E|}{|B||C|} &\ge a_1(y-c_1)+b_1(y-(1-c_1))+a_2(y-c_2)+a_3(y-c_3)+a_4 y\\
&\ge \left(\sum_{i=1}^3 a_i\right)(2y-1) + a_4 y + (2y-1+a_4-b_4)(y-1+c_1) \\
&= (2-3y)(2y-1) + (-2a_4+b_4)(2y-1)+a_4y + (2y-1+a_4-b_4)(y-1+c_1) \\
&= (2-3y)(2y-1) + a_4(1+c_1-2y)+b_4(y-c_1)+(2y-1)(y-1+c_1) \\
&\ge (2-3y)(2y-1)\end{aligned}$$ since all of the terms after the first one in the second-to-last line are non-negative since $1/2 < y < 2/3$ and $1-y < c_1 < y$.
Finally, suppose without loss of generality that $d_1 < y$, $d_2 < y$, $d_3 \ge y$. This is the final case since $c_i > 1-y$ for all $i$ implies $1 > 3-3y$, a contradiction. Then we have $$\begin{aligned}
b_2 &= (1-y)-(a_1+a_3+a_4) \\
&= (1-y)-(2-3y-a_4+b_4-a_2) \\
&= 2y-1+a_4-b_4+a_2\end{aligned}$$ and we have $$\begin{aligned}
\frac{|E|}{|B||C|} &\ge a_1(y-c_1)+b_1(y-(1-c_1))+a_2(y-c_2)+b_2(y-(1-c_2))+a_3(y-c_3)+a_4y \\
&\ge \left(\sum_{i=1}^3 a_i \right)(2y-1)+a_4 y + (2y-1+a_4-b_4)(y-1+c_1) + (2y-1+a_4-b_4)(y-1+c_2) \\
&= (2-3y)(2y-1) +a_4(c_1+c_2-y)+b_4(1-c_1-c_2)+(2y-1)(2y-2+c_1+c_2) \\
&\ge (2-3y)(2y-1)\end{aligned}$$ since $1/2 < y < 2/3$ and $1-y < c_1,c_2 < y$. This proves \[edgecounting\].
\[3/4ub4\] Suppose $x \ge 1/4$ and $y > 2/3$. Then $\psi(x,y) \ge 3/4$.
[[**Proof.**]{} ]{}Suppose not. We have that $y > 2-2y$ since $y > 2/3$, and also $y > (1-y)+1-x/(1-y)$ is equivalent to $x>2(1-y)^2$, which is implied by $x \ge 1/4$ and $y > 2/3$. Then, applying \[1-y\] to $N(v)$ for $k=2$ gives that for any $v \in C$: $$|N_A^2(v)|>(x+1/2)|A| \ge 3|A|/4$$ which is a contradiction. This proves \[3/4ub4\].
\[3/4ub5\] Suppose: $$2y+x-1>1-y+\max(1-y,1-x/(1-y))$$ Then $\psi(x,y) \ge 3/4$.
[[**Proof.**]{} ]{}We have that if $v_1,v_2 \in C$ with $|N(v_1) \cup N(v_2)| \le (1-x)|B|$ then $|N(v_1) \cap N(v_2)| \ge (2y+x-1)|B|$. Thus, \[1-y\] applied for $k=2$ tells us that, for all such $v_1,v_2 \in C$, more than $(x + 1/2)|A|$ vertices in $A$ have a neighbor in $N(v_1) \cap N(v_2)$. Now, let $H$ be the graph with vertices $C$ and an edge between $u,v$ if $|N(u) \cup N(v)| \le (1-x)|B|$. Then there is no stable set of size at least four in $H$, because if $\{v_1,v_2,v_3,v_4\}$ is such a stable set, then every pair of $v_i, v_j$ have second neighborhoods covering $A$, and thus every $v \in A$ is second-neighbors with at least three of the four $v_i$, so, by averaging, one of the $v_i$ has at least $3|A|/4$ second-neighbors in $A$, a contradiction.
It follows that you can find three vertices $v_1,v_2,v_3 \in C$ and a partition $C = C_1 \cup C_2 \cup C_3$ such that for all $v \in C_i$, we have $|N(v) \cup N(v_i)| \le (1-x)|B|$. Then suppose, without loss of generality, that $|C_1| \ge 1/3$. Then since $y > 2/3$, every vertex in $B$ has a neighbor in $C_1$. Then let $B_i= N(v_i)$, $A_i = N_A^2(v_i)$, and choose $u \in B \setminus B_1$ with more than $x|A|/(4-4y)$ neighbors in $A \setminus A_1$. Let $w \in C_1$ be a neighbor of $u$ in $C_1$. Then: $$\begin{aligned}
|N_A^2(w)| > (x+1/2+x/(4-4y))|A| > (1/2 + 7x/4)|A| \ge 3|A|/4\end{aligned}$$ since the second assumed condition is equivalent to: $$x > 3(1-y)^2 / (2-y) > 3/20 > 1/7$$ This proves \[3/4ub5\].
\[3/4from5.7\] If $y > 1/2$ and $16x^2y \ge (3-4x)^2$, then $\phi(x,y) \ge 3/4$.
[[**Proof.**]{} ]{}Apply \[5.7\] with $z = 3/4$.
\[3/4phiub\] If $y < 1/3$ and $$x > \frac{1-y}{1+y-3y^2}$$ then $\phi(x,y) \ge 3/4$.
[[**Proof.**]{} ]{}Let $$p = \frac{1-x-y}{1-3y}$$ and note that the assumption is equivalent to $p < xy$, since $0 < y < 1/3$.
Now, let $G$ be $(x,y)$-constrained via $(A,B,C)$. We can assume that $x,y \in \mathbb{Q}$, and also that every vertex in $A$ goes to strictly more than $x|B|$ vertices of $B$, by reducing $x$ and $y$ by a little bit and blowing up vertices, if necessary. Then by blowing up vertices if necessary we can also assume that $y|B| \in \mathbb{Z}$ and $p|B| \in \mathbb{Z}$. We also assume that $x + y \le 1$, since otherwise $\phi(x,y) = 1$.
Let $s \in [0,1]$. Now, choose $v_1 \in C$ with at least $y|B|$ neighbors in $B$, and let $B_1 \subseteq N(v_1)$ with $|B_1| = y|B|$. Then, choose $v_2 \in C$ such that $sb_0+b_2 \ge y(sy+(1-y))$, where $b_0|B| = |N(v_2) \cap B_1|$ and $b_2|B| = |N(v_2) \backslash B_1|$. I claim there exists such an $s \in [0,1]$ such that $b_0 + b_2 \ge p$, and $b_2 \ge 1 - x - y$. The first will be satisfied if $y(sy+(1-y)) \ge p$, since: $$\begin{aligned}
b_0 + b_2 &\ge sb_0 + b_2 \\
&\ge y(sy+(1-y)) \\
&\ge p\end{aligned}$$ The second condition will be satisfied if $y(sy+(1-y)) \ge sy+1-x-y$, since: $$\begin{aligned}
sy+b_2 &\ge sb_0+b_2 \\
&\ge y(sy+(1-y)) \\
&\ge sy+1-x-y\end{aligned}$$ So we want: $$\begin{aligned}
\frac{p-y(1-y)}{y^2} \le s \le \frac{2y-y^2+x-1}{y-y^2}\end{aligned}$$ and in order to be able to pick some $0 \le s \le 1$ to satisfy both these inequalities, we need both the lower bound to be at most one and the upper bound to be at least zero. Here, note that: $$\begin{aligned}
\frac{p-y(1-y)}{y^2} \le 1 \Leftrightarrow p \le y\end{aligned}$$ and: $$\begin{aligned}
\frac{2y-y^2+x-1}{y-y^2} \ge 0 \Leftrightarrow 2y-y^2+x-1 \ge 0\end{aligned}$$ so such an $s$ will exist if the following conditions hold: $$\begin{aligned}
p &\le \frac{xy}{1-y} \\
2y - y^2 + x - 1 &\ge 0 \\
p &\le y\end{aligned}$$ Note that $p \le \frac{xy}{1-y}$ and $x + y \le 1$ implies $p \le y$, so it follows that the only conditions we need are $p \le \frac{xy}{1-y}$ and $2y-y^2+x-1 \ge 0$. The first condition is implied by $p < xy$, and for the second condition note that $$x > \frac{1-y}{1+y-3y^2} \ge (1-y)^2$$ is true for $0 < y < 1/3$.
Let $t \in [0,1]$. Since $p \le \frac{xy}{1-y} \le y$, we can choose $X_1 \subseteq N(v_1)$ such that $|X_1| = p|B|$. Choose $X_2 \subseteq N(v_2)$ such that $|X_2| = p|B|$. Now, we proceed in a similar fashion. Let $c_0, c_2$ be such that $c_0|B| = |N(v_3) \cap (X_1 \cup X_2)|$ and $c_2|B| = |N(v_3) \cap (B \setminus (X_1 \cup X_2)|$. Choose $v_3 \in C$ such that $$tc_0|B| + c_2|B| \ge ty|X_1 \cup X_2| + y(|B| \setminus |X_1 \cup X_2|) = y(|B|+(t-1)|X_1 \cup X_2|) \ge y(1+(t-1)2p)|B|$$
I claim there exists $0 \le t \le 1$ such that the following two conditions hold: $c_0 + c_2 \ge p$, and $p + c_2 \ge 1 - x$. The first condition is satisfied if $y(1+(t-1)2p) \ge p$, since then: $$\begin{aligned}
c_0 + c_2 &\ge tc_0 + c_2 \\
&\ge y(1+(t-1)2p) \\
&\ge p\end{aligned}$$ The second condition is satisfied if $y(1+(t-1)2p) \ge 2tp + 1-x-p$, since then: $$\begin{aligned}
2tp + c_2 &\ge tc_0 + c_2 \\
&\ge y(1+(t-1)2p) \\
&\ge 2tp + 1 - x - p\end{aligned}$$ So it follows that these two condition are true if there exists $0 \le t \le 1$ such that: $$\begin{aligned}
\frac{2py+p-y}{2py} \le t \le \frac{y-2py+x+p-1}{2p(1-y)}\end{aligned}$$ Again, if the lower bound is at most the upper bound, the upper bound is at least $0$, and the lower bound is at most $1$, then there will exist such a $t \in [0,1]$. Now, note: $$\begin{aligned}
\frac{2py+p-y}{2py} \le 1 \Leftrightarrow p \le y\end{aligned}$$ and the upper bound being at least zero is equivalent to: $$\begin{aligned}
x+y \ge 1 - p(1-2y)\end{aligned}$$ and the lower bound being at most the upper bound simplifies to $p \le xy$. This means we want the three following conditions to hold: $$\begin{aligned}
p &\le y \\
p &\le xy \\
x + y &\ge 1 - p(1-2y)\end{aligned}$$ The first two condition are implied by $p < xy$, and the third is equivalent to $$\frac{y(1-x-y)}{1-3y} \ge 0$$ which is true since $x+y \le 1$ and $y < 1/3$.
Now, choose $X_3 \subset N(V_3)$ such that $|X_3|=q|B|$. Then, choose $v_4 \in C$ with at least $y(1-3p)|B|$ neighbors in $B \setminus (X_1 \cup X_2 \cup X_3)$. Then for all $1 \le i \le 3$ we have: $$|X_i \cup N(v_4)| \ge (p+y(1-3p))|B| \ge (1-x)|B|$$ by the definition of $p$. It follows that the union of the second-neighborhoods in $A$ of every pair of the $v_i$ is equal to $A$, and consequently that every vertex in $A$ is a second-neighbor of at least three of the $v_i$, and thus one of the $v_i$ has at least $3|A|/4$ second-neighbors in $A$, as desired. This proves \[3/4phiub\].
\[3/4psilb\] We have:
- $\psi(x,y)<3/4$ if $1/7\le y\le 1/6$ and $x+2y\le 1$;
- $\psi(x,y)<3/4$ if $1/7 \le x \le 1/6$ and $2x+y \le 1$;
- $\psi(x,y)<3/4$ if $0 \le x \le 1/7$ and $x+4y\le 3$, and $x+4y<3$ if x or y is irrational;
- $\psi(x,y)<3/4$ if $5/7 \le x \le 13/19$ and $x+3y \le 1$;
[[**Proof.**]{} ]{}The first bullet follows from applying the strict inequality version of \[psilb1\] with $z=3/4$ to the point $(3/5,1/5)$, which has $\phi(3/5,1/5) = 3/5$. The second bullet follows from applying the strict inequality version of \[psilb2\] with $z=3/4$ to the points from the first bullet of \[2/3psilb\]. The third bullet follows from applying the strict inequality version of \[psilb2\] with $z=3/4$ to the points from the second bullet of \[2/3psilb\]. Finally, the fourth bullet follows from applying the strict inequality version of \[psilb1\] with $z=3/4$ to the points from the third bullet of \[2/3psilb\]. This proves \[3/4psilb\].
\[3/4philb\] If $y \le 1/3$ and $$\frac{x}{1-x}+\frac{y}{1-3y} \le 3$$ then $\phi(x,y) < 3/4$.
[[**Proof.**]{} ]{}Apply \[philb\] with $z = 3/4$ to \[2/3philb\]. This proves \[3/4philb\].
The 2/5 Level
=============
Next, we analyze when $\psi \ge 2/5$, and similarly for $\phi$. The results are shown in Figure 2.
(0,0) – (7,0) node\[anchor=north\][$x$]{}; (0,0) – (0,7) node\[anchor=east\] [$y$]{}; at (5.5,5.3)
$(1/3,1/3)$
; (15\*2/5,15\*0/1)– (15\*39/98, 15\*0/1) – (15\*39/98, 15\*1/98) – (15\*37/93, 15\*1/98) – (15\*37/93, 15\*4/93) – (15\*35/88, 15\*4/93) – (15\*35/88, 15\*1/22) – (15\*29/73, 15\*1/22) – (15\*29/73, 15\*4/73) – (15\*21/53, 15\*4/73) – (15\*21/53, 15\*3/53) – (15\*19/48, 15\*3/53) – (15\*19/48, 15\*1/16) – (15\*17/43, 15\*1/16) – (15\*17/43, 15\*3/43) – (15\*13/33, 15\*3/43) – (15\*13/33, 15\*1/11) – (15\*8/21, 15\*1/11) – (15\*8/21, 15\*2/21) – (15\*3/8, 15\*2/21) – (15\*3/8, 15\*1/8) – (15\*7/20, 15\*1/8); (15\*1/3,15\*1/6) – (15\*1/3, 15\*1/3) – (15\*1/6,15\*1/3); (15\*1/8, 15\*7/20) – (15\*1/8, 15\*3/8) – (15\*2/21, 15\*3/8) – (15\*2/21, 15\*8/21) – (15\*1/11, 15\*8/21) – (15\*1/11, 15\*13/33) – (15\*3/43, 15\*13/33) – (15\*3/43, 15\*17/43) – (15\*1/16, 15\*17/43) – (15\*1/16, 15\*19/48) – (15\*3/53, 15\*19/48) – (15\*3/53, 15\*21/53) – (15\*4/73, 15\*21/53) – (15\*4/73, 15\*29/73) – (15\*1/22, 15\*29/73) – (15\*1/22, 15\*35/88) – (15\*4/93, 15\*35/88) – (15\*4/93, 15\*37/93) – (15\*1/98, 15\*37/93) – (15\*1/98, 15\*39/98) – (15\*0/1, 15\*39/98) –(15\*0/1,15\*2/5);
plot([15\*(2\*(1-)/5)]{},[15\*]{}); plot([15\*]{},[15\*(2\*(1-)/5)]{});
at (6.3,6.3) [ ]{}; at (3.2,3.2) [ ]{};
(15\*1/3,15\*1/3) – (15\*2/9,15\*1/3);
plot([15\*(2\*-3\*+1)]{},[15\*]{});
(15\*1/5,15\*.3469) – (15\*1/5,15\*2/5) – (15\*0,15\*2/5);
(15\*1/3,15\*1/3) – (15\*1/3,15\*1/4);
(15\*1/3,15\*1/4) – (15\*.35,15\*.25);
plot([15\*(.4 - /(10-20\*))]{},[15\*]{});
(15\*.3631,15\*.2123) – (15\*.4,15\*.2)–(15\*.4,15\*0);
at (-.8,6)
$y=2/5$
; at (6,-.33)
$x=2/5$
;
(r1) at (1.5, 7.1)
$\ref{maxbound}$
; (r1) to (1.5,6.1);
(r1) at (7.1, 1.5)
$\ref{maxbound}$
; (r1) to (6.1,1.5);
(r1) at (4, 6)
$\ref{2/5psiub1}$
; (r1) to (4,5.1); (r1) to (3.2,5.2);
(r1) at (6, 4)
$\ref{2/5psiub2}$
; (r1) to (5.1,4); (r1) to (5.5,3.5); (r1) to (5.8,3.2);
(r1) at (1.5, 4.5)
$\ref{2/5psilb}$
; (r1) to (2.1,5.1);
(r1) at (4.5, 1.5)
$\ref{2/5psilb}$
; (r1) to (5.1,2.1);
(0,0) – (7,0) node\[anchor=north\][$x$]{}; (0,0) – (0,7) node\[anchor=east\] [$y$]{}; at (5.5,5.3)
$(1/3,1/3)$
;
(15\*39/98, 15\*.04348) – (15\*39/98, 15\*1/11) – (15\*35/88, 15\*1/11) – (15\*35/88, 15\*1/9) – (15\*17/43, 15\*1/9) – (15\*17/43, 15\*1/8) – (15\*13/33, 15\*1/8) – (15\*13/33, 15\*2/15) – (15\*11/28, 15\*2/15) – (15\*11/28, 15\*1/7) – (15\*7/18, 15\*1/7) – (15\*7/18, 15\*2/13) – (15\*29/75,15\*2/13);
(15\*2/13, 15\*29/75) – (15\*2/13, 15\*7/18) – (15\*1/7, 15\*7/18) – (15\*1/7, 15\*11/28) – (15\*2/15, 15\*11/28) – (15\*2/15, 15\*13/33) – (15\*1/8, 15\*13/33) – (15\*1/8, 15\*17/43) – (15\*1/9, 15\*17/43) – (15\*1/9, 15\*35/88) – (15\*1/11, 15\*35/88) – (15\*1/11, 15\*39/98) – (15\*.04348, 15\*39/98);
(15\*9/25, 15\*17/75) – (15\*9/25, 15\*3/13) – (15\*5/14, 15\*3/13) – (15\*5/14, 15\*4/17) – (15\*11/31, 15\*4/17) – (15\*11/31, 15\*6/25) – (15\*6/17, 15\*6/25) – (15\*6/17, 15\*8/33) – (15\*7/20, 15\*8/33) – (15\*7/20, 15\*17/69) – (15\*9/26, 15\*17/69) – (15\*9/26, 15\*19/77) – (15\*.3387, 15\*19/77);
(15\*1/4,15\*1/3) – (15\*1/3,15\*1/3) – (15\*1/3,15\*1/4);
(15\*19/77,15\*.3387) – (15\*19/77, 15\*9/26) – (15\*17/69, 15\*9/26) – (15\*17/69, 15\*7/20) – (15\*8/33, 15\*7/20) – (15\*8/33, 15\*6/17) – (15\*6/25, 15\*6/17) – (15\*6/25, 15\*11/31) – (15\*4/17, 15\*11/31) – (15\*4/17, 15\*5/14) – (15\*3/13, 15\*5/14) – (15\*3/13, 15\*9/25) – (15\*17/75, 15\*9/25);
at (6.3,6.3) [ ]{}; at (3.2,3.2) [ ]{};
plot([15\*(2-5\*)/(7-17\*)]{},[15\*]{});
plot([15\*(2-5\*)/(7-17\*)]{},[15\*]{});
plot([15\*(2-5\*)/(7-17\*)]{},[15\*]{});
plot([15\*]{},[15\*(2-7\*)/(5-17\*)]{});
plot([15\*]{},[15\*(2-5\*)/(7-17\*)]{});
plot([15\*]{},[15\*(2-5\*)/(7-17\*)]{});
plot([15\*]{},[15\*(2-5\*)/(7-17\*)]{});
plot([15\*(2-7\*)/(5-17\*)]{},[15\*]{});
(15\*1/3,15\*1/3)–(15\*.297086,15\*1/3) – (15\*.297086,15\*.3811);
(15\*1/3,15\*1/3)–(15\*1/3,15\*.297086) – (15\*.3811,15\*.297086);
plot([15\*.2\*(6\*-2\*sqrt(3\*(3\*-5\*+5))-5\*+5)]{},[15\*]{});
plot([15\*]{},[15\*.2\*(6\*-2\*sqrt(3\*(3\*-5\*+5))-5\*+5)]{});
(15\*.275,15\*.4) – (15\*0,15\*.4);
(15\*.4,15\*.275)–(15\*.4,15\*0);
at (-.8,6)
$y=2/5$
; at (6,-.33)
$x=2/5$
;
(r1) at (2.25, 7.1)
$\ref{maxbound}$
; (r1) to (2.25,6.1);
(r1) at (7.1, 2.25)
$\ref{maxbound}$
; (r1) to (6.1,2.25);
(r1) at (6.85, 4.8)
$\ref{2/5phiub1}$
; (r1) to (6,4.3);
(r1) at (5.8, 4.85)
$\ref{2/5phiub2}$
; (r1) to (5.1,4.75);
(r1) at (4.75, 2.5)
$\ref{2/5philb}$
; (r1) to (5,3.6); (r1) to (5.5,3); (r1) to (5.9,.25);
\[2/5psiub1\] Suppose $x \ge 1/5$, $y > 1/3$, and $3y-2y^2>1-x$; then $\psi(x,y) \ge 2/5$.
[[**Proof.**]{} ]{}We may assume $x,y \in \mathbb{Q}$, and that $y|B| \in \mathbb{Z}$. Let $v_1 \in C$, and let $A_1 = N_A^2(v_1)$. If every $v \in C$ has a common neighbor with $v_1$ in $B$, then choose $v \in C$ with at least $y|A \setminus A_1|>3y|A|/5$ neighbors in $A \setminus A_1$. Then $$|N_A^2(v)| > 3y|A|/5+x|A|>2|A|/5$$ since $x \ge 1/5$, $y > 1/3$. Thus, we can assume there exists $v_2 \in C$ such that $|N(v_1) \cap N(v_2)| = 0$. Choose $B_i \subseteq N(v_i)$ such that $|B_i|=y|B|$. Then choose $v_3 \in C$ with at least $y(1-2y)|B|$ neighbors in $B \setminus (B_1 \cup B_2)$. Then by assumption it follows that $|N(v_1) \cup N(v_2) \cup N(v_3)| > (1-x)|B|$.
Thus, if $A_i = N_A^2(v_i)$, we have that $A = A_1 \cup A_2 \cup A_3$. Since $y > 1/3$, without loss of generality we have $|N(v_1) \cap N(v_2)| \ne 0$, which implies that $|A_1 \cap A_2| \ge x|A| \ge |A|/5$. But then we have $$|A_1|+|A_2|+|A_3| \ge |A_1 \cup A_2 \cup A_3| + |A_1 \cap A_2| \ge |A| + |A|/5 = 6|A|/5$$ and thus we have $|A_i| \ge 2|A|/5$ for some $i$. This proves \[2/5psiub1\].
\[2/5psiub2\] Suppose $x > 1/3$, $x+3y>1$, and either $y \ge 1/4$ or $x+y/(10(1-2y)) \ge 2/5$; then $\psi(x,y) \ge 2/5$.
[[**Proof.**]{} ]{}Let $v_1 \in C$, and let $A_1 = N_A^2(v_1)$. If every vertex in $C$ shares a neighbor with $v_1$ in $B$, then choose $w \in B \setminus N(v_1)$ with more than $3|A|x/(5(1-y))$ neighbors in $A \setminus A_1$. Let $v$ be a neighbor of $w$ in $C$. Then $$|N_A^2(v)| > x|A|+\frac{3x|A|}{5(1-y)}>\left(x+\frac{9x}{5(x+2)}\right)|A|>2|A|/5$$ since $x > 1/3$. So let $v_2 \in C$ be such that $|N(v_1) \cap N(v_2)| = 0$. If every $v \in C$ has a common neighbor in $B$ with at least one of $v_1, v_2$, then let $A_i = N_A^2(v_i)$, and choose $w \in B \setminus (N(v_1) \cup N(v_2))$ with more than $x|A|/(5(1-y))$ neighbors in $A \setminus (A_1 \cup A_2)$. Let $v \in C$ be a neighbor of $w$. Then $$|N_A^2(v)| > x|A| + \frac{x|A|}{5(1-y)} > \left(x+\frac{3x}{5(x+2)}\right)|A|>2|A|/5$$ since $x > 1/3$.
So let $v_3 \in C$ be such that $N(v_1),N(v_2),N(v_3)$ are pairwise disjoint. If there exists $v_4 \in C$ such that $N(v_4)$ is disjoint from $N(v_1) \cup N(v_2) \cup N(v_3)$, then since $x + 3y > 1$ it follows that every $v \in A$ is a second-neighbor of at least two of the $v_i$, so by averaging one of the $v_i$ has at least $|A|/2 > 2|A|/5$ second-neighbors in $A$. Thus, we can assume that every $v_4 \in C$ has a common neighbor with at least one of $v_1, v_2, v_3$. Now, let $A_i = N_A^2(v_i)$. Then we have $A = A_1 \cup A_2 \cup A_3$, and it follows that the pairwise unions of the $A_i$ each have size less than $|A|/5$, since $$6|A|/5 > |A_1|+|A_2|+|A_3| \ge |A_1 \cup A_2 \cup A_3| + |A_i \cup A_j| = |A| + |A_i \cup A_j|$$ for $i \ne j$. Then if $v_4$ shares a common neighbor with $v_i$ and $v_j$, we have $$|N_A^2(v_4)| > (2x-1/5)|A| > (2/3-1/5)|A|>2|A|/5$$ So we can assume that every $v_4 \in C$ has a common neighbor with exactly one of $v_1,v_2,v_3$. This implies that there are three connected components of $(B,C)$, which are furthermore complete. Let these components be $H_1,H_2,H_3$, and let $B_i = B \cap H_i$, $C_i = C \cap H_i$. Then without loss of generality, $|B_3|\le |B|/3$, so every vertex $v \in A$ has a neighbor in either $B_1$ or $B_2$. Let $b_i$ be such that $b_i |B| = |B_i|$, and $c_i$ such that $c_i |C| = |C_i|$. Suppose that a vertex $v \in A$ has at most $(b_i-y)|B|$ neighbors in $B_i$ for all $1 \le i \le 3$. Then $v$ has at most $(1-3y)|B| < x|B|$ neighbors in $B$, a contradiction. It follows that every vertex $v \in A$ has more than $(b_i-y)|B|$ neighbors in $B_i$ for at least one $i$, and consequently that $v$ is complete to $C_i$ for some $i$ (complete in the sense that $v$ is a second-neighbor of every vertex in $C_i$).
Now, if $x+y/(10(1-2y)) \ge 3/4$, then without loss of generality at least $|A|/2$ vertices in $A$ have a neighbor in $B_1$. Then at least $|A|/10$ vertices in $A \setminus A_1$ have a neighbor in $B_1$ and thus hit at least fraction $y/(1-2y)$ of the vertices in $A_1$, so there exists $v \in A_1$ with more than $y|A|/(10(1-2y))$ neighbors in $A \setminus A_1$. Then $$|N_A^2(v)| > x|A|+y|A|/(10(1-2y)) \ge 3|A|/4$$
So suppose $x + y/(10(1-2y)) < 3/4$, and thus $y \ge 1/4$. Now, partition $A = A_1^* \cup A_2^* \cup A_3^*$ such that $v \in A_i^*$ has more than $(b_i-y)|B|$ neighbors in $B_i$, and let $a_i^*$ be such that $a_i^* |A| = A_i^*$. Then $a_3^* < 2|A|/5$. Let $A_1,A_2 \subseteq A_3^*$ be the sets of vertices in $A_3^*$ with a neighbor in $B_1$ and $B_2$, respectively, and let $a_i = |A_i|/|A|$. Then every vertex in $A_i$ sees at least fraction $y/c_i$ of the vertices in $C_i$, so there exists a vertex $v \in C_i$ with at least $ya_i/c_i$ neighbors in $A_i$. It follows that for $i = 1,2$ $$a_i^*+y\frac{a_i}{c_i} < 2/5$$ Summing these inequalities gives $$a_1^* + a_2^* + y \left(\frac{a_1}{c_1}+\frac{a_2}{c_2}\right) < 4/5$$ and since $c_i \le 1-2y$ we have $$4/5 > a_1^*+a_2^*+(a_1+a_2)\frac{y}{1-2y} = a_1^*+a_2^*+\frac{y}{1-2y} a_3^* > \frac{3}{5}+\frac{2y}{5(1-2y)}$$ which gives $y<1/4$, a contradiction. This proves \[2/5psiub2\].
\[2/5phiub1\] Suppose $12x^2y \ge 5(1-x-y)^2$; then $\phi(x,y) \ge 2/5$.
[[**Proof.**]{} ]{}Suppose not. Then $\phi(x,y) = 1-(3/5+\epsilon)$ for some $\epsilon > 0$, so by rotating we have $\phi(x,3/5+\epsilon) \le 1-y$. But \[5.7\] gives $\phi(x,3/5+\epsilon) > 1-y$, a contradiction. This proves \[2/5phiub1\].
\[2/5phiub2\] Suppose $x > 1/3$ and $y \ge (5-\sqrt{3})/11$; then $\phi(x,y) \ge 2/5$.
Suppose not. Then $\phi(x,y) = 1 - (3/5+\epsilon)$ for some $\epsilon > 0$, so by rotating we have $\phi(3/5+\epsilon,y) \le 1-x < 2/3$, but \[2/3phiub\] gives that $3/5 \ge (1-y)^2/(1-2y^2)$ implies that $\phi(3/5+\epsilon,y) \ge 2/3$, a contradiction, since $3/5 \ge (1-y)^2/(1-2y^2)$ is equivalent to $y \ge (5-\sqrt{3})/11$. This proves \[2/5phiub2\].
\[2/5psilb\] Suppose $5x/2 + y \le 1$ and $x \ge 1/3$, or that $x+5y/2 \le 1$ and $y \ge 1/3$; then $\psi(x,y) < 2/5$.
[[**Proof.**]{} ]{}Apply \[3.3gen\] with $a/b = 2/5$. This proves \[2/5psilb\].
\[2/5philb\] Suppose $y \le 1/3$ and $x/(1-2x)+y/(1-3y) \le 2$; then $\phi(x,y) < 2/5$.
[[**Proof.**]{} ]{}Apply \[lowerboundextension\] to \[2/3philb\]. This proves \[2/5philb\].
The 3/5 Level
=============
Next, we analyze when $\psi \ge 3/5$, and similarly for $\phi$. The results are shown in the Figure 3.
(0,0) – (10,0) node\[anchor=north\][$x$]{}; (0,0) – (0,10) node\[anchor=east\] [$y$]{}; at (8.4,7.7)
$(1/2,1/2)$
; at (15\*1/1, 15\*0/1); at (15\*0/1, 15\*1/1); at (15\*58/97, 15\*1/97); at (15\*1/97, 15\*58/97); at (15\*55/92, 15\*3/92); at (15\*3/92, 15\*55/92); at (15\*49/82, 15\*2/41); at (15\*2/41, 15\*49/82); at (15\*43/72, 15\*1/18); at (15\*1/18, 15\*43/72); at (15\*31/52, 15\*3/52); at (15\*3/52, 15\*31/52); at (15\*28/47, 15\*3/47); at (15\*3/47, 15\*28/47); at (15\*25/42, 15\*1/14); at (15\*1/14, 15\*25/42); at (15\*19/32, 15\*3/32); at (15\*3/32, 15\*19/32); at (15\*11/19, 15\*2/19); at (15\*2/19, 15\*11/19); at (15\*4/7, 15\*1/7); at (15\*1/7, 15\*4/7); at (15\*1/2, 15\*1/2); at (15\*1/2, 15\*1/2); (15\*58/97, 15\*0/1) – (15\*58/97, 15\*1/97) – (15\*55/92, 15\*1/97) – (15\*55/92, 15\*3/92) – (15\*49/82, 15\*3/92) – (15\*49/82, 15\*2/41) – (15\*43/72, 15\*2/41) – (15\*43/72, 15\*1/18) – (15\*31/52, 15\*1/18) – (15\*31/52, 15\*3/52) – (15\*28/47, 15\*3/52) – (15\*28/47, 15\*3/47) – (15\*25/42, 15\*3/47) – (15\*25/42, 15\*1/14) – (15\*19/32, 15\*1/14) – (15\*19/32, 15\*3/32) – (15\*11/19, 15\*3/32) – (15\*11/19, 15\*2/19) – (15\*4/7, 15\*2/19) – (15\*4/7, 15\*1/7);
(15\*1/2, 15\*1/6) – (15\*1/2, 15\*1/2) – (15\*1/6, 15\*1/2);
(15\*1/7, 15\*4/7) – (15\*3/32, 15\*93/160) – (15\*3/32, 15\*19/32) – (15\*1/14, 15\*19/32) – (15\*1/14, 15\*25/42) – (15\*3/47, 15\*25/42) – (15\*3/47, 15\*28/47) – (15\*3/52, 15\*28/47) – (15\*3/52, 15\*31/52) – (15\*1/18, 15\*31/52) – (15\*1/18, 15\*43/72) – (15\*2/41, 15\*43/72) – (15\*2/41, 15\*49/82) – (15\*3/92, 15\*49/82) – (15\*3/92, 15\*55/92) – (15\*1/97, 15\*55/92) – (15\*1/97, 15\*58/97) – (15\*0/1, 15\*58/97);
plot ([15\*(1-3\*)]{},[15\*]{}); plot ([15\*()]{},[15\*(1-3\*)]{});
at (9,9) [ ]{}; at (5.3,5.3) [ ]{};
at (2.3,7)
$(1/6,1/2)$
; at (6.25,2.55)
$(1/2,1/6)$
; (15\*.2066, 15\*1/2) – (15\*1/2, 15\*1/2) – (15\*1/2, 15\*1/4); (15\*1/2, 15\*1/4) – (15\*3/5, 15\*1/5) – (15\*3/5, 0);
plot([15\*((5\*-6\*+6)/6 - sqrt(25\*-60\*+60\*)/6)]{},[15\*]{});
(15\*1/5, 15\*.5072) – (15\*1/5, 15\*3/5) – (0, 15\*3/5);
at (-0.8, 9)
$y=3/5$
;
(r1) at (2.25, 10.1)
$\ref{maxbound}$
; (r1) to (2.25,9.1);
(r1) at (10.1, 2.25)
$\ref{maxbound}$
; (r1) to (9.1,2.25);
(r1) at (4.5, 8.5)
$\ref{3/5psiub1}$
; (r1) to (4.5,7.6); (r1) to (3.1,7.6);
(r1) at (8.5, 4.5)
$\ref{3/5psiub2}$
; (r1) to (7.6,4.5); (r1) to (8.35,3.65);
(r1) at (1.25, 8)
$\ref{3/5psilb}$
; (r1) to (2.15,8);
(r1) at (8, 1.25)
$\ref{3/5psilb}$
; (r1) to (8,2.15);
(0,0) – (11,0) node\[anchor=north\][$x$]{}; (0,0) – (0,11) node\[anchor=east\] [$y$]{}; at (8.3,7.7)
$(1/2,1/2)$
; at (-1,9)
$y=3/5$
; at (5,5) ; at (9,9) [ ]{};
plot([15\*(3-11\*)/(5-18\*)]{}, [15\*]{});
plot([15\*(3-11\*)/(5-18\*)]{}, [15\*]{});
plot([15\*]{}, [15\*(3-11\*)/(5-18\*)]{});
plot([15\*]{}, [15\*(3-11\*)/(5-18\*)]{});
(15\*58/97, 15\*0/1) – (15\*58/97, 15\*1/11) – (15\*55/92, 15\*1/11) – (15\*55/92, 15\*1/10) – (15\*52/87, 15\*1/10) – (15\*52/87, 15\*1/9) – (15\*25/42, 15\*1/9) – (15\*25/42, 15\*1/8) – (15\*19/32, 15\*1/8) – (15\*19/32, 15\*2/15)– (15\*16/27, 15\*2/15) – (15\*16/27, 15\*1/7) – (15\*10/17, 15\*1/7) – (15\*10/17, 15\*2/13) – (15\*58/99, 15\*2/13);
(15\*11/20, 15\*22/97) – (15\*11/20, 15\*3/13) – (15\*6/11, 15\*3/13) – (15\*6/11, 15\*4/17) – (15\*13/24, 15\*4/17) – (15\*13/24, 15\*6/25) – (15\*7/13, 15\*6/25) – (15\*7/13, 15\*8/33) – (15\*8/15, 15\*8/33) – (15\*8/15, 15\*17/69) – (15\*9/17, 15\*17/69) – (15\*9/17, 15\*18/73) – (15\*10/19, 15\*18/73) – (15\*10/19, 15\*19/77) – (15\*.5116, 15\*19/77);
(15\*1/2, 15\*1/4) – (15\*1/2, 15\*1/2) – (15\*1/4, 15\*1/2);
(15\*19/77, 15\*.5116) – (15\*19/77, 15\*10/19) – (15\*18/73, 15\*10/19) – (15\*18/73, 15\*9/17) – (15\*17/69, 15\*9/17) – (15\*17/69, 15\*8/15) – (15\*8/33, 15\*8/15) – (15\*8/33, 15\*7/13) – (15\*6/25, 15\*7/13) – (15\*6/25, 15\*13/24) – (15\*4/17, 15\*13/24) – (15\*4/17, 15\*6/11) – (15\*3/13, 15\*6/11) – (15\*3/13, 15\*11/20) – (15\*22/97, 15\*11/20);
(15\*2/13, 15\*58/99) – (15\*2/13, 15\*10/17) – (15\*1/7, 15\*10/17) – (15\*1/7, 15\*16/27) – (15\*2/15, 15\*16/27) – (15\*2/15, 15\*19/32) – (15\*1/8, 15\*19/32) – (15\*1/8, 15\*25/42) – (15\*1/9, 15\*25/42) – (15\*1/9, 15\*52/87) – (15\*1/10, 15\*52/87) – (15\*1/10, 15\*55/92) – (15\*1/11, 15\*55/92) – (15\*1/11, 15\*58/97) – (15\*0/1, 15\*58/97);
(15\*.3167, 15\*1/2) – (15\*1/2, 15\*1/2) – (15\*1/2, 15\*.3167);
plot([15\*]{}, [15\*(5\*-3)\*(5\*-3)/(40\*)]{});
(0, 15\*.6) – (15\*.3031, 15\*.6);
plot([15\*(5\*-3)\*(5\*-3)/(40\*)]{}, [15\*]{});
(15\*.6, 0) – (15\*.6, 15\*.3031);
(r1) at (2.25, 10.1)
$\ref{maxbound}$
; (r1) to (2.25,9.1);
(r1) at (10.1, 2.25)
$\ref{maxbound}$
; (r1) to (9.1,2.25);
(r1) at (7.6, 1.5)
$\ref{3/5philb}$
; (r1) to (7.6,3.5); (r1) to (8.5,2.75); (r1) to (8.9,.5);
(r1) at (8.5, 6.25)
$\ref{3/5phiub1}$
; (r1) to (7.6,6.25); (r1) to (8.25,4.75);
\[3/5psiub1\] Suppose $y > 1/2$, $x \ge 1/5$, and $$2y-\frac{y^2(3-5x)}{3(1-x)}>1-x$$ Then $\psi(x,y) \ge 3/5$.
[[**Proof.**]{} ]{}Suppose not. Let $G$ be $(x,y)$-biconstrained via $(A,B,C)$ such that for all $v \in C$, $|N_A^2(v)|<3|A|/5$. Then by averaging, there exists $w \in A$ such that $|N_C^2(w)|<3|C|/5$. We can assume $x,y$ are rational, and by blowing up vertices if necessary, we can assume that both $x|B|$ and $|C|/5$ are integers. Choose $B_1 \subseteq N(w)$ such that $|B_1| = x|B|$, and choose $C_1 \subseteq C$ such that $N_C^2(w) \subseteq C_1$ and $|C_1|=3|C|/5$.
Now, choose $v_1 \in C_1$ and $v_2 \in C \setminus C_1$ at random. We compute the expectation of $S = |N(v_1) \cup N(v_2)|$. Let $t=2/5$. For $u \in B_1$, $u$ has at least $y|C|$ neighbors in $C_1$, so the probability it is in $S$ is at least $y/(1-t)$. For $u \in B \setminus B_1$, let $d(u)|C| = |N(u) \cap (C \setminus C_1)|$. Then the probability that $u$ is a neighbor of $v_2$ is $d(u)/t$, and the probability that $u$ is a neighbor of $v_1$ is at least $(y-d(u))/(1-t)$. Thus the expectation of $|S|$ is at least: $$\begin{aligned}
\left(\sum_{u \in B \setminus B_1} \frac{d(u)}{t}+\frac{y-d(u)}{1-t} - \frac{d(u)(y-d(u))}{t(1-t)}\right)+\sum_{u \in B_1} \frac{y}{1-t}\end{aligned}$$ I claim this quantity is more than $(1-x)|B|$. Note that each vertex $v \in C \setminus C_1$ has at least $y|B|$ neighbors in $B \setminus B_1$, so it follows that $$\sum_{u \in B \setminus B_1} d(u) = qt|C|$$ where $q \ge y$. Thus, this claim is equivalent to: $$qt|B|(1-2t-y)+\left(\sum_{u \in B \setminus B_1} d(u)^2 \right)+ yt|B| > t(1-t)(1-x)|B|$$ Now, since $\sum_{u \in B \setminus B_1} d(u) = qt|B|$ and $|B \setminus B_1| = (1-x)|B|$, it follows by Cauchy-Schwarz that $\sum_{u \in B \setminus B_1} d(v)^2 \ge q^2t^2|B|/(1-x)$. Thus to show the expectation of $S$ is greater than $(1-x)|B|$ it suffices to show: $$tq(1-2t-y)+\frac{q^2t^2}{1-x}+yt > t(1-t)(1-x)$$ Remembering that $t=2/5$, this is: $$2(1-5y)q+\frac{4q^2}{(1-x)}+10y > 6(1-x)$$ and the derivative of the left-hand side with respect to $q$ is $$2(1-5y)+\frac{8q}{1-x} \ge 2(1-5y)+\frac{8y}{4/5} = 2 > 0$$ for $q \ge y$. It follows that the left-hand side is minimized when $q = y$, so it suffices to show $$2(1-5y)y+\frac{4y^2}{1-x}+10y>6(1-x)$$ which is equivalent to the assumption. Thus, there exist $v_1, v_2 \in C$ such that $|N(v_1) \cup N(v_2)| > (1-x)|B|$. Then if $A_i = N_A^2(v_i)$, we have $A = A_1 \cup A_2$, and since $y > 1/2$ we have $N(v_1) \cap N(v_2) \ne \emptyset$, and consequently $|A_1 \cap A_2| \ge x|A| \ge |A|/5$. Then $$|A_1|+|A_2|=|A_1 \cup A_2| + |A_1 \cap A_2| \ge 6|A|/5$$ so we have $|A_i| \ge 3|A|/5$ for some $i$, as desired. This proves \[3/5psiub1\].
\[3/5psiub2\] Suppose $x > 1/2$ and $x+2y>1$. Then $\psi(x,y) \ge 3/5$.
[[**Proof.**]{} ]{}Let $G$ be $(x,y)$-biconstrained via $(A,B,C)$. Let the graph $H$ have $V(H)=V(C)$ and $(uv)\in E(H)$ if and only if $u$ and $v$ have a common neighbor in $B$. If there exists a stable set $\{v_1,v_2,v_3\}$ of size three in $H$, then every vertex $v \in A$ is a second-neighbor of at least two of the $v_i$, so by averaging one of the $v_i$ has at least $2|A|/3 > 3|A|/5$ second-neighbors in $A$, and we are done.
So we can suppose there is no stable set of size three in $H$. Take $v_1 \in C$, and suppose every $v_2 \in C$ has $|N(v_1) \cap N(v_2)| \ne 0$. Let $B_1 = N(v_1)$, $A_1 = N_A^2(v_1)$. We can assume $|A_1| < 3|A|/5$. Then choose $w \in B \setminus B_1$ such that $w$ has at least $x|A \setminus A_1|/(1-y) > 2x|A|/(5(1-y))$ neighbors in $A \setminus A_1$. Let $v_2 \in C$ be a neighbor of $w$. Then $$|N_A^2(v_2)| > x|A| + 2x|A|/(5(1-y)) > \left(x+\frac{4x}{5(x+1)}\right)|A| \ge 3|A|/5$$ since $x > 1/2$.
So there exists $v_2 \in C$ such that $|N(v_1) \cap N(v_2)| = 0$. Since there is no stable set of size three in $H$, every vertex $v_3 \in C$ shares a neighbor with at least one of $v_1, v_2$. Furthermore, since $x+2y>1$, letting $A_i = N_A^2(v_i)$, we have $A = A_1 \cup A_2$, so since $|A_i| < 3|A|/5$, we have $|A_1 \cap A_2| < |A|/5$. Then, if some $v_3 \in C$ shares a neighbor with both $v_1$ and $v_2$, it follows that $|N_A^2(v_3)| > 2x|A| - |A|/5 > 4|A|/5$, a contradiction.
Thus, every vertex $v_3 \in C$ shares a neighbor with exactly one of $v_1$ and $v_2$. It follows that there are exactly two connected components of the bipartite graph $(B,C)$. Let these components be $H_1, H_2$, and let $B_i = B \cap H_i$, $C_i = C \cap H_i$. Then without loss of generality we have $|B_1| \ge |B|/2$. We have $|C_i| \ge y|C|$, and thus $|C_i| \le (1-y)|C|$. It follows that every vertex $v \in A \setminus A_1$ has a neighbor in $B_1$ and consequently hits at least fraction $y/(1-y)$ of the vertices in $C_1$, so there exists $w \in C_1$ with more than $2|A|y/(5(1-y))$ neighbors in $A \setminus A_1$. Then, $w$ and $v_1$ share a common neighbor, so $$|N_A^2(w)| > \frac{2y|A|}{5(1-y)}+x|A| > \left(\frac{2y}{5(1-y)}+(1-2y)\right)|A| \ge 3|A|/5$$ since the last inequality is equivalent to $5y^2-5y+1 \ge 0$ which is true for $0 \le y \le 1/4$. This proves the claim for $x+2y>1$, $x > 1/2$, $0 \le y \le 1/4$, which implies the claim for $x+2y>1$, $x > 1/2$. This proves \[3/5psiub2\].
\[3/5phiub1\] Suppose $y > 1/2$ and $40x^2y \ge (3-5x)^2$. Then $\phi(x,y) \ge 3/5$.
[[**Proof.**]{} ]{}Apply \[5.7\] with $z = 3/5$. This proves \[3/5phiub1\].
\[3/5psilb\] We have:
- $\psi(x,y)<3/5$ if $1/7\le y\le 1/6$ and $x+3y\le 1$;
- $\psi(x,y)<3/5$ if $1/7 \le x \le 1/6$ and $3x+y \le 1$, with $3x+y < 1$ if $x$ or $y$ is irrational.
[[**Proof.**]{} ]{}The first bullet follows from applying the strict inequality version of \[psilb1\] with $z = 3/5$ to the point $(1/4,1/4)$.
For the second bullet, apply the strict inequality version of \[psilb2\] with $z = 3/5$ to the set of points $3x+y = 1$, for $1/4 \le x \le 5/18$ with $x,y$ rational (this suffices since if $x$ or $y$ is irrational, just increase them slightly so that they are rational). This proves \[3/5psilb\].
\[3/5philb\] Suppose $y < 1/3$ and $\frac{1-x}{2-3x}+\frac{y}{1-3y} \le 2$, with strict inequality if $x$ or $y$ is irrational; then $\phi(x,y) < 3/5$.
[[**Proof.**]{} ]{}We can assume $x,y \in \mathbb{Q}$. By \[1/kphilb\] for $k = 2$, we know that if $y < 1/2$, $x < 1/2$, and $\frac{x}{1-2x}+\frac{y}{1-2y} \le 1$ then $\phi(x,y) < 1/3$. Then applying \[philb\] for $z = 3/5$ gives that if $y < 1/3$ and $$\frac{2x-1}{2-3x}+\frac{y}{1-3y} \le 1$$ then $\phi(x,y) < 3/5$. This is equivalent to the assumption. This proves \[3/5philb\].
Acknowledgements
================
I would like to thank my thesis advisor, Paul Seymour, for suggesting the problems, as well as for the time he spent over the course of the year working with me. His advice and suggestions were invaluable.
[99]{} M. Chudnovsky, P. Hompe, A. Scott, P. Seymour, and S. Spirkl, “Concatenating bipartite graphs”, submitted for publication, <https://web.math.princeton.edu/~pds/papers/chain/paper.pdf>.
I vow that I did not violate the Princeton University Honor Code in the making of this paper.
|
---
abstract: 'An exact calculation of CaI $\lambda 6439.075 \; {\textup{\AA}}$ absorption profiles in the spectra of optical stars in low-mass X-ray binary systems with black holes (BH LMXBs) is carried out. We show that the approximation of a real Roche lobe filling star as disk with uniform local line profile and linear limb darkening law leads to overestimation of projected equatorial rotational velocity $V_{rot} \sin i$ and accordingly, underestimation of mass ratio $q=M_x/M_v$. Refined value of $q$ does not affect the mass of a black hole, but the mass of an optical star has shrunk $\sim 1.5$ times. We present refined components masses in low-mass X-ray binaries with black holes. Companion masses in BH LMXBs are found in the mass range 0.1 - 1.6 $M_{\odot}$ with the peak at $M_v \simeq 0.35 M_{\odot}$. This finding poses additional problem for the standard evolutionary scenarios of BH LMXBs formation with a common envelope phase (CE). We also discuss the implications of these masses into the evolutionary history of the binary.'
author:
- 'Petrov $^1$[ [^1] ]{}V. S., Antokhina$^1$ E. A., Cherepashchuk$^1$ A. M.'
bibliography:
- 'main.bib'
date: ' $^1$Sternberg Astronomical Institute, Moscow M.V. Lomonosov State University\'
title: 'Masses of optical components and black holes in x-ray novae: the effects of components proximity.'
---
Introduction
============
The mass ratio $q=M_x/M_v$ is an fundamental parameter of the evolution of the low-mass X-ray binary system. The binary mass ratio $q=M_x/M_v$ ($M_x$ is the mass of the black hole and $M_v$ is the mass of optical star) is obtained by measuring the rotational velocity of the secondary star $V_{rot} \sin i$ [@Wade_Horne_1988]. The procedure commonly used to measure of the low-mass X-ray binaries (LMXBs) secondary star rotational broadening is to convolve its spectrum with a limb-darkened standard rotation profile [@Collins_1995]. It used the observed line profiles for slowly rotating stars with close spectral types (reference stars) as profiles unbroadened by rotation. The spectra of the stars in the X-ray binaries and those of the single slowly rotating stars were obtained with the same spectral resolution. The rotational broadening of the spectra of the reference stars was modelled, assuming that these profiles would be the same in the absence of rotation, identifying the value of $V_{rot} \sin i$ for which the spectra of the single star and the star in the X-ray binary were in best agreement with $\chi^2$ criterion. In this way, the rotational velocity $V_{rot} \sin i$ of the star in the binary system was found. This approach made it possible not to correct the influence of the instrumental function of the spectrograph. In the up to date catalogue of black hole transients BlackCat [@Casares_Jonker_2014] authors shows the data about the rotational broadening of line profiles in the spectra of the optical components of X-ray binary systems/ obtaining by using classical rotational broadening model [@Collins_1995].
An exact calculation of CaI $\lambda 6439.075 \; {\textup{\AA}}$ absorption profiles in the spectra of optical stars in low mass X-ray binary systems is carried out @Petrov_2017. They showed that the Full Width Half Maximum ($FWHM$) of CaI $\lambda 6439.075 \; {\textup{\AA}}$ absorption profiles in tidally deformed Roche model [@Antokhina_Shim_2005] is completely higher than the $FWHM$ of CaI $\lambda 6439.075 \; {\textup{\AA}}$ absorption profiles in classical rotational broadening model [@Collins_1995]. It was shown that widely used approximation of optical star in LMXB as a disk with an uniform local profile and linear limb darkening law leads to overestimation of projected equatorial rotational velocity $V_{rot} \sin i$ and, accordingly, underestimation of mass ratio $q=M_x/M_v$. @Petrov_2017 presented an equation that allows to convert the mass ratio $q_{disk}$ obtained with the classical rotational broadening model to the mass ratio $q=M_x/M_v$ corresponding to Roche model [@Antokhina_Shim_2005]. There is a mismatch between modelled and observed distributions of optical stars and black holes masses in BH LMXBs (e.g. [@Podsiadlowski_2010]). For example, the BH mass spectrum is flat, with the gap in the mass range $2–5 M_{\odot}$ [@Ozel_2010; @Farr_2011; @Petrov_2014]. Companion masses in BH LMXB are found in the mass range 0.1 - 1.6 $M_{\odot}$ with the peak at 0.6 $M_{\odot}$. It seems rather difficult to explain the existence of such systems with the standard common envelope evolutionary scenarios: ejection of massive common envelope of such low-mass star proves rather difficult.@Wang_2016 proposed that it is possible to form BH LMXBs with the standard CE scenario if most BHs are born through failed supernovae with progenitor mass $ M<28 M_{\odot} $. But $\alpha$ - elements (O, Si, Mg e.t.c.) are found in atmospheres of at least two optical counterparts: GRO J1655-40 and SAX J1819.3-2525 [@Israelian_1999], so found evidence of the formation of the black holes in an explosive supernova (SN) event. In this paper we refined mass ratio $q$ for 9 BH LMXBs listed in BlackCAT catalogue [@Casares_Jonker_2014] using the approximation equation from Antokhina et al. [@Petrov_2017]. We assume that approximation equation based on CaI $\lambda 6439.075 \; {\textup{\AA}}$ absorption line profiles can also be applied to other lines that are not subject to strong Stark effects. The radial velocity curves of the optical stars in these systems differ from the radial velocity curves of their barycenters, due to the effects of tidal distortion, gravitational darkening, X-ray heating, etc. Hence, the mass functions $f_v(M)$ derived from these radial velocity curves will differ accordingly. We took into account the closeness of the components when determining masses of components using the K-correction method [@Petrov_2013]. Refined value of $q$ does not affect the mass of a black hole, but the mass of an optical star in most cases decreased up $\sim 1.5$ times.
Parameters of low mass X-ray binary systems containig black holes
=================================================================
The physical parameters of 9 BH LMXBs are listed in Table \[tabular:parameters\]. The spectra of optical counterparts received with intermediate resolution spectroscopy. This spectra allow to estimate rotational broadening of absorption lines, the radial velocity semi amplitudes, and the orbital inclination combined with a light curves.
In the Table \[tabular:parameters\] $V_{rot} \sin i$ is the projected equatorial rotational velocity, $K_v$ is the observed radial velocity semi-amplitude, $f_{v}(M)$ is the observed mass function, $i$ is the orbital inclination (see [@Lutiy_1973]).
Note, @Shahbaz2014 showed, that using relatively low-resolution spectroscopy can result in systemic uncertainties in the measured $V_{rot} \sin i$ values obtained classical rotational broadening method. The mass ratio $q=M_x/M_v$ we estimate with the following equation [@Wade_Horne_1988]: $$\begin{aligned}
\label{eq:Pach}
\frac{V_{rot}\sin i}{K_{c}}\simeq 0.462\; q^{-1/3}\left( 1+\frac{1}{q}\right)^{2/3}.\end{aligned}$$
It is necessary to mention that the radial velocity semi-amplitude of the optical stars $K_v$ are listed in Table \[tabular:parameters\] differ from the the radial velocity semi-amplitude of their barycenters $K_c$, due to the effects of tidal distortion, gravitational darkening, X-ray heating, etc. We took into account the closeness of the components using the K-correction method [@Wade_Horne_1988; @Petrov_2013]. The mass of black hole $M_x$ was determined using the equation: $$\begin{aligned}
\label{eq:massfun}
M_{x} &=& \frac{f_{v}(M)\left(1+q^{-1}\right)^{2}}{\sin^{3} i},\end{aligned}$$ and the optical star mass is: $$M_v=M_x/q.$$ Here $f_{v}(M) = 1.038 \cdot 10^{-7} K_v^{3} P_{orb}(1-e^2)^{3/2}$ is mass function, $K_v$ is observed radial velocity semi-amplitude of the optical stars, $e$ is the orbital eccentricity (for BH LMXBs $e=0$), $P_{orb}$ is the orbital period of the binary.
In the limit case where $q=M_x/M_v >> 1$ (this case corresponds to BH LMXB) mass ratio $q$ is given by: $$\begin{aligned}
\label{eq:big_q}
q \simeq \left( \frac{0.462 K_c}{V_{rot} \sin i } \right)^{3}.\end{aligned}$$ It follows from the expression (\[eq:big\_q\]) that small uncertainties in $V_{rot} \sin i$ lead to large uncertainties in $q$.
Refined black hole and optical star masses distribution
=======================================================
We use a method of mass ratio correction developed by @Petrov_2017 to study the black hole and optical star masses. As mentioned above we assume that approximation equation based on CaI $\lambda 6439.075 \; {\textup{\AA}}$ absorption line profiles can also be applied to other lines that are not subject to strong Stark effects. For BH LMXB in quiescent the effect of X-ray heating is ignored: $k_x=L_x/L_{v}^{bol}=0$ (the effect of heating by the X-ray radiation is described by @Petrov2015).
The method of mass ratio correction developed by @Petrov_2017 provides a corrected value of the binary mass ratio through the expression: $$\begin{aligned}
\label{eq:NonLinFitq}
q_{corr} = q_{disk} + \Delta q,\end{aligned}$$ where the $\Delta q$ correction is described by $$\begin{aligned}
\label{eq:NonLinFitDeltaq}
\Delta q = ( 0.41 \pm 0.01) \cdot {q_{disk}} ^{1.224 \pm 0.008}.\end{aligned}$$
@Petrov_2017 showed that equation (\[eq:NonLinFitq\]) is accurate better than $5 \; \%$ in effective temperature range $ 4000\; K$ - $8000 \; K$. We also took into account the closeness of the components when determining $M_x$. Hence, the mass functions $f_v(M)$ derived from observed radial velocity semi-amplitude of the optical stars will be changed with the following equation: $$\begin{aligned}
\label{eq:defnussfunc}
f_v^{corr}(M)=1.038\cdot 10^{-7} K_c^3P_{orb},\end{aligned}$$ where $K_c$ is the semi-amplitude of the radial velocity curve of the stellar barycenter. The $K_c$ is given by $K_c = K_v/K_{corr}(q,i)$ where $K_v$ is observed radial velocity semi-amplitude of the optical stars and $K_{corr}$ is the K-correction. For each system in Table \[tabular:parameters\], the corresponding K-correction was chosen from @Petrov_2013 [@Petrov2015]. In next step, we corrected masses of optical stars: $M_v=M_x/q$.
Initial and refined masses of black holes and optical stars are listed in Table \[tabular:Corparameters\]. We also present the refined distribution of the black hole masses in Fig. \[ris:Mx\_hist\] and the refined distribution of the optical star masses in Fig \[ris:Mv\_cor\_tot\]. Refined value of $q$ does not affect the mass of a black hole, but the mass of an optical star has in the most cases shrunk $\sim 1.5$ times (see Table \[tabular:Corparameters\]). The system GRO 1655-40 deserve special mention. The corrected black hole mass $M_x$ has the bias due the K-correction using.
Discussion
==========
We present the refined distribution of the donor masses in BH LMXBs with the peak at 0.35 $M_{\odot}$ (excluding the system GRO 1655-40). This distribution not only allow to estimate realistic binding energy of the common envelope $\lambda$, but also to bring several alternatives to the observations.
One of the alternative evolution scenario is that the secondaries in BH LMXBs enhanced the mass loss rate due to illumination of the stellar surface by the high energy radiation from the accretion disk around the BH during outburst [@Arons_1973; @Basko_1973; @Basko_1974; @Basko_1977]. This scenario produced a similar companion masses distribution with peak at $\sim 0.4 \;M_{\odot}$ [@Wiktorowicz_2014].
In the irradiation-induced wind scenario the mass transfer is non conservative [@ITY1995]. The black hole in an LMXB typically accretes only about 10% of the mass lost by the donor. A coronal wind carries about 90% mass lost by the donor away. Thus the increase of black hole mass should be negligible. Apparently, coronal wind velocity not enough to blow out the mass of the system and matter can form a disk-shaped circumbinary envelope.
Spectroscopic observations confirm fast orbital decays of black hole X-ray binaries: XTE J1118+480 ($\dot{P}=-1.90\pm 0.57$ ms/yr) and A0620-00 ($\dot{P}=-0.60\pm 0.08$ ms/yr) [@Gonzales_2014]. Angular momentum losses due to gravitational radiation and magnetic braking are unable to explain these large orbital decays in these two short-period black hole binaries. The fast spiral-in of the star in BH LMXB does not fit the rapid evaporation of stellar-mass black holes [@Postnov_2003] in multi-dimensional models of gravity [@Emparan_2003] on the RS brane [@Randall_1999]. @Chen_2015 showed that, for some reasonable parameters, tidal torque between the circumbinary disk and the binary can efficiently extract the orbital angular momentum from the binary. Observations have provided evidence that circumbinary disks around two compact black hole X-ray binaries may exist. @Muno_2006 have detected the blackbody spectrum of BHXBs A0620-00 and XTE J1118+480, and found that the inferred areas of mid-infrared 4.5-8 $\mu m$ excess emission are about two times larger than the binary orbital separations. A detection with the Wide-field Infrared Survey Explorer identified that XTE J1118+480 and A0620-00 are candidate circumbinary disk systems [@Wang_2014].
The extremely low-mass optical counterparts of BH LMXBs ( with masses $M_v < 0.35 \; M_{\odot}$) can result of evolution massive close binary accompanying at large distance by a G-K main sequence dwarf [@Eggleton_Verbunt_1986]. After the evolution of the close binary into an ordinary X-ray binary, the compact object is engulfed by its expanding massive companion, and spirals in to settle at its centre. The resulting Thorne-Zytkow supergiant gradually expands until it attains the size of the main sequence dwarf orbit. Then a second spiral-in phase ensues, leading to the formation of a low-mass close binary. Tidal capture formation scenario, also can correspond to this refined distribution [@Erez_2016]. In this formation channel, LMXBs are formed from wide binaries ($>1000$ au) with a BH component and a stellar companion. Resulting optical mass distribution is this scenario peaking at $0.4-0.6 \; M_{\odot}$. Note, that in this case the LMXBs orbit should not correlate with the spin of the BH.
Conclusions
===========
We present refined binary masses in low-mass X-ray binaries with black holes. The refined masses of an optical stars have shrunk $\sim 1.5$ times. In this way the refined distribution of the donor masses in BH LMXBs has the peak at 0.35 $M_{\odot}$ ($\bar{M}_v=0.31 \; M_{\odot}$ without GRO 1655-40). This distribution is well described by irradiation-induced wind evolution scenario. Furthermore, The extremely low-mass optical counterparts of BH LMXBs ($M_v<0.35 \; M_{\odot}$) can result of evolution massive close binary accompanying at large distance by a G-K main sequence dwarf [@Eggleton_Verbunt_1986] or tidal capture formation scenario [@Erez_2016].
We thank Dmitry V. Bisikalo, Konstantin A. Postnov and Chris Belczynski for their useful comments.
[lcccccc]{} System &$V_{rot} \sin i$, km/s &$K_v$, km/s &$f_v(M),\;M_{\odot}$ &$q=M_x/M_v$& $i$, deg.& Links\
A0620-00 &$83 \pm 5$&$437 \pm 2$&$2.80\pm 0.01$&$16 \pm 3$&$51 \pm 1$ &1,2\
GS 2023+338&$39.1 \pm 1.2$&$208.5\pm 0.7$&$6.08\pm 0.06$&$16.7 \pm 1.4$&$ 66-70 $&3\
GS 2000+251&$86 \pm 8$&$520 \pm 5$&$5.01\pm 0.12$&$24 \pm 7$&$ 43-74 $&4\
GRO 1655-40&$88\pm 5$&$226.1\pm 0.8$&$2.73\pm 0.09$&$3.0 \pm 0.3$&$70\pm 2$&5\
XTE J1118+48&$96^{+4}_{-11} $&$709 \pm 1$&$6.28\pm 0.04$&$41^{+5}_{-16}$&$68\pm 2$&6\
GRS 1915+105&$21 \pm 4 $&$126\pm 1$&$7.02 \pm 0.17$&$23 \pm 2$&$66\pm 2$&7\
GRS 1009-45&$ 86 \pm 5$&$475\pm 6$&$3.17\pm 0.12$&$18 \pm 3$&$37-80$&8\
GRO J0422+32&$ 90 \pm 22$&$378\pm 16$&$1.19\pm 0.02$&$11 \pm 6$&$45\pm 2$&9\
GRS 1124-68&$ 106 \pm 13$&$406 \pm 7$&$3.01\pm 0.15$&$13 \pm 1$&$54\pm 2$&10\
\
\
\
\
\
\
[lccccccc]{} System&$q_{corr}$&$K_c$, km/s &$f_v^{corr}(M),M_{\odot}$ &$M_x$, $ M_{\odot} $&$M_x^{corr}$, $ M_{\odot} $&$M_v$, $ M_{\odot} $&$M_v^{corr}$, $ M_{\odot} $\
A0620-00 & $26 \pm 5$&$438.4 \pm 2.0$& $2.82\pm 0.036$& $6.7\pm 0.3 $ & $6.5 \pm 0.3$ & $0.4 \pm 0.1 $&$ 0.25 \pm 0.1 $\
GS 2023+338& $26 \pm 2 $& $209.1\pm 0.7$& $6.14\pm 0.06$ & $8.2 -8.9$ & $7.9 - 8.6$&$0.5\pm 0.1 $& $0.3 \pm 0.1 $\
GS 2000+251& $39 \pm 8$& $521\pm 5$& $5.06\pm 0.12$ & $6.1 - 17.1 $ & $6.0-16.7$ &$0.2 \div 0.7 $&$0.1 \div 0.4$\
GRO 1655-40& $3.9\pm 0.5$& $229.6\pm 2.4$& $3.29\pm 0.09$&$5.8 \pm 0.4 $&$6.0 \pm 0.4$ & $ 1.9 \pm 0.2$ & $ 1.4 \pm 0.2 $\
XTE J1118+48& $73 ^{+10}_{-32} $& $710.2\pm 1.4$& $6.3 \pm 0.1$ &$8.2 \pm 0.4 $&$8.1 \pm 0.4$& $0.20 \pm 0.04$ &$ 0.10 \pm 0.04 $\
GRS 1915+105&$35 \pm 13 $&$126\pm 1$&$7.07\pm 0.17$&$10.1 \pm 0.6$&$9.8 \pm 0.6$&$0.4 \pm 0.1$ &$ 0.3 \pm 0.1$\
GRS 1009-45&$30 \pm 5$& $476\pm 4$&$3.20\pm 0.12$&$3.7 - 16.2$&$3.6-15.7$&$0.2 \div 0.9$ &$0.1 \div 0.5 $\
GRO J0422+32&$ 14 \pm 3$&$379\pm 16$&$1.20\pm 0.02$&$4.1 \pm 0.6$&$3.9 \pm 0.6$&$0.5 \pm 0.1$&$ 0.3 \pm 0.1$\
GRS 1124-68&$ 21 \pm 1$& $407.6 \pm 2.0$& $3.04 \pm 0.15$& $6.6 \pm 0.5$ & $6.3 \pm 0.5$&$ 0.5 \pm 0.1$&$ 0.3 \pm 0.1$\
\
\
\
\
[^1]: patrokl@gmail.com
|
---
abstract: |
In this paper we extend the results of the research started in [@YK2009] and [@YK2010], in which Karlin-McGregor diagonalization of certain reversible Markov chains over countably infinite general state spaces by orthogonal polynomials was used to estimate the rate of convergence to a stationary distribution.
We use a method of Koornwinder [@TK1984] to generate a large and interesting family of random walks which exhibits a lack of spectral gap, and a polynomial rate of convergence to the stationary distribution. For the Chebyshev type subfamily of Markov chains, we use asymptotic techniques to obtain an upper bound of order $O\left({\log{t} \over \sqrt{t}}\right)$ and a lower bound of order $O\left({1 \over \sqrt{t}}\right)$ on the distance to the stationary distribution regardless of the initial state. Due to the lack of a spectral gap, these results lie outside the scope of geometric ergodicity theory [@SMRT2009].
address:
- 'Department of Mathematics, Oregon State University, Corvallis, OR 97331'
- 'Department of Mathematics, Oregon State University, Corvallis, OR 97331'
author:
- Yevgeniy Kovchegov
- Nicholas Michalowski
title: A class of Markov chains with no spectral gap
---
Introduction
============
Let $P=\Big(p(i,j)\Big)_{i,j \in \Omega}$ be a reversible Markov chain over a sample space $\Omega$, that is, it must satisfy the following [*detailed balance conditions*]{}: $$\pi_i p(i,j)=\pi_jp(j,i) \qquad \forall i,j \in \Omega,$$ where $\pi$ is a non-trivial non-negative function over $\Omega$. If $P$ admits a unique stationary distribution $\nu$, then ${1 \over \sum\limits_{i \in \Omega} \pi_i}\pi=\nu$.
It can be shown that the reversible $P$ is a self-adjoint operator in $\ell^2(\pi)$, the space generated by the following inner product induced by $\pi$ $${\ensuremath{\left\langle f, g \right\rangle}}_{\pi}=\sum_{i \in \Omega} f(i)g(i)\pi_i$$ If $P$ is a tridiagonal operator (i.e. a nearest-neighbor random walk) on $\Omega=\{0,1,2,\dots\}$, then it must have a simple spectrum, and is diagonalizable via orthogonal polynomials as it was studied in the 50’s by Karlin and McGregor, see [@SKJM1959], [@GS1975], and [@MI2005]. There, the extended eigenfunctions $Q_j(\lambda)$ satisfying $Q_0 \equiv 1$ and $$P\begin{pmatrix}
Q_0(\lambda)\\
Q_1(\lambda)\\
Q_2(\lambda)\\
\vdots
\end{pmatrix} =
\lambda\begin{pmatrix}
Q_0(\lambda)\\
Q_1(\lambda)\\
Q_2(\lambda)\\
\vdots
\end{pmatrix}$$ are orthogonal polynomials with respect to a probability measure $\psi$. If we let $p_t(i,j)$ denote the entries of the operator $P^t$ that represent $t$ step transition probabilities from state $i$ to state $j$ then $$p_t(i,j)=\pi_j \int_{-1}^1 \lambda^t Q_i(\lambda) Q_j(\lambda) d\psi(\lambda)~~~\forall i,j \in \Omega,$$ where $\pi_j$ with $\pi_0=1$ is the reversibility measure of $P$.
We will use the following distance to measure the deviation from the stationary distribution on a scale from zero to one.
If $\mu$ and $\nu$ are two probability distributions over a sample space $\Omega$, then the [*total variation distance*]{} is $$\| \nu - \mu \|_{TV} = {1 \over 2} \sum_{x \in \Omega} |\nu(x)-\mu(x)|=\sup_{A \subset \Omega} |\nu(A)-\mu(A)|$$
Let $\rho=\sum_{k=0}^{\infty} \pi_k$. Observe that $\rho < \infty$ if and only if the random walk $P$ is positive recurrent. Recall that $\nu={1 \over \rho}\pi$ is the stationary probability distribution. If in addition to being positive recurrent, the aperiodic nearest neighbor Markov chain originates at site $j$, then the total variation distance between the distribution $\mu_t=\mu_0P^t$ and $\nu$ is given by $$\label{eqTV}
\left\|\nu - \mu_t \right\|_{TV} = {1 \over 2} \sum_{n \in \Omega} \pi_n \left|\int_{(-1,1)} \lambda^t Q_j(\lambda) Q_n(\lambda) d\psi(\lambda)\right|,$$ as measure $\psi$ contains a point mass of weight ${1 \over \rho}$ at $1$. See [@YK2009].
The rates of convergence are quantified via mixing times, which for an infinite state space with a unique stationary distribution are defined as follows. Here the notion of a mixing time depends on the state of origination $j$ of the Markov chain. See [@YK2010].
Suppose $P$ is a Markov chain with a stationary probability distribution $\nu$ that commences at $X_0=j$. Given an $\epsilon >0$, the mixing time $t_{mix}(\epsilon)$ is defined as $$t_{mix}(\epsilon)=\min\left\{t~:~\|\nu-\mu_t\|_{TV} \leq \epsilon \right\}$$
0.2 in In the case of a nearest-neighbor process on $\Omega=\{0,1,2,\dots\}$ commencing at $j$, the corresponding mixing time has the following simple expression in orthogonal polynomials $$t_{mix}(\epsilon)=\min\left\{t~:~\sum_{n} \pi_n \left|\int_{(-1,1)} \lambda^t Q_j(\lambda) Q_n(\lambda) d\psi(\lambda)\right| \leq 2 \epsilon \right\},$$
Investigations into the use of orthogonal polynomial techniques (see [@SKJM1959], [@GS1975]) in the estimation of mixing times and distance to the stationary distribution has been carried out in [@YK2010] for certain classes of random walks. In this paper we consider the problem from the other direction. Namely given a large class of orthogonal polynomials we outline how to find the corresponding random walk and estimate the rate for the distance to the stationary distribution.
More specifically beginning with the Jacobi polynomials, whose weight function lies in $(-1,1)$ we use Koornwinder’s techniques [@TK1984] to attach a point mass at $1$. For the class of Jacobi type polynomials $Q_n$ thus obtained, the three term recurrence relationship is understood [@HKJW1996]. The tridiagonal operator corresponding to these polynomials is not a Markov chain, however the operator can be deformed to become one. The corresponding changes in the polynomials are easy to trace. This gives a four parameter family of nearest neighbor Markov chains whose distance to the stationary distribution decays in a non-geometric way. In principle the asymptotic analysis presented in this paper can be applied to the entire four parameter family. We outline how this proceeds for Chebyshev-type subfamily consisting of taking $\alpha=\beta=-1/2$ in the Koornwinder class.
We would like to point out the important results of V. B. Uvarov [@VU1969] on transformation of orthogonal polynomial systems by attaching point masses to the orthogonality measure, predating the Koornwinder results by fifteen years. The results of V. B. Uvarov can potentially be used in order to significantly extend the scope of convergence rate problems covered in this current manuscript.
The paper is organized as follows. In Section \[sec:koorn\] we discuss constructing positive recurrent Markov chains from the Jacobi family of orthogonal polynomials adjusted by using Koornwinder’s techniques to place a point mass at $x=1$. Next, we derive an asymptotic upper bound on the total variation distance to the stationary distribution in the case of general $\alpha>-1$ and $\beta>-1$ in Section \[sec:asympt\]. Our main result, Theorem \[main\], is presented in Section \[sec:chebyshev\]. There, for the case of Chebyshev type polynomials corresponding to $\alpha=\beta=-1/2$, we produce both asymptotic lower and upper bounds for the total variation distance. Finally, in Section \[comparison\] we compare our main result to related results obtained by other techniques.
From Orthogonal Polynomials to Random Walks via Koornwinder {#sec:koorn}
===========================================================
T. Koornwinder [@TK1984] provides a method for finding the orthogonal polynomials whose weight distribution is obtained from the standard Jacobi weight functions $C_{\alpha,\beta}(1-x)^{\alpha}(1+x)^{\beta}$ by attaching weighted point masses at $-1$ and $1$. A spectral measure corresponding to a Markov chain contains a point mass at $-1$ if and only if the Markov chain is periodic. A spectral measure for an aperiodic Markov chain contains a point mass at $1$ if and only if it is positive recurrent. Thus in order to create a class of positive recurrent aperiodic Markov chains with a Koornwinder type orthogonal polynomial diagonalization we will only need to attach a point mass at $1$ and no point mass at $-1$.
Let $N\geq 0$ and let $\alpha$, $\beta>-1$. For $n=0, 1, 2, \ldots$ define $$\label{eq:Koornwinder}
P_n^{\alpha,\beta,N}(x)=\Big(\frac{(\alpha+\beta+2)_{n-1}}{n!}\Big)A_n\Big[-N
(1+x)\frac{d}{dx}+B_n\Big]P_n^{\alpha,\beta}(x),$$ where $$A_n=\frac{(\alpha+1)_n}{(\beta+1)_n},$$ $$B_n=\frac{(\beta+1)_nn!}{(\alpha+1)_n(\alpha+\beta+2)_{n-1}}+\frac{n(n+\alpha+\beta+1)N}{(\alpha+1)},$$ $P_n^{\alpha,\beta}$ is the standard Jacobi polynomials of degree $n$ and order $(\alpha,\beta)$, $(x)_n=x(x+1)\cdots(x+n-1)$. These polynomials form a system of orthogonal polynomials with respect to the probability measure $d\psi(x)={C_{\alpha,\beta}(1-x)^\alpha(1+x)^\beta dx+N\delta_1(x)
\over N+1}$, where $C_{\alpha,\beta}={1 \over
\mathcal{B}(\alpha+1,\beta+1)}$, $\mathcal{B}(\cdot,\cdot)$ is the beta function, and $\delta_1(x)$ denotes the a unit point mass measure at $x=1$. See T. Koornwinder [@TK1984]. Direct calculation shows that $P_n^{\alpha,\beta,N}(1)=\frac{(\alpha+1)_n}{n!}$, and so we normalize $Q_n(x)=n!P_n^{\alpha,\beta,N}(x)/(\alpha+1)_n$ which is the orthogonal set of polynomials with respect to $d\psi$ satisfying $Q_n(1)=1$.
As we have mentioned earlier, the tridiagonal operator $H$ corresponding to the recurrence relation of the orthogonal polynomials may not be a Markov chain operator. Let $p_i$, $r_i$ and $q_i$ denote the coefficients in the tridiagonal recursion $$p_i Q_{i+1}(x)+r_iQ_i(x)+q_i Q_{i-1}(x)=xQ_i(x),$$ for $i=0,1,2,\dots$, where we let $Q_{-1} \equiv 0$ as always.
Notice because the polynomials are normalized so that $Q_i(1)=1$ it follows immediately that $p_i+r_i+q_i=1$. However some of the coefficients $p_i$, $r_i$, or $q_i$ may turn out to be negative, in which case the rows of the tridiagonal operator $A$ would add up to one, but will not necessarily consist of all nonnegative entries.
In the case when all the negative entries are located on the main diagonal, this may be overcome by considering the operator $\frac{1}{\lambda+1}(H+\lambda I)$. For $\lambda \geq-\inf\limits_i r_i$ this ensures all entries in the matrix $\frac{1}{\lambda+1}(H+\lambda I)$ are nonnegative and hence can be thought of as transition probabilities. More generally, if a polynomial $p(\cdot)$ with coefficients adding up to one is found to satisfy $p(H) \geq 0$ coordinatewise, then such $p(H)$ would be a Markov chain.
An Asymptotic Upper Bound for Jacobi type Polynomials {#sec:asympt}
=====================================================
In this section we derive asymptotic estimates for the distance to the stationary distribution when our operator given by $P_{\lambda}=\frac{1}{\lambda+1}(H+\lambda I)$ is a Markov chain. In this case the Karlin-McGregor orthogonal polynomials for $P_{\lambda}$ are $Q_j\Big((1+\lambda)x-\lambda\Big)$ and the orthogonality probability measure is ${1 \over 1+\lambda}d\psi\Big((1+\lambda)x-\lambda\Big)$ over $\Big({\lambda-1 \over \lambda+1}, 1 \Big]$, where the $Q_j$ are the Jacobi type polynomials introduced by Koornwinder from the previous section.
Of course the new operator $P_{\lambda}$ is again tridiagonal. For the $n$-th row of $P_{\lambda}$, let us denote the $(n-1)$-st, $n$-th, and $(n+1)$-st entries by $q_n^\lambda$, $r_n^\lambda$, and $p_n^\lambda$ respectively. Here the entries of $P_{\lambda}$ can be expressed via the entries of $H$ as follows $$p_n^\lambda=\frac{p_n}{1+\lambda}, \qquad r_n^\lambda=\frac{r_n+\lambda}{1+\lambda}, \quad \text{ and } \quad q_n^\lambda=\frac{q_n}{1+\lambda}$$ Clearly we still have that $p_n^\lambda+r_n^\lambda+q_n^\lambda=1$.
With the probabilities in hand we now compute the corresponding reversibility function $\pi_n^\lambda$ of $P_{\lambda}$ which is equal to the corresponding function of $H$ defined as $\pi_n=\frac{p_0\cdots p_{n-1}}{q_1\cdots q_n}$. Here $\pi_0^\lambda=1=\pi_0$ and $\pi_n^\lambda=\frac{p_0^\lambda\cdots p_{n-1}^\lambda}{q_1^\lambda\cdots q_n^\lambda}=\frac{p_0\cdots p_{n-1}}{q_1\cdots q_n}=\pi_n$.
Changing variables in yields $$\|\nu-\mu_t\|_{TV}=\frac{1}{2}\sum_{n=0}^{\infty}\pi_n\bigg|\int_{(-1,1)}\Big(\frac{x}{1+\lambda}+\frac{\lambda}{1+\lambda}\Big)^tQ_j(x)Q_n(x)\,d\psi(x)\bigg|$$
\[lem:bounds\] Consider the case when $p_n>0$ and $q_n>0$ for all $n \geq 0$, and . Then, for the Jacobi type polynomials $Q_j$ the distance to the stationary distribution satisfies the following bound $$\label{jacobi_bound}
{\ensuremath{\left\| \nu-\mu_t \right\|}}_{TV}\leq\frac{C_{\alpha,\beta,\lambda}{\ensuremath{\left\| Q_j \right\|}}_\infty}{(t+1)^{1+\alpha}}\sum_{n=0}^{t+j}\pi_n{\ensuremath{\left\| Q_n \right\|}}_\infty+\frac{1}{2}\sum_{n=j+t+1}^{\infty}\pi_n$$ for a certain constant $C_{\alpha,\beta,\lambda}$.
For $n>j+t$, it follows from the orthogonality of the polynomials and our normalization $Q_i(1)=1$ that $$\int_{(-1,1)}\Big(\frac{x}{1+\lambda}+\frac{\lambda}{1+\lambda}\Big)^tQ_j(x)Q_n(x)\,d\psi(x)=1$$ It is then easy to see that ${\ensuremath{\left\| \nu-\mu \right\|}}_{TV}\leq I+II+\frac{1}{2}\sum_{n=j+t+1}^{\infty}\pi_n$, where $$I=\frac{1}{2}\sum_{n=0}^{j+t}\pi_n\int_{(-1,0)}{\ensuremath{\left|\Big(\frac{x+\lambda}{1+\lambda}\Big)^tQ_j(x)Q_n(x)\right|}}(1-x)^\alpha(1+x)^\beta\,dx$$ $$\text{ and } \quad II=\frac{1}{2}\sum_{n=0}^{j+t}\pi_n\int_{(0,1)}\Big(\frac{x+\lambda}{1+\lambda}\Big)^t\big|Q_j(x)Q_n(x)\big|(1-x)^\alpha(1+x)^\beta\,dx$$
To estimate $I$ notice that ${\ensuremath{\left|\frac{x+\lambda}{1+\lambda}\right|}}\leq
\max(\frac{\lambda}{1+\lambda}, {\ensuremath{\left|\frac{1-\lambda}{1+\lambda}\right|}})<1$ for $\lambda>0$. Hence for an appropriate polynomial $A_j(\cdot)$ such that $$\frac{1}{2}\|Q_j\|_{\infty}\sum_{n=0}^{j+t}\pi_n \|Q_n\|_{\infty}\int_{(-1,0)}(1-x)^\alpha(1+x)^\beta\,dx \leq A_j(|t|),$$ and $c=-\log\left\{\max(\frac{\lambda}{1+\lambda}, {\ensuremath{\left|\frac{1-\lambda}{1+\lambda}\right|}})\right\}$. Such polynomial $A_j$ exists since $\|Q_n\|_{\infty}$ grows polynomially in $n$ and $\pi_n$ is bounded. See formula 22.14.1 in Abramowitz and Stegun [@MAIS1972].
Thus $I$ is clearly bounded by the right hand side of .
For the second term, $II\leq \frac{1}{2}\sum_{n=0}^{j+t}\pi_n{\ensuremath{\left\| Q_nQ_j \right\|}}_{\infty}\int_0^1\Big(\frac{x+\lambda}{1+\lambda}\Big)^t\big(1-x)^\alpha(1+x)^\beta\,dx$. There we make the change of variables $s=-\log(\frac{x+\lambda}{1+\lambda})$, and for simplicity let $x(s)=(1+\lambda)e^{-s}-\lambda$. Then the integral reduces to $$(1+\lambda)^{1+\alpha}\int_0^{\log(\frac{1+\lambda}{\lambda})}e^{-s(t+1)}\big(1-\lambda+(1+\lambda)e^{-s})^\beta\big(1-e^{-s}\big)^\alpha\,ds$$
Using the fact that $(1-e^{-s})^\alpha=s^\alpha\Big(1+O(s)\Big)$ and $\big(1-\lambda+(1+\lambda)e^{-s})^\beta=2^\beta+O(s)$, the above integral becomes $$(1+\lambda)^{1+\alpha}\int_0^{\log(\frac{1+\lambda}{\lambda})}e^{-s(t+1)}\Big(2^\beta
s^\alpha+O(s^{\alpha+1})\Big)\,ds,$$ where the upper bounds $O(s)$ can be made specific. Next, applying the standard asymptotic methods of Laplace to this yields the following asymptotics $$\int_0^{\log(\frac{1+\lambda}{\lambda})}e^{-s(t+1)}
s^\alpha\,ds ~\asymp~ \frac{\Gamma(\alpha+1)}{(t+1)^{1+\alpha}}$$
Thus one can obtain a large enough constant $\widetilde{C}_{\alpha,\beta,\lambda}$ such that $$II\leq\
\frac{\widetilde{C}_{\alpha,\beta,\lambda}{\ensuremath{\left\| Q_j \right\|}}_\infty}{(t+1)^{1+\alpha}}\sum_{n=0}^{t+j}\pi_n{\ensuremath{\left\| Q_n \right\|}}_\infty$$
In order to derived effective bounds on ${\ensuremath{\left\| \nu-\mu_t \right\|}}_{TV}$ it is necessary to gain a more detailed understanding of $\pi_n$ and ${\ensuremath{\left\| Q_n \right\|}}_\infty$. When $\min(\alpha,\beta)\geq -\frac{1}{2}$, the ${\ensuremath{\left\| Q_n \right\|}}_\infty$ can be estimated using the known maximum for the Jacobi polynomials found in Lemma 4.2.1 on page 85 of [@MI2005] together with Koornwinder’s definition of these polynomials.
One way to derive estimates for $\pi_n$ is to use the expression $\pi_n$ in terms of $p_n$, $r_n$, and $q_n$. For Koorwinder’s class of polynomials these expressions are derived for all $\alpha, \beta, M,
N$ in [@HKJW1996]. It can be verified directly that in the case when $M=0$, then $p_0=\frac{2(\alpha+1)}{(1+N)(\alpha+\beta+2)}>0$. After taking into account the normilization $Q_n(1)=1$, and taking into account a small typo, it can be verified from equations (41)–(45) in [@HKJW1996] that $p_n$ and $q_n$ are positive for $n\geq 1$. Thus the conditions for Lemma \[lem:bounds\] are satisfied for all $\alpha, \beta >-1$. Furthermore, from (18), (19) and (32) in [@HKJW1996] it can be easily seen that $p_n\to \frac{1}{2}$ and $q_n\to \frac{1}{2}$ as $n\to\infty$, and hence $r_n=1-p_n-q_n\to 0$ as $n\to \infty$. Thus for $\lambda$ large enough the operator $P^\lambda$ corresponds to a Markov chain.
As the expressions for these quantities laborious to write down, instead we focus our attention on a specific case in which our calculations are easy to follow. Specifically we focus on the Chebyshev polynomials.
Chebyshev Polynomials: Upper and Lower Bounds {#sec:chebyshev}
=============================================
By applying Koorwinder’s results to the Chebyshev polynomials of the first kind which correspond to the case of $\alpha=\beta=-{1 \over 2}$, we arrive at a family of orthogonal polynomials with respect to the measure $\frac{1}{1+N}\Big(\frac{1}{\pi\sqrt{1-x^2}}dx+N\delta_1(x)\Big)$. Using we find that here, $$Q_n(x):=-N(x+1)U_{n-1}(x)+(1+2nN)T_n(x),$$ where $T_n$ and $U_n$ denote the Chebyshev polynomials of the first and second kind respectively. Notice that $U_n(1)=n+1$ and $T_n(1)=1$, which immediately to verify that $Q_n(1)=1$.
Once again we consider the operator $$H=\begin{pmatrix}r_0 & p_0 & 0 & 0 & 0 &\cdots\\
q_1 & r_1 & p_1 & 0 & 0 & \vdots\\
0 & q_2 & r_2 & p_2 & 0 & \ddots\\
0 & 0 & q_3 & r_3 & \ddots & \ddots\\
\vdots & \cdots & \ddots & \ddots & \ddots & \dots\\
\end{pmatrix},$$ on $\ell^2(\pi)$, so that vector $(Q_0(x), Q_1(x), Q_2(x),
\ldots)^T$ is an eigenvector with eigenvalue $x$.
Specifically the numbers $p_n$, $r_n$, and $q_n$ satisfy $p_0P_1(x)+r_0P_0(x)=x$ for $n=0$, and $$\label{eigvaleq}
p_nQ_{n+1}(x)+r_nQ_n(x)+q_nQ_{n-1}(x)=xQ_n(x)\qquad \text{for $n\geq 1$.}$$
Keisel and Wimp [@HKJW1996] give expressions for $p_n$, $r_n$ and $q_n$ for $n\geq 0$. To find the expressions directly in this case one could use to derive three linearly independent equations, and solve for $p_n$, $r_n$, and $q_n$.
For the case $n=0$ the equation immediately gives us that $p_0=\frac{1}{N+1}$ and $r_0=\frac{N}{N+1}$. Evaluating at convenient choices of $x$, such as $-1, 0, 1$, do not yield linearly independent equations for all $n$. One solution to this is to evaluate at $x=1,-1$ and differentiate and then evaluate at $x=0$. This gives three linearly independent equations and a direct calculation then shows that $$\begin{gathered}\label{eq:linearprobs}
p_n=\frac{1}{2}\cdot\frac{1+(2n-1)N}{1+(2n+1)N}, \quad
q_n=\frac{1}{2}\cdot\frac{1+(2n+1)N}{1+(2n-1)N}, \quad \text{and}\\
r_n=\frac{-2N^2}{(1+(2n-1)N)(1+(2n+1)N)}
\end{gathered}$$ As $r_n\leq 0$ the operator $H$ fails to correspond to a Markov chain. However this is the case we addressed at the end of Section \[sec:koorn\] of the current paper. Thus consider $P_\lambda=\frac{1}{1+\lambda}(H+\lambda I)$. Now, since $|r_n|$ is a decreasing sequence for $n\geq 1$. So provided that $\lambda\geq |r_1|=\frac{2N^2}{(1+N)(1+3N)}$, we then have $p_n^\lambda, r_n^\lambda, q_n^\lambda\geq 0$. Thus we can consider these coefficients $p_n^\lambda$, $r_n^\lambda$, and $q_n^\lambda$ as the transition probabilities in a nearest neighbor random walk.
Recall that $\pi_n^\lambda=\pi_n=\frac{p_0\cdots p_{n-1}}{q_1\cdots
q_n}$. Thus for $P_\lambda$ we can directly calculate $\pi_n$ from . We have that $p_0\cdots
p_{n-1}=\frac{1}{2^{n-1}}\frac{N}{1+(2n-1)N}$ and similarly $q_1\cdots q_n=\frac{1}{2^n}\frac{1+(2n+1)N}{1+N}$. Thus $\pi_n=\frac{2(1+N)N}{(1+(2n-1)N)(1+(2n+1)N)}$.
\[main\] Given $N>0$ and $\lambda\geq \frac{2N^2}{(1+N)(1+3N)}$. Consider the case of the Chebyshev-type random walks over $\Omega=\{0,1,2,\dots \}$ with probability operator $$P_\lambda=\begin{pmatrix}r_0^\lambda & p_0^\lambda & 0 & 0 & 0 &\cdots\\
q_1^\lambda & r_1^\lambda & p_1^\lambda & 0 & 0 & \vdots\\
0 & q_2^\lambda & r_2^\lambda & p_2^\lambda & 0 & \ddots\\
0 & 0 & q_3^\lambda & r_3^\lambda & \ddots & \ddots\\
\vdots & \cdots & \ddots & \ddots & \ddots & \dots\\
\end{pmatrix},$$ where $p_n^\lambda=\frac{1}{2(1+\lambda)}\cdot\frac{1+(2n-1)N}{1+(2n+1)N}$, $q_n^\lambda=\frac{1}{2(1+\lambda)}\cdot\frac{1+(2n+1)N}{1+(2n-1)N}$ and $r_n^\lambda=1-p_n^\lambda-q_n^\lambda$ for $n \geq 1$, with $p_0^\lambda=\frac{1}{(1+\lambda)(N+1)}=1-r_0^\lambda$.
Then for the random walk originating at some site $j \in \Omega$, there are positive constants $c$ and $C$ that depend on $j$, $N$ and $\lambda$ such that $$\frac{c}{\sqrt{t}}\leq {\ensuremath{\left\| \nu-\mu_t \right\|}}_{TV}\leq
C\frac{\log t}{\sqrt{t}}$$ for $t$ sufficiently large.
For the upper bound we simply need to estimate the sums appearing in Lemma \[lem:bounds\]. Since $\pi_n=O\big(\frac{1}{(n+1)^2}\big)$, it is easy to see that the second sum $\sum_{n=j+t+1}^{\infty}\pi_n$ is bounded by $C_N/(t+j+1)$. The main term turns out to be the first sum.
In the case of the Chebyshev type polynomials we have the bound . Thus the first sum in Lemma \[lem:bounds\] is bounded by $\hat C_{\alpha,\beta,\lambda,N}\frac{j\log(t+j+2)}{\sqrt t}$ for an appropriate constant $\hat C_{\alpha,\beta,\lambda,N}$. And so, for an appropriate $C$ and large $t$, $${\ensuremath{\left\| \nu-\mu_n \right\|}}_{TV}\leq C\frac{\log t}{\sqrt{t}}$$
On the other hand, recalling that $Q_0(x)=\pi_0=1$, we have that: $${\ensuremath{\left\| \nu-\mu_n \right\|}}_{TV}\geq {\ensuremath{\left|\int_{(-1,1)}\Big(\frac{x+\lambda}{1+\lambda}\Big)^tQ_j(x)(1+x^\beta)(1-x)^\alpha\,dx\right|}}$$ However we have already shown that for large enough $t$, the above right-hand side is asymptotic to $\frac{\tilde C}{\sqrt{1+t}}$.
We finish with some concluding remarks. At first the bound may appear somewhat imprecise since near $x=1$, we have that $Q_n(1)=1$. It is tempting to suggest that the correct asymptotic for the total variation norm is $C/\sqrt{t}$. However on closer examination in the neighborhood of $x=1$, $Q_n'(x)\approx n^3$. This $n^3$ causes the errors to be at least of the order of the main term. Overall it seems unlikely to the authors that $C/\sqrt{t}$ is the correct asymptotic for the Chebyshev-type polynomials.
Comparison to other methods {#comparison}
===========================
An ergodic Markov chain $P=\Big(p(i,j)\Big)_{i,j \in \Omega}$ with stationary distribution $\nu$ is said to be [*geometrically ergodic*]{} if and only if there exists $0<R<1$ and a function $M: \Omega \rightarrow \mathbb{R}_+$ such that for each initial state $i \in\Omega$, the total variation distance decreases exponentially as follows $$\|p_t(i,\cdot)-\nu(\cdot)\|_{TV}={1 \over 2} \sum\limits_{j \in \Omega} |p_t(i,j)-\nu(j)| \leq M(i) R^t$$ In other words, an ergodic Markov chain is geometric when the rate of convergence to stationary distribution is exponential. See [@SMRT2009] and references therein.
If the state space $\Omega$ is finite, $|\Omega|=d <\infty$, and Markov chain is irreducible and aperiodic, then $P$ will have eigenvalues that can be ordered as follows $$\lambda_1=1 >|\lambda_2| \geq \dots \geq |\lambda_d|$$ In which case, the Perron-Frobenious Theorem will imply geometric ergodicity with $$\|p_t(i,\cdot)-\nu(\cdot)\|_{TV}=O(t^{m_2-1} |\lambda_2|^t),$$ where $m_2$ is the algebraic multiplicity of $\lambda_2$. Here the existence of a positive [*spectral gap*]{}, $1-|\lambda_2|>0$, implies geometric ergodicity with the exponent $-\log |\lambda_2| \approx 1- |\lambda_2|$ whenever the spectral gap is small enough.
When dealing with Markov chains over general countably infinite state space $\Omega$, the existence of a positive spectral gap of the operator $P$ is essentially equivalent to the chain being geometrically ergodic. For instance, the orthogonal polynomial approach in [@YK2010] resulted in establishing the geometric rate $R=\max\left\{r+2\sqrt{pq},~{q \over q+r}\right\}$ for the Markov chain $$P=\left(\begin{array}{ccccc}0 & 1 & 0 & 0 & \dots \\q & r & p & 0 & \dots \\0 & q & r & p & \ddots \\0 & 0 & q & r & \ddots \\\vdots & \vdots & \ddots & \ddots & \ddots\end{array}\right) \qquad q>p,~~~r>0$$ over $\Omega=\mathbb{Z}_+$, together with establishing the value of the spectral gap, $1-r>0$.
As for the Markov chain $P_\lambda$ considered in Theorem \[main\] of this paper, its spectral measure ${1 \over 1+\lambda}d\psi\Big((1+\lambda)x-\lambda\Big)$ over $\Big({\lambda-1 \over \lambda+1}, 1 \Big]$ admits [*no*]{} spectral gap between the point mass at $1$ and the rest of the spectrum implying sub-geometric ergodicity. The sub-exponential rate in total variation norm is then estimated to be of polynomial order between $\frac{1}{\sqrt{t}}$ and $\frac{\log t}{\sqrt{t}}$.
In the field of probability and stochastic processes, there is a great interest in finding methods for analyzing Markov chains over general state space that have polynomial rates of convergence to stationary distribution. In Menshikov and Popov [@MMSP1995] a one dimensional version of Lamperti’s problem is considered. There, a class of ergodic Markov chains on countably infinite state space with sub-exponential convergence to the stationary probabilities is studied via probabilistic techniques. One of their results relates to our main result, Theorem \[main\]. Namely, Theorem 3.1 of [@MMSP1995] when applied to our case, implies for any $\varepsilon>0$ the existence of positive real constants $C_1$ and $C_2$ such that $$C_1t^{-{1 \over 2}-\varepsilon} \leq |\nu(0)-\mu_t(0)| \leq C_2t^{-{1 \over 2}+\varepsilon}$$ Thus for the Markov chain considered in Theorem \[main\], the orthogonal polynomials approach provides a closed form expression for the difference $\nu-\mu_t$, and a significantly sharper estimate on convergence of $\mu_t$ to the stationary distribution $\nu$, for both the single state distance $|\nu(0)-\mu_t(0)|$ and a much stronger total variation norm, $\|\nu-\mu_t\|_{TV}$.
Acknowledgments
===============
We would like to thank Yuan Xu of University of Oregon for his helpful comments that initiated this work. We would also like to thank Michael Anshelevich of Texas A & M for the feedback he provided during the conference on orthogonal polynomials in probability theory in July of 2010. We would like thank Andrew R. Wade of the University of Strathclyde for his helpful comments on the preprint of this paper. Finally, we would like to thank the anonymous referee for the many helpful corrections and suggestions.
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abstract: 'In this paper we investigate special arrangements of lines in multiprojective spaces. In particular, we characterize codimensional two arithmetically Cohen-Macaulay (ACM) varieties in $\mathbb P^1\times\mathbb P^1\times\mathbb P^1$, called *varieties of lines*. We also describe their ACM property from combinatorial algebra point of view.'
address:
- |
Dipartimento di Matematica e Informatica\
Viale A. Doria, 6 - 95100 - Catania, Italy
- |
Dipartimento di Matematica e Informatica\
Viale A. Doria, 6 - 95100 - Catania, Italy
- |
Dipartimento di Matematica e Informatica\
Viale A. Doria, 6 - 95100 - Catania, Italy
author:
- Giuseppe Favacchio
- Elena Guardo
- Beatrice Picone
date:
title: 'Special arrangements of lines: codimension two ACM varieties in $\mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^1$'
---
Introduction {#introduction .unnumbered}
============
Given a variety $X\subseteq \mathbb P^{a_1} \times \dots \times \mathbb P^{a_n}$ an interesting problem is the description of the homological invariants of the coordinate ring of $X$. This problem was especially studied for points and there is not a general answer in this direction. A great difficulty came from the fact that a set of distinct points $X\subseteq \mathbb P^{a_1} \times \dots \times \mathbb P^{a_n}$ is not necessarily arithmetically Cohen-Macaulay (ACM). See, for instance [@FG2016a; @GuVT2004; @GuVT2008b; @GuVT2012a; @GV-book; @GV15] for a short spectrum of the study on this topic, and [@FGM; @FM] for a recent characterization of the ACM property in $\mathbb P^1\times\cdots \times\mathbb P^1$ and, under certain conditions, in $\mathbb P^1\times\mathbb P^m.$ Recently, multiprojective spaces are getting more attention since many applications have been explored. For example, a specific value of the Hilbert function of a collection of (fat) points in a multiprojective space is related to a classical problem of algebraic geometry concerning the dimension of certain secant varieties of Segre varieties (see [@BBC; @CGG1; @CGG2] just to cite some of them). Or in [@BBCG; @CS] the readers could deduce new results about tensors, and in [@D], the author focus on the implicitization problem for tensor product surfaces.
In particular, it appears of interest in combinatorial algebraic geometry the study of finite arrangements of lines (see [@CHMN; @H] for recent developments in $\mathbb{P}^2$). A line arrangement over an algebraically closed field $K$ is a finite collection $L_1,\dots , L_d \subseteq \mathbb{P}^2$, $d > 1$, of distinct lines in the projective plane and their crossing points (i.e., the points of intersections of the lines). In this paper we investigate special arrangements of lines in multiprojective spaces by focusing on ACM codimensional two varieties in $\mathbb{P}^1\times\mathbb P^1\times\mathbb P^1$, called *varieties of lines*. In particular, we study special cases arising from their intersection points (see Theorem \[thmACM\]). These varieties can be viewed as special configurations of codimension two linear varieties in $\mathbb P^5$.
The paper is structured as follows. In Section \[sec:def\] we set up our notation and recall known results. In Section \[sec:comb\_char\], we describe a connection between ideals of varieties of lines and some squarefree monomial ideals (Lemma \[monomial\]). We introduce in Definition \[def:chordal\] the $Hyp_n({\star})$-property to give a combinatorial characterization of ACM varieties of lines using a well known property of chordal graphs (Theorem \[thm:ch1\]). In Section \[sec:num\_char\] we introduce a numerical way to check the ACM property for any varieties of lines. In Section \[sec:HF\] we describe the Hilbert function of Ferrers varieties of lines, a special ACM case. Finally, in Section \[sec:grids\] we start an investigation on varieties of lines whose crossing points set is a complete intersection of points in $\mathbb P^1\times\mathbb P^1\times\mathbb P^1$. We also characterize varieties of lines defined by a complete intersection ideal in $\mathbb P^1\times\mathbb P^1\times\mathbb P^1$ (Theorem \[thmcompleteintersection\]). We end the paper with two possible research topics to explore: (1) the connection between our varieties of lines in $\mathbb{P}^1 \times \mathbb{P}^1 \times \mathbb{P}^1$ and special configurations of lines of $\mathbb P^3$ and (2) the Hilbert function of any ACM variety of lines (Question \[HF\]).
**Acknowledgement.** The authors thank A. Van Tuyl for his comments in a previous version of the paper. They also thank the referee for his/her helpful suggestion. The second author would like to thank GNSAGA and Prin 2015. The software CoCoA [@cocoa] was indispensable for all the computations.
Notation and basic facts {#sec:def}
========================
Throughout the paper $\mathbb{N} := \{0,1,2, \dots \}$ denotes the set of non-negative integers and $\preceq$ denotes the natural partial order on the elements of $\mathbb{N}^3 := \mathbb{N} \times \mathbb{N} \times \mathbb{N}$ defined by $(a_1,a_2,a_3) \preceq (b_1,b_2,b_3)$ in $\mathbb{N}^3$ if and only if $a_i \le b_i \ \forall i =1,2,3$. Let $\{ \bf{e}_1,\bf{e}_2,\bf{e}_3 \}$ be the standard basis of $\mathbb N^3$. Let $R:=K[x_{1,0},x_{1,1},x_{2,0},x_{2,1},x_{3,0},x_{3,1}]$ be the polynomial ring over an algebraically closed field $K$ of characteristic zero. We induce a multi-grading by setting $\deg x_{i,j}=\bf{e}_i$ for $i\in \{1,2,3\}$. A monomial $m=x_{1,0}^{a_0}x_{1,1}^{a_1}x_{2,0}^{b_0}x_{2,1}^{b_1}x_{3,0}^{c_0}x_{3,1}^{c_1} \in R$ has *tridegree* (or simply, *degree*) $\deg m= (a_0+a_1, b_0 + b_1, c_0 +c_1)$. We make the convention that $0$ has degree $\deg 0=(i,j,k)$ for all $(i,j,k) \in \mathbb{N}^3$. Note that the elements of the field $K$ all have degree $(0,0,0)$. For each $(i,j,k) \in \mathbb{N}^3$, let $R_{i,j,k}$ denote the vector space over $K$ spanned by all the monomials of $R$ of degree $(i,j,k)$. The polynomial ring $R$ is then a *trigraded ring* because there exists a direct sum decomposition $$R = \bigoplus \limits_{(i,j,k)\in \mathbb{N}^3} R_{i,j,k}$$ such that $R_{i,j,k}R_{l,m,n} \subseteq R_{i+l,j+m,k+n}$ for all $(i,j,k),(l,m,n) \in \mathbb{N}^3$. An element $F \in R$ is *trihomogeneous* (or simply, *homogeneous*) if $F \in R_{i,j,k}$ for some $(i,j,k) \in \mathbb{N}^3$. An ideal $I = (F_1,F_2, \dots ,F_r) \subseteq R$ is a *(tri)homogeneous ideal* if $F_i$ is (tri)homogeneous for all $i=1,2, \dots ,r$.
Let $I \subseteq R$ be a homogeneous ideal, and we let $I_{i,j,k} := I \cap R_{i,j,k}$ for all $(i,j,k) \in \mathbb{N}^3$. Because $I$ is homogeneous, the quotient ring $R/I$ also inherists a graded ring structure. In particular, we have: $$R/I = \bigoplus \limits_{(i,j,k)\in \mathbb{N}^3} (R/ I)_{i,j,k} = \bigoplus \limits_{(i,j,k)\in \mathbb{N}^3} R_{i,j,k}/ I_{i,j,k}.$$
Let $I_X \subseteq R$ be the homogeneous ideal defining a variety $X \subseteq \mathbb{P}^1 \times \mathbb{P}^1 \times \mathbb{P}^1$. We say that $X$ is *arithmetically Cohen-Macaulay* (ACM) if $R/I_X$ is Cohen-Macaulay, i.e. $\text{depth}(R/I_X) = \text{Krull-dim}(R/I_X)$.
We say that a homogeneous ideal $J$ in a polynomial ring $S$ is *Cohen-Macaulay* (CM) if $S/J$ is Cohen-Macaulay.
A point in $\mathbb P^1\times\mathbb P^1\times\mathbb P^1$ is an ordered set of three points in $\mathbb P^1$. Say $P:=([a_0,a_1],[b_0,b_1],[c_0,c_1])\in \mathbb P^1\times\mathbb P^1\times\mathbb P^1,$ the defining ideal of $P$ is $I_P:=(a_1x_{1,0}-a_0x_{1,1},b_1x_{2,0}-b_0x_{2,1},c_1x_{3,0}-c_0x_{3,1})$. Note that $I_P$ is a height three prime ideal generated by homogeneous linear forms of different degree.
Throughout the paper, linear forms are denoted by capital letters. In particular, we use $A_i$ to denote a linear form of degree $(1,0,0)$, $B_j$ a linear form of degree $(0,1,0)$, and $C_k$ a linear form of degree $(0,0,1).$ We denote by $\mathcal{L}(A_i)$, $\mathcal{L}(B_j)$ and $\mathcal{L}(C_k)$ the respective hyperplanes of $\mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^1$, and we say that a hyperplane in $\mathbb P^1\times\mathbb P^1\times\mathbb P^1$ is of type $\bf{e}_i$ if it is defined by a form of degree $\bf{e}_i$.
We recall the following definition (see [@GV15], Definition 2.2).
Let $F, G \in R$ be two homogeneous linear forms of different degree. In $\mathbb P^1\times\mathbb P^1\times\mathbb P^1$ the variety $\mathcal{L}$ defined by the ideal $(F,G)\subseteq R$ is called a *line* of $\mathbb P^1\times\mathbb P^1\times\mathbb P^1$ and we denote it by $\mathcal{L}(F,G)$.
We say that a line $\mathcal{L}(F,G)$ is of type $\bf{e}_i+\bf{e}_j$, with $i\neq j,$ if $\{\deg F, \deg G\}=\{\bf{e}_i, \bf{e}_j\}$.
In particular, if $A\in R_{1,0,0}$, $B\in R_{0,1,0}$ and $C\in R_{0,0,1}$, then we denote by $\mathcal{L}(A,B)$ the variety in $\mathbb P^1\times\mathbb P^1\times\mathbb P^1$ defined by the ideal $(A,B)\subseteq R$ and we call it *line of type $(1, 1, 0)$*. Analogously, we call the variety $\mathcal{L}(A,C)$ *line of type $(1, 0,1)$* and the variety $\mathcal{L}(B,C)$ *line of type $(0, 1, 1)$*. We also refer to lines of type $\bf{e}_1+\bf{e}_2$, $\bf{e}_1+\bf{e}_3$ and $\bf{e}_2+\bf{e}_3$ by writing *lines having direction* $\bf{e_3}$, $\bf{e_2}$ and $\bf{e_1}$, respectively.
We say that $X \subseteq \mathbb P^1\times\mathbb P^1\times\mathbb P^1$ is a *variety of lines* if it is given by a finite union of distinct lines in $\mathbb P^1\times\mathbb P^1\times\mathbb P^1$.
\[Def:U\_i(X)\] Given $X\subseteq \mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^1$ a variety of lines, we denote by $\mathcal{H}_1(X):=\{\mathcal{L}(A_1),\ldots, \mathcal{L}(A_{d_1})\}$, $ \mathcal{H}_2(X):=\{\mathcal{L}(B_1),\ldots, \mathcal{L}(B_{d_2})\}$ and $\mathcal{H}_3(X):=\{\mathcal{L}(C_1),\ldots, \mathcal{L}(C_{d_3})\}$ the hyperplanes of $\mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^1$ containing some lines of $X$. In particular: $$X:= \bigcup_{(i,j)\in U_3(X)} \mathcal{L}(A_i,B_j)\cup\bigcup_{(i,k)\in U_2(X)} \mathcal{L}(A_i,C_k)\cup \bigcup_{(j,k)\in U_1(X)} \mathcal{L}(B_j,C_k)$$ where $U_3(X)\subseteq [d_1]\times[d_2], U_2(X)\subseteq [d_1]\times[d_3]$ and $U_1(X)\subseteq [d_2]\times[d_3]$ are sets of ordered pairs of integers, with $[n]:=\{1,2,\ldots, n\}\subset \mathbb{N}.$
For $i=1, 2, 3$, we denote by $X_i$ the [set of lines of $X$ having direction $\bf{e}_i$]{} and we call $U_i(X)$ the *index set of $X_i$*.
Thus, the ideal defining $X$ is $$I_X=\bigcap_{(i,j)\in U_3(X)} (A_i,B_j)\bigcap_{(i,k)\in U_2(X)} (A_i,C_k)\bigcap_{(j,k)\in U_1(X)} (B_j,C_k).$$
In this paper we are interested in a combinatorial characterization of ACM varieties of lines in $\mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^1$ and their Hilbert function. In $\mathbb P^1\times \mathbb P^1$, to describe combinatorially ACM sets of points, it was crucial the definition of the so called *Ferrers diagram* (see for instance [@GV-book]).
A tuple $\lambda = (\lambda_1 , \dots , \lambda_r)$ of positive integers is a *partition* of an integer $s$ if $\sum_{i=1}^{r} \lambda_i = s$ and $\lambda_i \geq \lambda_{i+1}$ for every $i$. We write $\lambda = (\lambda_1 , \dots , \lambda_r) \vdash s$.
\[d.Ferr\] To any partition $\lambda = (\lambda_1 , \lambda_2, \dots , \lambda_r) \vdash s$ we can associate the following diagram: on an $r \times \lambda_1$ grid, place $\lambda_1$ points on the first horizontal line, $\lambda_2$ points on the second, and so on, where the points are left justified. The resulting diagram is called the *Ferrers diagram* of the partition $\lambda$.
Let $Y$ be a finite set of points in $\mathbb{P}^1 \times \mathbb{P}^1$. We say that *$Y$ resembles a Ferrers diagram* if the set of points looks like a Ferrers diagram, i.e., after relabeling the horizontal and vertical rulings, we can assume that the first horizontal ruling contains the most number of points of $Y$, the second contains the same number or less of points of $Y$, and so on.
Applying Lemma 3.17, Theorem 3.21 and Theorem 4.11 in [@GV-book], we have
\[FD\] Let $Y$ be a finite set of points in $\mathbb{P}^1 \times \mathbb{P}^1$. $Y$ is ACM if and only if $Y$ resembles a Ferrers diagram.
We adapt Definition \[d.Ferr\] to our context.
\[construction\] Let $X$ be a variety of lines in $\mathbb{P}^1 \times \mathbb{P}^1 \times \mathbb{P}^1$ and consider the set $X_3$ of lines of $X$ of type $(1,1,0)$ indexed by $U_3(X)\subseteq [d_1]\times [d_2]$. We represent $X_3$ as a $d_1 \times d_2$ grid, where the horizontal lines are labeled by the $\mathcal{L}(A_i)$’s for $i=1,\ldots, d_1$ and the vertical lines by the $\mathcal{L}(B_j)$’s for $j=1,\ldots, d_2$. By abuse of notation, we denote the horizontal lines by $\mathcal{L}(A_i)$ and the vertical lines by $\mathcal{L}(B_j)$. Then, a line $\mathcal{L}(A_i,B_j) \in X_3$ is drawn as the intersection point of $\mathcal{L}(A_i)$ and $\mathcal{L}(B_j)$ in the grid. Similarly, we can construct a $d_1\times d_3$ grid representing $X_2$ and a $d_2\times d_3$ grid representing $X_1$.
\[defferrer\] Let $X$ be a variety of lines in $\mathbb{P}^1 \times \mathbb{P}^1 \times \mathbb{P}^1$ and $h\in\{1,2,3\}$. We say that $X$ *resembles a Ferrers diagram with respect to the direction $h$* if the grid representing the lines of $X_h$, constructed as above, resembles a Ferrers diagram.
\[d.U Ferr\] A finite subset $U = \{ (u_i,u_j) \} \subseteq \mathbb N^2$ *resembles a Ferrers diagram* if it satisfies the following property: $$(u_i,u_j) \in U \Rightarrow (u_h,u_k) \in U \ \ \forall \ 1 \leq h \leq i \ , \ 1 \leq k \leq j.$$
\[r. Ferr EQ\] Note that Definition \[defferrer\] is equivalent to say that the index set $U_h(X) \subset \mathbb{N}^2$ resembles a Ferrers diagram as Definition \[d.U Ferr\].
\[r.cono\] Construction \[construction\] makes clear the connection between $X_h$ ($h\in \{1,2,3\}$) and a set of points in $\mathbb{P}^1\times\mathbb{P}^1$. $X_h$ is a cone of a set of distinct points on a hyperplane of $\mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^1$. So, we can look at it as a set of points in $\mathbb{P}^1\times\mathbb{P}^1$ with associated grid as described in the construction.
\[e.Ferr\] Let $X$ be the following variety of 15 lines in $\mathbb{P}^1 \times \mathbb{P}^1 \times \mathbb{P}^1$: $$\begin{split} X = & \mathcal{L}(A_1,B_2) \cup \mathcal{L}(A_1,B_4) \cup \mathcal{L}(A_1,B_5) \cup \mathcal{L}(A_2,B_2) \cup \mathcal{L}(A_2,B_3) \cup \\ & \cup\mathcal{L}(A_2,B_4) \cup \mathcal{L}(A_2,B_5) \cup \mathcal{L}(A_3,B_1) \cup \mathcal{L}(A_3,B_2) \cup \mathcal{L}(A_3,B_3) \cup \\ &\cup \mathcal{L}(A_3,B_4) \cup \mathcal{L}(A_3,B_5) \cup \mathcal{L}(A_4,B_4) \cup \mathcal{L}(B_1,C_1) \cup \mathcal{L}(B_2,C_2).
\end{split}$$ Then, $$\begin{split}
X_3= & \{\mathcal{L}(A_1,B_2), \mathcal{L}(A_1,B_4), \mathcal{L}(A_1,B_5), \mathcal{L}(A_2,B_2), \mathcal{L}(A_2,B_3), \mathcal{L}(A_2,B_4),\mathcal{L}(A_2,B_5), \\ \phantom{=} & \ \mathcal{L}(A_3,B_1), \mathcal{L}(A_3,B_2), \mathcal{L}(A_3,B_3), \mathcal{L}(A_3,B_4), \mathcal{L}(A_3,B_5), \mathcal{L}(A_4,B_4)\}.
\end{split}$$ Using Construction \[construction\], $X_3$ is represented by a $4\times 5$ grid as Figure \[fig1\]. After renaming, we see that $X_3$ resembles a Ferrers diagram of type $(5,4,3,1)$. Then, using Lemma \[FD\], $X_3$ is ACM (Figure \[fig2\]).
(2,0.5) - - (2,9.5); (4,0.5) - - (4,9.5); (6,0.5) - - (6,9.5); (8,0.5) - - (8,9.5); (10,0.5) - - (10,9.5);
(0.5,2) - - (11.5,2); (0.5,4) - - (11.5,4); (0.5,6) - - (11.5,6); (0.5,8) - - (11.5,8); (8,2) circle (2mm); (2,4) circle (2mm); (10,6) circle (2mm);
(4,4) circle (2mm); (4,6) circle (2mm); (4,8) circle (2mm); (6,4) circle (2mm); (6,6) circle (2mm); (6,4) circle (2mm); (8,4) circle (2mm); (10,4) circle (2mm); (8,8) circle (2mm); (10,8) circle (2mm); (8,6) circle (2mm);
at (3,10) [ $\mathcal{L}(B_1)$]{};
at (5,10) [ $\mathcal{L}(B_2)$]{};
at (7,10) [ $\mathcal{L}(B_3)$]{};
at (9,10) [ $\mathcal{L}(B_4)$]{};
at (11,10) [ $\mathcal{L}(B_5)$]{};
at (13.4,2) [ $\mathcal{L}(A_4)$]{};
at (13.4,4) [ $\mathcal{L}(A_3)$]{};
at (13.4,6) [ $\mathcal{L}(A_2)$]{};
at (13.4,8) [ $\mathcal{L}(A_1)$]{};
(2,0.5) - - (2,9.5); (4,0.5) - - (4,9.5); (6,0.5) - - (6,9.5); (8,0.5) - - (8,9.5); (10,0.5) - - (10,9.5);
(0.5,2) - - (11.5,2); (0.5,4) - - (11.5,4); (0.5,6) - - (11.5,6); (0.5,8) - - (11.5,8); (2,2) circle (2mm); (2,4) circle (2mm); (2,6) circle (2mm); (2,8) circle (2mm); (4,4) circle (2mm); (4,6) circle (2mm); (4,8) circle (2mm); (6,4) circle (2mm); (6,6) circle (2mm); (6,8) circle (2mm);
(8,6) circle (2mm); (8,8) circle (2mm); (10,8) circle (2mm);
at (3,10) [ $\mathcal{L}(\overline{B}_1)$]{};
at (5,10) [ $\mathcal{L}(\overline{B}_2)$]{};
at (7,10) [ $\mathcal{L}(\overline{B}_3)$]{};
at (9,10) [ $\mathcal{L}(\overline{B}_4)$]{};
at (11,10) [ $\mathcal{L}(\overline{B}_5)$]{};
at (13.4,2) [ $\mathcal{L}(\overline{A}_4)$]{};
at (13.4,4) [ $\mathcal{L}(\overline{A}_3)$]{};
at (13.4,6) [ $\mathcal{L}(\overline{A}_2)$]{};
at (13.4,8) [ $\mathcal{L}(\overline{A}_1)$]{};
Let $X$ be the variety of lines as in Example \[e.Ferr\]. We have $X_1= \{\mathcal{L}(B_1,C_1), \mathcal{L}(B_2,C_2)\}$ and the $2\times 2$ grid representing $X_1$ does not resemble any Ferrers diagram (Figure \[fig3\]). Thus $X$ does not resemble a Ferrers diagram with respect to the direction $1$. Hence, from Lemma \[FD\], $X_1$ is not ACM.
(2,4.5) - - (2,9.5); (4,4.5) - - (4,9.5); (0.5,6) - - (6,6); (0.5,8) - - (6,8); (2,8) circle (2mm);
(4,6) circle (2mm); at (3,10) [ $\mathcal{L}(B_1)$]{};
at (5,10) [ $\mathcal{L}(B_2)$]{};
at (8,6) [ $\mathcal{L}(A_2)$]{};
at (8,8) [ $\mathcal{L}(A_1)$]{};
Since Ferrers diagrams play a crucial role in the characterization of the ACM property for a finite set of points in $\mathbb P^1\times \mathbb P^1$ (see for instance [@GV15]), it is natural for us to investigate the same property for a variety of lines $X\subseteq \mathbb P^1\times \mathbb P^1\times \mathbb P^1$ (since $X$ has also codimension 2). In the next section, we will show that the ACM property of $X$ depends on the $X_i$ (see Corollary \[cor. Xi\]), but the ACM-ness of the $X_i$ is not sufficient to ensure that $X$ is also ACM (see Remark \[rem. Xi\]).
A combinatorial characterization of ACM varieties of lines {#sec:comb_char}
===========================================================
In this section, we study the ACM property for varieties of lines from a combinatorial point of view. We refer to [@HH] for all the introductory material on monomial ideals.
The next lemma can be recovered from [@VT03], Proposition 3.2.
\[l.reg sequence\] Let $X\subseteq \mathbb P^{1}\times\mathbb P^{1}\times\mathbb P^{1}$ be a variety of lines. Then, there exist three forms $A,B$ and $C$ of degree $(1,0,0), (0,1,0)$ and $(0,0,1)$, respectively, such that $(\bar A,\bar B,\bar C)$ is a regular sequence in $R/I_X.$
Let $A\in R_{1,0,0}$ be such that $\mathcal{L}(A)\notin \mathcal H_1(X)$. We claim that $\bar A$ is a nonzero divisor of $R/I_X$. Indeed, take $F\in R$ a homogeneous form such that $AF\in I_X$. Then $AF\in I_{\mathcal L}$, for any line $\mathcal L\in X$. Since $I_{\mathcal L}$ is a prime ideal and $A\notin I_{\mathcal L}$, then we get $F\in I_{\mathcal L}$, for any $\mathcal L\in X$.
Now we prove the existence of the linear form $B.$ Since $X$ is ACM, then $J:=I_X+(A)$ is CM. Moreover, $J$ is homogeneous and its height is 3. Take the primary decomposition of $J$, say $J=\mathfrak q_1\cap \cdots \cap \mathfrak q_t $, and let $\mathfrak p_i:=\sqrt{\mathfrak q_i}$ for $i=1,\ldots, t.$ The set of the nonzero divisors of $R/J$ is then $\bigcup_{i} \bar{ \mathfrak p}_i$. In order to prove that there exists an element $B\in R_{0,1,0}$ nonzero divisor of $R/J$, it is enough to show that $(\bigcup_{i} { \mathfrak p}_i)_{0,1,0}\subsetneq R_{0,1,0}.$ Since $R_{0,1,0}$ is a $K$-vector space over an infinite field, it is not a union of a finite number of its proper subspaces, and so it is enough to show that $ (\mathfrak p_i)_{0,1,0}\subsetneq R_{0,1,0}$ for each $i=1,\ldots, t.$
Let $i\in \{1,\ldots, t\}$, then we have $I_X\subseteq J \subseteq \mathfrak p_i$. Therefore, there exists $\mathcal L\in X$ such that $I_X\subseteq I_{\mathcal L}\subseteq \mathfrak p_i$. This implies $\mathfrak p_i= I_{\mathcal L}+(A).$ Since $I_{\mathcal L}\neq R_{0,1,0}$ we are done.
Analogously we prove the existence of a form $C\in R_{0,0,1}.$
We set the notation for this section. Let $X$ be a variety of lines and $I_X$ its defining ideal $$I_X=\bigcap_{(i,j)\in U_3(X)} (A_i,B_j)\bigcap_{(i,k)\in U_2(X)} (A_i,C_k)\bigcap_{(j,k)\in U_1(X)} (B_j,C_k)\subseteq R.$$ We construct a new polynomial ring in $d_1 + d_2 + d_3$ variables each of them corresponding to a hyperplane containing some lines of $X.$ We denote by $S:=K[a_1, \ldots, a_{d_1}$, $b_1, \ldots, b_{d_2}, c_1, \ldots, c_{d_3}]$ the polynomial ring in $d_1 + d_2 + d_3$ variables and $\deg a_i=(1,0,0)$, $\deg b_j=(0,1,0)$, $\deg c_k=(0,0,1)$. We set $$J_X=\bigcap_{(i,j)\in U_3(X)} (a_i,b_j)\bigcap_{(i,k)\in U_2(X)} (a_i,c_k)\bigcap_{(j,k)\in U_1(X)} (b_j,c_k)\subseteq S.$$ $J_X$ is a height 2 monomial ideal of $S$ and its associated primes correspond to the components of $X$.
The next lemma is crucial since, as its consequence, we can connect homological invariants between ACM varieties of lines and some height 2 monomial ideals. Similar arguments were also used in [@FGM] (see proof of Theorem 3.2).
\[monomial\] Let $X$ be a variety of lines in $\mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^1$. Then $X$ is ACM if and only if $J_X \subseteq S $ is CM.
Set $T := S[x_{1,0}, x_{1,1}, x_{2,0}, x_{2,1}, x_{3,0}, x_{3,1}]$. Consider $J_X$ as an ideal, say $\overline{J}_X$, in the ring $T$. Since $J_X$ is a height 2 monomial ideal in $S$, then $\overline{J}_X$, being a cone, continues to be a height 2 monomial ideal. Moreover, $\overline{J}_X$ has the same primary decomposition as $J_X$. Consider the linear forms $a_i-A_i$, $b_j-B_j$, $c_k-C_k$ and let $L$ be the ideal generated by all these linear forms.
Assume $J_X$ is CM. Thus, in the quotient $T/(\overline{J}_X, L)$ we can view the addition of each linear form in $L$ as a proper hyperplane section. We have that $R/I_X$ and $T/(\overline{J}_X,L)$ both have height 2 and $ R/I_X\cong T/(\overline{J}_X, L)$. Then, since $J_X$ is CM, we get $X$ is ACM.
On the other hand, if $X$ is ACM, then, applying Lemma \[l.reg sequence\], there exists a sequence of linear forms $(A,B,C)\subseteq R$ that is regular in the quotient $R/I_X$. Let $\mathfrak q:= (A,B,C)\subseteq R$ be the ideal generated by these three linear forms. Consider the ideal $(I_X+\mathfrak q)/\mathfrak q \subseteq R/\mathfrak q$, that can be viewed as a codimension 2 monomial ideal in a polynomial ring in three variables. Since a Hilbert-Burch matrix of $I_X$ has the same structureas the Hilbert-Burch matrix of a monomial ideal, i.e. it is a matrix with only two non zero entries in each column (see for instance Lemma 3.21 in [@FRZ] or Theorem 1.5. in [@Naeem]), then $I_X$ is generated by some products among the linear forms defining the lines of $X$. Since the addition of each linear form in $L$ can be seen as a proper hyperplane section, we also have $ R/I_X\cong T/(\overline{J}_X, L)$. Then $J_X$ is CM.
\[c.monomial\] Let $X$ be an ACM variety of lines in $\mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^1$. Then $I_X$ is generated by products of linear forms.
As a consequence of Lemma \[monomial\], it is interesting to further investigate the structure of the monomial ideal $J_X$ associated to $X$. Now we recall only a few definitions we will use in the sequel. We refer to [@HH; @VT] for all preliminaries and for further results on graphs.
A (simple) graph $G$ is a pair $G =(V, E)$, where $V:=\{v_1,\ldots, v_N\}$ is a set of vertices of $G$ and $E$ is a collection of 2-subsets of $V,$ called the edges of $G$. The complementary graph of $G$, denoted by $G^c,$ is the graph $G^c = (V, E^c)$, where $E^c = \{\{v_i, v_j \}\ |\ \{v_i, v_j \} \notin E\}$. A sequence of vertices of $G$, $(v_1, v_2, \cdots, v_t)$, is a cycle of length $t$ if $\{v_1, v_2\},\{v_2, v_3\},\dots,\{v_t, v_1\}\in E$. A chord is an edge joining two not adjacent vertices in a cycle. A minimal cycle is a cycle without chords. A graph $G$ is called chordal when all its minimal cycles have length three. We associate to a graph $G=(V,E)$ two squarefree monomial ideals in the ring $K[v_1, \ldots, v_N]$, the face ideal of $G$ $$I(G)=(v_iv_j | \{v_i,v_j\}\in E)$$ and the cover ideal of $G$ $$J(G)=\bigcap\limits_{\{v_i,v_j\}\in E }(v_i,v_j).$$ It is a well known fact that $I(G)$ and $J(G)$ are the Alexander dual each of the other. In the sequel we will use the following results:
\[th.Fr\] Let $G$ be a graph. Then $I(G)$ has a linear resolution if and only if $G^c$ is a chordal graph.
\[th.ER\] Let $G$ be a graph. Then $I(G)$ has a linear resolution if and only if $J(G)$ is CM.
\[r.simpl compl\] Let $X$ be a variety of lines. Let $G_X=(V_X, E_X)$ be the graph with vertex set $$V_X:=\{a_1, \ldots, a_{d_1}, b_1, \ldots, b_{d_2}, c_1, \ldots, c_{d_3} \}$$ and edge set $$\begin{split}
E_X := & \big\{ \{a_i,b_j\}\subseteq V_X\ |\ \mathcal{L}(A_i,B_j)\in X_3\big\} \cup \\
\cup & \big\{ \{a_i,c_k\}\subseteq V_X\ |\ \mathcal{L}(A_i,C_k)\in X_2\big\}\cup \\
\cup & \big\{ \{b_j,c_k\}\subseteq V_X\ |\ \mathcal{L}(B_j,C_k)\in X_1\big\}.
\end{split}$$ Then, we note that the monomial ideal $J_X$ is the cover ideal of the graph $G_X$: $$J_X = J(G_X)\subseteq S,$$ that is the Stanley-Reisner ideal of the simplicial complex (see Lemma 1.5.4. in [@HH]) $$\Delta_X:= \langle V_X \setminus e\ |\ e \in E_X \rangle.$$
An useful application of Remark \[r.simpl compl\] is the following lemma.
\[remove-hyp\]Let $X$ be an ACM variety of lines in $\mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^1$ and let $\mathcal H\subseteq \mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^1$ be a hyperplane containing some lines of $X$. Then the variety of lines $Y= \{ \mathcal{L} \in X \ | \ \mathcal{L} \not\subset \mathcal{H}\}$ is ACM.
Let $H$ be the linear form defining $\mathcal H$. Denoted by $z$ the variable of $S$ corresponding to $H$ (the linear form $H$ is one of the forms $A_i, B_j, C_k$ and $z$ is the corresponding variable among $a_i, b_j, c_k$). We have
- $J_X:z= \bigcap\limits_{\tiny \begin{array}{c}
\mathfrak p\in \operatorname{ass}(J_X)\\ z\notin \mathfrak p\\
\end{array}}\!\!\!\mathfrak p.$ Both are monomial ideals, so the equality easily follows by checking the inclusions for monomials.
- $J_X:z$ is the Stanley-Reisner ideal of the simplicial complex $\operatorname{link}_{\Delta_X} z$ (see sections 1.5.2 and 8.1.1 in [@HH]). Indeed, the Stanley-Reisner ideal of the $link$ of $z$ in $\Delta_X$ is generated by monomials corresponding to the elements $F\subseteq V_X$ such that $\{z\}\cup F\notin \Delta_X$. All these monomials are in $J_X:z= I_{\Delta_X}:z$ and vice versa.
Then, in order to prove the statement, it is enough to show that $J_X:z$ is CM. From Lemma \[monomial\], we have that $J_X$ is CM, so the statement follows by Corollary 8.1.8 in [@HH].
\[cor. Xi\] If $X$ is an ACM variety of lines, then $X$ resembles a Ferrers diagram with respect to the direction $h$, for each $h = 1,2,3.$
We show that $U_1(X)$ resembles a Ferrers diagram. Analogously, one can show the same for $U_2(X)$ and $U_3(X)$. Let us consider the variety of lines $X_1$ consisting of the lines of $X$ of type $(0,1,1).$ Since $I_{X_1}= I_{X\setminus\{\mathcal L (A_1 ), \ldots, \mathcal{L}(A_{d_1}) \} }$, $X_1$ preserves the ACM property by Lemma \[remove-hyp\]. Moreover, $X_1=\bigcup\limits_{(j,k)\in U_1(X)}\mathcal{L}(B_j,C_k)$, i.e., it is a cone of an ACM set of distinct points on a hyperplane of $\mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^1$, see Remark \[r.cono\]. A well known characterization, see for instance Theorem 4.11 in [@GV-book], shows that this set of points resembles a Ferrers diagram. Using Remark \[r. Ferr EQ\], $U_1(X)$ resembles a Ferrers diagram. Then, the statement follows from Lemma \[FD\].
\[rem. Xi\] From previous corollary, if there exists $i \in \{1,2,3\}$ such that $X_i$ is not ACM, then $X$ is not ACM. The following example shows that even if all $X_i$ are ACM $X$ could be not ACM.
\[ex not acm\] Let us consider the following variety of lines in $\mathbb{P}^1 \times \mathbb{P}^1 \times \mathbb{P}^1$: $$\begin{split} X = \{ \mathcal{L}(A_1,B_1), \mathcal{L}(A_2,B_2), \mathcal{L}(B_3, C_3)\}.
\end{split}$$ It is clear that the sets $X_1, X_2$ and $X_3$ resemble a Ferrers diagram, so each of them is ACM. But, in this case, $X$ is not ACM. This follows for instance from Lemma \[monomial\] and from [@HH], Lemma 9.1.12.
The next definition introduces a property for varieties of lines in $\mathbb{P}^1 \times \mathbb{P}^1 \times \mathbb{P}^1$ in analogy to the known ($\star$)-property defined for sets of points in $\mathbb{P}^1 \times \mathbb{P}^1$ (see [@GuVT2012a]).
\[def:star\] Let $X\subseteq \mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^1$ be a variety of lines. We say that $X$ has the *($\star$)-property* (or explicitly, *star property*) if given any two lines $L_1$, $ L_2 \in X$, there exists $L_3 \in X$ such that $L_1, L_3$ and $L_2, L_3$ are coplanar.
We slight generalize this property for varieties of lines.
\[def:chordal\] Let $X\subseteq \mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^1$ be a variety of lines. Let $n \geq 4$, $n \in \mathbb{N}$, we say that $X$ has the *$n$-hyperplanes ($\star$) property* (for short, $Hyp_n(\star)$-property) if given $n$ hyperplanes $H_1, H_2, \ldots, H_n$ such that $\mathcal{L}(H_i, H_j)\in X$ for any $j\neq i-1, i, i+1$ then $\mathcal{L}(H_u, H_{u+1})\in X$ for some $u\in \{ 1,2, \dots, n \}$, where $H_0 = H_n$ and $H_{n+1} = H_1$.
\[n\*\] Note that if $n>6$, then $X$ has the $Hyp_n(\star)$-property. Indeed, among $n>6$ hyperplanes there are at least three of the same type and so the condition $\mathcal{L}(H_i, H_j)\in X$ for any $j\neq i-1, i, i+1$ (where $H_0=H_n$ and $H_{n+1} =H_1$) fails to be true.
Note that the $Hyp_4(\star)$-property is equivalent to $(\star)$-property as Definition \[def:star\].
Let us consider the following variety of lines in $\mathbb{P}^1 \times \mathbb{P}^1 \times \mathbb{P}^1$: $$\begin{split} X = & \mathcal{L}(A_1,B_1) \cup \mathcal{L}(A_1,B_2) \cup \mathcal{L}(A_1,B_3) \cup \mathcal{L}(A_2,B_1) \cup \\ \cup & \mathcal{L}(A_2,B_2) \cup \mathcal{L}(A_1,C_1) \cup \mathcal{L}(A_1,C_2) \cup \mathcal{L}(A_2,C_1) \cup \\ \cup & \mathcal{L}(B_1,C_1) \cup \mathcal{L}(B_1,C_2) \cup \mathcal{L}(B_2,C_1) \cup \mathcal{L}(B_3,C_1). \end{split}$$ $X$ has the $Hyp_4 (\star$)-property. Indeed, if we take the $4$ hyperplanes $\mathcal{L}(A_1)$, $ \mathcal{L}(A_2)$, $\mathcal{L}(B_1), \mathcal{L}(B_2)$, we have that $\mathcal{L}(A_1,B_1), \mathcal{L}(A_2,B_2) \in X$ and also $\mathcal{L}(A_1, B_2) \in X$; if we take the $4$ hyperplanes $\mathcal{L}(A_1), \mathcal{L}(A_2)$, $\mathcal{L}(B_1), \mathcal{L}(C_1)$, we have that $\mathcal{L}(A_1,B_1)$, $\mathcal{L}(A_2,C_1) \in X$ and also $\mathcal{L}(B_1, C_1) \in X$; and so on, if we take any two lines in $X$, there exists a third line in $X$ that is coplanar with the other two.
Let us consider the following variety of lines in $\mathbb{P}^1 \times \mathbb{P}^1 \times \mathbb{P}^1$: $$\begin{split} X = & \mathcal{L}(A_1,B_1) \cup \mathcal{L}(A_1,B_2) \cup \mathcal{L}(A_1,B_3) \cup \mathcal{L}(A_2,B_2) \cup \mathcal{L}(A_1,C_1) \\ \cup & \mathcal{L}(A_1,C_2) \cup \mathcal{L}(A_2,C_1) \cup \mathcal{L}(B_1,C_1) \cup \mathcal{L}(B_3,C_1).
\end{split}$$ $X$ has the $Hyp_5 (\star$)-property. Indeed, if we take the $5$ hyperplanes $\mathcal{L}(A_1)$, $ \mathcal{L}(A_2)$, $\mathcal{L}(B_1), \mathcal{L}(B_2)$, $\mathcal{L}(C_1)$ we have that the lines $\mathcal{L}(A_1,B_1), \mathcal{L}(A_1,B_2), \mathcal{L}(A_2,B_2)$, $\mathcal{L}(A_2,C_1)$, $\mathcal{L}(B_1,C_1) \in X$ and also $\mathcal{L}(A_1, C_1) \in X$; if we take the $5$ hyperplanes $\mathcal{L}(A_1), \mathcal{L}(A_2)$, $\mathcal{L}(B_3), \mathcal{L}(B_2), \mathcal{L}(C_1)$, we have that $\mathcal{L}(A_1,B_3)$, $\mathcal{L}(A_1,B_2)$, $\mathcal{L}(A_2,B_2)$, $\mathcal{L}(A_2,C_1)$, $\mathcal{L}(B_3,C_1) \in X$ and also $\mathcal{L}(A_1, C_1) \in X$; and so on, if we take any 5 hyperplanes $H_1, \dots , H_5$ among $\mathcal{L}(A_1), \mathcal{L}(A_2), \mathcal{L}(B_1), \mathcal{L}(B_2), \mathcal{L}(B_3), \mathcal{L}(C_1), \mathcal{L}(C_2)$ such that $\mathcal{L}(H_i, H_j)\in X$ for any $j\neq i-1, i, i+1$, then there exists $u\in \{ 1, \dots , 5 \}$ such that $\mathcal{L}(H_u, H_{u+1})\in X$, where $H_0 = H_5$ and $H_{6} = H_1$. Note that if we take $\mathcal{L}(B_1), \mathcal{L}(B_2), \mathcal{L}(B_3)$ among the 5 hyperplanes we choose, the condition $\mathcal{L}(H_i, H_j)\in X$ for any $j\neq i-1, i, i+1$ fails to be true and then there is nothing to verify.
The following theorem is the main result of this section.
\[thm:ch1\] Let $X$ be a variety of lines. Then $X$ is ACM if and only if $X$ has the $Hyp_n(\star)$-property for $n= 4,5,6$.
Let $I_X$ be the ideal defining the variety of lines $X \subseteq \mathbb{P}^1 \times \mathbb{P}^1 \times \mathbb{P}^1$. From Lemma \[monomial\], $X$ is ACM if and only if $J_X \subseteq S$ is CM. From Remark \[r.simpl compl\], the ideal $J_X$ is the cover ideal of the graph $G_X$, i.e., $J_X = J(G_X)$. From Theorem \[th.ER\], the face ideal $I(G_X)$ has a linear resolution and then, using Theorem \[th.Fr\], $G_X^c$ is a chordal graph, that is, $X$ has the $Hyp_n(\star)$-property for any $n$. Remark \[n\*\] completes the proof.
A numerical characterization of the ACM property {#sec:num_char}
================================================
Since we are interested in the study of the ACM property for varieties of lines $X$, from now on we assume that $U_h(X)$ resembles a Ferrers diagram for each $h=1,2,3.$ In order to give a characterization of the ACM property we introduce the following notation.
\[def:mi\] Let $P=P_{ijk}=\mathcal{L}(A_i)\cap\mathcal{L}(B_j)\cap\mathcal{L}(C_k)$ be a point of a variety of lines $X$, we call *multiplicity of $P$* the number of lines of $X$ passing through the point $P$ and we denote it by $\mu_{ijk}.$
Since at most three lines of $X$ (one of each type) pass through the point $P$, $\mu_{ijk} \leq 3$.
Given a variety of lines $X$, we define a 3-dimensional matrix $M_X:=(\mu_{ijk})\in \mathbb{N}^{d_1\times d_2\times d_3}$ whose $(i,j,k)$-entry is the multiplicity of $P_{i,j,k}.$ We call it the *matrix of the multiplicities* of $X$.
We also define
$M_X^{({3})}:=(\mu_{ij0})\in \mathbb{N}^{d_1\times d_2}$, where $$\mu_{ij0}:=\begin{cases}
1 & \ \text{if}\ (i,j) \in U_3(X),\ \text{i.e.,}\ \mathcal{L}(A_{i},B_{j})\in X\\
0 & \ \text{otherwise}.
\end{cases}$$
Analogously, $M_X^{(2)}:=(\mu_{i0 k})\in \mathbb{N}^{d_1\times d_3}$, where $\mu_{i0k}:=\begin{cases}
1 & \ \text{if}\ \mathcal{L}(A_{i},C_{k})\in X\\
0 & \ \text{otherwise}
\end{cases}$
and $M_X^{({1})}:=(\mu_{0 jk})\in \mathbb{N}^{d_2\times d_3}$, where $\mu_{0jk}:=\begin{cases}
1 & \ \text{if}\ \mathcal{L}(B_j, C_k)\in X\\
0 & \ \text{otherwise}
\end{cases}.$
\[examplemultiplicity\] Let us consider $X \subseteq \mathbb{P}^1 \times \mathbb{P}^1 \times \mathbb{P}^1$ as Figure \[figure6starproperty\] $$\begin{split} X = & \mathcal{L}(A_1,B_1) \cup \mathcal{L}(A_1,B_2) \cup \mathcal{L}(A_2,B_2) \cup \mathcal{L}(A_1,C_1) \cup \mathcal{L}(A_2,C_1) \\ \cup & \mathcal{L}(A_2,C_2) \cup \mathcal{L}(B_1,C_1) \cup \mathcal{L}(B_1,C_2) \cup \mathcal{L}(B_2,C_2).
\end{split}$$ We have $$\mu_{111} = \mu_{222} = 3 \ \ , \ \ \mu_{121} = \mu_{221} = \mu_{211} = \mu_{112} = \mu_{122} = \mu_{212} = 2$$ and $$M_X^{(3)} = \left( \begin{matrix} 1 & 1 \\ 0 & 1
\end{matrix} \right), \ \ M_X^{(2)} = \left( \begin{matrix} 1 & 0 \\ 1 & 1
\end{matrix} \right), \ \ M_X^{(1)} = \left( \begin{matrix} 1 & 1 \\ 0 & 1
\end{matrix} \right).$$
(2,0) - - (2,8.5); (5,1.3) - - (5,10); (8,0) - - (8,8.5); (11,1.3) - - (11,10);
(0,2) - - (10,2); (2.5,4) - - (13,4); (0,55/10) - - (10,55/10); (2.5,75/10) - - (13,75/10);
(0,2/3) - - (7,16/3); (0,25/6) - - (7,53/6); (6,2/3) - - (13,16/3); (6,25/6) - - (13,53/6); (2,2) circle (2mm); (8,2) circle (2mm); (8,5.5) circle (2mm); (2,5.5) circle (2mm); (5,4) circle (2mm); (11,4) circle (2mm); (11,7.5) circle (2mm); (5,7.5) circle (2mm);
at (4.5,4.5) [ $P_{111}$]{}; at (10.5,4.5) [ $P_{121}$]{}; at (1.5,2.5) [ $P_{211}$]{}; at (7.5,2.5) [ $P_{221}$]{}; at (4.5,8) [ $P_{112}$]{}; at (10.5,8) [ $P_{122}$]{}; at (1.5,6) [ $P_{212}$]{}; at (7.6,6) [ $P_{222}$]{};
Now we provide a criterion to establish if $X$ is ACM or not just looking at the matrices of the multiplicities $M_X$, $M_X^{(1)}$, $M_X^{(2)}$ and $M_X^{(3)}$.
\[6starproperty\] Let $X$ be a variety of lines. $X$ has the $Hyp_6(\star)$-property iff for all $a_1,a_2\in[d_1],b_1,b_2\in[d_2],c_1,c_2\in[d_3]$ $$\text{either }\ \left(\begin{array}{cc}
\mu_{a_1b_1c_1} & \mu_{a_1b_2c_1}\\
\mu_{a_2b_1c_1} & \mu_{a_2b_2c_1}\\
\end{array}\right)\neq\left(\begin{array}{cc}
3 & 2\\
2 & 2\\
\end{array}\right)\ \text{or} \ \left(\begin{array}{cc}
\mu_{a_1b_1c_2} & \mu_{a_1b_2c_2}\\
\mu_{a_2b_1c_2} & \mu_{a_2b_2c_2}\\
\end{array}\right)\neq \left(\begin{array}{cc}
2 & 2\\
2 & 3\\
\end{array}\right).$$
If $X$ does not have the $Hyp_6(\star)$-property then there exist six planes, say $\mathcal{L}(A_1), \mathcal{L}(A_2), \mathcal{L}(B_1), \mathcal{L}(B_2), \mathcal{L}(C_1), \mathcal{L}(C_2)$, such that the lines $\mathcal{L}(A_1,B_1),$ $\mathcal{L}(A_1,B_2),$ $\mathcal{L}(A_1,C_1), $ $\mathcal{L}(A_2,B_2),$ $\mathcal{L}(A_2,C_1),$ $\mathcal{L}(A_2,C_2), $ $\mathcal{L}(B_1,C_1), $ $\mathcal{L}(B_1,C_2), $ $\mathcal{L}(B_2,C_2)$ belong to $X$ and $\mathcal{L}(A_2,B_1), $ $\mathcal{L}(B_2,C_1), $ $\mathcal{L}(A_1,C_2)\notin X.$ Then we have that $$\left(\begin{array}{cc}
\mu_{111} & \mu_{121}\\
\mu_{211} & \mu_{221}\\
\end{array}\right)=\left(\begin{array}{cc}
3 & 2\\
2 & 2\\
\end{array}\right)\ \text{and} \ \left(\begin{array}{cc}
\mu_{112} & \mu_{122}\\
\mu_{212} & \mu_{222}\\
\end{array}\right)= \left(\begin{array}{cc}
2 & 2\\
2 & 3\\
\end{array}\right).$$ On the other hand if $\left(\begin{array}{cc}
\mu_{111} & \mu_{121}\\
\mu_{211} & \mu_{221}\\
\end{array}\right)=\left(\begin{array}{cc}
3 & 2\\
2 & 2\\
\end{array}\right)\ \text{and} \ \left(\begin{array}{cc}
\mu_{112} & \mu_{122}\\
\mu_{212} & \mu_{222}\\
\end{array}\right)= \left(\begin{array}{cc}
2 & 2\\
2 & 3\\
\end{array}\right)$ then it is easy to check that $X$ does not have the $Hyp_6(\star)$-property since the lines $\mathcal{L}(A_1,B_1),$ $\mathcal{L}(A_1,B_2),$ $\mathcal{L}(A_1,C_1), $ $\mathcal{L}(A_2,B_2),$ $\mathcal{L}(A_2,C_1),$ $\mathcal{L}(A_2,C_2), $ $\mathcal{L}(B_1,C_1), $ $\mathcal{L}(B_1,C_2), $ $\mathcal{L}(B_2,C_2)\in X $ and $\mathcal{L}(A_2,B_1), $ $\mathcal{L}(B_2,C_1), $ $\mathcal{L}(A_1,C_2)\notin X.$
\[5starproperty\] Let $X$ be a variety of lines. $X$ has the $Hyp_5(\star)$-property iff for all $a_1,a_2\in[d_1],b_1,b_2\in[d_2],c_1,c_2\in[d_3]$ the following three conditions hold: $$1) \text{either}\left(\begin{array}{cc}
\mu_{a_1b_1c_1} & \mu_{a_1b_2c_1}\\
\mu_{a_2b_1c_1} & \mu_{a_2b_2c_1}\\
\end{array}\right)\neq\left(\begin{array}{cc}
2 & 1\\
2 & 2\\
\end{array}\right)\ \text{or} \ \left(\begin{array}{cc}
\mu_{a_1b_10} & \mu_{a_1b_20}\\
\mu_{a_2b_10} & \mu_{a_2b_20}\\
\end{array}\right)\neq \left(\begin{array}{cc}
1 & 1\\
0 & 1\\
\end{array}\right),$$
$$2) \text{either}\left(\begin{array}{cc}
\mu_{a_1b_1c_1} & \mu_{a_1b_1c_2}\\
\mu_{a_2b_1c_1} & \mu_{a_2b_1c_2}\\
\end{array}\right)\neq\left(\begin{array}{cc}
2 & 1\\
2 & 2\\
\end{array}\right)\ \text{or} \ \left(\begin{array}{cc}
\mu_{a_10 c_1} & \mu_{a_10 c_2}\\
\mu_{a_20 c_1} & \mu_{a_20 c_2}\\
\end{array}\right)\neq \left(\begin{array}{cc}
1 & 1\\
0 & 1\\
\end{array}\right),$$
$$3) \text{either}\left(\begin{array}{cc}
\mu_{a_1b_1c_1} & \mu_{a_1b_1c_2}\\
\mu_{a_1b_2c_1} & \mu_{a_1b_2c_2}\\
\end{array}\right)\neq\left(\begin{array}{cc}
2 & 1\\
2 & 2\\
\end{array}\right)\ \text{or} \ \left(\begin{array}{cc}
\mu_{0 b_1c_1} & \mu_{0 b_1c_2}\\
\mu_{0 b_2c_1} & \mu_{0 b_2c_2}\\
\end{array}\right)\neq \left(\begin{array}{cc}
1 & 1\\
0 & 1\\
\end{array}\right).$$
If $X$ does not have the $Hyp_5(\star)$-property, we say, without loss of generality, that exist five planes $\mathcal{L}(A_1), \mathcal{L}(A_2), \mathcal{L}(B_1), \mathcal{L}(B_2), \mathcal{L}(C_1)$ such that, among all, only the lines $\mathcal L(A_2,B_1)$, $\mathcal L(A_1,C_1)$, $\mathcal L(B_2,C_1) \notin X.$ Then we have
$\left(\begin{array}{cc}
\mu_{111} & \mu_{121}\\
\mu_{211} & \mu_{221}\\
\end{array}\right)=\left(\begin{array}{cc}
2 & 1\\
2 & 2\\
\end{array}\right)\ \ \text{and} \ \left(\begin{array}{cc}
\mu_{110} & \mu_{120}\\
\mu_{210} & \mu_{220}\\
\end{array}\right)= \left(\begin{array}{cc}
1 & 1\\
0 & 1\\
\end{array}\right).$
On the other hand, assume, for instance, we have the following equalities
$\left(\begin{array}{cc}
\mu_{111} & \mu_{121}\\
\mu_{211} & \mu_{221}\\
\end{array}\right)=\left(\begin{array}{cc}
2 & 1\\
2 & 2\\
\end{array}\right)\ \ \text{and} \ \left(\begin{array}{cc}
\mu_{110} & \mu_{120}\\
\mu_{210} & \mu_{220}\\
\end{array}\right)= \left(\begin{array}{cc}
1 & 1\\
0 & 1\\
\end{array}\right).$
From the previous equalities, we get $\mathcal{L}(A_1,C_1)\notin X$ thus, since $\mu_{111}=2,$ we have $\mathcal{L}(A_1,B_1),\mathcal{L}(B_1,C_1)\in X.$ Analogously $\mathcal{L}(A_1,B_2)\in X$ and so, since $\mu_{121}=1,$ we have $\mathcal{L}(B_2,C_1)\notin X.$ Moreover $\mathcal{L}(A_2,B_2)\in X$ and so, since $\mu_{221} = 2$, we have $\mathcal L(A_2,C_1) \in X$. Finally $\mathcal L(A_2,B_1) \notin X$ since $\mu_{210}=0$. So $X$ does not have the $Hyp_5(\star)$-property.
\[4starproperty\] Let $X$ be a variety of lines. $X$ has the $Hyp_4(\star)$-property iff for all $a_1,a_2\in[d_1],b_1,b_2\in[d_2],c_1,c_2\in[d_3]$ the following three conditions hold: $$1) \text{either} \ \left(\begin{array}{c}
\mu_{a_1b_1c_1} \\
\mu_{a_2b_1c_1} \\
\end{array}\right)\neq\left(\begin{array}{cc}
1 \\
1 \\
\end{array}\right)\ \text{or} \ \left(\begin{array}{cc}
\mu_{a_1b_10}\\
\mu_{a_2b_10}\\
\end{array}\right)\neq \left(\begin{array}{cc}
1 \\
0 \\
\end{array}\right).$$
$$2) \text{either} \ \left(\begin{array}{c}
\mu_{a_1b_1c_1} \\
\mu_{a_1b_1c_2} \\
\end{array}\right)\neq\left(\begin{array}{cc}
1 \\
1 \\
\end{array}\right)\ \text{or} \ \left(\begin{array}{cc}
\mu_{a_10c_1}\\
\mu_{a_10c_2}\\
\end{array}\right)\neq \left(\begin{array}{cc}
1 \\
0 \\
\end{array}\right).$$
$$3) \text{either} \ \left(\begin{array}{c}
\mu_{a_1b_1c_1} \\
\mu_{a_1b_2c_1} \\
\end{array}\right)\neq\left(\begin{array}{cc}
1 \\
1 \\
\end{array}\right)\ \text{or} \ \left(\begin{array}{cc}
\mu_{0b_1c_1}\\
\mu_{0b_2c_1}\\
\end{array}\right)\neq \left(\begin{array}{cc}
1 \\
0 \\
\end{array}\right).$$
Suppose that $X$ does not have the $Hyp_4(\star)$-property. Since we are assuming do not exist four planes $\mathcal{L}(A_1), \mathcal{L}(A_2), \mathcal{L}(B_1), \mathcal{L}(B_2)$ such that $\mathcal L(A_1,B_1), \mathcal L(A_2,B_2) \in X$ and $\mathcal L(A_1, B_2)$ or $\mathcal L(A_2,B_1) \notin X$, then, $X$ fails the $Hyp_4(\star)$-property if, without loss of generality, there exist four planes $\mathcal{L}(A_1), \mathcal{L}(A_2), \mathcal{L}(B_1), \mathcal{L}(C_1)$ such that, among all, only the lines $\mathcal L(A_2,B_1),\mathcal L(B_1,C_1), \mathcal L(A_1,C_1)\notin X.$ Then we have
$\left(\begin{array}{cc}
\mu_{111} \\
\mu_{211} \\
\end{array}\right)=\left(\begin{array}{cc}
1 \\
1 \\
\end{array}\right)\ \ \text{and} \left(\begin{array}{cc}
\mu_{110} \\
\mu_{210} \\
\end{array}\right)= \left(\begin{array}{cc}
1 \\
0 \\
\end{array}\right).$
On the other hand, assume, for instance, we have the following equalities
$\left(\begin{array}{cc}
\mu_{111} \\
\mu_{211} \\
\end{array}\right)=\left(\begin{array}{cc}
1 \\
1 \\
\end{array}\right)\ \ \text{and} \ \left(\begin{array}{cc}
\mu_{110} \\
\mu_{210} \\
\end{array}\right)= \left(\begin{array}{cc}
1 \\
0 \\
\end{array}\right).$
From the previous equalities, we get $\mathcal{L}(A_1,B_1)\in X$ thus, since $\mu_{111}=1,$ we have $\mathcal{L}(A_1,C_1),\mathcal{L}(B_1,C_1)\notin X.$ Analogously $\mathcal{L}(A_2,B_1)\notin X$ and so, since $\mu_{211}=1,$ we have $\mathcal{L}(A_2,C_1)\in X$. So $X$ does not have the $Hyp_4(\star)$-property.
Let $X$ be as in Example \[examplemultiplicity\] (see Figure \[figure6starproperty\]). We observe that
$\left(\begin{array}{cc}
\mu_{111} & \mu_{121}\\
\mu_{211} & \mu_{221}\\
\end{array}\right)= \left(\begin{array}{cc}
3 & 2\\
2 & 2\\
\end{array}\right) \text{and} \ \left(\begin{array}{cc}
\mu_{112} & \mu_{122}\\
\mu_{212} & \mu_{222}\\
\end{array}\right)= \left(\begin{array}{cc}
2 & 2\\
2 & 3\\
\end{array}\right)$
and then, by Proposition \[6starproperty\], we have that $X$ does not have the $Hyp_6(\star)$-property and so, by Theorem \[thm:ch1\], $X$ is not ACM.
Let us consider the variety $W=X \cup \mathcal{L}(A_2,B_1)$, where $X$ is as Example \[examplemultiplicity\].
(2,0) - - (2,8.5); (5,1.3) - - (5,10); (8,0) - - (8,8.5); (11,1.3) - - (11,10);
(0,2) - - (10,2); (2.5,4) - - (13,4); (0,55/10) - - (10,55/10); (2.5,75/10) - - (13,75/10);
(0,2/3) - - (7,16/3); (0,25/6) - - (7,53/6); (6,2/3) - - (13,16/3); (6,25/6) - - (13,53/6);
(2,2) circle (2mm); (8,2) circle (2mm); (8,5.5) circle (2mm); (2,5.5) circle (2mm); (5,4) circle (2mm); (11,4) circle (2mm); (11,7.5) circle (2mm); (5,7.5) circle (2mm);
at (4.5,4.5) [ $P_{111}$]{}; at (10.5,4.5) [ $P_{121}$]{}; at (1.5,2.5) [ $P_{211}$]{}; at (7.5,2.5) [ $P_{221}$]{}; at (4.5,8) [ $P_{112}$]{}; at (10.5,8) [ $P_{122}$]{}; at (1.5,6) [ $P_{212}$]{}; at (7.6,6) [ $P_{222}$]{};
We have $ \mu_{111} = \mu_{222} = \mu_{211} = \mu_{212} =3, \ \mu_{121} = \mu_{221} = \mu_{112} = \mu_{122} = 2. $ And for all $ a_1,a_2\in[2],\ b_1,b_2\in[2],\ c_1,c_2 \in[2]$, we have: $$\left(\begin{array}{cc}
\mu_{a_1b_1c_1} & \mu_{a_1b_2c_1}\\
\mu_{a_2b_1c_1} & \mu_{a_2b_2c_1}\\
\end{array}\right)\neq \left(\begin{array}{cc}
3 & 2\\
2 & 2\\
\end{array}\right),
\left(\begin{array}{cc}
\mu_{a_1b_1c_1} & \mu_{a_1b_2c_1}\\
\mu_{a_2b_1c_1} & \mu_{a_2b_2c_1}\\
\end{array}\right)\neq \left(\begin{array}{cc}
2 & 1\\
2 & 2\\
\end{array}\right),$$
$$\left(\begin{array}{cc}
\mu_{a_1b_1c_1} & \mu_{a_1b_1c_2}\\
\mu_{a_2b_1c_1} & \mu_{a_2b_1c_2}\\
\end{array}\right)\neq \left(\begin{array}{cc}
2 & 1\\
2 & 2\\
\end{array}\right), \left(\begin{array}{cc}
\mu_{a_1b_1c_1} & \mu_{a_1b_1c_2}\\
\mu_{a_1b_2c_1} & \mu_{a_1b_2c_2}\\
\end{array}\right)\neq \left(\begin{array}{cc} 2 & 1\\
2 & 2\\
\end{array}\right),$$
$$\left(\begin{array}{c}
\mu_{a_1b_1c_1} \\
\mu_{a_2b_1c_1} \\
\end{array}\right)\neq \left(\begin{array}{cc}
1 \\
1 \\
\end{array}\right), \left(\begin{array}{c}
\mu_{a_1b_1c_1} \\
\mu_{a_1b_1c_2} \\
\end{array}\right)\neq \left(\begin{array}{cc}
1 \\
1 \\
\end{array}\right), \left(\begin{array}{c}
\mu_{a_1b_1c_1} \\
\mu_{a_1b_2c_1} \\
\end{array}\right)\neq \left(\begin{array}{cc}
1 \\
1 \\
\end{array}\right)$$ and then, by Propositions \[6starproperty\], \[5starproperty\] and \[4starproperty\], the variety of lines $W$ has the $Hyp_n(\star)$-property for $n=4, 5, 6$ and then, by Theorem \[thm:ch1\], $W$ is ACM.
The Hilbert function of ACM codimension two varieties in $\mathbb P^1 \times \mathbb P^1 \times \mathbb P^1$ {#sec:HF}
============================================================================================================
In this section we approach the study of the Hilbert function of these varieties. We start from the following specific case.
If $X$ is a variety of lines such that the index sets $U_1(X), U_2(X)$ and $ U_3(X)$ are Ferrers diagram, then we call $X$ a *Ferrers variety of lines*. That is, after renaming, we assume that if $\mathcal L(A_{i},B_{j})\in U_h(X)$ then $\mathcal L(A_{i'},B_{j'})\in U_h(X)$ for every $1 \le i'\le i,\ 1 \le j'\le j$ and for each direction $h=1,2,3$.
As a consequence of Theorem \[thm:ch1\], note that a Ferrers variety of lines is ACM.
Recall that given a homogeneous ideal $I\subseteq R$, the *Hilbert function of $R/I$* is the numerical function $$H_{R/I} : \mathbb{N}^3 \rightarrow \mathbb{N}$$ defined by $$H_{R/I}(i,j,k) := dim_K (R/I)_{i,j,k} = dim_K R_{i,j,k} - dim_K I_{i,j,k}.$$ The *first difference function of $H$*, denoted $\Delta H$, is the function $\Delta H : \mathbb{N}^3 \rightarrow \mathbb{N}$ defined by $$\Delta H(i,j,k) := \sum \limits_{(0,0,0) \le (l,m,n) \le (1,1,1)} (-1)^{l+m+n} H(i-l,j-m,k-n).$$
Now, let $X$ be a Ferrers variety of lines and let $X_3=\bigcup\limits_{(r,s)\in U_3(X)}\mathcal{L}(A_r,B_s)$ be the variety of lines consisting of the lines of $X$ of type $(1,1,0)$. Since $U_3(X)$ is a Ferrers diagram, the variety $X_3$ is ACM (in $\mathbb P^1\times \mathbb P^1$) and we can explicitly write out a set of minimal generators of $I_{X_3}$, see Remark \[r.cono\] and [@GV-book]. If $ \{ (a_{3,i}, b_{3,i})\} $ is the set of the degrees of these minimal generators, we denote by $D_3(X) := \{ (a_{3,i}, b_{3,i},0)\}$.
Analogously, if we consider the varieties of lines $X_1$ and $X_2$ consisting of the lines of $X$ of types $(0,1,1)$ and $(1,0,1)$, respectively, we obtain the sets of degrees $D_1(X) = \{ (0,b_{1,j}, c_{1,j})\} $ and $D_2(X) = \{ (a_{2,k}, 0, c_{2,k})\} $. Then we denote by
$$\begin{split}
D(X):= \{ & (\max\{a_{3,i}, a_{2,k}\}, \max\{b_{3,i}, b_{1,j}\}, \max\{c_{1,j}, c_{2,k}\} )\ |\ \forall \ (a_{3,i},b_{3,i},0)\in D_3(X), \\ \phantom{=} & (a_{2,k},0,c_{2,k})\in D_2(X), (0,b_{1,j},c_{1,j})\in D_1(X) \}. \nonumber
\end{split}$$
Finally, we denote by $\hat{D}(X)$ the set of the minimal elements of $D(X)$ with respect to the natural partial order $\preceq$ on the elements of $\mathbb{N}^3$.
\[beta\_0\] Let $X$ be a Ferrers variety of lines. Then $I_X$ is minimally generated by the following set of forms $$\left\{ \prod_{i\le a}A_i\prod_{j\le b}B_j\prod_{k\le c}C_k \ |\ \text{for each}\ (a,b,c)\in \hat{D}(X) \right \}.$$
First, we prove that if $ (a,b,c)\in \hat{D}(X)$, then $\prod_{i\le a}A_i\prod_{j\le b}B_j\prod_{k\le c}C_k\in I_X.$ Indeed $(a,b,c)\in \hat{D}(X)$ implies $\prod_{i\le a}A_i\prod_{j\le b}B_j \in I_{X_3}$, $\prod_{j\le b}B_j\prod_{k\le c}C_k \in I_{X_1}$, $\prod_{i\le a}A_i\prod_{k\le c}C_k \in I_{X_2}$ and they are not necessarily minimal elements of the respective ideal. Thus $\prod_{i \le a} A_i\prod_{j \le b} B_j\prod_{k \le c} C_k\in I_{X_1}\cap I_{X_2}\cap I_{X_3} = I_X$. Now, we show that if $ (a,b,c)\in \hat{D}(X)$ and $a>0$, then $\prod_{i\le a-1}A_i\prod_{j\le b}B_j\prod_{k\le c}C_k\notin I_X.$ This fact follows by contradiction. Indeed if $\prod_{i\le a-1}A_i\prod_{j\le b}B_j\prod_{k\le c}C_k\in I_X,$ then $(a-1,b,0), (a-1,0,c), (0,b,c)$ are degrees of some (not necessarily minimal) elements in the ideal and therefore there is an element in $D(X)$ less than or equal to $(a-1,b,c)$, contradicting the minimality of $(a,b,c)\in \hat{D}(X)$. Analogously, it can be easily showed that if $ (a,b,c)\in
\hat{D}(X)$ and $b>0$ (or $c>0$), then $\prod_{i\le
a}A_i\prod_{j\le b-1}B_j\prod_{k\le c}C_k\notin I_X$ (or $\prod_{i\le a}A_i\prod_{j\le b}B_j\prod_{k\le c-1}C_k\notin
I_X$). Finally, we claim that $I_X $ is minimally generated by the forms $ \prod_{i\le a}A_i\prod_{j\le b}B_j\prod_{k\le c}C_k$ with $(a,b,c)\in \hat{D}(X)$. Take a form $F\in I_X$, without loss of generality we can assume that $F:=\prod_{i\in \mathcal
A}A_i\prod_{j\in \mathcal B}B_j\prod_{k\in \mathcal C}C_k$ is product of linear forms. By contradiction we assume $A_i$ divides $F$ and $A_{i-1}$ does not divide $F$. Then $\prod_{i\in \mathcal
A}A_i\prod_{j\in \mathcal B}B_j \in I_{X_3}$. Then $F\in
(\prod_{i\le a'}A_i\prod_{j\le b'}B_j)$ for some $a',b'.$ Repeating the same argument with respect to the other two directions we get the proof. The minimality come from the minimality of the degrees in $\hat{D}(X)$.
The following corollary is an immediate consequence of Theorem \[beta\_0\] and the ACM property. Set $\langle \hat{D}(X) \rangle := \{(i,j,k)\ |\ (i,j,k) \ge (a,b,c),\ \text{for some}\ (a,b,c) \in \hat{D}(X) \}$.
\[GORFerrers\] Let $X$ be a Ferrers variety of lines. Then $$\Delta H_X{(i,j,k)}=\begin{cases}
0 & \text{if}\ (i,j,k) \in \langle \hat{D}(X) \rangle \\
1 & \text{otherwise}
\end{cases}.$$
Let us consider the following variety of lines $$X=\{\mathcal{L} (A_i, B_j) \cup \mathcal{L}(A_i,C_k) \cup \mathcal{L}(B_j,C_k) \ | \ 1\le i \le 4, \ 1\le j \le 3, \ 1\le k \le 2 \}.$$ In this case we have $D_3(X)=\{(4,0,0), (0,3,0) \}$, $D_2(X)=\{(4,0,0), (0,0,2) \}$ and $D_1(X)=\{(0,3,0), (0,0,2) \}$. Then $D(X)=\{(4,3,2), (4,3,0), (4,0,2), (0,3,2) \}$ and $\hat{D}(X)=\{(4,3,0), (4,0,2), (0,3,2) \}$. Therefore, from Theorem \[beta\_0\], a minimal set of generators of $I_X$ is given by: $$A_1A_2A_3A_4B_1B_2B_3, \ A_1A_2A_3A_4C_1C_2, \ B_1B_2B_3C_1C_2$$ and $$\Delta H_X{(i,j,k)}=\begin{cases}
0 & \text{if}\ (i,j,k) \ge (4,3,0) \ \text{or} \ (4,0,2) \ \text{or} \ (0,3,2) \\
1 & \text{otherwise}
\end{cases}.$$
Case study: grids of lines and complete intersections of lines {#sec:grids}
==============================================================
In the last section we focus on the study of special arrangements of lines in $\mathbb{P}^1 \times \mathbb{P}^1 \times \mathbb{P}^1$ having the ACM property. Recall that for a point $P\in \mathbb{P}^1 \times \mathbb{P}^1 \times \mathbb{P}^1$ there are exactly three lines passing through $P$, one for each direction. We have the following definition.
Let $\mathcal Y$ be a finite set of points in $\mathbb{P}^1 \times \mathbb{P}^1 \times \mathbb{P}^1$. We call *grid of lines* arising from $\mathcal Y$, and denote it by $X_{\mathcal Y}$, the set containing all the lines of $\mathbb{P}^1 \times \mathbb{P}^1 \times \mathbb{P}^1$ passing through some point of $\mathcal Y$.
In other words, if $\mathcal Y$ is a finite set of points in $\mathbb{P}^1 \times \mathbb{P}^1 \times \mathbb{P}^1$, then $$X_{\mathcal Y}:= \bigcup_{P_{ijk} \in \mathcal Y } \mathcal{L} (A_i, B_j) \cup \mathcal{L}(A_i,C_k) \cup \mathcal{L}(B_j,C_k)$$ where $P_{ijk}:=\mathcal{L}(A_i)\cap\mathcal{L}(B_j)\cap\mathcal{L}(C_k).$
The next example shows that, even if $\mathcal Y$ is an ACM set of points, $X_{\mathcal Y}$ could be not ACM.
Suppose $\mathcal Y:=\{P_{112},P_{122}, P_{121}, P_{212}\} \subseteq \mathbb{P}^1 \times \mathbb{P}^1 \times \mathbb{P}^1.$ According to [@FGM], $\mathcal Y$ is an ACM set of points. We have $\mathcal{L} (A_2, B_1),\mathcal{L}(B_2,C_1) \in X_{\mathcal Y}$ and $\mathcal L(A_2, B_2),\mathcal{L}(A_2,C_1)$, $\mathcal{L}(B_1,C_1) \notin X_{\mathcal Y} $, that is, $X_{\mathcal Y} $ has not the $Hyp_4(\star)$-property and then $X_{\mathcal Y}$ is not ACM.
It is interesting to ask which sets of points $\mathcal Y\subseteq \mathbb{P}^1 \times \mathbb{P}^1 \times \mathbb{P}^1$ lead to an ACM grid of lines $X_{\mathcal Y}.$ A special class of CM rings is represented by complete intersections. We recall their definitions and properties.
\[CI\] An ideal $I\subset R$ is a *complete intersection* if it is generated by a regular sequence.
As pointed out in [@GV-book], Lemma 2.25 a complete intersection is also Cohen-Macaulay.
\[d.CI\] In $ \mathbb{P}^1 \times \mathbb{P}^1 \times \mathbb{P}^1$, we say that a set of points $\mathcal C$ is a *complete intersection of points of type $(a_1,a_2,a_3)$* if $I_{\mathcal C} = (F_1, F_2, F_3)$ is a complete intersection and $\deg F_i = a_i \bf e_i$ for $i=1,2,3$.
Note that each $F_i$ in Definition \[d.CI\] is product of linear forms.
We say that a variety of lines $X$ is a *complete intersection of lines* in $\mathbb{P}^1 \times \mathbb{P}^1 \times \mathbb{P}^1$ if $I_X$ is a complete intersection.
\[thmACM\] Let $\mathcal C\subseteq \mathbb{P}^1 \times \mathbb{P}^1 \times \mathbb{P}^1$ be a complete intersection of points of type $(a,b,c).$ Then $X_{\mathcal C} $ is ACM and a trigraded minimal free resolution of $I_{X_{\mathcal C}}$ is $$0 \rightarrow R^2(-a,-b,-c) \rightarrow R(-a,-b,0) \oplus R(-a,0,-c) \oplus R(0,-b,-c) \rightarrow I_{X_{\mathcal C}} \rightarrow 0.$$
The grid of lines $X:=X_{\mathcal C}$ has the $Hyp_n(\star)$-property for $n=4, 5, 6$ and then, by Theorem \[thm:ch1\], $X$ is ACM. Moreover, by Corollary \[c.monomial\] the generators of $I_X$ are product of linear forms, so $$I_X= \left(\prod \limits _{\substack{ i \in [a] }} A_i \prod \limits_{\substack{j \in [b]}} B_j \ , \ \prod \limits _{\substack{i \in [a]}} A_i \prod \limits_{\substack{k \in [c]}} C_k \ , \ \prod \limits _{\substack{j \in [b]}} B_j \prod \limits_{\substack{k \in [c]}} C_k \right).$$ Then a Hilbert-Burch matrix of $I_X$ is $$\left( \begin{matrix} \prod \limits _{\substack{ i \in [a]}} A_i & \prod \limits _{\substack{ i \in [a]}} A_i \\ \prod \limits_{\substack{j \in [b]}} B_j & 0 \\ 0 & \prod \limits_{\substack{k \in [c]}} C_k
\end{matrix} \right).$$
If $\mathcal C \subseteq \mathbb{P}^1 \times \mathbb{P}^1 \times
\mathbb{P}^1$ is a complete intersection of points of type $(2,3,2)$, then the grid $X_{\mathcal C}$ is formed by 6 lines of type $(1,1,0)$, 4 lines of type $(1,0,1)$ and 6 lines of type $(0,1,1)$:
(2,0) - - (2,8.5); (5,1.3) - - (5,10); (8,0) - - (8,8.5); (11,1.3) - - (11,10);
(14,0) - - (14,8.5); (17,1.3) - - (17,10);
(0,2) - - (16,2); (2.5,4) - - (19,4); (0,55/10) - - (16,55/10); (2.5,75/10) - - (19,75/10);
(0,2/3) - - (7,16/3); (0,25/6) - - (7,53/6); (6,2/3) - - (13,16/3); (6,25/6) - - (13,53/6);
(12,2/3) - - (19, 16/3); (12,25/6) - - (19, 53/6);
(2,2) circle (1.5mm); (8,2) circle (1.5mm); (14,2) circle (1.5mm); (8,5.5) circle (1.5mm); (2,5.5) circle (1.5mm); (14, 5.5) circle (1.5mm); (5,4) circle (1.5mm); (11,4) circle (1.5mm); (17,4) circle (1.5mm); (11,7.5) circle (1.5mm); (5,7.5) circle (1.5mm); (17, 7.5) circle (1.5mm);
In particular, $ I_{X_{\mathcal C}}$ has a trigraded minimal free resolution of the following type $$0 \rightarrow R^2(-2,-3,-2) \rightarrow R(-2,-3,0) \oplus R(-2,0,-2) \oplus R(0,-3,-2) \rightarrow I_{X_{\mathcal C}} \rightarrow 0.$$
The following example shows that there exists an ACM grid of lines $X_{\mathcal Y}$ arising from a not ACM set of points $\mathcal Y$.
The following set of points $\mathcal Y:=\{ P_{111}, P_{121},P_{211}, P_{122},P_{212}, P_{222}\}$ is not an ACM set of points in $\mathbb{P}^1 \times \mathbb{P}^1 \times \mathbb{P}^1$ (see [@FGM]). However, $X_{\mathcal Y}=X_{\mathcal C}$ where $\mathcal C:=\{ P_{ijk}\ |\ 1\le i,j,k \le 2\}$, and then $X_{\mathcal Y}$ is an ACM grid of lines.
From Theorem \[thmACM\], we note that the ideal $I_{X_{\mathcal
C}}$ is generated by three forms that do not form a regular sequence. That is, even if $\mathcal C$ is a complete intersection of points, then its associated $X_{\mathcal C}$ variety of lines is not a complete intersection of lines. Thus, it is natural to study which varieties of lines are defined by a complete intersection, i.e., their defining ideal has only two generators. Theorem \[thmcompleteintersection\] and Remark \[CIlines\] will describe complete intersections of lines in $\mathbb{P}^1
\times \mathbb{P}^1 \times \mathbb{P}^1$.
\[rem:dgens\] If $X$ is an ACM variety of lines, from Corollary \[c.monomial\], $I_X$ is generated by products of linear forms. Then $$I_X\supseteq \left(\prod \limits _{\substack{ i \in [a]}} A_i \prod \limits_{\substack{j \in [b]}} B_j \ , \ \prod \limits _{\substack{i \in [a]}} A_i \prod \limits_{\substack{k \in [c]}} C_k \ , \ \prod \limits _{\substack{j \in [b]}} B_j \prod \limits_{\substack{k \in [c]}} C_k \right).$$ So any set of minimal generators of $I_X$ contains one element of degree $(a_3, b_3, 0)$, one element of degree $(a_2,0,c_2)$ and one element of degree $(0,b_1,c_1)$.
\[thmcompleteintersection\] Let $X$ be a variety of lines of $\mathbb{P}^1 \times \mathbb{P}^1 \times \mathbb{P}^1$. Then the ideal $I_X$ is a complete intersection if and only if $I_X= (F_1,F_2)$, with $\deg F_1=a \textbf{e}_i$ and $\deg F_2=b\textbf{e}_{j}+c\textbf{e}_{k}$ with $j,k \neq i$, for some $a,b,c \in \mathbb{N}.$
One implication is trivial. Let $I_X$ be a complete intersection, i.e. $I_X$ is generated by a regular sequence of length 2, then $X$ is ACM. So, from Remark \[rem:dgens\], any set of minimal generators of $I_X$ contains one element $G_1$ of degree $(0, b_1, c_1)$, one element $G_2$ of degree $(a_2, 0, c_2)$ and one element $G_3$ of degree $(a_3, b_3, 0)$ for some integers $a_i,b_j,c_k$. Since $I_X$ is a complete intersection, one of these three generators say, without loss of generality, the one of degree $(0, b_1, c_1)$, is not minimal, i.e. $G_1\in (G_2, G_3)$. This easily implies $a_2a_3=0.$
\[CIlines\] From Theorem \[thmcompleteintersection\], a complete intersection of lines $X$ is then obtained from a grid arising from a complete intersection of points by removing either all the lines having direction $\textbf{e}_i$ for some $i$, or all the lines having direction $\textbf{e}_i$ and $\textbf{e}_j$ with $i\neq j$. Indeed, from Remark \[rem:dgens\], we have for instance $$I_X=\left(\prod \limits _{\substack{ i \in [a]}} A_i , \ \prod \limits _{\substack{j \in [b]}} B_j \prod \limits_{\substack{k \in [c]}} C_k \right)= \bigcap_{i \in [a] \atop j \in [b]}(A_i,B_j) \cap \bigcap_{i \in [a] \atop k \in [c]}(A_i,C_k).$$
Let $X$ be the set of lines of $\mathbb{P}^1 \times \mathbb{P}^1 \times \mathbb{P}^1$ obtained by a grid of lines $X_{\mathcal C}$ arising from a complete intersection $\mathcal C$ of type $(4,3,2)$ removing all the lines having direction $\bf{e_2}$: $$X= \bigcup_{i \in [4] \atop j \in [3]} \mathcal{L} (A_i, B_j) \bigcup_{j \in [3] \atop k \in [2]} \mathcal{L}(B_j,C_k).$$ Then the ideal $I_X$ is a complete intersection and it is generated by the regular sequence $F_1=B_1B_2B_3$ and $F_2=A_1A_2A_3A_4C_1C_2$ of degree $(0,3,0)$ and $(4,0,2)$, respectively.
We end the paper with two research topics that are still under our investigation.
1. Guida, Orecchia and Ramella, in [@GOR], studied the *complete grids* of lines in $\mathbb{P}^3$, whose defining ideal is the 1-lifting ideal of a specific monomial ideal $J$ in a polynomial ring $S$ in three variables. In particular, from Example 4.9 in [@GOR] and Corollary \[GORFerrers\], we noted that the first difference of the Hilbert function of the ideal $I_{X_{\mathcal C}}$ of a grid of lines arising from a complete intersection of points of type $(2,2,2)$ in $\mathbb{P}^1 \times \mathbb{P}^1 \times
\mathbb{P}^1$ in degree $(i,j,k)$ is equal to $1$ if and only if $(i,j,k)$ belongs to the order ideal $N(J) \subseteq
\mathbb{N}^3$ of the specific monomial ideal $J= (x_{1}^2x_2^2,
x_1^2x_3^2,x_2^2x_3^2)$ in $S$.
2. Let us consider the ACM varieties of lines $X$ and the Ferrers variety of lines $X'$ as in Figure \[fig7\] and Figure \[fig8\], respectively. We have that, for each $h=1,2,3$, $X_h$ and $X'_h$ have the same Hilbert functions. We also get $H_X=H_{X'}$.
(5,1.3) - - (5,10); (2.5,4) - - (13,4); (6,2/3) - - (13,16/3); (5,4) circle (2mm); (11,4) circle (2mm); at (2.8,10) [ $\bf{\mathcal{L}(A_1,B_1)}$]{}; at (0.4,4) [ $\bf{\mathcal{L}(A_1,C_1)}$]{}; at (7.9,2/3) [ $\bf{\mathcal{L}(B_2, C_1)}$]{};
(5,1.3) - - (5,10); (2.5,4) - - (13,4); (0,2/3) - - (7,16/3); (5,4) circle (2mm); at (2.8,10) [ $\bf{\mathcal{L}(A_1,B_1)}$]{}; at (0.4,4) [ $\bf{\mathcal{L}(A_1,C_1)}$]{}; at (2,2/3) [ $\bf{\mathcal{L}(B_1, C_1)}$]{};
According to many experimental computations using CoCoA, [@cocoa], we ask the following question:
\[HF\] Let $X$ be an ACM variety of lines and $X'$ be a Ferrers variety of lines such that, for $h=1,2,3,$ $X_h$ and $X'_h$ have the same Hilbert functions. Is it true that $H_X=H_{X'}?$
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---
abstract: 'We show equivalence between the massive Thirring model and the sine-Gordon theory by gauge fixing a wider gauge invariant theory in two different ways. The exact derivation of the equivalence hinges on the existence of an underlying conformal symmetry. Previous derivations were all perturbative in mass (althought to all orders).'
---
[**The Role of Conformal Symmetry in Abelian Bosonization of the Massive Thirring Model\
**]{} Aleksandar Bogojevi' c, Branislav Sazdovi' c\
[*Institute of Physics\
P.O.Box 57, Belgrade 11001, Yugoslavia\
*]{} Olivera Miškovi' c\
[*Institute of Nuclear Sciences “Vinča"\
Department for Theoretical Physics\
P.O.Box 522, Belgrade 11001, Yugoslavia\
*]{}
In a previous paper [@bogojevic] we have derived a model which under two different gauge fixings goes over into the massive Thirring and sine-Gordon models respectively. Rather than doing this, here we directly present the wider model. It is given in terms of scalar fields $\phi$, $\varphi$, spinor field $\psi$ and gauge field $A_\mu$, living in two dimensional Euclidian space. The generating functional and Lagrangian are $$\begin{aligned}
Z&=&\int{\cal D}\bar\psi{\cal D}\psi{\cal D}A{\cal D}\varphi{\cal D}\phi
\,e^{-\int d^2x{\cal L}}\nonumber\\
{\cal L} &=& \bar\psi\gamma_\mu\partial_\mu\psi+
\bar\psi\gamma_\mu\psi \, A_\mu-\frac{1}{2g}\,A_\mu^2+
\frac{1}{2g}(\partial_\mu\varphi)^2+\frac{i}{g}\,A_\mu\partial_\mu\varphi-
\nonumber\\
\label{pocetak}&&-\frac{\pi}{2g}\,(\partial_\mu\phi)^2
-\frac{2\pi}{g\beta}\,\varepsilon_{\mu\nu}\,A_\mu\partial_\nu\phi
+m\,\bar\psi\psi\cos \beta\phi+im\,\bar\psi\gamma_5\psi\sin \beta\phi\ ,\end{aligned}$$ where $\frac{4\pi}{\beta^2}=1+\frac{g}{\pi}$. We can check that the Lagrangian $\cal L$ and measure ${\cal D}\bar\psi{\cal D}\psi{\cal D}A{\cal D}\varphi{\cal D}\phi$ are invariant under local vector transformations:
$$\begin{aligned}
\psi &\to& \psi^\omega=e^{i\omega}\,\psi\nonumber\\
\bar\psi &\to& \bar\psi^\omega=e^{-i\omega}\,\bar\psi\nonumber\\
A_\mu &\to& A_\mu^\omega=A_\mu-i\partial_\mu\omega\nonumber\\
\varphi &\to& \varphi^\omega=\varphi-\omega\nonumber\\
\phi &\to& \phi^\omega=\phi\ .\end{aligned}$$
The invariance of the generating functional follows. On the other hand, under local axial-vector transformations: $$\begin{aligned}
\psi &\to& \psi^\lambda=e^{i\lambda\gamma_5}\,\psi\nonumber\\
\bar\psi &\to& \bar\psi^\lambda=\bar\psi e^{i\lambda\gamma_5}\nonumber\\
A_\mu &\to& A_\mu^\lambda=A_\mu+\varepsilon_{\mu\nu}\partial_\nu\lambda
\nonumber\\
\varphi &\to& \varphi^\lambda=\varphi\nonumber\\
\phi &\to& \phi^\lambda=\phi-\frac{2}{\beta}\, \lambda\ , \end{aligned}$$ the Lagrangian and the measure are not invariant. They transform according to $$\begin{aligned}
{\cal L}^\lambda &=&{\cal L}+\frac{1}{2\pi}\, (\partial_\mu\lambda)^2
+\frac{1}{\pi}\, \varepsilon_{\mu\nu}A_\mu\partial_\nu\lambda\nonumber\\
({\cal D}\bar\psi{\cal D}\psi)^\lambda&=&{\cal D}\bar\psi{\cal D}\psi\,
\exp \int d^2x\left[ \frac{1}{2\pi}(\partial_\mu\lambda)^2+
\frac{1}{\pi}\varepsilon_{\mu\nu}A_\mu\partial_\nu\lambda\right]\nonumber\\
({\cal D}A{\cal D}\varphi{\cal D}\phi)^\lambda&=&
{\cal D}A{\cal D}\varphi{\cal D}\phi\ .\end{aligned}$$ The transformation law of ${\cal D}\bar\psi{\cal D}\psi$ is the well known axial anomaly calculated by Fujikawa [@fujikawa]. Taken together, the effects of the non invariant terms in the Lagrangian and the measure cancel, and we are left with an invariant generating functional. $A_\mu$ is an auxilliary field and integrating it out, we find $$\begin{aligned}
\label{pomocno}
{\cal L} &=& \bar\psi\gamma_\mu\partial_\mu\psi+
\frac{1}{2}\,g\, \left(\bar\psi\gamma_\mu\psi\right)^2+
m\,\bar\psi\psi\cos \beta\phi+im\,\bar\psi\gamma_5\psi\sin \beta\phi+
\nonumber \\
&&+i\bar\psi\gamma_\mu\psi\,\partial_\mu\varphi+\frac{1}{2}\,
(\partial_\mu\phi)^2-\frac{2\pi}{\beta}\,
\varepsilon_{\mu\nu}\,\bar\psi\gamma_\mu\psi\,\partial_\nu\phi\ .\end{aligned}$$ Now we fix local vector and axial-vector symmetry of $Z$. The first way to do it is to set $\varphi=0,\enspace \phi=0$. Then (\[pomocno\]) becomes $${\cal L}_\mathrm{MTM}=\bar\psi(\gamma_\mu\partial_\mu+m)\psi+
\frac{1}{2}\,g\left( \bar\psi\gamma_\mu\psi\right)^2\ .$$ This is the famous massive Thirring model [@thirring], a pure fermionic theory, equivalent to our starting model (\[pocetak\]).
A second way to gauge fix (\[pomocno\]) is to take $\psi_1^\dagger=\psi_1,\enspace
\psi_2^\dagger=\psi_2$, where $\psi=(\psi_1\, , \,\psi_2)^T $. We then have $\bar\psi\gamma_5
\psi=\bar\psi\gamma_\mu\psi=0$ and (\[pomocno\]) becomes $$\label{konformna}
\widetilde{\cal L}= \bar\psi\gamma_\mu\partial_\mu\psi+\frac{1}{2}\,
(\partial_\mu\phi)^2+m\,\bar\psi\psi\,\cos\beta\phi\ .$$ We haven’t obtained a purely bosonic theory despite having fixed both vector and axial-vector symmetries. However, we still have at our disposal an additional conformal symmetry. To see this it is easier look at the Lagrangian (\[konformna\]) in the operator formalism, with normal ordered operator fields. Conformal transformations are given by $z\to z'=f(z)$ and $\bar z\to \bar z'=\bar f(\bar z)$, where we have introduced complex coordinates $z=x_0+ix_1$ and $\bar z=x_0-ix_1$. Spinors transform according to $$\begin{aligned}
\label{spinori}
\psi_1(z)\to \psi_1'(z')&=&
\left(\frac{df}{dz}\right)^{-\frac{1}{2}}\psi_1(z)\nonumber\\
\psi_2(\bar z)\to \psi_2'(\bar z')&=&
\left(\frac{d\bar f}{d\bar z}\right)^{-\frac{1}{2}}\psi_2(\bar z)\ .\end{aligned}$$ The conformal weights can be read off directly from the correlators for spinor fields $\left\langle\psi_1(z)
\psi_1^\dagger(\zeta)\right\rangle=\frac{1}{2\pi}\,\frac{1}{z-\zeta}$, and similary for $\psi_2(\bar z)$. From (\[spinori\]) we easily find transformation laws of all fermionic terms, writing them in components. On the other hand, it is known [@klaiber] that the free correlator for a two dimensional massless scalar field is $$\label{korelator}
\left\langle\phi(x)\phi(y)\right\rangle=-\frac{1}{\beta^2}\,
\ln \mu^2(x-y)^2\ ,$$ where $\mu$ is an infra red regulator with dimension of mass. The scale $\mu$ plays a central role. It is, in fact, advantageous to write the scalar field as $\phi(x|\mu)$. At the end of all calculations we take the $\mu \to 0$ limit. From the above correlator we see that $\phi$ transforms in a complicated way under conformal transformations. However, as is well known, its derivative has a simple transformation law: $$\begin{aligned}
\partial_z\phi(x|\mu)&\to&
\partial'_z\phi'(x'|\mu)\,=\,\left(\frac{df}{dz}\right)^{-1}
\partial_z\phi(x|\mu)\nonumber \\
\partial_{\bar z}\phi(x|\mu)&\to&
\partial'_{\bar z}\phi'(x'|\mu)\,=\,\left(\frac{d\bar f}{d\bar z}\right)^{-1}
\partial_{\bar z}\phi(x|\mu)\ .\end{aligned}$$ Another set of objects made out of $\phi$ transforming in such a simple way are the normal ordered exponentials of $\phi$. Using the identity $\left\langle :e^A:\,:e^B:\right\rangle =e^{\left\langle
AB\right\rangle}$ which is valid when $[A,B]$ is a $c$-number, we have $\left\langle :e^{i\beta\phi(x|\mu)}:\,:e^{i\beta\phi(y|\mu)}:
\right\rangle =0$, and $\mu^2\left\langle :e^{i\beta\phi(x|\mu)}:\,
:e^{-i\beta\phi(y|\mu)}:\right\rangle = \frac{1}{(x-y)^2}$ in the $\mu\to 0$ limit. As a consequence, for the cosine we get $$\label{kosinus}
\mu\,:\cos\beta\phi(x|\mu):\,\to \,
\left(\frac{df}{dz}\right)^{-\frac{1}{2}}
\left(\frac{d\bar f}{d\bar z}\right)^{-\frac{1}{2}}
\mu\,:\cos\beta\phi(x|\mu):\ .$$ Therefore, the whole Lagrangian density transforms like $\widetilde{\cal L}'(x')=\left(\frac{df}{dz}\right)^{-1}
\left(\frac{d\bar f}{d\bar z}\right)^{-1} \widetilde{\cal L}(x)$ and, because of $d^2x'\equiv dz'd\bar z'=\frac{df}{dz}\, \frac{d\bar f}{d\bar z}
\, d^2x$, we have the conformal invariant quantum action $\int d^2x
\widetilde{\cal L}$. Fixing the conformal simmetry by $\psi_1=
\theta \left(\frac{df}{dz}\right)^{\frac{1}{2}}, \enspace \psi_2=\bar\theta
\left(\frac{d\bar f}{d\bar z}\right)^{\frac{1}{2}}$, where $\theta$ and $\bar\theta$ are Grassmann constants normalized by $\bar\theta\theta
=-\frac{i\alpha}{2m\beta^2}=$ Const, we find that $m\,\bar\psi\psi =
\frac{\alpha} {\beta^2}=$ Const. Then (\[konformna\]) becomes $${\cal L}_\mathrm{SG}=\frac{1}{2}\, (\partial_\mu\phi)^2+\frac{\alpha}{\beta^2}
\, \cos \beta\phi\ ,$$ where the free fermionic Lagrangian was integrated out. ${\cal L}_\mathrm{SG}$ is the well known sine-Gordon model, a pure bosonic theory.
In this paper we have re-derived Abelian bosonization results of Coleman [@coleman], Mandelstam [@mandelstam] and others [@dorn] - [@damgaard], concerning the equivalence between the massive Thirring and sine-Gordon models. Contrary to our derivation, all the previous results were perturbative (to all orders) in mass $m$. As we have seen, the central point in the above equivalence is the existence of [*two*]{} mass scales $m$ and $\mu$, and the fact that in (\[konformna\]) they enter solely through their ratio $\frac{m}{\mu}$.
[2]{}
A. Bogojevi' c, O. Mišković, B. Sazdović, [*“Abelian Bosonization, the Wess-Zumino Functional and Conformal Symmetry"*]{}, Institute of Physics preprint IP-HET-98/14 K. Fujikawa, Phys. Rev. [**D21**]{}, 2848 (1980) W. E. Thirring, Ann. Phys. [**3**]{}, 91 (1958) B. Klaiber, [*“Lectures in Theoretical Physics"*]{} [**10A**]{} 141-176 (1967) (Gordon and Breach, New York, 1968) S. Coleman, Phys. Rev. [**D11**]{}, 2088 (1975) S. Mandelstam, Phys. Rev. [**D11**]{}, 3026 (1975) H. Dorn, Phys. Lett. [**B167**]{}, 86 (1986) T. Ikehashi, Phys. Lett. [**B313**]{}, 103 (1993) C. M. Naón, Phys. Rev. [**D31**]{}, 2035 (1985) P. H. Damgaard, H. B. Nielsen, R. Sollacher, preprint CERN-TH-6460/92 (1992); preprint CERN-TH-6486/92 (1992); Phys. Lett. [**B296**]{}, 132 (1992)
|
---
abstract: 'An equation describing the evolution of phenotypic distribution is derived using methods developed in statistical physics. The equation is solved by using the singular perturbation method, and assuming that the number of bases in the genetic sequence is large. Applying the equation to the mutation-selection model by Eigen provides the critical mutation rate for the error catastrophe. Phenotypic fluctuation of clones (individuals sharing the same gene) is introduced into this evolution equation. With this formalism, it is found that the critical mutation rate is sometimes increased by the phenotypic fluctuations, i.e., noise can enhance robustness of a fitted state to mutation. Our formalism is systematic and general, while approximations to derive more tractable evolution equations are also discussed.'
address:
- ' $^1$ Complex Systems Biology Project, ERATO JST'
- ' $^2$ Department of Pure and Applied Sciences, University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo 153-8902, Japan '
author:
- 'Katsuhiko Sato$^1$ and Kunihiko Kaneko$^{1,2}$'
title: 'Evolution Equation of Phenotype Distribution: General Formulation and Application to Error Catastrophe'
---
Introduction
============
For decades, quantitative studies of evolution in laboratories have used bacteria and other microorganisms[@Lenski; @Lenski-Rose; @Kishony]. Changes in phenotypes, such as enzyme activity and gene expressions introduced by mutations in genes, are measured along with the changes in their population distribution in phenotypes [@Bactreia-many-generations; @Dekel-Alon; @Kashiwagi-Noumachi; @Ito]. Following such experimental advances, it is important to analyze the evolution equation of population distribution of concerned genotypes and phenotypes.
In general, fitness for reproduction is given by a phenotype, not directly by a genetic sequence. Here, we consider evolution in a fixed environment, so that the fitness is given as a fixed function of the phenotype. A phenotype is determined by mapping a genetic sequence. This phenotype is typically represented by a continuous (scalar) variable, such as enzyme activity, protein abundances, and body size. For studying the evolution of a phenotype, it is essential to establish a description of the distribution function for a continuous phenotypic variable, where the fitness for survival, given as a function of such a continuous variable, determines population distribution changes over generations.
However, since a gene is originally encoded on a base sequence (such as AGCTGCTT in DNA), it is represented by a symbol sequence of a large number of discrete elements. Mutation in a sequence is not originally represented by a continuous change. Since the fitness is given as a function of phenotype, we need to map base sequences of a large number of elements onto a continuous phenotypic variable $x$, where the fitness is represented as a function of $x$, instead of the base sequence itself. A theoretical technique and careful analysis are needed to project a discrete symbol sequence onto a continuous variable.
Mutation in a nucleotide sequence is random, and is represented by a stochastic process. Thus, a method of deriving a diffusion equation from a random walk is often applied. However, the selection process depends on the phenotype. If a phenotype is given as a function of a sequence, the fitness is represented by a continuous variable mapped from a base sequence. Since the population changes through the selection of fitness, the distribution of the phenotype changes accordingly. If the mapping to the phenotype variable is represented properly, the evolutionary process will be described by the dynamics of the distribution of the variable, akin to a Fokker-Planck equation.
In fact, there have been several approaches to representing the gene with a continuous variable [@footnote] . Kimura[@Kimura2] developed the population distribution of a continuous fitness. Also, for certain conditions, a Fokker-Planck type equation has been analyzed by Levine[@Levine]. Generalizing these studies provides a systematic derivation of an equation describing the evolution of the distribution of the phenotypic variable. We adopt selection-mutation models describing the molecular biological evolution discussed by Eigen[@Eigen], Kauffman[@kauffmann-book], and others, and take a continuum limit assuming that the number of bases $N$ in the genetic sequence is large, and derive the evolution equation systematically in terms of the expansion of $1/N$.
In particular, we refer to Eigen’s equation[@Eigen], originally introduced for the evolution of RNA, where the fitness is given as a function of a sequence. Mutation into a sequence is formulated by a master equation, which is transformed to a diffusion-like equation. With this representation, population dynamics over a large number of species is reduced to one simple integro-differential equation with one variable. Although the equation obtained is a non-linear equation for the distribution, we can adopt techniques developed in the analysis of the (linear) Fokker-Planck equation, such as the eigenfunction expansion and perturbation methods.
So far, we have assumed a fixed, unique mapping from a genotype to a phenotype. However, there are phenotypic fluctuations in individuals sharing the same genotype, which has recently been measured quantitatively as a stochastic gene expression [@Koshland; @Elowitz; @Kaern-Collins; @Collins; @Furusawa; @Ueda; @noise-review]. Relevance of such fluctuations to evolution has also been discussed[@SatoPNAS; @kaneko-book; @KKFurusawaJTB; @Ancel]. In this case, mapping from a gene gives the average of the phenotype, but phenotype of each individual fluctuates around the average. In the second part of the present paper, we introduce this isogenic phenotypic fluctuation into our evolution equation. Indeed, our framework of Fokker-Planck type equations is fitted to include such fluctuations, so that one can discuss the effect of isogenic phenotypic fluctuations on the evolution.
The outline of the present paper is as follows: We first establish a sequence model in section (\[31:setup-and-derivation\]). For deriving the evolution equation from the sequence model, we postulate the assumption that the transition probability of phenotype values is uniquely determined by the original phenotype value. The assumption may appear too demanding at a first sight, but we show that it is not unnatural from the viewpoint of evolutionary biology. In fact, most models studied so far satisfy this postulate. With this assumption, we derive a Fokker-Planck type equation of phenotypic distribution using the Kramers-Moyal expansion method from statistical physics[@vanKampen; @KuboMatsuoKitahara]. We discuss the validity of this expansion method to derive the equation, also from a biological point of view.
As an example of the application of our formulation, we study the Eigen’s model in section (\[20\]), and estimate the critical mutation rates at which error catastrophe occurs, using a singular perturbation method. In section (\[32:discussion\]), we discuss the range of the applicability of our method and discuss possible extensions to it.
Following the formulation and application of the Fokker-Planck type equations for evolution, we study the effect of isogenic phenotypic fluctuations. While fluctuation in the mapping from a genotype to phenotype modifies the fitness function in the equation, our formulation itself is applicable. We will also discuss how this fluctuation changes the conditions for the error catastrophe, by adopting Eigen’s model.
For concluding the paper, we discuss generality of our formulation, and the relevance of isogenic phenotypic fluctuation to evolution.
Derivation of evolution equation {#31:setup-and-derivation}
================================
We consider a population of individuals having a haploid genotype, which is encoded on a sequence consisting of $N$ sites (consider, for example, DNA or RNA). The gene is represented by this symbol sequence, which is assigned from a set of numbers, such as $\{-1,1\}$. This set of numbers is denoted by $S$. By denoting the state value of the $i$th site by $s_i$ ($ \in S$), the configuration of the sequence is represented by the ordered set $s=\{s_1,...,s_N\}$.
We assume that a scalar phenotype variable $x$ is assigned for each sequence $s$. This mapping from sequence to phenotype is given as function $x(s)$. Examples of the phenotype include the activity of some enzyme (protein), infection rate of bacteria virus, and replication rate of RNA. In general, the function $x(s)$ is a degenerate function, i.e., many different sequences are mapped onto the same phenotypic value $x$.
Each sequence is reproduced with rate $A$, which is assumed to depend only on the phenotypic value $x$, as $A(x)$; this assumption may be justified by choosing the phenotypic value $x$ to relate to the replication. For example, if a protein concerns with the metabolism of a replicating cell, its activity may affect the replication rate of the cell and of the protein itself.
In the replication of the sequence, mutation generally occurs; for simplicity, we consider only the substitution of $s(i)$. With a given constant mutation rate $\mu$ over all sites in the sequence, the state $s'_i$ of the daughter sequence is changed from $s_i$ of the mother sequence, where the value $s'_i$ is assigned from the members of the set $S$ with an equal probability. We call this type of mutation symmetric mutation [@Baake2]. The mutation is represented by the transition probability $Q(s \rightarrow s')$, from the mother $s$ to the daughter sequence $s'$. The probability $Q$ is uniquely determined from the sequence $s$, the mutation rate $\mu$, and the number of members of $S$. The setup so far is essentially the same as adopted by Eigen et al.[@Eigen], where the fitness is given as a function of the RNA sequence or DNA sequence of virus.
Now, we assume that the transition probability depends only on the phenotypic value $x$, i.e., the function $Q$ can be written in terms of a probability function $W$, which depends only on $x$, $W(x \rightarrow x')$, as $$\sum_{s' \in \{ s'|x'=x(s') \}} Q(s \rightarrow
s')=W(x(s) \rightarrow x') .
\label{1}$$
This assumption may appear too demanding. However, most models of sequence evolution somehow adopt this assumption. For example, in Eigen’s model, fitness is given as a function of the Hamming distance from a given optimal sequence. By assigning a phenotype $x$ as the Hamming distance, the above condition is satisfied (this will be discussed later). In Kauffman’s NK model, if we set $N \gg 1$, $K \gg 1$, and $K/N \ll 1$, this assumption is also satisfied (see Appendix \[29\]). For the RNA secondary structure model[@Waterman], this assumption seems to hold approximately, from statistical estimates through numerical simulations. Some simulations on a cell model with chemical reaction networks[@Furusawa; @Furusawa-KK] also support the assumption. In fact, a similar assumption has been made in evolution theory with a gene substitution process[@Gillespie; @Orr].
The validity of this assumption in experiments has to be confirmed. Consider a selection experiment to enhance some function through mutation, such as the evolution of a certain protein to enhance its activity[@Ito]. In this case, the assumption means that the activity distribution over the mutant proteins is statistically similar as long as they have the same activity, even though their mother protein sequences are different.
With the above setup, we consider the population of these sequences and their dynamics, allowing for overlap between generations, by taking a continuous-time model[@Baake2]. We do not consider the death rate of the sequence explicitly since its consideration introduces only an additional term, as will be shown later. The time-evolution equation of the probability distribution $\hat{P}(s,t)$ of the sequence $s$ is given by: $${\frac{\partial \hat{P}(s,t)}{\partial t}} = -\bar{A}(t) \hat{P}(s,t) +
\sum_{s'} A(x(s')) Q(s' \rightarrow s) \hat{P}(s',t), \label{3}$$ as specified by Eigen[@Eigen]. Here the quantity $\bar{A}(t)$ is the average fitness of the population at time $t$, defined by $\bar{A}(t)=\sum_{s} A(x(s)) \hat{P}(s,t)$ and $Q$ is the transition probability satisfying $\sum_{s} Q(s' \rightarrow s)=1$ for any $s'$.
According to the assumption (\[1\]), eq. (\[3\]) is transformed into the equation for $P(x,t)$, which is the probability distribution of the sequences having the phenotypic value $x$, defined by $P(x,t)=\sum_{ s \in \{s | x = x(s)\}} \hat{P}(s,t)$. The equation is given by $${\frac{\partial P(x,t)}{\partial t}} = - \bar{A}(t) P(x,t) + \sum_{x'} A(x')
W(x' \rightarrow x) P(x',t), \label{4}$$ where the function $W$ satisfies $$\sum_{x} W(x' \rightarrow x)=1 \qquad \mbox{for any $x'$,}
\label{2}$$ as shown.
Since $N$ is sufficiently large, the variable $x$ is regarded as a continuous variable. By using the Kramers-Moyal expansion[@vanKampen; @KuboMatsuoKitahara; @Haken], with the help of property (\[2\]), we obtain: $${\frac{\partial P(x,t)}{\partial t}} = (A(x)-\bar{A}(t)) P(x,t) + \sum_{n=1}^{{\infty}} \frac{(-1)^n }{n!}
{\frac{\partial {}^n}{\partial x^n}} m_n(x) A(x) P(x,t),
\label{5}$$ where $m_n(x)$ is the $n$th moment about the value $x$, defined by $m_n(x)= \int (x'-x)^n W(x \rightarrow x') dx' $.
Let us discuss the conditions for the convergence of expansion (\[5\]), without mathematical rigor. For convergence, it is natural to assume that the function $W(x'
\rightarrow x)$ decays sufficiently fast as $x$ gets far from $x'$, by the definition of the moment.
Here, the transition $W(x' \rightarrow x)$ is a result of $n$ point mutants of the original sequence $s'$ for $n=0,1,2,...,N$. Accordingly, we introduce a set of quantities, $w_n(x(s') \rightarrow x)$, as the fitness distribution of $n$ point mutants of the original sequence $s'$ (Naturally, $w_0(x(s') \rightarrow x)=\delta(x(s')-x)$, which does not contribute to the $n$th moment $m_n$ ($n \geq 1$)). Next, we introduce the probability $p_n$ that a daughter sequence is an $n$ point mutant $(n=0,1,2,...,N)$ from her mother sequence, which are determined only by the mutation rate $\mu$ and the sequence length $N$. Indeed, ${p_n}'s$ form a binomial distribution, characterized by $\mu$ and $N$.
In terms of the quantities $w_n$ and $p_n$, we are able to write down the transition probability $W$ as $$W(x(s') \rightarrow x)=\sum_{n=0}^{N} p_n w_n(x(s') \rightarrow x).\label{6}$$ Now, we discuss if $W(x(s') \rightarrow x)$ decays sufficiently fast with $|x(s')-x|$. First, we note that the width of the domain, in which $w_n(x(s') \rightarrow x)$ is not close to zero, increases with $n$ since $n$-point mutants involve increasing number of changes in the phenotype with larger values of $n$. Then, to satisfy the condition for $W(x(s') \rightarrow x)$, at least the single-point-mutant transition $w_1(x(s')
\rightarrow x)$ has to decay sufficiently fast with $|x(s')-x|$. In other words, the phenotypic value of a single-point mutant $s$ of the mother sequence $s'$ must not vary much from that of the original sequence, i.e., $|x(s')-x(s)|$ should not be large (“continuity condition").
In general, the domain $|x-x(s')|$, in which $w_n(x(s') \rightarrow x) \neq 0$, increases with $n$. On the other hand, the term $p_n$ decreases with $n$ and with the power of $\mu^n$. Hence, as long as the mutation rate is not large, the contribution of $w_n$ to $W$ is expected to decay with $n$. Thus, if the continuity condition with regards to a single-point mutant and a sufficiently low mutation rate are satisfied, the requirement on $W(x(s') \rightarrow x)$ should be fulfilled. Hence, the convergence of the expansion is expected.
Following the argument, we further restrict our study to the case with a small mutation rate $\mu$ such that $\mu N \ll 1$ holds. The transition probability $W$ in eq. (\[6\]) is written as $$W(x(s') \rightarrow x) \simeq (1-\mu N)
\delta(x(s')-x) + \mu N w_1(x(s') \rightarrow x),\label{7}$$ where we have used the property that ${p_n}'s$ form the binomial distribution characterized by $\mu$ and $N$. Introducing a new parameter, $\gamma$ ($\gamma=\mu N$), that gives the average of the number of changed sites at a single-point mutant, and using the transition probability (\[7\]), we obtain $${\frac{\partial P(x,t)}{\partial t}} =
(A(x)-\bar{A}(t)) P(x,t) + \gamma \sum_{n=1}^{{\infty}} \frac{(-1)^n
} {n!} {\frac{\partial {}^n}{\partial x^n}} m_n^{(1)}(x) A(x) P(x,t), \label{8}$$ where $m_n^{(1)}(x)$ is the $n$th moment of $w_{1}(x \rightarrow x')$, i.e., $m_n^{(1)}(x)=\int (x'-x)^n w_1(x \rightarrow x') dx'$.
When we stop the expansion at the second order, as is often adopted in statistical physics, we obtain $${\frac{\partial P(x,t)}{\partial t}} = (A(x)-\bar{A}(t)) P(x,t) + \gamma {\frac{\partial {}}{\partial x}}
\left[ - m_1^{(1)}(x) + \frac{1}{2} {\frac{\partial {}}{\partial x}} m_2^{(1)}(x) \right]
A(x) P(x,t). \label{9}$$ Eqs. (\[8\]) and (\[9\]) are basic equations for the evolution of distribution function. Eq. (\[9\]) is an approximation. However, it is often more tractable, with the help of techniques developed for solving the Fokker-Planck equation ( see Appendix \[10\] and [@PhysicalBiology]), while there is no established standard method for solving eq. (\[8\]).
At the boundary condition we naturally impose that there are no probability flux, which is given by $$\left.
\sum_{n=1}^{{\infty}} \frac{(-1)^n } {n!} {\frac{\partial {}^{(n-1)}}{\partial x^{(n-1)}}} m_n^{(1)}(x) A(x)
P(x,t) \right|_{x=x_1, x_2} =0, \label{26}$$ in the case of (\[8\]) and $$\left. \left[ - m_1^{(1)}(x) + \frac{1}{2} {\frac{\partial {}}{\partial x}} m_2^{(1)}(x) \right] A(x) P(x,t)
\right|_{x=x_1, x_2} =0
\label{27}$$ in the case of (\[9\]), where $x_1$ and $x_2$ are the values of the left and right boundaries, respectively.
Next, as an example of the application of our formula, we derive the evolution equation for Eigen’s model, and estimate the error threshold, with the help of a singular perturbation theory. Through this application, we can see the validity of eq. (\[9\]) as an approximation of eq. (\[8\]).
Two additional remarks: First, introduction of the death of individuals is rather straightforward. By including the death rate $D(x)$ into the evolution equation, the first term in eq. (\[8\]) (or eq. (\[9\])) is replaced by $\left[(A(x)-D(x))-(\bar{A}(t)-\bar{D}(t))\right] P(x,t)$, where $\bar{D}(t) \equiv \int
D(x) P(x,t) dx$. Second, instead of deriving each term in eq. (\[9\]) from microscopic models, it may be possible to adopt it as a phenomenological equation, with parameters (or functions) to be determined heuristically from experiments.
Application of error threshold in Eigen model {#20}
=============================================
In the Eigen model[@Eigen], the set $S$ of the site state values is given by $\{-1,1\}$, and the fitness (replication rate) of the sequence is given as a function of its Hamming distance from the target sequence $\{1,...,1\}$, i.e., the fitness of an individual sequence is given as a function of the number $n$ of the sites of the sequence having value $1$. Hence it is appropriate to define a phenotypic value $x$ in the Eigen model as a monotonic function of the number $n$; we determine it as $x=\frac{2n-N}{N}$, in the range $[-1,1]$. Accordingly, the replication rate $A$ of the sequence can be written as a function of $x$, i.e., $A(x)$; it is natural to postulate that $A$ is a non-negative and bounded function over the whole domain. If the sequence length $N$ is sufficiently large, the phenotypic variable $x$ can be regarded as a continuous variable, since the step size of $x$ ($\Delta x=\frac{2}{N}$) approaches 0 as $N$ goes to infinity.
In order to derive the evolution equation of form (\[8\]) corresponding to the Eigen model, we only need to know the function $w_1$ in that model. (Recall that in our formulation the mutation rate $\mu$ is assumed to be so small that only a single-point mutation is considered.) Due to the assumption of the symmetric mutation, this distribution function is obtained as $w_1(x \rightarrow x - \Delta x)=\frac{1+x}{2}$, $w_1(x \rightarrow x + \Delta x)=\frac{1-x}{2}$, and $w_1(x \rightarrow x')=0$ for any other $x'$. Accordingly, the $n$th moment is given by $m_n^{(1)}(x)= \frac{1+x}{2}
(-\Delta x)^n + \frac{1-x}{2} (\Delta x)^n$. Now, we obtain $${\frac{\partial P(x,t)}{\partial t}} = (A(x)-\bar{A}(t)) P(x,t) + \gamma \sum_{n=1}^{{\infty}} \frac{1} {n!}
{\frac{\partial {}^n}{\partial x^n}} \left[ \frac{1+x}{2} \left( \frac{2}{N} \right)^n + \frac{1-x}{2}
\left(- \frac{2}{N} \right)^n \right] A(x) P(x,t) \label{12}$$ where $\gamma=N \mu$, the mutation rate per sequence. When we ignore the moment terms higher than the second order, we have $${\frac{\partial P(x,t)}{\partial t}} = (A(x)-\bar{A}(t)) P(x,t) + \frac{2
\gamma}{N} {\frac{\partial }{\partial x}} \left[ x + \frac{1}{N}
{\frac{\partial }{\partial x}} \right]A(x) P(x,t).
\label{11}$$
In fact, if we focus on a change near $x\sim 0$ ( to be specific $x \sim O(1/\sqrt{N})$), the truncation of the expansion up to the second order is validated (Or equivalently, if we define $x'=(2n-N)/\sqrt{N}$ instead of $(2n-N)/N$, and expand eq.(3) by $1/\sqrt{N}$ instead of $1/N$, terms higher than the second order are negligible, as is also discussed in [@Levine]. However, in this case, the validity is restricted to $x' \sim O(1)$ (i.e., $(n-N/2) \sim O(1)$), which means $x\sim O(1/\sqrt{N})$ in the original variable).
Now, we solve the eq. (\[11\]) with a standard singular perturbation method (see Appendix \[10\]), and then return to eq. (\[12\]). According to the analysis in Appendix \[10\], the stationary solution of the equation of form (\[11\]) is given by the eigenfunction corresponding to the largest eigenvalue of the linear operator $L$ defined by $L=A(x)+2 \gamma \varepsilon {\frac{\partial }{\partial x}} \left[ x + \varepsilon {\frac{\partial }{\partial x}}
\right]A(x)$ with $\varepsilon=\frac{1}{N}$. Now we consider the eigenvalue problem $$A(x) P(x) + 2 \gamma \varepsilon {\frac{\partial }{\partial x}} \left[ x + \varepsilon
{\frac{\partial }{\partial x}} \right]A(x) P(x) = \lambda P(x) \label{23},$$ where $P(x) \geq 0$, with $\lambda$ to be determined.
Since $\varepsilon$ is very small (because $N$ is sufficiently large), a singular perturbation method, the WKB approximation[@Morse-book], is applied. Let us put $$P(x)=e^{\frac{1}{\varepsilon}\int_{x0}^{x} R(\varepsilon,x') dx'},
\label{28}$$ where $x_0$ is some constant and $R$ is a function of $\varepsilon$ and $x$, which is expanded with respect to $\varepsilon$ as $$R(\varepsilon,x)=R_0(x)+\varepsilon R_1(x)+\varepsilon^2 R_2(x)+...
\label{22}$$ Retaining only the zeroth order terms in $\varepsilon$ in eq. (\[23\]), we get $$A(x) + 2 \gamma \left[ x R_0(x) + R_0^2(x) \right]
A(x) =\lambda,
\label{24}$$ which is formally solved for $R_0$ as $R_0^{(\pm)}(x)= \frac{-x \pm \sqrt{g(x)}}{2}$ where $g(x)= x^2+\frac{2}{\gamma}
(\frac{\lambda}{A(x)}-1)$. Hence the general solution of eq. (\[23\]) up to the zeroth order in $\varepsilon$ is given by $P(x)=\alpha e^{\frac{1}{\varepsilon} \int_{x_0}^{x}
R_0^{(+)}(x')dx'} +\beta e^{\frac{1}{\varepsilon} \int_{x_0}^{x} R_0^{(-)}(x')dx'} $ with $\alpha$ and $\beta$ constants to be determined.
Now, recall the boundary conditions (\[27\]); $P$ has to take the two branches in $R_0$ as $ P(x)=\alpha e^{\frac{1}{\varepsilon} \int_{x_b}^{x} R_0^{(+)}(x')dx'} $ for $x < x_b$ and $ P(x)= \beta e^{\frac{1}{\varepsilon} \int_{x_b}^{x} R_0^{(-)}(x')dx'} $ for $x >
x_b$, where $x_b$ is defined as the value at which $g(x)$ has the minimum value. Next, from the continuity of $P$ at $x_b$, $\alpha=\beta$ follows, while from the continuity of ${\frac{\partial P}{\partial x}}$ at $x_b$, the function $g$ has to vanish at $x=x_b$. This requirement $g(x_b)=0$ determines the value of the unknown parameter $\lambda$ as $$\lambda=A(x_b) (1-\frac{\gamma}{2} {x_b}^2).
\label{18:approximated-eigenvalue}$$ From function $P$, we find that $P$ has its peak at the point $x=x_p$, where $R_0(x)$ vanishes, i.e., at $ A(x_p)=\lambda $. Then, $P(x)$ approaches $\delta(x-x_p)$ in the limit $\varepsilon \rightarrow +0$. These results are consistent with the requirement that the mean replication rate in the steady state be equal to the largest eigenvalue of the system (see Appendix \[10\]).
The stationary solution of eq.(\[12\]) is obtained by following the same procedure of singular perturbation. Consider the eigenvalue problem $$A(x)P(x)+\gamma \sum_{n=1}^{{\infty}} \frac{1} {n!}
{\frac{\partial {}^n}{\partial x^n}} \left[ \frac{1+x}{2} \left( 2
\varepsilon \right)^n + \frac{1-x}{2} \left(- 2
\varepsilon \right)^n \right] A(x) P(x)=\lambda
P(x). \label{25}$$ By putting $P(x)=e^{\frac{1}{\varepsilon}\int_{x0}^{x} R_0(x')
dx'}$ and taking only the zeroth order terms in $\varepsilon$, we obtain $$A(x)+\gamma \left[
\frac{1+x}{2} \left( e^{2 R_0(x)} - 1 \right) +
\frac{1-x}{2} \left( e^{-2 R_0(x)} -1 \right) \right]
A(x) =\lambda ,$$ which gives $$R_0^{(\pm)}(x)=\frac{1}{2} \log
\frac{1+\frac{1}{\gamma} (\frac{\lambda}{A(x)}-1) \pm
\sqrt{ \hat{g}(x)}}{1+x}$$ with $\hat{g}(x)=
(1+\frac{1}{\gamma} (\frac{\lambda}{A(x)}-1))^2-(1-x^2)
$.
By defining again the value $x=x_b$ at which $\hat{g}(x)$ takes the minimum, $P$ is represented as $ P(x)=\alpha e^{\frac{1}{\varepsilon} \int_{x_b}^{x} R_0^{(+)}(x')dx'} $ for $x < x_b$ and $ P(x)= \beta e^{\frac{1}{\varepsilon} \int_{x_b}^{x} R_0^{(-)}(x')dx'}
$ for $x > x_b$. The continuity of ${\frac{\partial P}{\partial x}}$ at $x=x_b$ requires $\hat{g}(x_b)=0$, which determines the value of $\lambda$ as $$\lambda=A(x_b) \left[1-\gamma
\left(1-\sqrt{1-{x_b}^2}\right) \right].
\label{15:more-exact-eigenvalue}$$ Again, $P(x)=\delta(x-x_p)$, in the limit $\varepsilon \rightarrow +0$, with $x_p$ given by the condition $A(x_p)=\lambda$. When $|x_b| \ll 1$, the form (\[15:more-exact-eigenvalue\]) approaches eq. (\[18:approximated-eigenvalue\]) asymptotically. This implies that the time evolution equation (\[8\]), if restricted to $|x| \ll 1$, is accurately approximated by eq.(\[9\]) that keeps the terms only up to the second moment.
Let us estimate the threshold mutation rate for error catastrophe. This error threshold is defined as the critical mutation rate $\gamma^{*}$ at which the peak position $x_p$ of the stationary distribution drops from $x_p\neq 0$ to $x_p =0$, with an increase of $\gamma$. We use the following procedure to obtain the critical value $\gamma^{*}$.
First consider an evaluation function whose form corresponds to that of eigenvalue (\[15:more-exact-eigenvalue\]) as $$f(x)=A(x) \left[1-\gamma \left(1-\sqrt{1-{x}^2}\right)
\right], \label{30:more-exact-evaluation-function}$$ and find the position at which the function $f(x)$ takes the maximum value. This procedure is equivalent to obtaining $x_b$ in the above analysis, since the relation $f(x)=\lambda-\frac{\gamma^2 A^2(x)}{\lambda-A(x) \left( 1-\gamma
\left(1+\sqrt{1-x^2}\right) \right)} \hat{g}(x)$ and the requirement on $x_b$ that $\hat{g}(x_b)=0$ and $\left. \frac{d \hat{g}(x)}{dx} \right|_{x=x_b}=0$ lead to $\left. \frac{df(x)}{dx} \right|_{x=x_b}=0$. Obviously, $x_b$ is given as a function of $\gamma$, thus, we denote it by $x_b(\gamma)$. The position $x_b$ determines the position $x_p$ of the stationary distribution through the relation $A(x_p)=\lambda=f(x_b)$ as in the above analysis. If $A$ has flat parts around $x=0$ and higher parts in the region ($x
> 0$), $x_p(\gamma)$ discontinuously changes from $x_p \neq 0$ to $x_p = 0$ at some critical mutation rate $\gamma^{*}$, when $\gamma$ increases from zero. A schematic illustration of this transition is given in Fig.(\[33:fig:schematical-explaination-of-estimation\]).
As a simple example of this estimate of error threshold, let us consider the case $$A(x)=1+A_0 \Theta(x-x_0),
\label{14:step}$$ with $A_0>0$ and $0<x_0<1$, and $\Theta$ as the Heaviside step function, defined as $\Theta(x)=0$ for $x <
0$ and $\Theta(x)=1$ for $x \geq 0$. According to the procedure given above, the critical mutation rate is straightforwardly obtained as $\gamma^{*}=\frac{A_0}{(1+A_0)\left(1-\sqrt{1-{x_0}^2}\right)}$, for $\gamma<\gamma^{*}$, $x_p=x_0$ and for $\gamma > \gamma^{*}$, $x_p=0$.
[*Remark*]{}
An exact transformation from the sequence model (Eigen model[@Eigen]) into a class of Ising models[@Leuthausser; @Baake] has recently been reported, such that the sequence model is treated analytically with methods developed in statistical physics. Rigorous estimation of the error threshold for various fitness landscapes[@Baake2; @Taiwan] and relaxation times of species distribution have been obtained[@Taiwan2]. In fact, our estimate (above) agrees with that given by their analysis.
Their method is indeed powerful when a microscopic model is prescribed in correspondence with a spin model. However, even if such microscopic model is not given, our formulation with a Fokker-Planck type equation will be applicable because it only requires estimation of moments in the fitness landscape. Alternatively, by giving a phenomenological model describing the fitness without microscopic process, it is possible to derive the evolution equation of population distribution. Hence, our formulation has a broad range of potential applications.
Consideration of phenotypic fluctuation
=======================================
In this section, we include the fluctuation in the mapping from genetic sequence to the phenotype into our formula, and examine how it influences the error catastrophe. We first explain the term “phenotypic fluctuation” briefly, and show that in its presence our formulation (\[8\]) remains valid by redefining the function $A(x)$. By applying the formulation, we study how the introduction of the phenotypic fluctuation changes the critical mutation rate $\gamma^{*}$ for the error catastrophe.
In general, even for individuals with identical gene sequences in a fixed environment, the phenotypic values are distributed. Some examples are the activities of proteins synthesized from the identical DNA [@Yang-et-al], the shapes of RNA molecules of identical sequences [@ancel-fontana], and the numbers of specific proteins for isogenic bacteria [@Elowitz; @Kaern-Collins; @Collins; @Furusawa]. Next, the phenotype $x$ from each individual with the sequence $s$ is distributed, which is denoted by $P_{phe}(s,x)$.
We assume that the form of distribution $P_{phe}$ is characterized only in terms of its mean value, i.e., the distributions ${P_{phe}}'s$ having the same mean value $X$ take the same form. By representing the mean value of the phenotype $x$ by $\bar{x}(s)$, the distribution $P_{phe}$ is written as $P_{phe}(s,x)=\hat{P}_{phe}(\bar{x}(s),x)$, where $\hat{P}_{phe}$ is a function of $\bar{x}$ and $x$, which is normalized with respect to $x$, i.e., satisfying $\int \hat{P}_{phe}(\bar{x},x) dx =1$.
In our formulation, the replication rate $A$ of the sequence with the phenotypic value $x$ is given by a function of phenotypic value $x$, denoted by $A(x)$. The mean replication rate $\hat{A}$ of the species $s$ is calculated by $$\hat{A}(\bar{x}(s))=\int \hat{P}_{phe}(\bar{x}(s),x) A(x)
dx. \label{phe:mean}$$
As in the case of (\[1\]), we assume that the transition probability from $s$ to $s'$ during the replication is represented only by its mean values $\bar{x}(s)$ and $\bar{x}(s')$, i.e., the transition probability function is written as $W(\bar{x}(s)
\rightarrow \bar{x}(s'))$. With this setup, the population dynamics of the whole sequences is represented in terms of the distribution of the mean value $\bar{x}$ only, so that we can use our formulation (\[8\]) even when the phenotypic fluctuation is taken into account; we need only replace the replication rate $A$ in (\[8\]) by the mean replication rate $\hat{A}$ obtained from eq. (\[phe:mean\]).
Now, we can study the influence of phenotypic fluctuation on the error threshold by taking the step fitness function $A(x)$ of eq. (\[14:step\]) and including the phenotypic fluctuation as given in eq.(\[phe:mean\] ). We consider a simple case where the form of $\hat{P}_{phe}$ is given by a constant function within a given range (we call this the piecewise flat case). Our aim is to illustrate the effect of the phenotypic fluctuation on the error threshold, so we evaluate the critical mutation rate $\gamma^{*}$ using the simpler form $f(x)=A(x)(1-\frac{\gamma}{2} x^2)$ from eq.(\[18:approximated-eigenvalue\]), while the use of the form (\[30:more-exact-evaluation-function\]) gives the same qualitative result. With this simpler evaluation function, the critical mutation rate $\gamma^{*}$ is given by $$\gamma_0^{*}=\frac{2 A_0}{(1+A_0) {x_0}^2},
\label{34:gamma-zero}$$ in the case without phenotypic fluctuation. Here we examine if this critical value $\gamma^{*}_0$ increases under isogenic phenotypic fluctuation.
We make two further technical assumptions in the following analysis: first we assume that $A_0$ in the form (\[14:step\]) is sufficiently small, so that the value of critical $\gamma^{*}$ is not large. Second, we extend the range of $x$ to $[-{\infty},{\infty}]$ for simplicity. This does not cause problems because we have set the range of $x_0$ to $(0,1)$. Hence, the stationary distribution has its peak around the range $0 \leq x < 1$; everywhere outside this range, the distribution vanishes.
We consider the case in which distribution $\hat{P}_{ phe }$ of the phenotype of the species $s$ is given by $$\hat{P}_{ phe }^{(F)}(\bar{x}(s), x) = \left\{
\begin{array}{ll} 0 & \quad \mbox{for $ x
<\bar{x}-\ell$}\\ \frac{1}{2 \ell} & \quad \mbox{for $ \bar{x}-\ell \leq x \leq \bar{x} +
\ell$}\\ 0 & \quad \mbox{for $ \bar{x} + \ell < x $, }
\end{array} \right.
\label{36:flat-case}$$ where $\ell$ gives the half-width of the distribution. ($(F)$ represents the piecewise-flat distribution case). Then, $\hat{A}$ is calculated by $$\hat{A}^{(F)}(x) = \left\{ \begin{array}{ll} 1 & \quad \mbox{for $x<x_0-\ell$}\\ 1+
\frac{A_0}{2 \ell} (x-(x_0-\ell)) & \quad \mbox{for $x_0-\ell \leq x \leq x_0 + \ell$}\\
1+ A_0 & \quad \mbox{for $x_0 + \ell < x$. } \end{array} \right.$$ An example of $\hat{A}^{(F)}(x)$ is shown in Fig. (\[35:fig:profile-of-A\]). The evaluation function $f$ in section (\[20\]) is given by $ f^{(F)}(x)=\hat{A}^{(F)}(x) (1-\frac{\gamma}{2}
x^2) $.
We study the case where the position ${x_b^{*}}^{(F)} (\equiv {x_b^{(F)}}(\gamma^{*}))$ is within the range $[x_0-\ell,x_0]$ because the profile of $\hat{A}^{(F)}$ shows that ${\gamma^{*}}^{(F)}$ is smaller than $\gamma^{*}_0$ if ${x_b^{*}}^{(F)}>x_0$. If $\frac{x_0}{2+A_0} \leq \ell < x_0$, the position ${x_b^{*}}^{(F)}$ is within the range $[x_0-\ell,x_0]$. In that case, ${\gamma^{*}}^{(F)}$ is given by ${\gamma^{*}}^{(F)}
\simeq \frac{A_0}{4 \ell (x_0-\ell)} $ to the first order of $A_0$. Comparing ${\gamma^{*}}^{(F)}$ with $\gamma^{*}_0$ in (\[34:gamma-zero\]), we conclude that ${\gamma^{*}}^{(F)} < \gamma^{*}_0$ for $ 0 <\ell<\frac{2+\sqrt{2}}{4} x_0 $, and ${\gamma^{*}}^{(F)} > \gamma^{*}_0$ for $ \frac{2+\sqrt{2}}{4} x_0 <\ell< x_0$. Hence, when the half width $\ell$ of the distribution $P_{phe}$ is within the range $(\frac{2+\sqrt{2}}{4} x_0,x_0)$, the critical mutation rate for the error catastrophe threshold is increased. In other words, the isogenic phenotypic fluctuation increases the robustness of high fitness state against mutation.
We also studied the case in which $ \hat{P}_{ phe } (\bar{x}, x)$ decreases linearly around its peak, i.e., with a triangular form. In this case, the phenotypic fluctuation decreases the critical mutation rate as long as $A_0$ is small, while it can increase for sufficiently large values of $A_0$, for a certain range of the values of width of phenotypic fluctuation.
Discussion {#32:discussion}
==========
In the present paper, we have presented a general formulation to describe the evolution of phenotype distribution. A partial differential equation describing the temporal evolution of phenotype distribution is presented with a self-consistently determined growth term. Once a microscopic model is provided, each term in this evolution equation is explicitly determined so that one can derive the evolution of phenotype distribution straightforwardly. This eq. (\[8\]) is obtained as a result of Kramers-Moyal expansion, which includes infinite order of derivatives. However, this expansion is often summed to a single term in the large number limit of base sequences, with the aid of singular perturbation.
If the value of a phenotype variable $|x|$ is much smaller than unity (which is the maximal possible value giving rise to the fittest state), the terms higher than the second order can be neglected, so that a Fokker-Planck type equation with a self-consistent growth term is derived. The validity of this truncation is confirmed by putting $x'=(2n-N)/\sqrt{N}$ and verifying that the third or higher order moment is negligible compared with the second-order moment. Thus the equation up to its second order, (\[9\]), is relevant to analyzing the initial stage of evolution starting from a low-fitness value.
As a starting point for our formalism, we adopted eq. (\[3\]), which is called the “coupled” mutation-selection equation[@Hofbauer]. Although it is a natural and general choice for studying the evolution, a simpler and approximate form may be used if the mutation rate and the selection pressure are sufficiently small. This form given by ${\frac{\partial \hat{P}(s,t)}{\partial t}} = -\bar{A}(t) \hat{P}(s,t) + \sum_{s'} Q(s' \rightarrow s)
\hat{P}(s',t)$, is called the “parallel” mutation-selection equation[@kimura1970; @Akin]. It approaches the coupled mutation-selection equation (\[3\]), in the limits of small mutation rate and selection pressure, as shown in [@Hofbauer]. If we start from this approximate, parallel mutation-selection equation, and follow the procedure presented in this paper, we obtain ${\frac{\partial P(x,t)}{\partial t}} =
(A(x)-\bar{A}(t)) P(x,t) + \gamma {\frac{\partial {}}{\partial x}} \left[ - m_1^{(1)}(x) + \frac{1}{2}
{\frac{\partial {}}{\partial x}} m_2^{(1)}(x) \right] P(x,t)$.
In general, this equation is more tractable than eq. (\[9\]), as the techniques developed in Fokker-Planck equations are straightforwardly applied as discussed in [@PhysicalBiology], and it is also useful in describing of evolution. Setting $A(x)=x^2$ and replacing $m_1^{(1)}$ and $m_2^{(1)}$ with some constants, the equation is reduced to that introduced by Kimura[@Kimura2]; while setting $A(x)=x$, $m_1^{(1)}(x)
\propto x$, and replacing $m_2^{(1)}$ with some constant derives the equation by Levine[@Levine]. Because our formalism is general, these earlier studies are derived by approximating our evolution equation suitably.
Besides the generality, another merit of our formulation lies in its use of the phenotype as a variable describing the distribution, rather than the fitness (as adopted by Kimura). Whereas the phenotype is an inherent variable directly mapped from the genetic sequence, the fitness is a function of the phenotype and environment, and strongly influenced by environmental conditions. The evaluation of the transition matrix by mutation in eq.(\[8\]) would be more complicated if we used the fitness as a variable, due to crucial dependence of fitness values on the environmental conditions. In the formalism by phenotype distribution, environmental change is feasible by changing the growth term $A(x)$ accordingly. Our formalism does include the fitness-based equation as a special case, by setting $A(x)=x$.
Another merit in our formulation is that it easily takes isogenic phenotypic fluctuation into account without changing the form of the equation, but only by modifying $A(x)$. By applying this equation, we obtained the influence of isogenic phenotype fluctuations on error catastrophe. The critical mutation rate for the error catastrophe increases because of the fluctuation, in a certain case. This implies that the fluctuation can enhance the robustness of a high-fitness state against mutation.
In fact, the relevance of isogenic phenotypic fluctuations on evolution has been recently proposed[@SatoPNAS; @kaneko-book; @KKFurusawaJTB], and change in phenotypic fluctuation through evolution has been experimentally verified[@Ito; @SatoPNAS]. In general, phenotypic fluctuations and a mutation-selection process for artificial evolution have been extensively studied recently. The present formulation will be useful in analysing such experimental data, as well as in elucidating the relevance of phenotypic fluctuations to evolution.
[**Figures**]{}
The authors would like to thank P. Marcq, S. Sasa, and T. Yomo for useful discussion.
Estimation of the transition probability in the NK model {#29}
========================================================
In the NK model[@well-written-NK; @kauffmann-book], the fitness $f$ of a sequence $s$ is given by $$f(s)=\frac{1}{N} \sum_{i=1}^{N} \omega_i(s) ,$$ where $\omega_i$ is the contribution of the $i$th site to the fitness, which is a function of $s_i$ and the state values of other $K$ sites. The function $\omega_i$ takes a value chosen uniformly from $[0,1]$ at random. We assume that the phenotype $x$ of the sequence $s$ is given by $x=f(s)$.
When $N \gg 1$, $K \gg 1$, and $K/N \ll 1$, the phenotype distribution of mutants of a given sequence $s$ (whose phenotype is $x$) is characterized only by the phenotype $x$ (without the need to specify the sequence $s$). For showing this, we first examine the one-point mutant case.
We consider the “number of changed sites” of sites at which $\omega's$ are changed due to a single-point mutation. By assuming that the average number of changed sites is $K$, the distribution of the number of changed sites $n$, denoted by $P_{site}(n)$, is approximately given by $$P_{site}(n) \simeq e^{-\frac{(n-K)^2}{2 K }},
\label{1:appen:site}$$ with the help of the limiting form of binomial distribution. Here, we have omitted the normalization constant.
Next, we study the distribution of the difference between the phenotype $x$ of the original sequence and the phenotype $x'$ of its one-point mutant, given the number $n$ of changed sites of the single-point mutant. We denote the distribution as $P_{dif \!
f}(n;X)$, where $X=x'-x$. Here the average of $x'$ is $x(N-n)/N$, since $(N-n)$ sites are unchanged. Thus, according to the central limit theorem, the distribution is estimated as $$P_{dif \! f}(n;X) \simeq \exp \left[ {-\frac{(X+\frac{n}{N}
x)^2}{2 n \frac{\sigma^2}{N^2} }} \right] ,
\label{2:appen:diff}$$ where $\sigma^2$ is the variance of the distribution of the value of $\omega$. This variance is estimated from the probability distribution $P_{(s,\{\omega_i\})}(\omega)$ that the sequence $\omega$ is generated.. Although the explicit form of $P_{(s,\{\omega_i\})}$ is hard to obtain unless $\{\omega_i\}$ and $s$ are given, it is estimated by means of the “most probable distribution,” obtained as follows: Find the distribution that maximizes the evaluation function $S$ (called “entropy”) defined by $S=-\int_{0}^{1} P(\omega) \log P(\omega) d\omega$ under the conditions $\int_{0}^{1} P(\omega) d\omega=1$ and $\int_{0}^{1} \omega P(\omega)
d\omega=x$. Accordingly the variance $\sigma^2$ may depend on $x$.
Combining these distributions (\[1:appen:site\]) and (\[2:appen:diff\]) gives the distribution of $X$ without constraint on the number of changed sites: $$P(X) = \sum_{n=1}^{N} P_{site}(n) P_{dif \! f}(n;X) \simeq \exp \left[
{-\frac{(X+\frac{K}{N} x)^2}{2 K \frac{(\sigma^2+x^2)}{N^2}}} \right] .$$ This result indicates that the phenotype distribution of single-point mutants from the original sequence $s$ having the phenotype $x$ is characterized by its phenotype $x$ only; $s$ is not necessary. Similarly, one can show that phenotype distribution of $n$-point mutants is also characterized only by $x$. Hence, the transition probability in the NK model is described only in terms of the phenotypes of the sequences, when $N \gg 1$, $K \gg 1$, and $K/N \ll 1$.
Mathematical structure of the equation of form (\[9\]) {#10}
=======================================================
We first rewrite eq. (\[9\]) as $${\frac{\partial P(x,t)}{\partial t}} = -\bar{A}(t) P(x,t) + L(x) P(x,t),
\label{17}$$ where $L$ is a linear operator, defined by $L(x)=A(x) +{\frac{\partial {}}{\partial x}} f(x) +
{\frac{\partial {}^2}{\partial x^2}} g(x) $ with $f(x)= - \gamma m_1^{(1)}(x) A(x)$ and $g(x)=
\frac{\gamma}{2} m_2^{(1)}(x) A(x)$.
The linear operator $L$ is transformed to an Hermite operator using variable transformations (see below) so that $L$ is represented by a complete set of eigenfunctions and corresponding eigenvalues, which are denoted by $\{\phi_i(x)\}$ and $\{\lambda_i\}$ ($i=0,1,2,...$), respectively. Eigenvalues are real and not degenerated, so that they are arranged as $\lambda_0 > \lambda_1 > \lambda_2 >...$.
According to [@PhysicalBiology], $P(x,t)$ is expanded as $$P(x,t)=\sum_{i=0}^{{\infty}} a_i(t) \phi_i(x), \label{13}$$ where $a_i$ satisfies $$\frac{d a_i(t)}{dt}= a_i(t)
(\lambda_i-{\sum_{j=0}^{{\infty}}}' a_j(t) \lambda_j). \label{19}$$ The prime over the sum symbol indicates that the summation is taken except for those of noncontributing eigenfunctions as defined in [@PhysicalBiology].
Stationary solutions of eq. (\[19\]) are given by $\{ a_{k}=1$ and $a_i=0$ for $i \ne k
\}$. Among these stationary solutions, only the solution $\{a_0=1$ and $a_i=0$ for $i \ne
0\}$ is stable. Hence, the eigenfunction for the largest eigenvalue (the largest replication rate) gives the stationary distribution function. Now it is important to obtain eigenfunctions and eigenvalues of $L$, in particular the largest eigenvalue $\lambda_0$ and its corresponding eigenfunction $\phi_0$. Hence, we focus our attention on the eigenvalue problem $$\left[ A(x) + {\frac{\partial {}}{\partial x}} f(x) + {\frac{\partial {}^2}{\partial x^2}} g(x)
\right] P(x) =\lambda P(x), \label{16}$$ where $\lambda$ is a constant and P is a function of $x$.
We can transform eigenvalue problem (\[16\]) to a Schroedinger equation-type eigenvalue problem as follows: First we introduce a new variable $y$ related to $x$ as $y(x)=\int_{x_0}^{x} \sqrt{\frac{h}{g(x')}} dx'$ where $x_0$ and $h$ are constants. Next, we consider a new function $\Psi(y)$ related to $P(x)$ as $$\Psi(y)= \left. \sqrt{\frac{g(x)}{h}} e^{{\int_{y_0}^{y} \frac{\hat{f}(y')}{2 h} dy'}}
P(x) \right|_{x=x(y)}$$ where $y_0$ is some constant, $x(y)$ the inverse function of $y(x)$, and $\hat{f}$ a function of $y$ defined by $$\hat{f}(y)= \left. \sqrt{\frac{h}{g(x)}} (f(x)+\frac{1}{2} \frac{d
g(x)}{dx}) \right|_{x=x(y)} .$$
Using these new quantities $y$ and $\Psi$ and rewriting eigenvalue problem (\[16\]) suitably, we get $$\left[ V(y) + h {\frac{\partial {}^2}{\partial y^2}} \right] \Psi(y) =\lambda
\Psi(y), \label{21}$$ where $V(y)=\hat{A}(y) + \frac{\frac{d \hat{f} (y)}{d
y}}{2} - \frac{\hat{f}^2(y)}{4 h}$ with $\hat{A}(y)=A(x(y))$.
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---
abstract: 'I have used a sample of long Gamma Ray Bursts (GRBs) common to both [*Swift*]{} and [*Fermi*]{} to re-derive the parameters of the Yonetoku correlation. This allowed me to self-consistently estimate pseudo redshifts of all the bursts with unknown redshifts. This is the first time such a large sample of GRBs from these two instruments are used, both individually and in conjunction, to model the long GRB luminosity function. The GRB formation rate is modelled as the product of the cosmic star formation rate and a GRB formation efficiency for a given stellar mass. An exponential cut-off powerlaw luminosity function fits the data reasonably well, with $\nu = 0.6$ and $ L_b = 5.4 \times10^{52} \, \rm{erg.s^{-1}},$ and does not require a cosmological evolution. In the case of a broken powerlaw, it is required to incorporate a sharp evolution of the break given by $L_{b}\sim0.3\times10^{52}\left(1+z\right)^{2.90} \, \rm{erg.s^{-1}},$ and the GRB formation efficiency (degenerate up to a beaming factor of GRBs) decreases with redshift as $\propto\left(1+z\right)^{-0.80}.$ However it is not possible to distinguish between the two models. The derived models are then used as templates to predict the distribution of GRBs detectable by CZTI on board $\AS$, as a function of redshift and luminosity. This demonstrates that via a quick localization and redshift measurement of even a few CZTI GRBs, $\AS$ will help in improving the statistics of GRBs both typical and peculiar.'
author:
- |
Debdutta Paul[^1]\
Tata Institute of Fundamental Research, India
title: 'Modelling the luminosity function of long Gamma Ray Bursts using [*Swift*]{} and [*Fermi*]{}'
---
gamma ray burst: general – astronomical databases: miscellaneous – methods: statistical – cosmology: miscellaneous.
Introduction
============
For any detector of gamma ray bursts (GRBs), an interesting estimable quantity is the number of GRBs observed, as a function of measurable parameters. This depends on instrumental parameters like duration-of-operation, $T$ and field-of-view $\Delta \Omega$, as well as the observed GRB production-rate and the distribution over intrinsic properties of GRBs. Following the formalism outlined in @Tan_et_al.-2013-ApJL, let us assume that the rate of GRBs beamed towards an observer on earth from an infinitesimal co-moving volume $dV,$ is given by $\overset{.}{R}\left(z\right)\frac{dV}{1+z},$ where $z$ denotes the redshift, and the factor $(1+z)^{-1}$ takes care of the cosmological time dilation.
Most generally, the number of GRBs detected by the instrument in the luminosity ($L$) range $L_{1}$ to $L_{2}$ and redshift range $z_{1}$ to $z_{2}$ is given by,
$$N(L_{1},L_{2};z_{1},z_{2})=T\,\dfrac{\Delta\Omega}{4\pi}\,\intop_{max[L_1,\, L_c]}^{L_2}\Phi_z(L)dL\,\intop_{z_{1}}^{z_{2}}\dfrac{\overset{.}{R}(z)}{1+z}dV,\label{eq:definition_of_phi}$$
where $L_{c}$ denotes a lower-cutoff in the intrinsic luminosity of GRBs (see Section \[sec:The-estimated-luminosities\]). The function $\Phi_z(L)$ is formally called the ‘luminosity function’ (henceforth LF), having the units of $\rm{ ( erg.s^{-1} )^{-1} }, $ the subscript refering to an implicit dependence on the redshift. In view of the fact that GRBs are end-products of massive stars in galaxies, the GRB formation rate $\overset{.}{R}\left(z\right)$ can be written as $$\overset{.}{R}\left(z\right)=f_{B}C\,\overset{.}{\rho_{\star}},\label{eq:R_dot}$$
where $\overset{.}{\rho_{\star}}$ gives the cosmic star-formation rate (CSFR) in ${\rm M_{\odot}Gpc^{-3}yr^{-1}},$ $C$ gives the efficiency of GRB production given a certain stellar mass, in units of ${\rm M_{\odot}^{-1}},$ and $f_{B}$ encodes the beaming effect of the relativistic jets producing the burst. All of these terms may be functions of the redshift.
The dependence of the detected number distribution of a certain class of astrophysical object on its luminosity function, is quite general. It has been extensively studied in the context of galaxies, galaxy clusters (see @Galaxy_cluster_LF-2017-MNRAS and references therein), white dwarfs (see @White_dwarf_LF-2016-review for a recent review), quasars (see @Quasar_LF_in_UV-2017-MNRAS and references therein), high mass Xray binaries (see @High_mass_XRB_LF-2017-MNRAS and references therein) etc. The methodologies depend on the observational window available for the study of the particular objects of interest (e.g. while @Galaxy_LF_using_WISE-2017-AJ, @Galaxy_LF_in_Kband-2017-MNRAS, @Galaxy_nearby_LF-2017-ApJ etc. use the infrared bands to calculate the absolute magnitude of galaxies, use the optical B-band, and @Galaxy_LF_in_UV_at_Cosmic_High_Noon-2017-ApJ, @Galaxy_primeval_LF_in_UV-2017-arXiv, etc. use the UV bands), but the central theme is similar for all of the objects – to measure the intrinsic properties of the sources and statistically study their cosmological evolution. Moreover, the LF of the various objects are related to each other, making this a difficult quantity to measure. For example, the cosmic star-formation rate (CSFR) derived from the galaxy LF, and the GRB LF, are related via Equations \[eq:definition\_of\_phi\] and \[eq:R\_dot\]. This will be discussed in more details below.
The measurement of the redshift (hence distance) of a GRB is essential for measuring its intrinsic luminosity. In the era of the Burst and Transient Source Experiment (BATSE) on board the *Compton Gamma Ray Observatory* (CGRO), which detected around $2700$ GRBs in a span of $9$ years (approximately one GRB per day, see <https://heasarc.gsfc.nasa.gov/docs/cgro/batse/>), the measurement of redshift of a detected GRB depended on coincident detection by other instruments with greater localization capabilities. In 1997, the Italian-Dutch satellite BeppoSAX provided the redshift of a burst for the first time via afterglow observations, that of GRB970508 (see @Costa-1997-Natur, @Bloom_et_al.-1998-ApJ, @Fruchter_et_al.-2000-ApJ and references therein). However, the number of GRBs with redshifts measured by BeppoSAX remained only a handful over the years . The situation changed entirely with the advent of the Burst Alert Telescope (BAT) on board [*Swift*]{} [@Barthelmy_et_al._2005], launched in 2004. In addition to detecting roughly $1$ GRB every $3$ days, it has fast on board algorithms to localize the burst and follow it up swiftly with other on-board instruments, the X-Ray Telescope (XRT) and UltraViolet/Optical Telescope (UVOT), as well as other ground-based missions. This provides redshifts via emission lines, absorption lines and photometry of the host-galaxies and/or afterglow, for roughly $\frac{1}{3}^{{\rm rd}}$ of the [*Swift*]{} GRBs, making it possible to study the intrinsic properties of $\sim300$ GRBs till date (<https://swift.gsfc.nasa.gov/archive/grb_table/>).
@Yonetoku_et_al.-2004-ApJ used the measured redshift and spectral parameters of 12 *BeppoSax* GRBs from and an additional 11 GRBs detected by BATSE to derive the ‘Yonetoku correlation’ between the 1-sec peak luminosity and the spectral energy break in the source frame. Using this correlation, they estimated the ‘pseudo redshift’ of 689 BATSE long GRBs with unknown redshifts. Subsequently, they discussed the GRB formation rate and found that a constant LF does not fit the data.
@Daigne_et_al.-2006-MNRAS studied the logN-logP diagram of BATSE GRBs and the peak-energy distribution of bright BATSE and HETE-2 GRBs, as well as carried out extensive simulations for [*Swift*]{} GRBs and applied them to early [*Swift*]{} data to predict that long GRBs show significant cosmological evolution. @Salvaterra_et_al.-2007-ApJ and @Salvaterra_et_al.-2009-MNRAS investigated the peak-flux distribution of BATSE GRBs in different scenarios regarding the CSFR, the evolution of the GRB LF, and the metallicity of the GRB formation environments. They then compared the predicted peak-flux distribution of [*Swift*]{} GRBs primarily with $z>2$ with available data to conclude that the two satellites observe the same distribution of GRBs, the GRB LF shows significant cosmological evolution, and that the GRB formation is limited to low metallicity environments.
Since then, [*Swift*]{} GRBs with measured redshifts have been studied extensively to model the long GRB LF. To do this, directly inverted the observed luminosity-redshift relationship. @Cao_et_al.-2011-MNRAS carried out a phenomenological study of the observational biases on doing this, concluding that a broken powerlaw model of the long GRB LF is preferred, with pre and post break luminosity of $2.5\times10^{52}\,{\rm erg.s^{-1}}$ given by $1.72$ and $1.98$ respectively. They also point to the requirement of cosmological evolution of the LF at high metallicity environments. @Salvaterra_et_al.-2012-ApJ used a flux-complete sample of 58 [*Swift*]{} GRBs, with a redshift completeness of $90\%,$ to conclude that the broken powerlaw model is degenerate with the exponential cut-off powerlaw model. They also conclude that the GRB LF evolves with redshift, claiming that this conclusion is independent of the used model. @Robertson_and_Ellis-2012-ApJ however used a sample of 112 [*Swift*]{} GRBs to disfavour strong cosmological evolution of the formation rates of GRBs at $z<4,$ and concluded that the best-fit trend of the evolution strongly over-predicts the CSFR at $z>4$ when compared to UV-selected galaxies, alluding to unclear effects in addition to metallicity and the GRB formation environment. @Howell_et_al.-2014-MNRAS used two new observation-time relations and accounted for the complex triggering algorithm of [*Swift*]{}-BAT to reduce the degeneracy between the CSFR and the GRB LF. They satisfactorily fit a non-evolving LF with a powerlaw broken at $0.80\pm0.40\times10^{52}\,{\rm erg.s^{-1}}$ by pre and post indices of $0.95\pm0.09$ and $2.59\pm0.93$ respectively, while not entirely ruling out the possibility of an evolution in the break luminosity. @Petrosian_et_al.-2015-ApJ used a sample of 200 redshift measured [*Swift*]{} ** GRBs to carry out a non-parametric determination of the quantities related to the CSFR and the GRB LF. They claimed that the LF evolves strongly with $z,$ satisfactorily fit to a broken powerlaw model with pre and post break indices $1.5$ and $3.2$ respectively. They also estimated a GRB formation rate an order of magnitude higher than that expected from CSFR at redshifts $z<1,$ but matching with the CSFR at higher redshifts, contrary to all previous studies. On the other hand, @Deng_et_al.-2016-ApJ carried out an extensive study of the observational biases on the flux-truncation, trigger probability, redshift measurement, etc. with 258 [*Swift*]{} GRBs, concluding that it is not possible to argue for a robust cosmological evolution of the long GRB LF. The major limitations in the study of the GRB LF with [*Swift*]{} GRBs is the narrow energy band of BAT, which does not allow an accurate determination of the spectral parameters of the GRBs, since most of the bursts have spectral cutoffs at energies greater than the BAT high-energy threshold of $150$ keV. The conclusions of several of these studies are moreover in contradiction to each other. Regardless, several authors have discussed the implications of these results in the context of the structure of the GRB jets, for both BATSE (@Firmani_et_al.-2004-ApJ, @Guetta_Granot_Begelman-2005-ApJ, @Guetta_Piran_Waxman-2005-ApJ) and [*Swift*]{} GRBs [@Pescalli_et_al.-2015-MNRAS]. The redshift distribution of [*Swift*]{} bursts emerging from the study of the LFs, assuming different metallicity environments of GRBs, has been discussed by @Natarajan_et_al.-2005-MNRAS.
The two major limitations of studies that use GRBs with measured redshifts to constrain the GRB LF are: (1) the number of such available sources is rather small to tightly constrain the LF or statistically answer questions related to its evolution with redshift, leading to a variety of conclusions; (2) the measurement of redshifts always suffers from observational biases. To overcome these limitations, @Lloyd-Ronning_et_al.-2002-ApJ used 220 BATSE GRBs with redshifts inferred from an empirical luminosity-variability relation [@Fenimore_and_Ramirez-Ruiz-2000-arXiv]. This was extended by @Firmani_et_al.-2004-ApJ who carried out a joint fit of these GRBs along with the observed peak-flux distribution of more than 3300 *Ulysses*/BATSE GRBs. The conclusions always favoured a cosmological evolution of the GRB LF, although the data was not able to distinguish between single powerlaw and double-powerlaw models. @Shahmoradi-2013-ApJ proposed a multivariate log-normal distribution which he fitted for 2130 BATSE GRBs. on the other hand used an empirical lag-luminosity correlation to constrain the GRB LF and the CSFR independently from the study of 900 GRBs, favouring a single powerlaw fit to the GRB LF. Incidentally, similar methods have also been applied for galaxies to study the galaxy LF (see @Galaxy_LF_pseudo_redshift-2017-arXiv and references therein).
@Tan_et_al.-2013-ApJL uses the Yonetoku correlation to estimate the pseudo redshifts of 498 GRBs. This avoids the observational bias of the redshift measurements, and the flux truncation is corrected for during the modelling. First they test the correlation parameters by comparing the redshift distribution of 172 [*Swift*]{} GRB whose redshifts are known. They find that the best-fit parameters do not predict the redshift distribution of this sub-sample well. So they choose the values for which the distributions of known and pseudo redshifts of these GRBs are statistically similar. Since the [*Swift*]{} bandpass is too narrow to determine the spectral parameters of [*Swift*]{} GRBs, they use the @Butler_et_al.-2007-ApJ catalog in which the Band function [@Band_et_al.-1993-ApJ] parameters are estimated with a Bayesian technique. They conclude that the GRB LF is inconsistent with a simple powerlaw, demanding a fit with a broken powerlaw with pre and post break indices given by $0.8$ and $2.0$ respectively. In addition, the break itself evolves cosmologically as $L_{b}=1.2\times10^{51}\,{\rm erg.s^{-1}}(1+z)^{2},$ and the GRB formation rate evolves as $\propto(1+z)^{-1}$ in contradiction to all previous studies. They do not look into the accuracy of pseudo redshifts of the GRBs individually, and the analysis is entirely based only on a statistical comparison of the redshift distributions.
In the present work, I carry out a detailed study of the estimation of pseudo redshifts, using long GRBs that have firm redshift measures from [*Swift*]{}, as well as spectral parameter measurements from [*Fermi*]{}. The reason such a sample is useful is because it combines the wide spectral coverage of [*Fermi*]{} (which however does not provide redshift) with the redshift measurements from [*Swift*]{} ** follow-ups. This reduces the errors on the correlated quantities compared to the Butler catalog, which allows me to test the correlation itself, and also carefully examine the accuracy of the pseudo redshifts estimated from the correlation. I then use it to estimate the pseudo redshift of all [*Fermi*]{} and [*Swift*]{} GRBs, and place constraints on the long LF from a combined study of all these $2067$ GRBs. Previously, @Yu_et_al.-2015-ApJS has used a combined sample of 127 long GRBs with spectra from [*Fermi*]{} and Konus-*Wind*, and redshift from [*Swift*]{}, to independently model the CSFR and the GRB LF. They used the GRBs irrespective of whether the spectral peak is actually seen in the instrumental waveband. In the present work, I choose only those bursts in which the spectral peak is accurately modelled, to re-derive the parameters of the Yonetoku correlation, which is then used to include a much larger number of sources.
This paper is organized as follows. In Section \[sec:The-Yonetoku-correlation\], the Yonetoku correlation is re-derived. In Section \[sec:The-estimated-luminosities\], I describe the use of the correlation to generate pseudo redshifts of all remaining [*Fermi*]{} and [*Swift*]{} GRBs. The GRB LF is modelled in Section \[sec:Modeling-the-GRB-LF\], and in Section \[sec:Conclusions\], I present concluding remarks. Throughout this paper, a standard $\Lambda$CDM cosmology with $ H_0 = 72 \, \rm{km.s^{-1}.Mpc^{-1}} ,$ $ \Omega_m = 0.27 $ and $ \Omega_{\Lambda} = 0.73 $ has been assumed. All the scripts used and important databases generated in the work are publicly available at <https://github.com/DebduttaPaul/luminosity_function_of_LGRBs_using_Swift_and_Fermi>.
The Yonetoku correlation {#sec:The-Yonetoku-correlation}
========================
It is the correlation seen between the peak luminosity $L_{p}$ and the spectral energy break $E_{p}$ [@Band_et_al.-1993-ApJ] in the source frame.
The peak luminosity is defined as
$$L_{p}=P.\,4\pi d_{L}(z)^{2}\times k(z;\,{\rm spectrum}),\label{eq:Luminosity_formula}$$
where $P$ denotes the peak flux modelled by the Band function during the burst duration, given in ${\rm erg.cm^{-2}s^{-1},}$ and
$$k(z)=\dfrac{\int_{1\,{\rm keV}}^{10^{4}\,{\rm keV}}E.S(E)dE}{\int_{(1+z)E_{min}}^{(1+z)E_{max}}E.S(E)dE}
\label{eq:definition_of_k--Fermi}$$
for [*Fermi*]{} GRBs. In case of [*Swift*]{} bursts, where the peak flux is given in ${\rm ph.cm^{-2}s^{-1},}$
$$k(z)=\dfrac{\int_{1\,{\rm keV}}^{10^{4}\,{\rm keV}}E.S(E)dE}{\int_{(1+z)E_{min}}^{(1+z)E_{max}}S(E)dE}\,.\label{eq:definition_of_k---Swift}$$
![The $k$-correction factors for [*Fermi*]{} (left) and [*Swift*]{} (right) assuming average spectral parameters as derived from the sample of [*Fermi*]{} bursts: $<E_{p}>\,=181.3$ keV, $<\alpha>\,=-0.566,$ $<\beta>\,=-2.823.$ Since these average numbers are used, they do not include uncertainties. Note that the units of $k$ are different for [*Fermi*]{} and [*Swift*]{}, owing to Equations \[eq:definition\_of\_k–Fermi\] and \[eq:definition\_of\_k—Swift\]. One also notices the striking difference in the scales: whereas it is much close to unity for [*Fermi*]{} which is a wide-band detector, for [*Swift*]{} it is much larger for large redshifts than its value at the local universe, because of [*Swift*]{}’s limited energy-range. \[fig:k-correction\]](k_correction--Fermi.pdf "fig:")![The $k$-correction factors for [*Fermi*]{} (left) and [*Swift*]{} (right) assuming average spectral parameters as derived from the sample of [*Fermi*]{} bursts: $<E_{p}>\,=181.3$ keV, $<\alpha>\,=-0.566,$ $<\beta>\,=-2.823.$ Since these average numbers are used, they do not include uncertainties. Note that the units of $k$ are different for [*Fermi*]{} and [*Swift*]{}, owing to Equations \[eq:definition\_of\_k–Fermi\] and \[eq:definition\_of\_k—Swift\]. One also notices the striking difference in the scales: whereas it is much close to unity for [*Fermi*]{} which is a wide-band detector, for [*Swift*]{} it is much larger for large redshifts than its value at the local universe, because of [*Swift*]{}’s limited energy-range. \[fig:k-correction\]](k_correction--Swift.pdf "fig:")
To accurately derive the Yonetoku correlation, I first select the sub-sample of all [*Fermi*]{} and [*Swift*]{} bursts that have accurate estimations of the Band function [@Band_et_al.-1993-ApJ] parameters during the prompt emission, by [*Fermi*]{}, as well as accurate redshift measurement by [*Swift*]{} follow-up. Previous works have relied on modeling the spectral parameters by [*Swift*]{}, which suffers from the limited wavelength range of BAT. I use the accurate spectral parameters from [*Fermi*]{} instead, reducing the inaccuracy of the estimates of luminosity. Moreover, due to the same reason, I also notice that the $k$ correction is very close to unity for these bursts, unless the redshift is not too large (even for $z=10,$ the factor is less than $1.5$). This is illustrated in left of Fig \[fig:k-correction\]. In comparison, the k-correction of [*Swift*]{} is much larger for larger redshifts.
Selecting the common GRBs
-------------------------
![Despite the expected correlation between the [*Fermi*]{} and [*Swift*]{} $\T$s, some GRBs have systematically smaller $\T$ in [*Fermi*]{} than [*Swift*]{}.\[fig:T90\_comparison\]](comparing_T90s_of_common_GRBS--all.pdf)
The updated list of [*Fermi*]{} GRBs are selected from the [*Fermi*]{} catalog[^2] till GRB170501467, which includes 2070 GRBs. Firstly, I choose only those bursts from the catalog that have spectroscopically measured parameters for the GRB Band function, which includes 1729 such cases. Then only those with small errors on the spectral parameters are chosen. For this, it is noted that the primary parameter that drive the error estimates in the luminosity is the $\Ep.$ Choosing only those with errors less than $100\%$ in $E_p,$ 1566 bursts are retrieved.
The updated list of [*Swift*]{} GRBs are selected from the [*Swift*]{} catalog[^3] till GRB 170428A. The total number is 1021, out of which those with firm redshift measure are 312.
Since the nomenclature of [*Fermi*]{} and [*Swift*]{} GRBs are different, I select the following criteria for selecting the common GRBs. The difference between the trigger times are selected to be less than 10 minutes, and they are restricted to within $10^{\circ}\times10^{\circ}$ in RA and Dec for the two instruments. These numbers are empirically chosen, such that the common number of GRBs converge within a reasonable range of these cutoffs. This ensures I do not mistake two GRBs which are well separated in time and space to be the same GRB. Consequently I get 68 common GRBs. Applying the $\T$ criterion for identifying short versus long bursts [@Kouveliotou_et_al.-1993-ApJ] separately for the two missions, I note that 65 are long according to both [*Fermi*]{} and [*Swift*]{}, two are short in both, while only one is short only in [*Fermi*]{}, GRB090927422 ([*Fermi*]{} nomenclature). Its [*Fermi*]{}-$\T$ is $0.512\pm0.231$ sec while that of [*Swift*]{} is $2.2$ sec. [*Fermi*]{}-$\T$s are calculated at higher energies and hence known to be systematically smaller in a handful of GRBs. Fig \[fig:T90\_comparison\] illustrates this effect. Hence, I choose this as a long burst. Moreover, this also gives me confidence to make the distinction between long and short GRBs based on the [*Swift*]{}-criterion whenever it is available, i.e. for the other common GRBs (without redshift estimates from [*Swift*]{}). For the ones that are detected only by [*Fermi*]{}, I resort to applying the criterion based on the [*Fermi*]{}-$\T.$
Testing the correlation
-----------------------
![The Yonetoku correlation as seen from the data of 66 long GRBs with accurate Band parameters from [*Fermi*]{} and redshift measurement from [*Swift*]{}. The parameters of the correlation, from various studies, are over-plotted. I get the best-fit parameters of $A=4.783\pm1.026$ and $\eta=1.227\pm0.038$ for the the correlation defined in Equation \[eq:Yonetoku\_correlation\]. (A coloured version of this figure is available in the online journal.) \[fig:Yonetoku\_correlation\]](L_vs_Ep_1+z--correlations--my_bestfit.pdf)
![The redshift distribution for the 66 long GRBs chosen in our sample. *Left:* Individual comparison, the line indicating the expected relationship if the method was successful in predicting the pseudo redshifts accurately. *Right:* Statistical comparison: filled (cyan) histogram shows the observed distribution, hatched (black) histogram shows the pseudo redshift distribution. The small discrepancies, specially at higher redshifts, can be easily understood to be due to the errors on the pseudo redshifts. (A coloured version of this figure is available in the online journal.) \[fig:redshift\_distribution–bestfit\]](redshift_comparison.pdf "fig:")![The redshift distribution for the 66 long GRBs chosen in our sample. *Left:* Individual comparison, the line indicating the expected relationship if the method was successful in predicting the pseudo redshifts accurately. *Right:* Statistical comparison: filled (cyan) histogram shows the observed distribution, hatched (black) histogram shows the pseudo redshift distribution. The small discrepancies, specially at higher redshifts, can be easily understood to be due to the errors on the pseudo redshifts. (A coloured version of this figure is available in the online journal.) \[fig:redshift\_distribution–bestfit\]](distribution--my_bestfit.pdf "fig:")
When I plot $L_{p}$ versus $E_{p}(1+z)$ (the factor of $(1+z)$ takes care of the transformation into the co-moving frame) for all the 68 GRBs, I notice that the only burst with systematically smaller $L_{p}$ than the rest, is a short burst. Moreover, the sample of short GRBs with accurate spectral and redshift measures consists of only two cases. Hence, I do not attempt to study the correlation for short bursts separately. Moreover, I do not find any burst with luminosity lower than $ 10^{49} \rm{ \, erg.s^{-1} }, $ nor with $\T > 10^3 \, \rm{sec}, $ and hence I do not attempt to segregate the possible separate classes of low-luminosity long GRBs (see e.g. @Liang_et_al.-2007-ApJ), or ultra-long GRBs (e.g. @Levan_et_al.-2014-ApJ).
I retrieve the Yonetoku correlation from the $66$ long bursts to a high degree of confidence (a null-hypothesis of the Spearman correlation co-efficient of $0.623$ being false, ruled out with $p=2.368\times10^{-8}$), as shown in Fig \[fig:Yonetoku\_correlation\]. The errors on $L_{p}$ consist of errors in the flux as well as a conservative estimate of $70\%$ systematic error added to all bursts, to take care of the inaccuracy in the spectral parameters. These parameters are non-linear and hence the errors cannot be calculated directly. The systematic error is chosen conservatively, since the changes in the spectral parameters always affect the estimates in $L_{p}$ within a factor of $1.5$ even for the highest redshift bursts (see Fig \[fig:k-correction\] for reference). Also, if linear errors are propagated, the mean errors are again of the same order.
![A strong anti-correlation is seen against the measured redshift, for the ratio between the luminosities predicted (from the best-fit Yonetoku correlation) and that measured directly.\[fig:correlation\_of\_ratio\_with\_measured\_z\]](scatter_with_measured_redshift--my_bestfit.pdf)
For the Yonetoku correlation defined as $$\dfrac{L_{p}}{10^{52}{\rm \, erg.s^{-1}}}=A.\left[\dfrac{E_{p}}{{\rm MeV}}(1+z)\right]^{\eta},\label{eq:Yonetoku_correlation}$$ I get the best-fit parameters of $A=4.780\pm0.123$ and $\eta=1.229\pm0.037.$ The corresponding redshift distributions for the same GRBs, both statistically and individually, are shown in Fig \[fig:redshift\_distribution–bestfit\]. It is noticed that although the method does not reproduce the redshifts on an individual basis, it is statistically reliable. The pseudo and observed redshifts has a median ratio of $1.002 \pm 0.721$, i.e. the number is consistent with unity. This is not an effect of normalization, as all the normalization factors are defined explicitly via Equation \[eq:Yonetoku\_correlation\]. The reason of it being statistically reliable is that, the method produces the pseudo redshifts of a larger sample by assuming gross parameters from a smaller sample which is however unbiased. The systematic discrepancies for individual bursts can be ascribed to the scatter around the Yonetoku correlation, as discussed below.
@Tan_et_al.-2013-ApJL uses the set of parameters that reduce the discrepancy between the distributions of the observed and pseudo redshifts. This method tries to reconcile the problem by changing the parameters, while circumventing the actual problem, that the Yonetoku correlation is intrinsic scattered. This is best illustrated by the left panel of Fig \[fig:redshift\_distribution–bestfit\]. Moreover to verify their method, I run it on the current dataset, to find no global minimum of the discrepancy between the distributions. Hence, instead of modifying the parameters, I investigate the possible reasons for the scatter.
To investigate the presence of systematics in the discrepancy between the observed and the pseudo redshifts, I look for possible correlations of the ratio of the predicted luminosity from the Yonetoku correlation with the physical parameters $E_{p}(1+z)$ and the measured redshift. No correlation is found with the former, which confirms that the scatter in the Yonetoku correlation is intrinsic. However, I find a strong anti-correlation between the ratio and the measured redshift, as shown in Fig \[fig:correlation\_of\_ratio\_with\_measured\_z\], with a null hypothesis of the Spearman correlation co-efficient of $-0.533$ being false, ruled out with $p=4.056\times10^{-6}.$ The following qualitative hypothesis is proposed to explain this trend. The luminosities predicted by the best-fit parameters of the observed correlation are the better physical estimates of the luminosity, physically correlating with the spectral peak. The scatter in the observed correlation between the quantities $L_{p}$ and $E_{p}$ (in the source frame) is due to the inadequacy of the definition of the luminosity, which needs to be corrected for physical factors like the beaming of the burst and the burst environment. This explanation, however, is qualitative and requires an in-depth analysis via modeling the possible physical effects, not attempted in the current work.
The estimated luminosities {#sec:The-estimated-luminosities}
==========================
I next calculate the luminosities of all the [*Fermi*]{} detected bursts. This includes the 66 GRBs already used in Section \[sec:The-Yonetoku-correlation\], and the rest with spectral estimates from [*Fermi*]{} but without redshift estimates from [*Swift*]{} (irrespective of they are detected by [*Swift*]{}). For the latter cases, pseudo redshifts are predicted via the Yonetoku correlation, using [*Fermi*]{} flux and $k$-corrections. However the [*Swift*]{}-$\T$ criterion is applied to those with [*Swift*]{}-detections to distinguish between the short and long classes. For the GRBs with only [*Swift*]{} detections along with measured redshifts, I directly calculate the luminosity from the flux and redshifts from the same catalog, and the [*Swift*]{} $k$-corrections derived from the Band function parameters fixed at the average values of the [*Fermi*]{} distribution, given by $<E_{p}>\,=181.3$ keV, $<\alpha>\,=-0.566,$ $<\beta>\,=-2.823.$ It is to be noted that the $k$-correction is not sensitive to these parameters, as long as they are within a reasonable range (see e.g. @Preece_et_al.-2000-ApJS for the study of BATSE bursts). For those bursts detected only by [*Swift*]{} and further lacking redshift measurements, I estimate the pseudo redshifts via the [*Swift*]{} $k$-corrections and the Yonetoku correlation. Since $E_{p}$ features explicitly in the correlation, they are randomly sampled from the distribution of the [*Fermi*]{} bursts. The justification for such an approach is again that the [*Fermi*]{} being a wide-band detector, samples out all possible values of $E_{p}.$
In Fig \[fig:pseudo\_redshifts\_and\_luminosities\] is shown the $L$-$z$ distribution of all these cases. The instrumental sensitivities are given by Equation \[eq:Luminosity\_formula\] with $P=8.0\times10^{-8}\,{\rm erg.cm^{-2}.s^{-1}}$ for [*Fermi*]{} and $P=0.2\,{\rm ph.cm^{-2}.s^{-1}}$ for [*Swift*]{} (for a $100$ keV photon, this is equivalent to $3.2\times10^{-8}\,{\rm erg.cm^{-2}.s^{-1}}$). These numbers are chosen empirically from the respective catalogs, and describe the lower cutoff well. This places confidence on the used method and the estimated luminosities, and I proceed to use them for modeling the luminosity function (in Section \[sec:Modeling-the-GRB-LF\]). The slopes of the two correlations are $1.584\pm0.002$ for [*Fermi*]{}and $1.834\pm0.002$ for [*Swift*]{}. A few bursts (eight) fall below the sensitivity line, which may be ascribed to the fact that the spectral parameters are sampled randomly from the [*Fermi*]{} distribution, whereas the flux is measured by [*Swift*]{}; also, the $k$-correction increases sharply with $z$ for [*Swift*]{}. These bursts are removed from the sample for subsequent analysis.
type redshift measured number modelled as
---------------------------------- ------------------- -------- -------------
both [*Fermi*]{} and [*Swift*]{} yes 66
only [*Fermi*]{}, or both no 1278
only [*Swift*]{} no 499
only [*Swift*]{} yes 224
: The type of [*Fermi*]{} and [*Swift*]{} long GRBs used for modeling, and how they are referred. The total number is 2067.\[tab:GRB\_numbers\]
On an average, the pseudo redshifts have $ \sim 20 \% $ errors and the luminosities calculated from them have $ \sim 40 \% $ errors, after propagating errors in all the estimation steps. Theoretically, the redshifts and hence luminosities of the [*Swift*]{} bursts have much larger uncertainties, because their $E_p$s are not known, but this fact is ignored, to use these bursts in the statistical sense, laying no claim to the accuracy of the individual pseudo redshifts.
I also note that the distribution of pseudo redshifts and corresponding luminosities are relatively insensitive to the exact value of the parameters used for the Yonetoku correlation, as long as they are not significantly different from the best-fit estimates. The advantage of using this method lies in the fact that it evades the complex observational biases that plague and limit the study of redshift measured bursts. Also, it allows the model to take care of the instrumental thresholds while modeling the luminosity function via Equation \[eq:definition\_of\_phi\], to which I turn next.
Modeling the long GRB luminosity function {#sec:Modeling-the-GRB-LF}
=========================================
![The luminosity versus redshifts of all GRBs. The red points are for those with redshift measurements, while black points are for those whose pseudo redshifts are derived as described in the text. The dotted lines show the corresponding instrumental sensitivity limits. The errors are not shown for the purpose of better visibility. *Left:* For the GRBs that are detected by [*Fermi*]{}, irrespective of [*Swift*]{} detection, including those with known redshifts (the 66 cases considered to study the correlation). For all these bursts, the [*Fermi*]{} $k$-correction is used, whereas the [*Swift*]{}-$\T$ criterion is applied for those available. *Right:* For the bursts with detection only by [*Swift*]{}, including those with measured redshifts. See Table \[tab:GRB\_numbers\] for more details on the nomenclature. (A coloured version of this figure is available in the online journal.) \[fig:pseudo\_redshifts\_and\_luminosities\]](L_vs_z--Fermi_long_all.pdf "fig:")![The luminosity versus redshifts of all GRBs. The red points are for those with redshift measurements, while black points are for those whose pseudo redshifts are derived as described in the text. The dotted lines show the corresponding instrumental sensitivity limits. The errors are not shown for the purpose of better visibility. *Left:* For the GRBs that are detected by [*Fermi*]{}, irrespective of [*Swift*]{} detection, including those with known redshifts (the 66 cases considered to study the correlation). For all these bursts, the [*Fermi*]{} $k$-correction is used, whereas the [*Swift*]{}-$\T$ criterion is applied for those available. *Right:* For the bursts with detection only by [*Swift*]{}, including those with measured redshifts. See Table \[tab:GRB\_numbers\] for more details on the nomenclature. (A coloured version of this figure is available in the online journal.) \[fig:pseudo\_redshifts\_and\_luminosities\]](L_vs_z--Swift_long_all.pdf "fig:")
For the purpose of modeling the luminosity function, the GRBs that have pseudo redshift greater than $10$ are not considered (27). The final number of GRBs used are showed in Table \[tab:GRB\_numbers\]. Also, the modeling is carried out separately for [*Fermi*]{} and [*Swift*]{}, since the cut-off luminosities which feature in the model, via Equation \[eq:definition\_of\_phi\], are different for the two instruments, as discussed in Section \[sec:The-estimated-luminosities\]. For each instrument, I bin the data into three equipopulous redshift bins: $0<z<1.538,$ $1.538\leq z<2.657,$ $2.657\leq z<10.0$ for [*Fermi*]{}, and $0<z<1.809,$ $1.809\leq z<3.455,$ $3.455\leq z<10.0$ for [*Swift*]{}. It is to be noted that the errors on $N(L)$ are proportionally large, due to the large percentage errors on the derived luminosities, which are propagated across the bins.
In the most recent work on GRB LF, @Amaral-Rogers_et_al.-2017-MNRAS discusses various kinds of models. In particular, they test models in which the GRB formation rate is tied to a single population of progenitors via the cosmic star formation rate, another similar but distinct model where low and high luminosity GRBs are separated into two distinct classes, and a third kind where no assumption of the GRB formation rates are made. They conclude that a clear distinction between the three kinds of models cannot be asserted however. In the present work, I do not attempt to classify low and high luminosity GRBs for the reason that there is no clear evidence from the study in Section \[sec:The-Yonetoku-correlation\]. Moreover, I assume that the GRB formation rate is proportional to the star-formation rate, because after all it is massive stars formed in the galaxies that later end their lives in GRBs. There may be an additional dependence on the redshift: most generally represented via Equation \[eq:R\_dot\]. I take the cosmic star-formation rate $\overset{.}{\rho_{\star}}\left(z\right)$ from @Bouwens_et_al.-2015-ApJ (see references therein for the values at different redshifts), and model additional dependencies of the normalization, that is the GRB formation rate per unit cosmic star formation rate (or the GRB formation efficiency), as
$$f_{B}C(z)\propto\left(1+z\right)^{\epsilon}.\label{eq:fB.C_of_z}$$
It is to noted that the detailed processes involved in the formation of GRBs do not affect this treatment, which is similar to that followed by @Tan_et_al.-2013-ApJL. Within this framework, I attempt to fit two models: the exponential cut-off powerlaw (ECPL) model, described by
$$\Phi_z(L)=\Phi_{0}
\left(\frac{L}{L_{b}}\right)^{-\nu} \exp\left[- \left(\frac{L}{L_{b}}\right) \right],
\label{eq:The-ECPL-model}$$
and the broken powerlaw (BPL) model, given as
$$\Phi_z(L)=\Phi_{0}\begin{cases}
\left(\frac{L}{L_{b}}\right)^{-\nu_{1}}, & L\leq L_{b}\\
\left(\frac{L}{L_{b}}\right)^{-\nu_{2}}, & L>L_{b}.
\end{cases}
\label{eq:The-BPL-model}$$
Moreover, most generally the ‘break-luminosity’ $L_{b}$ is allowed to vary with redshift, as
$$L_{b}=L_{b,0}\left(1+z\right)^{\delta},\label{eq:evolution_of_break_luminosity}$$
with the quantity $L_{b,0}$ describing the normalization at zero redshift, and $\delta$ describing the evolution with redshift. The quantity $\Phi_{0}$ normalizes the probability density function $\Phi(L),$ and is an implicit function of the redshift $z$ via the dependence on $L_{b}.$ The models are then described by Equations \[eq:definition\_of\_phi\], \[eq:R\_dot\], \[eq:fB.C\_of\_z\], \[eq:The-ECPL-model\], \[eq:The-BPL-model\] and \[eq:evolution\_of\_break\_luminosity\], along with $\overset{.}{\rho_{\star}}\left(z\right)$ extracted numerically from @Bouwens_et_al.-2015-ApJ.
parameter present work @Amaral-Rogers_et_al.-2017-MNRAS
----------- ---------------------- ---------------------------------- --
$\nu$ $0.60 \pm 0.1 $ $0.71 \pm 0.07$
$L_{b,0}$ $5.40_{-1.5}^{+2.0}$ $4.02_{-0.96}^{+1.52}$
: The best-fit parameters for the ECPL model, as found by search in the $2$-dimensional space of $\nu$ and $L_{b,0}$. As a comparison, I show the best-fit parameters for the equivalent model of the recent works of @Amaral-Rogers_et_al.-2017-MNRAS. We see an overall agreement between the values. \[tab:ECPL\_model\_parameters\]
I look for the best-fit parameters of each model for [*Fermi*]{} and [*Swift*]{} GRBs separately, because they have different $L_{cut}\left(z\right)$ as shown in Fig \[fig:pseudo\_redshifts\_and\_luminosities\] (refer to Table \[tab:GRB\_numbers\] for the classes). For the case of the ECPL, it is noticed that any non-zero values of $\delta$ or $\chi$ (or both) decreases the quality of fit, for both [*Fermi*]{} and [*Swift*]{}. This allows me to decrease the parameter-space into a 2-dimensional space of $\nu$ and $L_{b,0}$ (which is equal to $L_b$ for $\delta = 0.)$ In the case of the BPL however, the data strongly requires the inclusion of a positive-definite $\delta$ and a negative-definite $\chi.$ It is to be noted that the ECPL has one parameter less than the BPL, but allows the break to vary naturally, explaining why the data requires the additional dependencies on the parameters $\delta$ and $\chi$ for the BPL model.
I search for the solutions by computing $d_{z}^{2}=\sum_{L}\left[N_{{\rm model}}\left(L,z\right)-N_{{\rm observed}}\left(L,z\right)\right]^{2}$ for each redshift bin, then evaluating the discrepancy $d^{2}=\sum_{z}d_{z}^{2},$ and finally looking for the model parameters that reduces $d^{2}.$ I optimize the search by first choosing a large grid of parameters with sufficiently small bins, and then gradually converge on the best-fit parameters by decreasing the search-space and bin-size at each run.
In the case of the ECPL, both the [*Fermi*]{} and [*Swift*]{} runs converge to similar values of parameters, and are consistent within the deduced errors. The fits are generally poorer for the latter case, and also because [*Swift*]{} detects a larger number of GRBs at higher redshifts due to its higher sensitivity compared to [*Fermi*]{}. This, however, is not directly taken into account in the modelling, being a limitation of the present work. This is because the exact mathematical form of the detection probabilities at various fluxes is not known. Hence, I tabulate the parameters from only the [*Fermi*]{} fits, in Table \[tab:ECPL\_model\_parameters\]. The data is generally over-fitted, with the the $\red$ for the two instruments being $0.116$ for [*Fermi*]{} and $0.539$ for [*Swift*]{}. This is because of the large number of bursts with similar luminosities, all with similarly large uncertainties.
In the case of the BPL, there is no oscillation of any of the five parameters, justifying that the solutions are global. However it is found that [*Fermi*]{} and [*Swift*]{} have different best-fits, significant differences being only in the related parameters $\nu_{2},$ $L_{b,0}$ and $\delta.$ The [*Swift*]{} solutions require extreme evolution of the break luminosity $\left(\delta=3.95\right),$ and raises suspicion of being an artefact of unaccounted systematics. To understand this, I model the detection probabilities of the two instruments by a simple flux powerlaw model, and plugging in the retrieved parameters of the two instruments, find that the difference can be explained by the variation of the detection probabilities with redshift and luminosity. On further investigation, I find that the [*Swift*]{} solutions are in fact degenerate with the [*Fermi*]{} solutions. The $d^{2}$ contours in the $L_{b,0}$-$\delta$ space have similar global shapes, and also behave similar locally around the [*Fermi*]{} solutions. Thus I conclude that the best-fit solutions obtained for [*Swift*]{} are driven by complications of its detection probability, and hence choose the [*Fermi*]{} best-fits as the accepted solutions, thus breaking the degeneracy. These are tabulated in Table \[tab:BPL\_model\_parameters\]. The corresponding fits for the two instruments are shown in Fig \[fig:bestfit\_BPL\_models\]. The larger proportional errors for [*Swift*]{} make the $\red$ comparable for the two instruments however, $0.362$ for [*Fermi*]{} and $0.364$ for [*Swift*]{}. This demonstrates that the use of [*Fermi*]{} bursts helps in solving the degeneracy of the parameter space of the model.
parameter present work @Amaral-Rogers_et_al.-2017-MNRAS @Tan_et_al.-2013-ApJL
------------ ------------------------ ---------------------------------- -----------------------
$\nu_{1}$ $0.65_{-0.3}^{+0.1}$ $0.69 \pm 0.09$ $0.8$
$\nu_{2}$ $3.10_{-0.4}^{+0.5}$ $1.88 \pm 0.25$ $2.0$
$L_{b,0}$ $0.30_{-0.1}^{+0.15}$ $0.15_{-0.09}^{+0.20}$ $0.12$
$\delta$ $2.90_{-0.50}^{+0.25}$ $2.04 \pm 0.45$ $2$
$\epsilon$ $-0.80_{-1.0}^{+0.75}$ - $-1.0$
Since the constant in the RHS of Equation \[eq:fB.C\_of\_z\] is not known a priori, it is calculated via the solutions of the models. It is known that for [*Fermi*]{}, $T\sim8.5$ yr and for [*Swift*]{}, $T\sim12$ yr. I assume $\frac{\Delta\Omega}{4\pi}\sim\frac{1}{3}$ for [*Fermi*]{} and $\frac{1}{10}$ for [*Swift*]{}, to get ratios of the observed and modelled numbers, which are converted to get
$$f_{B}C(0)=\begin{cases}
12.329\times10^{-8}\,{\rm M_{\odot}^{-1}}, & Fermi,\\
12.842\times10^{-8}\,{\rm M_{\odot}^{-1}}, & Swift.
\end{cases}\label{eq:fB.C_of_zero_ECPL}$$
for the ECPL model, and
$$f_{B}C(0)=\begin{cases}
7.498\times10^{-8}\,{\rm M_{\odot}^{-1}}, & Fermi,\\
8.200\times10^{-8}\,{\rm M_{\odot}^{-1}}, & Swift.
\end{cases}\label{eq:fB.C_of_zero_BPL}$$
for the BPL model.
These numbers are consistent with each other, and in rough agreement with those quoted by @Tan_et_al.-2013-ApJL.
The ECPL shows agreement with the most recent work of @Amaral-Rogers_et_al.-2017-MNRAS. The BPL model shows a sharp change at its break, which itself evolves quite strongly with redshift as $L_{b}\sim0.3\times10^{52}\left(1+z\right)^{2.90} \, \rm{erg.s^{-1}},$ in general agreement with @Amaral-Rogers_et_al.-2017-MNRAS. The GRB formation rate for a given star-formation rate decreases with increasing redshift as $f_{B}C \propto (1+z)^{-0.80}$ (the normalization is given by Equation \[eq:fB.C\_of\_zero\_BPL\]), in agreement to the reports of @Tan_et_al.-2013-ApJL. Whereas the ECPL automatically takes into account the variation of the break, this needs to be incorporated via strong evolutions with redshift in the BPL model. However, it is not possible to distinguish between the two models based on the fits. One of the reasons is that the data is generally over-fitted due to the large uncertainties, and another possible reason being that the discrepancies between data and model could be a result of the complex nature of detection probabilities of the instruments, which I have not attempted to model directly.
It is to be noted that the present work is empirical; it does not attempt to provide an understanding of the models used, nor of the derived values of the parameters. A thorough understanding of the observed GRB number distribution requires one to justify the models via the phenomenology of long GRBs, taking into consideration the beaming of GRB jets and the GRB formation environment. This the scope of future work.
Predictions for CZTI {#predictions-for-czti .unnumbered}
--------------------
The CZT Imager or CZTI [@Bhalerao_et_al.-2016-arXiv-JAA], on the Indian multi-wavelength observatory $\AS$ [@Rao_et_al.-2016-arXiv-Astrosat] is capable of detecting transients at wide off-axis angles, localizing them to a few degrees, and carrying out spectroscopic and polarization studies of GRBs, as demonstrated in @Rao_et_al.-2016-ApJ. A preliminary analysis done with the weakest GRB detected by CZTI suggests that it is at least as sensitive as [*Fermi*]{}, which detects roughly $3$ times the number of GRBs per year compared to [*Swift*]{}. Similar to [*Fermi*]{}, the CZTI is also a wide-field detector. Moreover, it covers a wide energy range, being the most sensitive between $50$ and $200$ keV. Thus, it is reasonable to assume that its GRB detection rate is at least comparable to that of [*Swift*]{}. Assuming this, I make predictions for CZTI over the redshift bins that were chosen for [*Fermi*]{}. The best-fit ECPL model predicts that CZTI should detect $150$ GRBs per year. The best-fit BPL model predicts detection-rate of around $140$ GRBs per year, with the [*Fermi*]{} equipopulous redshift bins almost equipopulous for CZTI as well. In $\sim1.3$ years of operation, $\sim 120$ GRBs has been detected by CZTI by triggered searches alone,[^4] however the exact number is subjective. An automated algorithm to detect GRBs is being thoroughly tested and implemented, the details of which will be reported elsewhere. In the view of this, the predictions point out the fact $\sim 20$-$30$ GRBs are yet undiscovered from the CZTI data. This is encouraging for the efforts on automatic detection, as well as that of the quick localization and follow-up, which will also be reported elsewhere.
Conclusions {#sec:Conclusions}
===========
Previously, BATSE and [*Swift*]{} GRBs have been used to constrain the GRB luminosity function. Only a few BATSE GRBs had redshift measurements, so indirect methods were used to study the luminosity function of these GRBs. On the other hand, about $30\%$ of the [*Swift*]{} GRBs have redshift measurements. However, the measurement of the spectral parameters are also crucial to the measurement of the luminosity, via the $k$-correction factor. Being limited in the energy coverage, estimates of the [*Swift*]{} spectral parameters have large uncertainties. Moreover, the number of [*Swift*]{} GRBs with redshift measures are not as large as the entire BATSE sample. [*Fermi*]{} is a GRB detector with large sky coverage, a detection rate roughly $3$ times more than [*Swift*]{}, and wide energy coverage, thus measuring the broad-band spectrum of a large fraction ($\sim75\%$) of the detected GRBs to sufficient accuracy. However, its poor localization capabilities makes it impossible to make [*Swift*]{}-like follow up observations, and hence the measurement of redshifts.
In this work, I show that one of the methods proposed to solve the absence of redshift measures for BATSE GRBs can be used self-consistently to estimate the luminosities of [*Swift*]{} and [*Fermi*]{} GRBs without redshift measurements. This method works on the premise that the ‘Yonetoku correlation’ is applicable to all GRBs. For this, I have first used the most updated common sample of $66$ long GRBs detected by these two instruments, to re-derive the parameters of this correlation. By a careful study of the discrepancies, I find a significant trend between the ratio of the observed and predicted luminosities with the measured redshift. The exact reason for this trend is not clear, but it highlights the fact that the weakness of the correlation is intrinsic, being driven by physical effects and not measurement uncertainties. I conclude that although the large scatter in the Yonetoku correlation rules out the possibility of using GRBs as distance-indicators, the statistical distribution of observed redshifts is reproduced well, and there is no need to modify the extraction of the correlation parameters as has been suggested previously [@Tan_et_al.-2013-ApJL].
Next, the method is shown to self-consistently predict ‘pseudo redshifts’ of all long GRBs without redshift measurements. This allows calculation of the luminosities of a total of $2067$ GRBs from these instruments, including the subsample (of $66$ bursts) that has direct measurements of both redshift and spectra. I then use this large sample to model the GRB luminosity function, and place constraints on two models. The GRB formation rate is assumed to be a product of the cosmic star formation rate and a GRB formation efficiency for a given stellar mass. Whereas an exponential cut-off powerlaw model does not require a cosmological evolution, a broken powerlaw model requires strong cosmological evolution of both the break as well as the GRB formation efficiency (degenerate upto the beaming factor of GRBs). This is the first time [*Fermi*]{} GRBs have been used independent of measured redshifts from [*Swift*]{} to study the long GRB luminosity function. Moreover, this is the first time such a large sample of [*Swift*]{} GRBs have been used. The use of the large sample of [*Fermi*]{} GRBs helps in placing sufficient confidence on the derived parameters of the broken powerlaw model, when [*Swift*]{} GRBs alone suffer from degeneracies and observational biases. Comparison with recent studies shows reasonable agreement for both the models, however it is not possible to distinguish between them.
@Amaral-Rogers_et_al.-2017-MNRAS has proposed on increasing the sample of GRBs by taking individual pulses of the same bursts as physically separate entities. In the future, perhaps a conglomeration of their method with the one here can be implemented to increase the sample size even further, to further test the parameters of the models and also carry out an in-depth analysis of the detection probabilities of the two instruments, which is presently quite a daunting task. This work also does not attempt to provide a physical understanding of the empirical models or the parameter values derived, which should be addressed in future works.
Finally, I have used the derived models as templates to make predictions about the detection rate of GRBs by CZTI on board $\AS.$ The predictions are encouraging for the ongoing efforts of this collaboration. The quick localization of the few bursts that are predicted to be detected only by CZTI can increase the GRB database even further, and reveal interesting answers about the GRB phenomenon in both the local and the distant universe.
Acknowledgements {#sec:Acknowledgements .unnumbered}
================
I sincerely thank my Ph.D. advisor A. R. Rao for helpful discussions and suggestions during the entire course of the work; Pawan Kumar for his comments on the manuscript, thus improving its quality; and the referee for his critical comments which immensely improved the quality of the work.
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[^1]: debdutta.paul@tifr.res.in
[^2]: <https://heasarc.gsfc.nasa.gov/W3Browse/fermi/fermigbrst.html>
[^3]: <https://swift.gsfc.nasa.gov/archive/grb_table/>
[^4]: See a comprehensive list at <http://astrosat.iucaa.in/czti/?q=grb>.
|
---
abstract: 'We investigate 738 complete genomes of viruses to detect the presence of short inverted repeats. The number of inverted repeats found is compared with the prediction obtained for a Bernoullian and for a Markovian control model. We find as a statistical regularity that the number of observed inverted repeats is often greater than the one expected in terms of a Bernoullian or Markovian model in several of the viruses and in almost all those with a genome longer than 30,000 bp.'
title: INVERTED REPEATS IN VIRAL GENOMES
---
Introduction
============
In the last few years there has been a progressively growing interest about the role of noncoding RNA (ncRNA) sequences producing functional RNA molecules having regulatory roles [@Eddy]. Prominent examples of these new regulatory RNA families are microRNA (miRNA) [@Lagos; @Lau; @Lee] and small interference RNA (siRNA) [@Hamilton; @Hutvagner]. Most of these structures shares the property of being characterized by a hairpin secondary structure. DNA or RNA short sequences that may be associated to RNA secondary structures are present in genomes of different species of phages, viruses, bacteria and eukaryotes. Indication about the potential existence of RNA secondary structures can be inferred throughout the detection of short pair sequences having the characteristic of inverted repeats (IRs) in the investigated genomes [@Schroth].
In the present study we systematically investigate all the complete genomes of viruses publicly available at on April 2003 to detect the presence of short IRs. The complete list containing the accession numbers of the investigated genome sequences is accessible at the web-page: . The number of IRs found for different classes of structures and for each set of control parameters is compared with the prediction obtained for a Bernoullian (i.e. indipendent and identically distributed nucleotide occurence) and for a Markovian control models.
With this technique we are able to evaluate the presence of a large number of IRs that cannot be explained in terms of simple control models therefore indicating their potential biological role. For each virus, the study is performed – (i) over the entire genome and (ii) in its coding and noncoding regions.
Viral Genomes
=============
During the past years about heigth hundred viral genomes have been completely sequenced. The complete sequence of their genomes is publicly accessible at specialized web pages. The database comprises different classes of viruses characterized by single stranded or double stranded nucleic acids, different infected organisms etc. In the present study we search the genomes for the presence of short subsequences, which might be associated with the existence of a secondary structure in regions of RNA originating from that subsequences.
Hairpin structures can occur when an IR is present in the nucleic acid sequence. For example, the DNA sequence 5’aGGAATCGATCTTaacgAAGATCGATTCCa3’ is a sequence having a sub-sequence GGAATCGATCTT which is the IR of AAGATCGATTCC. IRs of this type can form a hairpin having a stem of length 12 nucleotides and a loop (aacg) of length 4 nucleotide in the transcribed RNA. The number of IRs in complete genomes was first investigated with bioinformatics methods in long DNA sequences of eukaryotic (human and yeast) and bacterial ([*E.coli*]{}) DNA [@Schroth]. Successive studies have considered complete genomes such as the complete genomes of eubacterium [*Haemophilus influenzae*]{} [@Smith], archaebacterium [*Methanococcus jannaschii*]{} and cyanobacterium [*Synechocystis*]{} sp. PCC6803 [@Cox]. A Comparative genomic study of inverted repeats in prokaryotes has been investigated in Ref. [@Lillo].
An example of hairpin is shown in Fig. 1 for illustrative purposes. The figure caption describes in detail all the different part of such secondary structure. The example also presents a region of the stem with mismatches.
The simplest type of IR is the one without mismatches with the additional condition that the base pair before and after the stem are not complementary base pairs. Within this definition, Ref. [@Lillo] has shown that the number of IRs expected in a genome under the simplest assumption of a random Bernoullian DNA is given by the equation $$n_{ex}(\ell,m)=N (1-2P_aP_t-2P_cP_g)^2 (2P_aP_t+2P_cP_g)^{\ell},$$ where $N$ is the number of nucleotides in the genome sequence and $P_a$, $P_c$, $P_g$ and $P_t$ are the observed frequencies of nucleotides. Eq. (1) shows that the number of expected IRs is independent of $m$ whereas it depends on the CG content of the genome. The CG content vary considerably across different genomes and for long genomes also across different regions of the same genome.
In the present study we are interested in a wider class of IRs in which the presence of mismatches is allowed, because these are present in many IRs with known biological role. Specifically, we detect all the IRs present in the complete genomes of viruses characterized by a stem length $\ell$ ranging from 6 to 20 and a loop length $m$ ranging from 3 to 10. Inside the stem up to 2 mismatches are allowed provided that the number of links between complementary nucleotides inside the stem is always equal or larger than 6. With these constraints and with the additional requirement of avoiding to count the same substructure as a portion of differently classified structures we focus on three different classes of IRs defined as follows. Example of the three different classes of inverted repeats detected are shown in the schematic drawing of Fig. 2. The first one is characterized by a stem with no mismatches with the additional check that the three base pairs before and after the stem are not complementary base pairs (see scheme at Fig. 2(a)). The second one is a stem with one mismatch inside and with the additional check that the two base pairs before and after the stem are not complementary base pairs (Fig. 2(b)). Finally the third one is a stem with two mismatches inside and with the additional condition that the base pair before and after the stem are not complementary base pairs (Fig. 2(c)). In Ref. [@Lillo2] we generalize Eq. (1) for the three classes of structures considered here.
These equations allow therefore to perform a statistical test of the null hypothesis that the number of detected IRs is compatible with the assumption that viral genomes are Bernoullian symbolic sequences. This null hypothesis is equivalent to the assumption that IRs are observed in the genomes just by pure chance.
$\chi^2$ tests
==============
We perform several $\chi^2$ test [@Test]. In all cases the $\chi^2$ test is performed by comparing the number of IRs of the three classes described in Fig. 2 for different structures. Each structure is defined by a single value of ${\ell}$ ranging from 6 to 20. For each value of ${\ell}$, the loop length $m$ is varying from 3 to 10 and we verify the additional condition of observing at least 6 links within the stem. When the expected number of IR of a structure defined as before is larger than 5 we consider the structure of that kind as a degree of freedom of the $\chi^2$ test. When the number is smaller than 5 we aggregate different structures together until the number of expected IRs of the aggregated structures is larger than 5. In our procedure, the test cannot be performed for a certain number of viruses not reaching the threshold value of 5 for at least one type of the three investigated structures. Due to the wide range of lengths of the investigated genomes, the $\chi^2$ test is realized with a variable number of degree of freedom. We set the confidence threshold of the $\chi^2$ test at 0.05. This implies that the null hypothesis is rejected when the p-value is smaller than 0.05.
We have performed the $\chi^2$ test of the hypothesis that genomes are described by a Bernoullian sequence on 736 genomes of viruses. In this set, 324 genomes pass the test whereas in the remaining 412 (56 %) the Bernoullian hypothesis must be rejected. In 409 cases out of 412 the statistical test is not passed due to an excess of the number of detected IRs.
Having verified that the null hypothesis of IRs compatible with a Bernoullian DNA sequence is falsified in a large number of viruses of our set, we have repeated the same test in the coding and non coding regions of the genomes separately. Specifically, we label as coding regions all the regions of viral nucleic acids that are annotated in the databases as sequences coding for aminoacids in proteins. We label as noncoding nucleic acid regions the remaining regions of the genomes, therefore including nucleic acid regions producing different kind of RNA.
When we investigate coding regions we are able to perform the test on 719 viruses. Within this set 356 (50 %) viruses do not pass the test whereas the remaining 363 do pass it. In all 356 cases the statistical test is not passed due to an excess of the number of detected IRs. Moving to the noncoding regions, the test can be performed on 540 viruses. The lower number is due to the fact that noncoding regions are typically just 10 percent of the viral genomes and therefore the number of expected IRs under the Bernoullian hypotesis is roughly a tenth of the number expected for the coding regions. Within this set of 540 viruses, 165 (31 %) do not pass the test whereas the remaining 375 do pass it. In 162 cases out of 165 the statistical test is not passed due to an excess of the number of detected IRs with respect to the expected ones in term of the Bernoullian hypothesis.
We have therefore verified that the Bernoullian hypothesis saying IRs are present in viral genomes just by chance is not passed in a significant fraction of the investigated viral genomes. Moreover, the tests performed separately in coding and noncoding regions show that the excess of IRs is observed both in coding and noncoding regions.
To shed light on the parameters influencing the validation or falsification of the null hypothesis in our viral database, we have investigated the number of viruses passing the Bernoullian test conditioned to the length of the viral genome. In this investigation we aim to check the characteristic of viruses in the entire genome and both in the coding and noncoding regions. For this reason we have selected the viruses where tests are possible both in the coding and in the noncoding regions. This set is composed of 524 viruses. Results are summarized in panel a) of Fig. \[fig3\]. In the figure we show the value of the mean value $E[k|length]$ of the parameter $k$ conditioned to the length of the viral genome. The parameter $k$ assumes the value $k=1$ if the genome passes the Bernoullian test or the value $k=0$ in the opposite case.
The three curves obtained respectively for the entire genome (triangles), its coding (circles) and noncoding (squares) regions share the same global behavior. The percentage of viruses passing the test is higher for shorter viruses and this percentage is steeply declining for length longer than 30,000 nucleotides. The conditional mean value reaches approximately the zero value for the longest genomes. The figure also shows that at fixed value of the genome length the test is passed with a higher percentage in noncoding than in coding regions. Global exceedence of the number of detected IRs with respect to the number of IRs expected in terms of a Bernoullian hypothesis is detected in a large number of viruses both in coding and in noncoding regions. The exceedence is progressively more pronounced in longer viral genomes.
The results we have obtained are not due to the fact that Bernoullian hypothesis is a zero–order null hypothesis not well reproducing the statistical properties of DNA and RNA sequences. To prove this sentence we have repeated the test by comparing the number of detected IRs in each complete genome with the number of IRs observed in a numerically simulated genome generated according to a 1-order Markov chain. The expected number of IRs is computed by simulating 100 different realizations of each genome with the same measured Markovian transition matrix. We have first measured the empirical Markovian transition matrix and therefore performed the test in all viral genomes. The $\chi^2$ test with the Markovian hypothesys is performed on 738 viral genomes. Within this set of 738 viruses, 309 (42 %) do not pass the test whereas the remaining 429 do pass it. In 306 cases out of 309 the statistical test is not passed due to an excess of the number of detected IRs. This result shows that the Bernoullian assumption does not give results too different from the more accurate Markovian one. Panel b) of Fig. \[fig3\] shows the mean of the $k$ parameter conditioned to the length of viral genomes for the Bernoullian and the Markovian hypotheses. The investigated sets comprises 736 viruses for the Bernoullian case and 738 for the Markovian one. In the rest of this paper we will present results obtained by using as a testing assumption the Bernoullian model. This choice is motivated by the fact that under the Bernoullian assumption we are able to use an analytical estimation of the expected number of IRs for each kind of the investigated structure whereas the expected number of IRs under the Markovian assumption can be obtained only on a statistical basis by performing numerical simulations.
The next step is then to investigate how uniform is the localization of each kind of structure in each virus with respect to the kind of considered coding or noncoding region. This new test is done separately for each structure identified as type a, b or c structure according to the classification of Fig. \[fig2\] and by its stem length $\ell$. The null hypothesis is done in terms of a Bernoullian sequence to take advantage of the knowledge of analytical relations for the number of IRs expected for each structure. The test is devised to evaluate the degree of uniformity of the localization of observed structure of IRs inside each genome. The estimation of the number of expected IRs in coding $N^{cr}_{uni}(t,\ell)$ and noncoding $N^{ncr}_{uni}(t,\ell)$ regions under the assumption of not preferential (uniform) localization is done by using both the information about the total number of detected structures in the genome $N_{obs} (t,\ell)$ and the frequencies expected in terms of the Bernoullian hypothesis through the equations $$N^{cr}_{uni}(t,\ell)=
N_{obs} \frac{N^{cr}_{Ber} (t,\ell)}{N^{cr}_{Ber} (t,\ell)+N^{ncr}_{Ber}(t,\ell)}, \label{this}$$ and $$N^{ncr}_{uni}(t,\ell)=
N_{obs} \frac{N^{ncr}_{Ber} (t,\ell)}{N^{cr}_{Ber} (t,\ell)+N^{ncr}_{Ber}(t,\ell)}, \label{that}$$ where $t$ indicates the type of the structure (a, b or c) and $\ell \ge 6$ the stem length. These equations take into account the possibility that coding and noncoding regions might have different nucleotide frequencies.
We are able to perform the test in 1316 structures of 524 different viruses. Among these structures 1070 of 423 distinct viruses are consistent with the assumptions used in the test, which are (i) no preferential location between coding and noncoding and (ii) frequency of the IRs proportional to a Bernoullian expectation. Only in the remaining 246 (19 %) structures of 104 distinct viruses the statistical test is not passed and therefore in this restrict number of cases there might be a preferential location of these structure in one of the two consider regions. By looking at the specific contributions to the $\chi^2$ test we note that in 204 cases of the 246 considered the detected structures are preferentially located in the noncoding regions. At first sight this result can be seen as not consistent with the results summarized in Fig. 3a where the Bernoullian test is more easily passed in noncoding rather than coding regions. But indeed there is no contradiction. In fact, among the 104 viruses having the 246 structures that do not pass the Bernoullian test and show a preferential location in the noncoding regions, 79 of them are longer than 10,000 bp. Moreover, these 79 viruses contains 220 of the 246 considered structures. For viruses of such length the difference between the conditional mean value of the test indicator $k$ for coding and noncoding regions is much less pronounced than for viruses shorter than 10,000 bp and tend to disappear for longer viruses. The last $\chi^2$ test is preferentially passed by structures located in longest viruses.
Conclusions
===========
The present study has sistematically investigated the presence of IRs in 738 complete genomes of viruses. The investigated IRs can be both with a perfect matching of links within the stem and with the presence of mismatches up to a maximal value of 2. The empirically observed inverted repeats are compared with the values expected in terms of Bernoullian or Markovian model of genomes.
We find as a statistical regularity that the number of observed IRs are often greater than the one expected in terms of a Bernoullian or Markovian model in several of the viruses and in almost all those with a genome longer than 30,000 bp.
There is not a pronounced preferential location of these IRs in coding or noncoding regions in the majority of the considered viruses. This result is different from the one obtained by investigating complete genomes of bacteria [@Lillo] where a distinct preferential location of long structure of IRs in noncoding regions was observed.
We have therefore devised a methodology to detect sets of IRs whose existence cannot be explained in terms of simple random models of genome sequences. The selection of these sets of IRs may allow the future analysis of these secondary structure finalized to several distinct goals such as, for example, the detection of the degree of homology among them both in terms of sequence similarity or in terms of free energy of the secondary structures.
Acknowledgements {#acknowledgements .unnumbered}
================
This work has been funded by the MIUR project “A new approach to drug design. From quantum mechanics to the screening of anti-viral drugs". M.S. acknowledges a fellowship within this project. S.M. acknowledges support from FIRB project “Cellular Self-Organizing nets and chaotic nonlinear dynamics to model and control complex system".
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|
---
abstract: 'The method of potential solutions of Fokker-Planck equations is used to develop a transport equation for the joint probability of $N$ stochastic variables with Lochner’s generalized Dirichlet distribution [@Lochner_75] as its asymptotic solution. Individual samples of a discrete ensemble, obtained from the system of stochastic differential equations, equivalent to the Fokker-Planck equation developed here, satisfy a unit-sum constraint at all times and ensure a bounded sample space, similarly to the process developed in [@Bakosi_dir] for the Dirichlet distribution. Consequently, the generalized Dirichlet diffusion process may be used to represent realizations of a fluctuating ensemble of $N$ variables subject to a conservation principle. Compared to the Dirichlet distribution and process, the additional parameters of the generalized Dirichlet distribution allow a more general class of physical processes to be modeled with a more general covariance matrix.'
author:
- |
[J. Bakosi]{}, [J.R. Ristorcelli]{}\
`{jbakosi,jrrj}@lanl.gov`\
Los Alamos National Laboratory, Los Alamos, NM 87545, USA\
bibliography:
- 'jbakosi.bib'
title: |
[A stochastic diffusion process for Lochner’s generalized Dirichlet distribution]{}\
`LA-UR 13-21573`\
`Accepted for publication in Journal of Mathematical Physics, September 12, 2013`\
---
Introduction {#sec:introduction}
============
We develop a Fokker-Planck equation whose statistically stationary solution (i.e. invariant) is Lochner’s generalized Dirichlet distribution [@Lochner_75; @Connor_69; @Wong_98].
The (standard) Dirichlet distribution [@Johnson_60; @Mosimann_62; @Kotz_00] has been used to represent a set of non-negative fluctuating variables subject to a unit-sum requirement in a variety of fields, including evolutionary theory [@Pearson_1896], Bayesian statistics [@Paulino_95], geology [@Chayes_62; @Martin_65], forensics [@Lange_95], econometrics [@Gourieroux_06], turbulent combustion [@Girimaji_91], and population biology [@Steinrucken_2013]. Following the method of potential solutions, applied in [@Bakosi_dir], we derive a system of coupled stochastic differential equations (SDE) whose (Wiener-process) diffusion terms are nonlinearly coupled and whose invariant is Lochner’s generalized Dirichlet distribution.
The standard Dirichlet distribution can only represent non-positive covariances [@Mosimann_62], which limits its application to a specific class of processes. The stochastic system whose invariant is the generalized Dirichlet distribution allows for a more general class of physical processes with a more general covariance matrix. The process may be stationary or non-stationary, not limited to non-positive covariances, and satisfies the unit-sum requirement at all times, necessary for variables that obey a conservation principle.
Preview of results
==================
The generalized Dirichlet distribution for a set of scalars $0 \le Y_i$, $i=1,\dots,K$, $\sum_{i=1}^KY_i\le1$, and parameters, $\alpha_i>0$, $\beta_i>0$, as given by Lochner [@Lochner_75] reads $$\mathscr{G}({{\mbox{\boldmath$\mathbf{Y}$}}},{{\mbox{\boldmath$\mathbf{\alpha}$}}},{{\mbox{\boldmath$\mathbf{\beta}$}}}) =
\prod_{i=1}^K\frac{\Gamma(\alpha_i+\beta_i)}{\Gamma(\alpha_i)\Gamma(\beta_i)}
Y_i^{\alpha_i-1} \mathcal{Y}_i^{\gamma_i} \qquad \mathrm{with} \qquad
\mathcal{Y}_i = 1-\sum_{k=1}^i Y_k,
\label{eq:GD}$$ and $\gamma_i=\beta_i-\alpha_{i+1}-\beta_{i+1}$ for $i=1,\dots,K-1$, and $\gamma_K=\beta_K-1$. Here $\Gamma(\cdot)$ denotes the gamma function. We derive the stochastic diffusion process, governing the scalars, $Y_i$, $$\begin{split}
\mathrm{d}Y_i(t) = \frac{\mathcal{U}_i}{2}\left\{ b_i\Big[S_i \mathcal{Y}_K -
(1-S_i)Y_i\Big] + Y_i\mathcal{Y}_K
\sum_{j=i}^{K-1}\frac{c_{ij}}{\mathcal{Y}_j}\right\}\mathrm{d}t +
\sqrt{\kappa_i Y_i \mathcal{Y}_K \mathcal{U}_i}\mathrm{d}W_i(t), \\
\qquad i=1,\dots,K,
\end{split}
\label{eq:iSDE}$$ where $\mathrm{d}W_i(t)$ is an isotropic vector-valued Wiener process with independent increments [@Gardiner_09] and $\mathcal{U}_i = \prod_{j=1}^{K-i}
\mathcal{Y}_{K-j}^{-1}$. We show that the statistically stationary solution of the coupled system of nonlinear stochastic differential equations in [(\[eq:iSDE\])]{} is the generalized Dirichlet distribution, [Eq. (\[eq:GD\])]{}, provided the coefficients, $b_i\!>\!0$, $\kappa_i\!>\!0$, $0\!<\!S_i\!<\!1$, and $c_{ij}$, with $c_{ij}\!=\!0$ for $i\!>\!j$, $i,j\!=\!1,\dots,K\!-\!1$, satisfy $$\begin{aligned}
\alpha_i & = \frac{b_i}{\kappa_i}S_i, \qquad i=1,\dots,K,\\
1-\gamma_i & = \frac{c_{1i}}{\kappa_1} = \dots = \frac{c_{ii}}{\kappa_i},
\qquad i=1,\dots,K-1,\\
1+\gamma_K & = \frac{b_1}{\kappa_1}(1-S_1) = \dots =
\frac{b_K}{\kappa_K}(1-S_K).\end{aligned}$$ The restriction on the coefficients ensure reflection towards the interior of the sample space, which together with the specification $Y_N=\mathcal{Y}_K$ ensures $$\sum_{i=1}^NY_i=1. \label{eq:sum}$$ Indeed, if for example $Y_1\!=\!0$, the diffusion in [Eq. (\[eq:iSDE\])]{} is zero and the drift is strictly positive, while if $Y_1\!=\!1$, the diffusion is zero (as $\mathcal{Y}_K\mathcal{U}_1\!\rightarrow\!0$) and the drift is strictly negative.
Development of the diffusion process {#sec:sde}
====================================
The diffusion process, [Eq. (\[eq:iSDE\])]{}, is developed by the method of potential solutions. The steps below closely follow the methodology introduced in [@Bakosi_dir], used to derive a diffusion process for the Dirichlet distribution.
We start from the Itô diffusion process [@Gardiner_09] for the stochastic vector, $Y_i$, $$\mathrm{d}Y_i(t) = a_i({{\mbox{\boldmath$\mathbf{Y}$}}})\mathrm{d}t + \sum_{j=1}^K
b_{ij}({{\mbox{\boldmath$\mathbf{Y}$}}})\mathrm{d}W_j(t),
\qquad\quad i = 1,\dots,K,\label{eq:Ito}$$ with drift, $a_i({{\mbox{\boldmath$\mathbf{Y}$}}})$, diffusion, $b_{ij}({{\mbox{\boldmath$\mathbf{Y}$}}})$, and the isotropic vector-valued Wiener process, $\mathrm{d}W_j(t)$. Using standard methods given in [@Gardiner_09] the equivalent Fokker-Planck equation governing the joint probability, $\mathscr{F}({{\mbox{\boldmath$\mathbf{Y}$}}},t)$, derived from [Eq. (\[eq:Ito\])]{}, is $$\frac{\partial\mathscr{F}}{\partial t} = - \sum_{i=1}^K \frac{\partial}{\partial
Y_i}\big[a_i({{\mbox{\boldmath$\mathbf{Y}$}}})\mathscr{F}\big] + \frac{1}{2}\sum_{i=1}^K\sum_{j=1}^K
\frac{\partial^2}{\partial Y_i \partial
Y_j}\big[B_{ij}({{\mbox{\boldmath$\mathbf{Y}$}}})\mathscr{F}\big], \qquad\quad
B_{ij} = \sum_{k=1}^K b_{ik} b_{kj}.
\label{eq:FP}$$ As the drift and diffusion coefficients are time-homogeneous, $a_i({{\mbox{\boldmath$\mathbf{Y}$}}},t)\!=\!a_i({{\mbox{\boldmath$\mathbf{Y}$}}})$ and $B_{ij}({{\mbox{\boldmath$\mathbf{Y}$}}},t)\!=\!B_{ij}({{\mbox{\boldmath$\mathbf{Y}$}}})$, [Eq. (\[eq:Ito\])]{} is a statistically stationary process and the solution of [Eq. (\[eq:FP\])]{} converges to a stationary distribution [@Gardiner_09], Sec.6.2.2. Our task is to specify the functional forms of $a_i({{\mbox{\boldmath$\mathbf{Y}$}}})$ and $b_{ij}({{\mbox{\boldmath$\mathbf{Y}$}}})$ so that the stationary solution of [Eq. (\[eq:FP\])]{} is $\mathscr{G}({{\mbox{\boldmath$\mathbf{Y}$}}})$, defined by [Eq. (\[eq:GD\])]{}.
A potential solution of [Eq. (\[eq:FP\])]{} exists if $$\frac{\partial\ln\mathscr{F}}{\partial Y_j} = \sum_{i=1}^K B_{ij}^{-1}\left(2a_i
- \sum_{k=1}^K \frac{\partial B_{ik}}{\partial Y_k}\right) \equiv
-\frac{\partial\phi}{\partial Y_j}, \qquad\quad j = 1,\dots,K,
\label{eq:solution}$$ is satisfied, [@Gardiner_09] Sec. 6.2.2. Since the left hand side of [Eq. (\[eq:solution\])]{} is a gradient, the expression on the right must also be a gradient and can therefore be obtained from a scalar potential denoted by $\phi({{\mbox{\boldmath$\mathbf{Y}$}}})$. This puts a constraint on the possible choices of $a_i$ and $B_{ij}$ and on the potential, as $\phi,_{ij}=\phi,_{ji}$ must also be satisfied. The potential solution is $$\mathscr{F}({{\mbox{\boldmath$\mathbf{Y}$}}}) = \exp[-\phi({{\mbox{\boldmath$\mathbf{Y}$}}})].\label{eq:phi}$$ Now functional forms of $a_i({{\mbox{\boldmath$\mathbf{Y}$}}})$ and $B_{ij}({{\mbox{\boldmath$\mathbf{Y}$}}})$ that satisfy [Eq. (\[eq:solution\])]{}, with $\mathscr{F}({{\mbox{\boldmath$\mathbf{Y}$}}}) \equiv \mathscr{G}({{\mbox{\boldmath$\mathbf{Y}$}}})$ are sought. The mathematical constraints on the specification of $a_i$ and $B_{ij}$ are as follows:
$B_{ij}$ must be symmetric positive semi-definite. This is to ensure that
the square-root of $B_{ij}$ (e.g. the Cholesky-decomposition, $b_{ij}$) exists, required by the correspondence of the stochastic equation [(\[eq:Ito\])]{} and the Fokker-Planck equation [(\[eq:FP\])]{},
[Eq. (\[eq:Ito\])]{} represents a diffusion, and
$\det(B_{ij})\ne0$, required by the existence of the inverse in [Eq. (\[eq:solution\])]{}.
For a potential solution to exist [Eq. (\[eq:solution\])]{} must be satisfied.
With $\mathscr{F}({{\mbox{\boldmath$\mathbf{Y}$}}}) \equiv \mathscr{G}({{\mbox{\boldmath$\mathbf{Y}$}}})$ [Eq. (\[eq:phi\])]{} shows that the scalar potential must be $$-\phi({{\mbox{\boldmath$\mathbf{Y}$}}}) = \sum_{i=1}^K(\alpha_i-1)\ln Y_i + \sum_{i=1}^K\gamma_i\ln
\mathcal{Y}_i.\label{eq:gphi}$$ It is straightforward to verify that the specifications [$$\begin{aligned}
a_i({{\mbox{\boldmath$\mathbf{Y}$}}}) & = \frac{\mathcal{U}_i}{2}\left\{ b_i\Big[S_i \mathcal{Y}_K -
(1-S_i)Y_i\Big] + Y_i\mathcal{Y}_K \sum_{j=i}^{K-1}
\frac{c_{ij}}{\mathcal{Y}_j}\right\}, \label{eq:ga}\\
B_{ij}({{\mbox{\boldmath$\mathbf{Y}$}}}) & = \left\{ \begin{array}{lr} \kappa_i Y_i \mathcal{Y}_K
\mathcal{U}_i & \quad \mathrm{for} \quad i = j,\\\noalign{\smallskip} 0 & \quad
\mathrm{for} \quad i \ne j, \end{array} \right.\label{eq:gB}\end{aligned}$$]{}satisfy the above mathematical constraints, 1. and 2. Here $\mathcal{U}_i\!=\!\prod_{j=1}^{K-i} \mathcal{Y}_{K-j}^{-1}$, where an empty product is assumed to be unity, while an empty sum is zero. In addition to the coefficients $b_i\!>\!0$, $\kappa_i\!>\!0$, and $0\!<\!S_i\!<\!1$, governing the Dirichlet diffusion process [@Bakosi_dir], the drift now has the additional (not all independent) ones, denoted by $c_{ij}$, with $c_{ij}=0$ for $i>j$, $i,j=1,\dots,K-1$.
Substituting Eqs. (\[eq:gphi\]–\[eq:gB\]) into [Eq. (\[eq:solution\])]{} yields a system with the same functions on both sides with different coefficients, yielding the correspondence between the parameters of the generalized Dirichlet distribution, [Eq. (\[eq:GD\])]{}, and the Fokker-Planck equation [(\[eq:FP\])]{} with Eqs. (\[eq:ga\]–\[eq:gB\]) as $$\begin{aligned}
\alpha_i & = \frac{b_i}{\kappa_i}S_i, \qquad i=1,\dots,K,\label{eq:galphai}\\
1-\gamma_i & = \frac{c_{1i}}{\kappa_1} = \dots = \frac{c_{ii}}{\kappa_i},
\qquad i=1,\dots,K-1,\label{eq:gammai}\\
1+\gamma_K & = \frac{b_1}{\kappa_1}(1-S_1) = \dots =
\frac{b_K}{\kappa_K}(1-S_K).\label{eq:gammaN}\end{aligned}$$ The above result is arrived at inductively based on the special case of $K=3$ in Appendix A. If Eqs. (\[eq:galphai\]–\[eq:gammaN\]) hold, the stationary solution of the Fokker-Planck equation [(\[eq:FP\])]{} with drift [(\[eq:ga\])]{} and diffusion [(\[eq:gB\])]{} is the generalized Dirichlet distribution, [Eq. (\[eq:GD\])]{}. The same methodology was applied to the Dirichlet case in [@Bakosi_dir]. Eqs. (\[eq:galphai\]–\[eq:gammaN\]) specify the correspondence between the coefficients of the stochastic system [(\[eq:Ito\])]{} with drift [(\[eq:ga\])]{} and diffusion [(\[eq:gB\])]{} and the generalized Dirichlet distribution, [Eq. (\[eq:GD\])]{}. With $\gamma_i=\beta_i-\alpha_{i+1}-\beta_{i+1}$, $i=1,\dots,K-1$, and $\gamma_K=\beta_K-1$, the correspondence between ($\alpha_i,\beta_i$) and ($b_i,S_i,\kappa_i,c_{ij}$) is also complete. Note that Eqs.(\[eq:ga\]–\[eq:gB\]) are one possible way of specifying drift and diffusion to arrive at a generalized Dirichlet distribution; other functional forms may be possible. It is straightforward to verify, that setting $c_{1i}/\kappa_i=\dots=c_{ii}/\kappa_i=1$ for $i=1,\dots,K\!-\!1$, i.e., $\gamma_1=\dots=\gamma_{K-1}=0$, in [Eqs. (\[eq:ga\])]{} and [(\[eq:gB\])]{} yields the same system in [Eq. (\[eq:solution\])]{} as with $a_i$ and $B_{ij}$ specified for the (standard) Dirichlet distribution, see Appendix A for $K=3$. The shape of the generalized Dirichlet distribution, [Eq. (\[eq:GD\])]{}, is determined by the $2K$ coefficients, $\alpha_i$, $\beta_i$. Eqs. (\[eq:galphai\]–\[eq:gammaN\]) show that in the stochastic system, different combinations of $b_i$, $S_i$, $\kappa_i$, and $c_{ij}$ may yield the same $\alpha_i$, $\beta_i$ and that not all of $b_i$, $S_i$, $\kappa_i$, and $c_{ij}$ may be chosen independently to make the invariant generalized Dirichlet. In other words, a unique set of SDE coefficients always corresponds to a unique set of distribution parameters, but the converse is not true: a set of distribution parameters do not uniquely determine all the SDE coefficients, for a given specific asymptotic generalized Dirichlet distribution.
Properties of Dirichlet distributions
=====================================
It is useful to show how the generalized Dirichlet distribution, [Eq. (\[eq:GD\])]{}, reduces to standard Dirichlet, and their univariate case, the beta distribution.
Density functions
-----------------
Setting $\gamma_1=\dots=\gamma_{K-1}=0$ in [Eq. (\[eq:GD\])]{} yields the (standard) Dirichlet distribution $$\mathscr{D}({{\mbox{\boldmath$\mathbf{Y}$}}},{{\mbox{\boldmath$\mathbf{\omega}$}}}) =
\frac{\Gamma\left(\sum_{i=1}^N\omega_i\right)}
{\prod_{i=1}^N\Gamma(\omega_i)}\prod_{i=1}^N
Y_i^{\omega_i-1},\label{eq:D}$$ with $\omega_i=\alpha_i$, $i=1,\dots,K=N-1$, $\omega_N=\beta_K$, and $Y_N=1-\sum_{j=1}^KY_j$. In the univariate case, $K=N-1=1$, ${{\mbox{\boldmath$\mathbf{Y}$}}}=(Y_1,Y_2)=(Y,1\!-\!Y)$, both $\mathscr{G}$ and $\mathscr{D}$ yield the beta distribution $$\mathscr{B}(Y,\alpha,\beta) =
\frac{\Gamma(\alpha+\beta)}{\Gamma(\alpha)\Gamma(\beta)}
Y^{\alpha-1}(1-Y)^{\beta-1},$$ with $\omega_1=\alpha$ and $\omega_2=\beta$.
$\mathscr{G}$, $\mathscr{D}$, and $\mathscr{B}$ are zero outside the $K$-dimensional generalized triangle; the sample spaces are bounded. Compared to $\mathscr{D}$, there are $K-1$ additional parameters in $\mathscr{G}$ for a set of $K$ scalars.
Moments
-------
All moments of the generalized Dirichlet distribution, [Eq. (\[eq:GD\])]{}, can be obtained from $\alpha_i$ and $\beta_i$ of which the first two are [@Connor_69; @Wong_98] [$$\begin{aligned}
{{\langle{Y_i}\rangle}} & = \int Y_i\mathscr{G}({{\mbox{\boldmath$\mathbf{Y}$}}})\mathrm{d}{{\mbox{\boldmath$\mathbf{Y}$}}} =
\frac{\alpha_i}{\alpha_i+\beta_i} \prod_{j=1}^{i-1}
\frac{\beta_j}{\alpha_j+\beta_j},\label{eq:GDmeans}\\
{{\langle{y_iy_j}\rangle}} & = {{\langle{(Y_i-{{\left\langle{Y_i}\right\rangle}})(Y_j-{{\left\langle{Y_j}\right\rangle}})}\rangle}} = \left\{
\begin{array}{lr}
\displaystyle{{\langle{Y_i}\rangle}}\left(\frac{\alpha_i+1}{\alpha_i+\beta_i+1} M_{i-1} -
{{\langle{Y_i}\rangle}}\right) &\qquad \mathrm{for} \quad i = j,\vspace{6pt}\\
\displaystyle{{\langle{Y_j}\rangle}}\left(\frac{\alpha_i}{\alpha_i+\beta_i+1} M_{i-1} -
{{\langle{Y_i}\rangle}}\right) & \qquad \mathrm{for} \quad i \ne j,
\end{array}
\right.\label{eq:GDcovariances}\\
& \qquad\qquad\qquad\qquad\qquad\qquad\qquad i,j = 1,\dots,K,\nonumber\end{aligned}$$]{}where $M_{i-1} = \prod_{k=1}^{i-1} (\beta_k+1)/(\alpha_k+\beta_k+1)$. Setting $\gamma_1=\dots=\gamma_{K-1}=0$, with $\omega_i=\alpha_i$, $i=1,\dots,K=N-1$, $\omega_N=\beta_K$, in Eqs. (\[eq:GDmeans\]–\[eq:GDcovariances\]) reduces to the first two moments of the Dirichlet distribution, $$\begin{aligned}
{{\langle{Y_i}\rangle}} & = \frac{\omega_i}{\omega},\label{eq:Dmeans}\\
{{\langle{y_i y_j}\rangle}} & = \left\{
\begin{array}{lr}
\displaystyle\frac{\omega_i(\omega-\omega_i)}{\omega^2(\omega+1)} &
\quad \mathrm{for} \quad i = j,\vspace{6pt}\\
\displaystyle\frac{-\omega_i \omega_j}{\omega^2(\omega+1)} & \quad
\mathrm{for} \quad i \ne j,
\end{array}\label{eq:Dcovariances}
\right.\\
& \qquad\qquad\qquad i,j = 1,\dots,K,\nonumber\end{aligned}$$ where $\omega\!=\!\sum_{j=1}^N\omega_j$. [Eq. (\[eq:GDcovariances\])]{} shows that in the generalized Dirichlet distribution $Y_1$ is always negatively correlated with the other scalars. However, $Y_j$ and $Y_m$ can be positively correlated for $j,m>1$, see also [@Lochner_75]. According to Wong [@Wong_98], *“If there exists some $m>j$ such that $Y_j$ and $Y_m$ are positively (negatively) correlated, then $Y_j$ will be positively (negatively) correlated with $Y_n$ for all $n>j$.”* This can be seen from [Eq. (\[eq:GDcovariances\])]{}: the sign of ${{\langle{y_my_j}\rangle}}$ is independent of $j$, so the sign of ${{\langle{y_my_j}\rangle}}$, $m>j$ will imply the signs of all ${{\langle{y_ny_j}\rangle}}$, $n>j$. This is in contrast with the Dirichlet distribution, [Eq. (\[eq:D\])]{}, whose covariances are always non-positive as can be seen from [Eq. (\[eq:Dcovariances\])]{}.
In the univariate case, $K=N-1=1$, ${{\mbox{\boldmath$\mathbf{Y}$}}}=(Y_1,Y_2)=(Y,1\!-\!Y)$, the first two moments of both the generalized and the standard Dirichlet distributions, Eqs.(\[eq:GDmeans\]–\[eq:GDcovariances\]) and Eqs.(\[eq:Dmeans\]–\[eq:Dcovariances\]), respectively, reduce to the moments of the beta distribution, with $\omega_1=\alpha$ and $\omega_2=\beta$, $$\begin{aligned}
{{\langle{Y}\rangle}} & = \frac{\alpha}{\alpha+\beta},\\
{{\langle{y^2}\rangle}} & = \frac{\alpha \beta}{(\alpha+\beta)^2(\alpha+\beta+1)}.\end{aligned}$$
Relation to other diffusion processes
=====================================
It also useful to relate the generalized Dirichlet process, [Eq. (\[eq:iSDE\])]{}, to other multivariate diffusion processes with linear drift and quadratic diffusion.
Setting $c_{1i}/\kappa_i=\dots=c_{ii}/\kappa_i=1$ for $i=1,\dots,K\!-\!1$, in [Eq. (\[eq:iSDE\])]{} yields $$\mathrm{d}Y_i(t) = \frac{b_i}{2} \big[S_i Y_N - (1-S_i)Y_i\big] \mathrm{d}t +
\sqrt{\kappa_i Y_i Y_N} \mathrm{d}W_i(t), \qquad i=1,\dots,K=N-1,
\label{eq:DSDE}$$ with $Y_N\!=\!1\!-\!\sum_{j=1}^{N-1}Y_j$ whose invariant is the (standard) Dirichlet distribution, [Eq. (\[eq:D\])]{}. [Eq. (\[eq:DSDE\])]{} is discussed in [@Bakosi_dir]. Another diffusion process whose invariant is also Dirichlet is the multivariate Wright-Fisher process [@Steinrucken_2013], $$\mathrm{d}Y_i(t) = \frac{1}{2} (\omega_i-\omega Y_i) \mathrm{d}t
+ \sum_{j=1}^{K} \sqrt{Y_i(\delta_{ij}-Y_j)}
\mathrm{d}W_{ij}(t), \qquad i = 1,\dots,K=N-1,
\label{eq:WF}$$ where $\delta_{ij}$ is Kronecker’s delta. Another process similar to [Eqs. (\[eq:iSDE\])]{}, [(\[eq:DSDE\])]{}, and [(\[eq:WF\])]{} is the multivariate Jacobi process, used in econometrics, $$\mathrm{d}Y_i(t) = a(Y_i-\pi_i)\mathrm{d}t + \sqrt{cY_i}\mathrm{d}W_i(t) -
\sum_{j=1}^{N-1} Y_i\sqrt{cY_j}\mathrm{d}W_j(t), \qquad i = 1,\dots,N
\label{eq:Jacobi}$$ of Gourieroux & Jasiak [@Gourieroux_06] with $a<0$, $c>0$, $\pi_\alpha>0$, and $\sum_{j=1}^N\pi_j=1$.
In the univariate case, $K=N-1=1$, ${{\mbox{\boldmath$\mathbf{Y}$}}}=(Y_1,Y_2)=(Y,1\!-\!Y)$, the generalized Dirichlet, Dirichlet, Wright-Fisher, and Jacobi diffusions, [Eqs. (\[eq:iSDE\])]{}, [(\[eq:DSDE\])]{}, [(\[eq:WF\])]{}, [(\[eq:Jacobi\])]{}, respectively, all reduce to $$\mathrm{d}Y(t) = \frac{b}{2} (S-Y)\mathrm{d}t +
\sqrt{\kappa Y(1-Y)}\mathrm{d}W(t),
\label{eq:beta}$$ see also [@Bakosi_beta], whose invariant is the beta distribution, which belongs to the family of Pearson diffusions, discussed in detail by Forman & Sorensen [@Forman_08].
Summary {#sec:summary}
=======
Following the development in [@Bakosi_dir] we started with a multivariate distribution for a set of stochastic variables that satisfies a conservation principle in which all variables sum to unity. Applying the constraints on the existence of potential solutions of Fokker-Planck equations, we derived a system of stochastic differential equations [(\[eq:iSDE\])]{} whose joint distribution in the statistically stationary state is Lochner’s generalized Dirichlet distribution, [Eq. (\[eq:GD\])]{}. [Eq. (\[eq:iSDE\])]{} is a generalization of the Dirichlet diffusion process developed in [@Bakosi_dir]. Compared to the standard Dirichlet process, the generalized diffusion allows for representing a more general class of stochastic processes with a more general covariance matrix. The process may be stationary or non-stationary, not limited to non-positive covariances, and satisfies the unit-sum requirement, [Eq. (\[eq:sum\])]{}, at all times, necessary for variables that obey a conservation principle.
Appendix A: Inductive proof of Eqs.(\[eq:galphai\]–\[eq:gammaN\]) based on ${{\mbox{\boldmath$\mathbf{K=3}$}}}$ {#appendix-a-inductive-proof-of-eqs.eqgalphaieqgamman-based-on-mboxboldmathmathbfk3 .unnumbered}
===============================================================================================================
Eqs. (\[eq:galphai\]–\[eq:gammaN\]) are now arrived at for $K=3$, yielding the correspondence of the generalized Dirichlet distribution, [Eq. (\[eq:GD\])]{}, and its stochastic process, [Eq. (\[eq:iSDE\])]{}, for $K=3$. The procedure generalizes to arbitrary $K>3$.
From [Eq. (\[eq:gphi\])]{} the scalar potential for $K=3$ is $$\begin{aligned}
-\phi(Y_1,Y_2,Y_3) & = (\alpha_1-1)\ln Y_1 +
(\alpha_2-1)\ln Y_2 +
(\alpha_3-1)\ln Y_3 \nonumber \\
& \quad + \gamma_1\ln(1-Y_1) +
\gamma_2\ln(1-Y_1-Y_2) +
\gamma_3\ln(1-Y_1-Y_2-Y_3).
\label{eq:gphi3}\end{aligned}$$ From Eqs. (\[eq:ga\]–\[eq:gB\]) the drift and diffusion for $K=3$ are $$\begin{aligned}
a_1 & = \frac{b_1/2}{(1-Y_1)(1-Y_1-Y_2)}
\Big[S_1(1-Y_1-Y_2-Y_3) - (1-S_1)Y_1\Big] \nonumber \\
& \qquad + \frac{Y_1(1-Y_1-Y_2-Y_3)}{(1-Y_1)(1-Y_1-Y_2)}
\left[\frac{c_{11}/2}{1-Y_1} + \frac{c_{12}/2}{1-Y_1-Y_2} \right],
\\[0.3cm]
a_2 & = \frac{b_2/2}{1-Y_1-Y_2}
\Big[S_2(1-Y_1-Y_2-Y_3) - (1-S_2)Y_2\Big]
+ \frac{c_{22}}{2}\!\cdot\!\frac{Y_2(1-Y_1-Y_2-Y_3)}{(1-Y_1-Y_2)^2},
\\[0.3cm]
a_3 & = \frac{b_3}{2}\Big[S_3(1-Y_1-Y_2-Y_3) - (1-S_3)Y_3\Big], \\[0.3cm]
B_{11} & = \kappa_1 \frac{Y_1(1-Y_1-Y_2-Y_3)}{(1-Y_1)(1-Y_1-Y_2)}, \\[0.3cm]
B_{22} & = \kappa_2 \frac{Y_2(1-Y_1-Y_2-Y_3)}{1-Y_1-Y_2}, \\[0.3cm]
B_{33} & = \kappa_3 Y_3 (1-Y_1-Y_2-Y_3), \\[0.3cm]
B_{12} & = B_{23} = B_{13} = 0, \label{eq:offDiagDiff}\end{aligned}$$ Substituting Eqs. (\[eq:gphi3\]–\[eq:offDiagDiff\]) into [Eq. (\[eq:solution\])]{} for $K\!=\!3$ yields $$\begin{aligned}
\frac{\alpha_1-1}{Y_1}
- \frac{\gamma_1}{1-Y_1}
- \frac{\gamma_2}{1-Y_1-Y_2}
- \frac{\gamma_3}{1-Y_1-Y_2-Y_3} = \nonumber
\qquad\qquad\qquad\qquad\qquad\qquad\qquad \\
= \left(\frac{b_1}{\kappa_1}S_1-1\right) \frac{1}{Y_1}
+ \left(\frac{c_{11}}{\kappa_1}-1\right) \frac{1}{1-Y_1}
+ \left(\frac{c_{12}}{\kappa_1}-1\right) \frac{1}{1-Y_1-Y_2} \nonumber \\
+ \left[1-\frac{b_1}{\kappa_1}(1-S_1) \right] \frac{1}{1-Y_1-Y_2-Y_3},
\label{eq:corr1}\\[0.3cm]
\frac{\alpha_2-1}{Y_2}
- \frac{\gamma_2}{1-Y_1-Y_2}
- \frac{\gamma_3}{1-Y_1-Y_2-Y_3} = \nonumber
\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad \\
= \left(\frac{b_2}{\kappa_2}S_2-1\right) \frac{1}{Y_2}
+ \left(\frac{c_{22}}{\kappa_2}-1\right) \frac{1}{1-Y_1-Y_2}
+ \left[1-\frac{b_2}{\kappa_2}(1-S_2) \right] \frac{1}{1-Y_1-Y_2-Y_3},
\label{eq:corr2}\\[0.3cm]
\frac{\alpha_3-1}{Y_3}
- \frac{\gamma_3}{1-Y_1-Y_2-Y_3} =
\left(\frac{b_3}{\kappa_3}S_3-1\right) \frac{1}{Y_3}
+ \left[1-\frac{b_3}{\kappa_3}(1-S_3) \right] \frac{1}{1-Y_1-Y_2-Y_3},\end{aligned}$$ which shows that if $$\begin{aligned}
\alpha_1 & = \frac{b_1}{\kappa_1}S_1, \label{eq:a1} \\
\alpha_2 & = \frac{b_2}{\kappa_2}S_2, \\
\alpha_3 & = \frac{b_3}{\kappa_3}S_3, \\
1-\gamma_1 & = \frac{c_{11}}{\kappa_1}, \\
1-\gamma_2 & = \frac{c_{12}}{\kappa_1} = \frac{c_{22}}{\kappa_2}, \\
1+\gamma_3 & = \frac{b_1}{\kappa_1}(1-S_1)
= \frac{b_2}{\kappa_2}(1-S_2)
= \frac{b_3}{\kappa_3}(1-S_3), \label{eq:g3}\end{aligned}$$ all hold, the invariant of [Eq. (\[eq:iSDE\])]{} is [Eq. (\[eq:GD\])]{} for $K=3$, $$\begin{aligned}
\mathscr{G}(Y_1,Y_2,Y_3,\alpha_1,\alpha_2,\alpha_3,\beta_1,\beta_2,\beta_3) =
\qquad &\nonumber \\
\frac{\Gamma(\alpha_1+\beta_1) \Gamma(\alpha_2+\beta_2)\Gamma(\alpha_3+\beta_3)}
{\Gamma(\alpha_1)\Gamma(\beta_1)
\Gamma(\alpha_2)\Gamma(\beta_2)
\Gamma(\alpha_3)\Gamma(\beta_3)}
\label{eq:GD3}
\times &\\
\times Y_1^{\alpha_1-1} Y_2^{\alpha_2-1} Y_3^{\alpha_3-1}
(1-Y_1)^{\gamma_1} & (1-Y_1-Y_2)^{\gamma_2} (1-Y_1-Y_2-Y_3)^{\gamma_3},\nonumber\end{aligned}$$ with $$\gamma_1 = \beta_1 - \alpha_2 - \beta_2, \qquad
\gamma_2 = \beta_2 - \alpha_3 - \beta_3, \qquad
\gamma_3 = \beta_3 - 1. \label{eq:g}$$ Eqs. (\[eq:a1\]–\[eq:g3\]) give the correspondence between the coefficients of the stochastic system, [Eq. (\[eq:iSDE\])]{}, and its invariant, [Eq. (\[eq:GD\])]{}, for $K=3$. With [Eq. (\[eq:g\])]{} the correspondence between the parameters of the joint probability density function (PDF), ($\alpha_1,\alpha_2,\alpha_3,\beta_1,\beta_2,\beta_3$), and the coefficients of the stochastic system, ($b_1,b_2,b_3,$ $S_1,S_2,S_3,$ $\kappa_1,\kappa_2,\kappa_3,c_{11},c_{12},c_{22}$), is also given.
It is straightforward to verify that setting $\gamma_1 = \gamma_2 = 0$ in [Eq. (\[eq:GD3\])]{} yields the Dirichlet distribution, [Eq. (\[eq:D\])]{}, for $K=3$ ($N=4)$, $$\begin{aligned}
\mathscr{D}(Y_1,Y_2,Y_3,\omega_1,\omega_2,\omega_3,\omega_4) =
&\nonumber \\
\frac{\Gamma(\omega_1+\omega_2+\omega_3+\omega_4)}
{\Gamma(\omega_1)\Gamma(\omega_2)\Gamma(\omega_3)\Gamma(\omega_4)} &
Y_1^{\omega_1-1} Y_2^{\omega_2-1} Y_3^{\omega_3-1} (1-Y_1-Y_2-Y_3)^{\omega_4-1}\end{aligned}$$ with $$\omega_1 = \alpha_1, \qquad
\omega_2 = \alpha_2, \qquad
\omega_3 = \alpha_3, \qquad
\omega_4 = \beta_3.$$ Similarly, setting $c_{11}/\kappa_1 = c_{12}/\kappa_1 = c_{22}/\kappa_2 = 1$ in Eqs. (\[eq:corr1\]–\[eq:corr2\]) reduces to the system corresponding that of the Dirichlet case [@Bakosi_dir].
Appendix B: Numerical simulation: The effect of the extra coefficient for ${{\mbox{\boldmath$\mathbf{K=2}$}}}$ {#appendix-b-numerical-simulation-the-effect-of-the-extra-coefficient-for-mboxboldmathmathbfk2 .unnumbered}
==============================================================================================================
Numerical simulations are used to demonstrate the effect of the extra coefficient, $c_{11}$, compared to the standard Dirichlet case, given in [@Bakosi_dir].
The time-evolution of an ensemble of $10,\!000$ particles has been numerically computed by integrating the system [(\[eq:Ito\])]{}, with drift and diffusion (\[eq:ga\]–\[eq:gB\]), for $K=2$, i.e., ($Y_1, Y_2, Y_3 = 1 - Y_1
- Y_2$), $$\begin{aligned}
\mathrm{d}Y_1 & = \frac{b_1/2}{1-Y_1} \big[S_1 Y_3 - (1-S_1)Y_1\big]\mathrm{d}t
+ \frac{Y_1Y_3}{1-Y_1} \cdot \frac{c_{11}/2}{1-Y_1} \mathrm{d}t + \sqrt{\kappa_1
\frac{Y_1 Y_3}{1-Y_1}}\mathrm{d}W_1, \label{eq:Ito1}\\
\mathrm{d}Y_2 & = \frac{b_2}{2}\big[S_2 Y_3 - (1-S_2)Y_2\big]\mathrm{d}t +
\sqrt{\kappa_2 Y_2 Y_3}\mathrm{d}W_2, \label{eq:Ito2}\\
Y_3 & = 1 - Y_1 - Y_2. \label{eq:Ito3}\end{aligned}$$
------------------------------------------------------------------------
[0.51]{} Asymptotic moments for $K=2$,\
see Eqs. (\[eq:GDmeans\]–\[eq:GDcovariances\]) $$\begin{aligned}
{{\left\langle{Y_1}\right\rangle}} &= \frac{\alpha_1}{\alpha_1 + \beta_1} \\
{{\left\langle{Y_2}\right\rangle}} &= \frac{\alpha_2}{\alpha_2 + \beta_2} \cdot
\frac{\alpha_1}{\alpha_1 + \beta_1} \\
{{{\left\langle{y_1^2}\right\rangle}}} &= {{\left\langle{Y_1}\right\rangle}}\left( \frac{\alpha_1 + 1}{\alpha_1 + \beta_1 + 1}
- {{\left\langle{Y_1}\right\rangle}} \right) \\
{{{\left\langle{y_2^2}\right\rangle}}} &= {{\left\langle{Y_2}\right\rangle}}\left( \frac{\alpha_2 + 1}{\alpha_2 + \beta_2 + 1}
\cdot \frac{\alpha_1 + 1}{\alpha_1 + \beta_1 + 1} - {{\left\langle{Y_2}\right\rangle}} \right)\\
{{\left\langle{y_1y_2}\right\rangle}} &= {{\left\langle{Y_2}\right\rangle}}\left( \frac{\alpha_1}{\alpha_1 + \beta_1 +
1} - {{\left\langle{Y_1}\right\rangle}} \right)
\end{aligned}$$
[0.48]{} Dirichlet SDE coefficients (common to all cases) $$\begin{aligned}[c]
b_1 &= 1/10 \\
S_1 &= 5/8 \\
\kappa_1 &= 1/80 \\
\end{aligned} \qquad
\begin{aligned}[c]
b_2 &= 3/2 \\
S_2 &= 2/5 \\
\kappa_2 &= 3/10
\end{aligned}$$\
Generalized Dirichlet SDE coefficients $$\begin{aligned}[c]
c_{11} &= \kappa_{11} = 1/80 \\
c_{11} &= -1/80 \\
c_{11} &= -1/4 \\
\end{aligned} \quad
\begin{aligned}[c]
\textrm{(case 1)} \\
\textrm{(case 2)} \\
\textrm{(case 3)}
\end{aligned}$$
\
[0.5]{} PDF parameters from the SDE coefficients,\
see Eqs. (\[eq:galphai\]–\[eq:gammaN\]) $$\begin{aligned}
\alpha_1 &= \frac{b_1}{\kappa_1}S_1 \\
\alpha_2 &= \frac{b_2}{\kappa_2}S_2 \\
1-\gamma_1 &= \frac{c_{11}}{\kappa_1} \\
1+\gamma_2 &= \frac{b_1}{\kappa_1}(1-S_1) = \frac{b_2}{\kappa_2}(1-S_2)\\
\beta_2 &= 1 + \gamma_2 = \frac{b_1}{\kappa_1}(1-S_1) =
\frac{b_2}{\kappa_2}(1-S_2) \\
\beta_1 &= \gamma_1 + \alpha_2 + \beta_2 = 1 - \frac{c_{11}}{\kappa_1}
+ \alpha_2 + \beta_2 \\
\end{aligned}$$
[0.49]{} SDE asymptotic moments for cases 1, 2, 3 $$\begin{aligned}
c_{11} &= \frac{1}{80} \\
\alpha_1 &= 5 \\
\alpha_2 &= 2 \\
\beta_2 &= 3 \\
\beta_1 &= 5 \\
{{\left\langle{Y_1}\right\rangle}} &= \frac{1}{2} \\
{{\left\langle{Y_2}\right\rangle}} &= \frac{1}{5} \\
{{{\left\langle{y_1^2}\right\rangle}}} &= \frac{1}{44} \\
{{{\left\langle{y_2^2}\right\rangle}}} &= \frac{4}{275} \\
{{\left\langle{y_1y_2}\right\rangle}} &= -\frac{1}{110}
\end{aligned} \quad
\begin{aligned}
c_{11} &= -\frac{1}{80} \\
\alpha_1 &= 5 \\
\alpha_2 &= 2 \\
\beta_2 &= 3 \\
\beta_1 &= 7 \\
{{\left\langle{Y_1}\right\rangle}} &= \frac{5}{12} \\
{{\left\langle{Y_2}\right\rangle}} &= \frac{7}{30} \\
{{{\left\langle{y_1^2}\right\rangle}}} &= \frac{35}{1872} \\
{{{\left\langle{y_2^2}\right\rangle}}} &= \frac{609}{35100} \\
{{\left\langle{y_1y_2}\right\rangle}} &= -\frac{35}{4680}
\end{aligned} \quad
\begin{aligned}
c_{11} &= -\frac{1}{4} \\
\alpha_1 &= 5 \\
\alpha_2 &= 2 \\
\beta_2 &= 3 \\
\beta_1 &= 26 \\
{{\left\langle{Y_1}\right\rangle}} &= \frac{5}{31} \\
{{\left\langle{Y_2}\right\rangle}} &= \frac{52}{155} \\
{{{\left\langle{y_1^2}\right\rangle}}} &= \frac{65}{15376} \\
{{{\left\langle{y_2^2}\right\rangle}}} &= \frac{11141}{384400} \\
{{\left\langle{y_1y_2}\right\rangle}} &= -\frac{13}{7688}
\end{aligned}$$
\
------------------------------------------------------------------------
In Eqs. (\[eq:Ito1\]–\[eq:Ito2\]) $\mathrm{d}W_1$ and $\mathrm{d}W_2$ are independent Wiener processes, sampled from Gaussian streams of random numbers with mean ${{\langle{\mathrm{d}W_i}\rangle}}\!=\!0$ and covariance ${{\langle{\mathrm{d} W_i
\mathrm{d} W_j}\rangle}}\!= \!\delta_{ij} \mathrm{d}t$. Eqs.(\[eq:Ito1\]–\[eq:Ito3\]) were advanced in time with the Euler-Maruyama scheme [@Kloeden_99] with time step $\Delta t\!=\!0.025$. The coefficients of the stochastic system (\[eq:Ito1\]–\[eq:Ito3\]), the corresponding parameters and the first two moments of the asymptotic generalized Dirichlet distributions for $K\!=\!2$ are shown in Table \[tab:coeff\]. Three different cases were simulated. Here the initial condition of $(Y_1,Y_2) \equiv 0$ was used. The initial PDF in all cases is the same: all samples are zero and the PDF is therefore not Dirichlet nor Generalized Dirichlet, see also [@Bakosi_dir] for nonzero but different non-Dirichlet initial conditions. Our motivation is two-fold: (1) to show that the solution approaches the invariant, and (2) to show how the new additional parameter in the generalized Dirichlet SDE affects the dynamics. Had the initial conditions coincided with the given invariant, the PDF (and its statistics) would not have changed in time – as has been demonstrated mathematically. The SDE coefficients in the three simulations only differ in the extra generalized Dirichlet coefficient, $c_{11}$, otherwise, the setup corresponds to the example in [@Bakosi_dir]. In the first simulation $c_{11}\!=\kappa_1\!=\!1/80$, i.e., $c_{11}$ is *not* a free coefficient and is chosen to yield an asymptotic solution that is a (standard) Dirichlet, the same as in [@Bakosi_dir]. In the second and third simulations $c_{11}$ are freely chosen and thus yield generalized Dirichlet solutions. Figure \[fig:sim\] shows the evolutions of the first two moments in time for the three cases.
|
---
abstract: 'We study stochastic bifurcation for a system under multiplicative stable Lévy noise (an important class of non-Gaussian noise), by examining the qualitative changes of equilibrium states in its most probable phase portraits. We have found some peculiar bifurcation phenomena in contrast to the deterministic counterpart: (i) When the non-Gaussianity parameter in Lévy noise varies, there is either one, two or none backward pitchfork type bifurcations; (ii) When a parameter in the vector field varies, there are two or three forward pitchfork bifurcations; (iii) The non-Gaussian Lévy noise clearly leads to fundamentally more complex bifurcation scenarios, since in the special case of Gaussian noise, there is only one pitchfork bifurcation which is reminiscent of the deterministic situation.'
address:
- |
Center for Mathematical Sciences $\&$ School of Mathematics and Statistics\
$\&$ Hubei Key Laboratory for Engineering Modeling and Scientific Computing,\
Huazhong University of Science and Technology, Wuhan 430074, China\
huiwheda@hust.edu.cn, xlchen@hust.edu.cn
- |
Department of Applied Mathematics, Illinois Institute of Technology, Chicago, IL 60616, USA\
$\&$ Center for Mathematical Sciences,\
Huazhong University of Science and Technology, Wuhan 430074, China\
duan@iit.edu
author:
- 'Hui Wang, Xiaoli Chen'
- Jinqiao Duan
title: |
A Stochastic Pitchfork Bifurcation\
in Most Probable Phase Portraits
---
Introduction
============
Despite the rapid development in many aspects of stochastic dynamical systems, the investigation of stochastic bifurcation is still in its infancy. A stochastic bifurcation may be defined as a qualitative change in the evolution of a stochastic dynamical system, as a parameter varies. Stochastic bifurcations have been observed in a wide range of nonlinear systems in physical science and engineering. The existing works on stochastic bifurcation mostly are for stochastic dynamical systems with Gaussian noise and focus on the qualitative changes in stationary probability densities [@Sri1990] as solutions of steady Fokker-Planck equations, invariant measures (together with their supports and Lyapunov spectra) and random point attractors [@Arnold2003], or Conley index [@Chen2009].
Random fluctuations are often assumed to have Gaussian distributions [@Gui2016; @Suel2006; @Hasty2000b; @Liu2004; @Li2014] and are represented by Brownian motion. But the fluctuations in some complex systems, such as temperature evolution in paleoclimate ice-core records [@Ditlevsen1999] and bursty transition in gene expression [@Kumar2015; @Dar2012], are not Gaussian. Then it is more appropriate to model these random fluctuations by a non-Gaussian Lévy motion (i.e., $\alpha-$stable Lévy motion) with heavy tails and bursting sample paths [@Zheng2016; @Klafter2011; @Woyczynski2001; @Chechkin2007].
A bifurcation in deterministic low dimensional dynamical systems often appears as a qualitative change in phase portraits in state space, and is usually illustrated via a bifurcation diagram in a ‘parameter-steady state plane’ [@Guckenheimer1983; @Wiggins2003; @Strogatz1994].
In this present work, we study stochastic bifurcation in a kind of stochastic phase portraits. However, phase portraits for stochastic differential equations are delicate objects. It turns out that the phase portraits in terms of most probable orbits [@Duan2015; @Cheng2016] offer a promising option. Thus we propose here to study stochastic bifurcation by examining the qualitative changes (especially the changes in the number and stability type for equilibrium states) in most probable phase portraits. To this end, we consider bifurcation for the prototypical scalar stochastic differential equation with multiplicative $\alpha-$stable Lévy motion $$dX_t = (r X_t -X_t^3) dt + X_t dL_t^\alpha,$$ where $r$ is a real parameter and the parameter $\alpha$ is in the interval $(0, 2)$. The $\alpha-$stable Lévy motion $L_t^\alpha$ will be reviewed in the next section.
The deterministic counterpart $\dot x= r x-x^3$ has the well-known (forward) ‘pitchfork’ bifurcation [@Guckenheimer1983], as the parameter $r$ increases.
![ (Color online) Bifurcation diagram for deterministic dynamical system $\dot x= r x-x^3$: Equilibrium states vs. parameter $r$. This is a pitchfork bifurcation at $r=0$.[]{data-label="Fig.0"}](determin.eps){width="45.00000%"}
Figure \[Fig.0\] is the bifurcation diagram for this deterministic pitchfork system. For $r \leq 0$, $x=0$ is the only equilibrium state which is stable. While for $ r>0 $, there exist two stable equilibrium states $ \sqrt r$ and $ -\sqrt r$ and one unstable equilibrium state $x=0$. The bifurcation parameter value is at $r=0$.
This paper is organized as follows. In Section 2, we review the definition of a scalar stable Lévy motion $L_t^{\alpha}$ , the most probable phase portraits, and the numerical methods for bifurcation diagrams. In Section 3, we show bifurcation diagrams for a stochastic pitchfork bifurcation under multiplicative stable Lévy motion. Finally, we summarize our results in Section 4.
Methods
=======
Stable Lévy motion
-------------------
A scalar stable Lévy motion $L_t^{\alpha}$, for $0<\alpha<2$, is a non-Gaussian stochastic process with the following properties [@Duan2015; @Applebaum2009; @Sato1999; @Samorodnitsky1994]:\
(i) $L_0^{\alpha} = 0$, almost surely (a.s.);\
(ii) $L_t^{\alpha}$ has independent increments;\
(iii)$L_t^{\alpha}$ has stationary increments: $L_t^{\alpha}-L_s^{\alpha} $ has probability distribution $S_\alpha((t-s)^\frac{1}{\alpha}, 0, 0)$ for $ s \leq t $; in particular, $L_t^{\alpha}$ has distribution $S_\alpha(t^\frac{1}{\alpha}, 0, 0)$;\
(iv) $L_t^{\alpha}$ has stochastically continuous sample paths, i.e., $L_t^{\alpha} \rightarrow L_s^{\alpha}$ in probability, as $t\rightarrow s$.
Here $S_{\alpha}(\sigma,\beta,\mu)$ is the so-called stable distribution [@Samorodnitsky1994; @Duan2015] and is determined by four parameters, non-Gaussianity index $\alpha (0 < \alpha < 2)$, skewness index $\beta (-1\leq \beta \leq 1)$, shift index $\mu (-\infty < \mu < +\infty)$ and scale index $\sigma (\sigma \geq 0)$.
The stable Lévy motion $L_t^\alpha$ has the jump measure $$\nu_{\alpha}(dy)=C_\alpha |y|^{-(1+\alpha)}\, dy,$$ where the coefficient $$C_{\alpha} =
\frac{\alpha}{2^{1-\alpha}\sqrt{\pi}}
\frac{\Gamma(\frac{1+\alpha}{2})}{\Gamma(1-\frac{\alpha}{2})}.$$
Note that the well-known Brownian motion $B_t$ is a special case corresponding to $\alpha=2$. Brownian motion $B_t$ has independent and stationary increments, and has continuous sample paths (a.s.). Moreover, $B_t -B_s $ has normal distribution $ \mathcal{N}(t-s, 0)$ for $ s \leq t $. In particular, $B_t$ has normal distribution $ \mathcal{N}(t, 0)$. That is, Brownian motion is a Gaussian process.
Nonlocal Fokker-Planck equation and numerical methods
-----------------------------------------------------
Consider a scalar stochastic differential equation with multiplicative Lévy noise $$\label{sde}
d X_t = f(X_t) dt + \sigma (X_t) d L_t^\alpha, \;\; X_0= x_0,$$ where $f$ is a given vector field (or drift) and $\sigma$ is the noise intensity.
The generator for this stochastic differential equation is $$A\varphi(x)=f(x)\varphi'(x) + \int_{\mathbb{R}^{1}\backslash \{0\}}[\varphi(x + y\sigma(x)) - \varphi(x)] \nu_\alpha(dy). \label{gener1}$$ Let $ z= y\sigma(x)$. The generator becomes $$A\varphi(x)=f(x)\varphi'(x) + |\sigma(x)|^\alpha \int_{\mathbb{R}^{1}\backslash \{0\}}[\varphi(x + z) - \varphi(x)]\nu_\alpha(dz). \label{gener2}$$
The Fokker-Planck equation for this stochastic differential equation, i.e., the probability density $p(x,t)$ for the solution process $X_t $ with initial condition $X_0=x_0$ is [@Duan2015] $$\label{fpe}
p_t = A^* p, \;\; p(x,0)=\delta(x-x_0),$$ where $A^*$ is the adjoint operator of the generator $A$ in Hilbert space $ L^2(R^1) $, as defined by $$\int_{\mathbb{R}^{1}\backslash \{0\}} A\varphi(x)u(x)dx = \int_{\mathbb{R}^{1}\backslash \{0\}}\varphi(x)A^*u(x)dx.$$ Then via integration by parts, we get the adjoint operator for $A$ $$A^*u(x)=\int_{\mathbb{R}^{1}\backslash \{0\}} [|\sigma(x-y)|^\alpha u(x-y)- |\sigma(x)|^\alpha u(x)] \; \nu_\alpha(dy). \label{hilbert}$$ Thus we have the nonlocal Fokker-Planck equation $$\label{fpe2}
p_t = - (f(x)p(x, t))_x +\int_{\mathbb{R}^{1}\backslash \{0\}} [|\sigma(x-y)|^\alpha p(x-y, t)- |\sigma(x)|^\alpha p(x, t)] \; \nu_\alpha(dy).$$
When the stable Lévy motion is replaced by Brownian motion, we have the following stochastic differential equation $$\label{sde2}
d X_t = f(X_t) dt + \sigma (X_t) d B_t, \;\; X_0= x_0.$$ The corresponding Fokker-Planck equation is a local partial differential equation $$\label{fpe3}
p_t =-(f(x)p(x, t))_x + \frac12 (\sigma^2(x) p(x, t))_{xx}, \;\; p(x,0)=\delta(x-x_0).$$
We use a numerical finite difference method developed in Gao et al. [@Gao2016] to simulate the nonlocal Fokker-Planck equation (\[fpe2\]) and use the standard finite difference method to simulate the local Fokker-Planck equation (\[fpe3\]).
Most probable phase portraits
-----------------------------
As the solution of the Fokker-Planck equation, the probability density function $p(x,t)$ is a surface in the $(x,t,p)-$space. At a given time instant $t$, the maximizer $x_m(t)$ for $p(x,t)$ indicates the most probable (i.e., maximal likely) location of this orbit at time $t$. The orbit traced out by $x_m(t)$ is called a most probable orbit starting at $x_0$. Thus, the deterministic orbit $x_m(t)$ follows the top ridge of the surface in the $(x,t,p)-$space as time goes on. For more information, see [@Duan2015; @Cheng2016].
*Definition*: A most probable equilibrium state is a state which either attracts or repels all nearby orbits. When it attracts all nearby orbits, it is called a most probable *stable* equilibrium state, while if it repels all nearby orbits, it is called a most probable *unstable* equilibrium state.
A phase portrait for a stochastic dynamical system, in the sense of most probable orbits, consists of representative orbits (including invariant objects such as most probable equilibrium states) in the state space. Both most probable phase portraits and most probable equilibrium states are deterministic geometric objects. As in the study of bifurcation for deterministic dynamical systems [@Guckenheimer1983; @Wiggins2003; @Strogatz1994], we examine the qualitative changes in the most probable phase portraits as a parameter varies. A simple qualitative change is the change in the ‘number’ and ‘stability type’ of ‘most probable equilibrium states’.
Results
=======
We now investigate the bifurcation for the scalar stochastic differential equation with multiplicative Lévy noise $$\label{pitchfork}
d X_t = f(r, X_t) dt + X_t \; d L_t^\alpha,$$ where $ f(r, X_t)= r X_t-X_t^3 $, $r$ is a real parameter, and the non-Gaussianity parameter $\alpha \in (0, 2)$. We also compare this bifurcation diagram with that of the same system under multiplicative Brownian noise $$\label{pitchforkBM}
d X_t = f(r, X_t) dt + X_t \; d B_t.$$
*The existing relevant works*. The stochastic bifurcation for $d X_t = f(r, X_t) dt +B_t$, with *additive* Brownian noise, was studied in [@Crauel1998; @Callaway2017] by examining the qualitative changes in invariant measure and their spectral stability. The stochastic bifurcation for $d X_t = f(r, X_t) dt + X_t \; B_t$, with *multiplicative* Brownian noise, was considered in [@Xu1995] by examining the qualitative changes in invariant measures with supports, and in [@Wang2015] by examining the qualitative changes in random complete quasi-solutions. Moreover, the stochastic bifurcation for $d X_t = f(r, X_t) dt +L_t^\alpha$, with *additive* Lévy noise, was studied in [@ChenHQ] by considering steady probability distributions for the solutions.
Bifurcation diagram: System under stable Lévy motion $L_t^\alpha$
-----------------------------------------------------------------
In the present work, we consider the case for a stochastic bifurcation in system (\[pitchfork\]), with *multiplicative* $\alpha-$ stable Lévy motion, using most probable phase portraits (especially most probable equilibrium states) as a parameter $r$ in vector field or the non-Gaussianity parameter $\alpha$ varies. As the analytical results for most probable equilibrium states are lacking at this time [@Cheng2016], we conduct numerical simulations to generate bifurcation diagrams.
For this system (\[pitchfork\]), $0$ is always a most probable equilibrium state. Figure \[Fig\_1\] shows the most probable orbits, starting from several initial points, with one or two most probable equilibrium states. To generate a bifurcation diagram, we plot all possible equilibrium states versus a parameter $r$ or $\alpha$ in the ‘parameter-equilibrium states plane’.
Figure \[Fig\_2\] shows the most probable equilibrium states with respect to $\alpha$. We divide the real line $r$ into five intervals, in each interval the system (\[pitchfork\]) has the same bifurcation phenomenon.\
(a) For $ r \lesssim -0.5$, the system (\[pitchfork\]) has only the stable equilibrium state $0$ with all $\alpha$ and there is no bifurcation.\
(b) For $-0.5 \lesssim r \lesssim -0.2$, there is a backward pitchfork bifurcation at $\alpha_1 \thickapprox 0.93$: with two stable equilibrium states and one unstable equilibrium state $0$ when $\alpha < \alpha_1$ but only one unstable equilibrium state $0$.\
(c) For $ -0.2 < r \lesssim 0.2$, there is a backward pitchfork bifurcation at $\alpha_{21} \thickapprox 1.07$ (two stable equilibrium states and one unstable equilibrium state $0$). Then there is a ‘collapsing’ bifurcation (as if three equilibrium states collapse into one) at $\alpha_{22} \thickapprox 0.23$ when two stable equilibrium states disappear but the equilibrium state $0$ remains and becomes stable.\
(d) For $ 0.2 \lesssim r < 2.5 $, there is a backward pitchfork bifurcation at $\alpha_{31} \thickapprox 0.69$, a forward pitchfork bifurcation at $\alpha_{32} \thickapprox 0.95$, and finally a ‘collapsing’ bifurcation at $\alpha_{33} \thickapprox 1.5$ when two stable equilibrium states disappear but the equilibrium state $0$ remains and becomes stable.\
(e) For $ r \gtrsim 2.5$, the system (\[pitchfork\]) has two stable equilibrium states and one unstable equilibrium state $0$ and there is no bifurcation.\
Figure \[Fig\_3\] shows the most probable equilibrium states with respect to $r$. The parameter $\alpha$ can be divided into two parts, with $\alpha=1$ as the critical or borderline value.\
(a) For $ 0 < \alpha \lesssim 1 $, the stochastic dynamical system (\[pitchfork\]) has a forward pitchfork bifurcation at $r_{11} \thickapprox -0.25$, a ‘collapsing’ bifurcation at $r_{12} \thickapprox 0.73$ when two stable equilibrium states disappear and the equilibrium state $0$ becomes stable, and finally a forward pitchfork bifurcation at $r_{13} \thickapprox 1.05$.\
(b) For $1 < \alpha < 2 $ , there is a forward pitchfork bifurcation at $r_{21} \thickapprox 0.20$ and then a ‘collapsing’ bifurcation at $r_{22} \thickapprox 1.17 $, suddenly a small forward pitchfork bifurcation at $r_{23} \thickapprox 1.29 $, again a ‘collapsing’ bifurcation at $r_{24} \thickapprox 1.31 $, and finally a forward pitchfork bifurcation at $r_{25} \thickapprox 1.48 $.\
Bifurcation diagram: System under Brownian motion $B_t$
-------------------------------------------------------
![(Color online) Bifurcation diagram for system (\[pitchforkBM\]) with multiplicative Brown noise: Stochastic pitchfork bifurcation at $r \thickapprox -1.6$ .[]{data-label="Fig.4"}](rbm3.eps){width="45.00000%"}
Figure \[Fig.4\] shows the bifurcation diagram, i.e., the most probable equilibrium states versus parameter $r$, for the system (\[pitchforkBM\]) with multiplicative Brownian motion. There is a pitchfork bifurcation at $r \thickapprox -1.6$, and this bifurcation was also detected in [@Xu1995 Fig. 2(b)] by examining the support of the invariant measures. This bifurcation diagram is qualitatively the same as the bifurcation diagram in Figure \[Fig.0\] for the corresponding deterministic system $\dot x = rx -x^3$, although the bifurcation value is different due to the effect of noise. More significantly, this bifurcation is fundamentally different from the bifurcations under $\alpha$-stable Lévy noise, as shown in Figure \[Fig\_3\].
Conclusion
==========
Although bifurcation studies for deterministic dynamical systems have a long history, the stochastic bifurcation investigation is still in its early stage. One reason for this slow development in stochastic bifurcation is due to the lack of appropriate phase portraits, in contrast to deterministic dynamical systems.
One promising option for phase portraits of stochastic dynamical systems is the so-called most probable phase portraits [@Duan2015; @Cheng2016]. We thus conduct stochastic bifurcation study with the help of these phase portraits.
To demonstrate this stochastic bifurcation approach, we study the bifurcation for a system under multiplicative stable Lévy noise (non-Gaussian). The deterministic counterpart of this system has the well-known pitchfork bifurcation. The existing works in this topic is for the case of Brownian noise (Gaussian) and in terms of the qualitative changes of invariant measures or point attractors. But analytical studies of invariant measures, together with their spectra and supports, are not easily available for stochastic dynamical systems with Lévy noise. This also motivates us to investigate stochastic bifurcation by most probable phase portraits, especially their invariant structures such as most probable equilibrium states. By numerically examining the qualitative changes of equilibrium states in its most probable phase portraits, we have detected some bifurcation phenomena such as the double or triple pitchfork bifurcation and a collapsing bifurcation, when a parameter in the vector field or in Lévy noise varies.
We would like to thank Xiujun Cheng and Jian Ren for helpful discussions. This work was partly supported by the National Science Foundation grant 1620449, and the National Natural Science Foundation of China grants 11531006 and 11771449.
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---
abstract: 'Primordial black holes (PBHs) are a profound signature of primordial cosmological structures and provide a theoretical tool to study nontrivial physics of the early Universe. The mechanisms of PBH formation are discussed and observational constraints on the PBH spectrum, or effects of PBH evaporation, are shown to restrict a wide range of particle physics models, predicting an enhancement of the ultraviolet part of the spectrum of density perturbations, early dust-like stages, first order phase transitions and stages of superheavy metastable particle dominance in the early Universe. The mechanism of closed wall contraction can lead, in the inflationary Universe, to a new approach to galaxy formation, involving primordial clouds of massive BHs created around the intermediate mass or supermassive BH and playing the role of galactic seeds.'
author:
- 'Maxim Yu. Khlopov$^{1,2,3}$[^1]'
title: Primordial Black Holes
---
[*$^{1}$ Centre for Cosmoparticle Physics “Cosmion”\
Moscow, 125047, Miusskaya pl. 4\
$^{2}$ Moscow State Engineering Physics Institute,\
Kashirskoe Sh., 31, Moscow 115409, Russia, and\
$^{3}$ APC laboratory 10, rue Alice Domon et Léonie Duquet\
75205 Paris Cedex 13, France*]{}
Introduction
============
The convergence of the frontiers of our knowledge in micro- and macro- worlds leads to the wrong circle of problems, illustrated by the mystical Uhroboros (self-eating-snake). The Uhroboros puzzle may be formulated as follows: [*The theory of the Universe is based on the predictions of particle theory, that need cosmology for their test*]{}. Cosmoparticle physics [@ADS; @MKH; @book; @book3; @Khlopov:2004jb; @bled] offers the way out of this wrong circle. It studies the fundamental basis and mutual relationship between micro-and macro-worlds in the proper combination of physical, astrophysical and cosmological signatures. Some aspects of this relationship, which arise in the astrophysical problem of Primordial Black Holes (PBH) is the subject of this review.
In particle theory Noether’s theorem relates the exact symmetry to conservation of respective charge. Extensions of the standard model imply new symmetries and new particle states. The respective symmetry breaking induces new fundamental physical scales in particle theory. If the symmetry is strict, its existence implies new conserved charge. The lightest particle, bearing this charge, is stable. It gives rise to the deep relationship between dark matter candidates and particle symmetry beyond the Standard model.
The mechanism of spontaneous breaking of particle symmetry also has cosmological impact. Heating of condensed matter leads to restoration of its symmetry. When the heated matter cools down, phase transition to the phase of broken symmetry takes place. In the course of the phase transitions, corresponding to given type of symmetry breaking, topological defects can form. One can directly observe formation of such defects in liquid crystals or in superfluid He. In the same manner the mechanism of spontaneous breaking of particle symmetry implies restoration of the underlying symmetry. When temperature decreases in the course of cosmological expansion, transitions to the phase of broken symmetry can lead, depending on the symmetry breaking pattern, to formation of topological defects in very early Universe. Defects can represent new forms of stable particles (as it is in the case of magnetic monopoles [@t'Hooft; @polyakov; @kz; @Priroda; @preskill; @SAOmonop]), or extended structures, such as cosmic strings [@zv1; @zv2] or cosmic walls [@okun].
In the old Big bang scenario cosmological expansion and its initial conditions were given [*a priori*]{} [@Weinberg; @ZNSEU]. In the modern cosmology expansion of Universe and its initial conditions are related to inflation [@Star80; @Guthinfl; @Linde:1981mu; @Albrecht; @Linde:1983gd], baryosynthesis and nonbaryonic dark matter (see review in [@Lindebook; @Kolbbook]). Physics, underlying inflation, baryosynthesis and dark matter, is referred to extensions of the standard model, and variety of such extensions makes the whole picture in general ambiguous. However, in a framework of each particular physical realization of inflationary model with baryosynthesis and dark matter the corresponding model dependent cosmological scenario can be specified in all details. In such scenario main stages of cosmological evolution, structure and physical content of the Universe reflect structure of the underlying physical model. The latter should include with necessity the standard model, describing properties of baryonic matter, and its extensions, responsible for inflation, baryosynthesis and dark matter. In no case cosmological impact of such extensions is reduced to reproduction of these three phenomena only. A nontrivial path of cosmological evolution, specific for each particular realization of inflational model with baryosynthesis and nonbaryonic dark matter, always contains some additional model dependent cosmologically viable predictions, which can be confronted with astrophysical data. Here we concentrate on Primordial Black Holes as profound signature of such phenomena.
It was probably Pierre-Simon Laplace [@Laplace] in the beginning of XIX century, who noted first that in very massive stars escape velocity can exceed the speed of light and light can not come from such stars. This conclusion made in the framework of Newton mechanics and Newton corpuscular theory of light has further transformed into the notion of “black hole” in the framework of general relativity and electromagnetic theory. Any object of mass $M$ can become a black hole, being put within its gravitational radius $r_g=2 G M/c^2.$ At present time black holes (BH) can be created only by a gravitational collapse of compact objects with mass more than about three Solar mass [@1; @ZNRA]. It can be a natural end of massive stars or can result from evolution of dense stellar clusters. However in the early Universe there were no limits on the mass of BH. Ya.B. Zeldovich and I.D. Novikov [@ZN] noticed that if cosmological expansion stops in some region, black hole can be formed in this region within the cosmological horizon. It corresponds to strong deviation from general expansion and reflects strong inhomogeneity in the early Universe. There are several mechanisms for such strong inhomogeneity and we’ll trace their links to cosmological consequences of particle theory.
Primordial Black Holes (PBHs) are a very sensitive cosmological probe for physics phenomena occurring in the early Universe. They could be formed by many different mechanisms, [*e.g.*]{}, initial density inhomogeneities [@hawking1; @hawkingCarr] and non-linear metric perturbations [@Bullock:1996at; @Ivanov:1997ia; @Bullock:1998mi], blue spectra of density fluctuations [@Khlopov:1984wc; @polnarev; @Lidsey:1995ir; @Kotok:1998rp; @Dubrovich02; @Sendouda:2006nu], a softening of the equation of state [@canuto; @Khlopov:1984wc; @polnarev], development of gravitational instability on early dust-like stages of dominance of supermassive particles and scalar fields [@khlopov0; @polnarev0; @polnarev1; @khlopov1] and evolution of gravitationally bound objects formed at these stages [@Kalashnikov; @Kadnikov], collapse of cosmic strings [@hawking2; @Polnarev:1988dh; @Hansen:2000jv; @Cheng:1996du; @Nagasawa:2005hv] and necklaces [@Matsuda:2005ez], a double inflation scenario [@nas; @Kim:1999xg; @Yamaguchi:2001zh; @Yamaguchi:2002sp], first order phase transitions [@hawking3; @pt1Jedamzik; @kkrs; @kkrs1; @kkrs2], a step in the power spectrum [@Sakharov0; @polarski1], etc. (see [@polnarev; @book; @book3; @Carr:2003bj; @book2] for a review).
Being formed, PBHs should retain in the Universe and, if survive to the present time, represent a specific form of dark matter [@khlopov7; @Ivanov:1994pa; @book; @book3; @Blais:2002nd; @Chavda:2002cj; @Afshordi:2003zb; @book2; @Chen:2004ft]. Effect of PBH evaporation by S.W.Hawking [@hawking4] makes evaporating PBHs a source of fluxes of products of evaporation, particularly of $\gamma$ radiation [@Page:1976wx]. MiniPBHs with mass below $10^{14}$ g evaporate completely and do not survive to the present time. However, effect of their evaporation should cause influence on physical processes in the early Universe, thus providing a test for their existence by methods of cosmoarcheology [@Cosmoarcheology], studying cosmological imprints of new physics in astrophysical data. In a wide range of parameters the predicted effect of PBHs contradicts the data and it puts restrictions on mechanism of PBH formation and the underlying physics of very early Universe. On the other hand, at some fixed values of parameters, PBHs or effects of their evaporation can provide a nontrivial solution for astrophysical problems.
Various aspects of PBH physics, mechanisms of their formation, evolution and effects are discussed in [@carr1; @carrMG; @LGreen; @khlopov6; @polnarev; @Grillo:1980uj; @Chapline:1975tn; @Hayward:1989jq; @Yokoyama:1995ex; @Kim:1996hr; @Heckler:1997jv; @MacGibbon:2007yq; @Page:2007yr; @green; @Niemeyer:1997mt; @Kribs:1999bs; @Green:2004wb; @Yokoyama:1998pt; @Yokoyama:1998xd; @Yokoyama:1999xi; @Bringmann:2001yp; @Dimopoulos:2003ce; @Nozari:2007kv; @LythMalik; @Zaballa:2006kh; @Harada:2004pf; @Custodio:2005en; @Bousso:1995cc; @Bousso:1996wy; @Elizalde:1999dw; @Nojiri:1999vv; @Bousso:1999iq; @Silk:2000em; @Polarski:2001jk; @Barrow:1996jk; @Paul:2000jb; @Paul:2001yt; @Paul:2005bk; @Polarski:2001yn; @Carr:1993aq; @Yokoyama:1998qw; @Kaloper:2004yj; @Pelliccia:2007fh; @Stojkovic:2005zh; @Soda; @Ahn:2006uc; @babichev4; @babichev5; @babichev6; @Guariento:2007bs] particularly specifying PBH formation and effects in braneworld cosmology [@Guedens:2002km; @Guedens:2002sd; @Clancy:2003zd; @Tikhomirov:2005bt], on inflationary preheating [@Bassett:2000ha], formation of PBHs in QCD phase transition [@Jedamzik:1998hc; @Widerin:1998my], properties of superhorizon BHs [@Harada:2005sc; @Harada:2006gn], role of PBHs in baryosynthesis [@Grillo:1980rt; @Barrow:1990he; @Turner:1979bt; @Upadhyay:1999vk; @Bugaev:2001xr], effects of PBH evaporation in the early Universe and in modern cosmic ray, neutrino and gamma fluxes [@mujana; @Fegan:1978zn; @Green:2001kw; @Frampton:2005fk; @MacGibbon:1990zk; @MacGibbon:1991tj; @Halzen:1991uw; @Halzen:1995hu; @Bugaev:2000bz; @Bugaev:2002yt; @Volkova:1994fb; @Gibilisco:1996ft; @Golubkov:2000qy; @He:2002vz; @Gibilisco:1996dk; @Custodio:2002jv; @Sendouda:2003dc; @Maki:1995pa; @barraupbar; @Barrau:2002mc; @Wells:1998jv; @Cline:1996uk; @Xu:1998hn; @Cline:1998fx; @Sendouda:2006yc; @Barrau:1999sk; @Derishev:1999xn; @Tikhomirov:2004rs; @Seto; @barrau; @barraugamma; @barrauprd], in creation of hypothetical particles [@Bell:1998jk; @lemoine; @green1; @barrau2], PBH clustering and creation of supermassive BHs [@Bean:2002kx; @Duechting:2004dk; @Chisholm:2005vm; @Dokuchaev:2004kr; @Mack:2006gz; @Rubin:2005pq], effects in cosmic rays and colliders from PBHs in low scale gravity models [@barrauADDBH; @barrauADDac]. Here we outline the role of PBHs as a link in cosmoarcheoLOGICAL chain, connecting cosmological predictions of particle theory with observational data. We discuss the way, in which spectrum of PBHs reflects properties of superheavy metastable particles and of phase transitions on inflationary and post-inflationary stages. We briefly review possible cosmological reflections of particle physics (section \[Cosmophenomenology\]), illustrate in section \[dust\] some mechanisms of PBH formation on stage of dominance of superheavy particles and fields (subsection \[particles\]) and from second order phase transition on inflationary stage. Effective mechanism of BH formation during bubble nucleation provides a sensitive tool to probe existence of cosmological first order phase transitions by PBHs (section \[phasetransitions\]). Existence of stable remnants of PBH evaporation can strongly increase the sensitivity of such probe and we demonstrate this possibility in section \[gravitino\] on an example of gravitino production in PBH evaporation. Being formed within cosmological horizon, PBHs seem to have masses much less than the mass of stars, constrained by small size of horizon in very early Universe. However, if phase transition takes place on inflationary stage, closed walls of practically any size can be formed (subsection \[walls\]) and their successive collapse can give rise to clouds of massive black holes, which can play the role of seeds for galaxies (section \[MBHwalls\]). The impact of constraints and cosmological scenarios, involving primordial black holes, is briefly discussed in section \[Discussion\].
PBHs as cosmological reflection of new physics {#Cosmophenomenology}
==============================================
The simplest primordial form of new physics is a gas of new stable massive particles, originated from early Universe. For particles with mass $m$, at high temperature $T>m$ the equilibrium condition, $n \cdot \sigma v \cdot t > 1$ is valid, if their annihilation cross section $\sigma > 1/(m m_{pl})$ is sufficiently large to establish equilibrium. At $T<m$ such particles go out of equilibrium and their relative concentration freezes out. Weakly interacting species decouple from plasma and radiation at $T>m$, when $n \cdot \sigma v
\cdot t \sim 1$, i.e. at $T_{dec} \sim (\sigma m_{pl})^{-1}$. This is the main idea of calculation of primordial abundance for WIMP-like dark matter candidates (see e.g. [@book; @book3; @Cosmoarcheology] for details). The maximal temperature, which is reached in inflationary Universe, is the reheating temperature, $T_{r}$, after inflation. So, very weakly interacting particles with annihilation cross section $\sigma <
1/(T_{r} m_{pl})$, as well as very heavy particles with mass $m \gg
T_{r}$ can not be in thermal equilibrium, and the detailed mechanism of their production should be considered to calculate their primordial abundance.
Decaying particles with lifetime $\tau$, exceeding the age of the Universe, $t_{U}$, $\tau > t_{U}$, can be treated as stable. By definition, primordial stable particles survive to the present time and should be present in the modern Universe. The net effect of their existence is given by their contribution into the total cosmological density. They can dominate in the total density being the dominant form of cosmological dark matter, or they can represent its subdominant fraction. In the latter case more detailed analysis of their distribution in space, of their condensation in galaxies, of their capture by stars, Sun and Earth, as well as effects of their interaction with matter and of their annihilation provides more sensitive probes for their existence. In particular, hypothetical stable neutrinos of 4th generation with mass about 50 GeV are predicted to form the subdominant form of modern dark matter, contributing less than 0,1 % to the total density [@ZKKC; @DKKM]. However, direct experimental search for cosmic fluxes of weakly interacting massive particles (WIMPs) may be sensitive to existence of such component (see [@DAMA; @DAMA-review; @CDMS; @CDMS2] and references therein). It was shown in [@Fargion99; @Grossi; @Belotsky; @Belotsky2] that annihilation of 4th neutrinos and their antineutrinos in the Galaxy can explain the galactic gamma-background, measured by EGRET in the range above 1 GeV, and that it can give some clue to explanation of cosmic positron anomaly, claimed to be found by HEAT. 4th neutrino annihilation inside the Earth should lead to the flux of underground monochromatic neutrinos of known types, which can be traced in the analysis of the already existing and future data of underground neutrino detectors [@Belotsky; @BKS1; @BKS2; @BKS3].
New particles with electric charge and/or strong interaction can form anomalous atoms and contain in the ordinary matter as anomalous isotopes. For example, if the lightest quark of 4th generation is stable, it can form stable charged hadrons, serving as nuclei of anomalous atoms of e.g. anomalous helium [@BKS; @BKSR; @BKSR1; @BKSR2; @BKSR3; @BKSR4].
Primordial unstable particles with lifetime, less than the age of the Universe, $\tau < t_{U}$, can not survive to the present time. But, if their lifetime is sufficiently large to satisfy the condition $\tau \gg (m_{pl}/m) \cdot (1/m)$, their existence in early Universe can lead to direct or indirect traces. Cosmological flux of decay products contributing into the cosmic and gamma ray backgrounds represents the direct trace of unstable particles. If the decay products do not survive to the present time their interaction with matter and radiation can cause indirect trace in the light element abundance or in the fluctuations of thermal radiation.
If particle lifetime is much less than $1$ s multi-step indirect traces are possible, provided that particles dominate in the Universe before their decay. On dust-like stage of their dominance black hole formation takes place, and spectrum of such primordial black holes traces particle properties (mass, frozen concentration, lifetime) [@polnarev]. Particle decay in the end of dust like stage influences the baryon asymmetry of the Universe. In any way cosmophenomenoLOGICAL chains link the predicted properties of even unstable new particles to the effects accessible in astronomical observations. Such effects may be important in analysis of the observational data.
Parameters of new stable and metastable particles are also determined by a pattern of particle symmetry breaking. This pattern is reflected in a succession of phase transitions in the early Universe. First order phase transitions proceed through bubble nucleation, which can result in black hole formation (see e.g. [@kkrs] and [@book2] for review and references). Phase transitions of the second order can lead to formation of topological defects, such as walls, string or monopoles. The observational data put severe constraints on magnetic monopole [@kz] and cosmic wall production [@okun], as well as on the parameters of cosmic strings [@zv1; @zv2]. Structure of cosmological defects can be changed in succession of phase transitions. More complicated forms like walls-surrounded-by-strings can appear. Such structures can be unstable, but their existence can leave a trace in nonhomogeneous distribution of dark matter and give rise to large scale structures of nonhomogeneous dark matter like [*archioles*]{} [@Sakharov2; @kss; @kss2]. Primordial Black Holes represent a profound signature of such structures.
PBHs from early dust-like stages {#dust}
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A possibility to form a black hole is highly improbable in homogeneous expanding Universe, since it implies metric fluctuations of order 1. For metric fluctuations distributed according to Gaussian law with dispersion $$\label{DispBH}\left\langle \delta^2 \right\rangle \ll
1$$ a probability for fluctuation of order 1 is determined by exponentially small tail of high amplitude part of this distribution. This probability can be even more suppressed in a case of non-Gaussian flutuations [@Bullock:1996at].
In the Universe with equation of state $$\label{EqState}p=\gamma \epsilon,$$ with numerical factor $\gamma$ being in the range $$\label{FacState}0 \le \gamma \le 1$$ a probability to form black hole from fluctuation within cosmological horizon is given by (see e.g. [@book; @book3] for review and references) $$\label{ProbBH}W_{PBH} \propto \exp \left(-\frac{\gamma^2}{2
\left\langle \delta^2 \right\rangle}\right).$$ It provides exponential sensitivity of PBH spectrum to softening of equation of state in early Universe ($\gamma \rightarrow 0$) or to increase of ultraviolet part of spectrum of density fluctuations ($\left\langle \delta^2 \right\rangle \rightarrow 1$). These phenomena can appear as cosmological consequence of particle theory.
Dominance of superheavy particles in early Universe {#particles}
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Superheavy particles can not be studied at accelerators directly. If they are stable, their existence can be probed by cosmological tests, but there is no direct link between astrophysical data and existence of superheavy metastable particles with lifetime $\tau \ll
1s$. It was first noticed in [@khlopov0] that dominance of such particles in the Universe before their decay at $t \le \tau$ can result in formation of PBHs, retaining in Universe after the particles decay and keeping some information on particle properties in their spectrum. It provided though indirect but still a possibility to probe existence of such particles in astrophysical observations. Even the absence of observational evidences for PBHs is important. It puts restrictions on allowed properties of superheavy metastable particles, which might form such PBHs on a stage of particle dominance, and thus constrains parameters of models, predicting these particles.
After reheating, at $$\label{Eareq}T <
T_0=rm$$ particles with mass $m$ and relative abundance $r=n/n_r$ (where $n$ is frozen out concentration of particles and $n_r$ is concentration of relativistic species) must dominate in the Universe before their decay. Dominance of these nonrelativistic particles at $t>t_0$, where $$\label{EarMD}t_0=\frac{m_{pl}}{T_0^2},$$ corresponds to dust like stage with equation of state $p=0,$ at which particle density fluctuations grow as$$\label{dens}\delta(t)=\frac{\delta \rho}{\rho} \propto t^{2/3}$$ and development of gravitational instability results in formation of gravitationally bound systems, which decouple at $$\label{decoup}t \sim t_f \approx t_i
\delta(t_i)^{-3/2}$$ from general cosmological expansion, when $\delta(t_f)\sim 1$ for fluctuations, entering horizon at $t=t_i>t_0$ with amplitude $\delta(t_i)$.
Formation of these systems can result in black hole formation either immediately after the system decouples from expansion or in result of evolution of initially formed nonrelativistic gravitationally bound system.
If density fluctuation is especially homogeneous and isotropic, it directly collapses to BH as soon as the amplitude of fluctuation grows to 1 and the system decouples from expansion. A probability for direct BH formation in collapse of such homogeneous and isotropic configurations gives minimal estimation of BH formation on dust-like stage.
This probability was calculated in [@khlopov0] with the use of the following arguments. In the period $t \sim t_f$, when fluctuation decouples from expansion, its configuration is defined by averaged density $\rho_1$, size $r_1$, deviation from sphericity $s$ and by inhomogeneity $u$ of internal density distribution within the fluctuation. Having decoupled from expansion, the configuration contracts and the minimal size to which it can contract is $$\label{sphcontr}r_{min} \sim s r_1,$$ being determined by a deviation from sphericity $$\label{spheric}s=\max\{\left\vert\gamma_1-\gamma_2\right\vert,\left\vert\gamma_1-
\gamma_3\right\vert,\left\vert\gamma_3-\gamma_2\right\vert\},$$ where $\gamma_1$, $\gamma_2$ and $\gamma_3$ define a deformation of configuration along its three main orthogonal axes. It was first noticed in [@khlopov0] that to form a black hole in result of such contraction it is sufficient that configuration returns to the size $$\label{rminBH}r_{min} \sim r_g \sim t_i \sim
\delta(t_i) r_1,$$ which had the initial fluctuation $\delta(t_i)$, when it entered horizon at cosmological time $t_i$. If $$\label{spher}s \le \delta(t_i),$$ configuration is sufficiently isotropic to concentrate its mass in the course of collapse within its gravitational radius, but such concentration also implies sufficient homogeneity of configuration. Density gradients can result in gradients of pressure, which can prevent collapse to BH. This effect does not take place for contracting collisionless gas of weakly interacting massive particles, but due to inhomogeneity of collapse the particles, which have already passed the caustics can free stream beyond the gravitational radius, before the whole mass is concentrated within it. Collapse of nearly spherically symmetric dust configuration is described by Tolmen solution. It’s analysis [@polnarev0; @polnarev1; @KP; @polnarev] has provided a constraint on the inhomogeneity $u=\delta \rho_1/\rho_1$ within the configuration. It was shown that both for collisionless and interacting particles the condition $$\label{inhom}u<\delta(t_i)^{3/2}$$ is sufficient for configuration to contract within its gravitational radius.
A probability for direct BH formation is then determined by a product of probability for sufficient initial sphericity $W_s$ and homogeneity $W_u$ of configuration, which is determined by the phase space for such configurations. In a calculation of $W_s$ one should take into account that the condition (\[spher\]) implies 5 conditions for independent components of tensor of deformation before its diagonalization (2 conditions for three diagonal components to be close to each other and 3 conditions for nondiagonal components to be small). Therefore, the probability of sufficient sphericity is given by [@khlopov0; @polnarev0; @polnarev1; @KP; @polnarev] $$\label{Wspher}W_s \sim \delta(t_i)^{5}$$ and together with the probability for sufficient homogeneity $$\label{Winhom}W_u \sim \delta(t_i)^{3/2}$$ results in the strong power-law suppression of probability for direct BH formation \[WPBH\] W\_[PBH]{} = W\_s W\_u \~(t\_i)\^[13/2]{}.Though this calculation was originally done in [@khlopov0; @polnarev0; @polnarev1; @KP; @polnarev] for Gaussian distribution of fluctuations, it does not imply specific form of high amplitude tail of this distribution and thus should not change strongly in a case of non-Gaussian fluctuations [@Bullock:1996at].
The mechanism [@khlopov0; @polnarev0; @polnarev1; @KP; @polnarev; @book; @book3] is effective for formation of PBHs with mass in an interval \[Mint\]M\_0 M M\_[bhmax]{}.The minimal mass corresponds to the mass within cosmological horizon in the period $t
\sim t_0,$ when particles start to dominate in the Universe and it is equal to [@khlopov0; @polnarev0; @polnarev1; @KP; @polnarev; @book; @book3]\[MBHmin\] M\_[0]{} = t\^3\_0 m\_[pl]{}()\^2. The maximal mass is indirectly determined by the condition \[Mconmax\]= t(M\_[bhmax]{}) (M\_[bhmax]{})\^[-3/2]{}that fluctuation in the considered scale $M_{bhmax}$, entering the horizon at $t(M_{bhmax})$ with an amplitude $\delta(M_{bhmax})$ can manage to grow up to nonlinear stage, decouple and collapse before particles decay at $t=\tau.$ For scale invariant spectrum $\delta(M)=\delta_0$ the maximal mass is given by [@book2]\[MBHmax\] M\_[bhmax]{} = m\_[pl]{} \_0\^[3/2]{} =m\_[pl]{}\^2 \_0\^[3/2]{}.The probability, given by Eq.(\[WPBH\]), is also appropriate for formation of PBHs on dust-like preheating stage after inflation [@khlopov1; @book; @book3]. The simplest example of such stage can be given with the use of a model of homogeneous massive scalar field [@book; @book3]. Slow rolling of the field in the period $t \ll 1/m$ (where $m$ is the mass of field) provides chaotic inflation scenario, while at $t
> 1/m$ the field oscillates with period $1/m$. Coherent oscillations of the field correspond to an averaged over period of oscillations dust-like equation of state $p=0,$ at which gravitational instability can develop. The minimal mass in this case corresponds to the Jeans mass of scalar field, while the maximal mass is also determined by a condition that fluctuation grows and collapses before the scalar field decays and reheats the Universe.
The probability $W_{PBH}(M)$ determines the fraction of total density \[beta\](M)= W\_[PBH]{}(M),corresponding to PBHs with mass $M$. For $\delta(M) \ll 1$ this fraction, given by Eq.(\[WPBH\]), is small. It means that the bulk of particles do not collapse directly in black holes, but form gravitationally bound systems. Evolution of these systems can give much larger amount of PBHs, but it strongly depends on particle properties.
Superweakly interacting particles form gravitationally bound systems of collisionless gas, which remind modern galaxies with collisionless gas of stars. Such system can finally collapse to black hole, but energy dissipation in it and consequently its evolution is a relatively slow process [@ZPod; @book; @book3]. The evolution of these systems is dominantly determined by evaporation of particles, which gain velocities, exceeding the parabolic velocity of system. In the case of binary collisions the evolution timescale can be roughly estimated [@ZPod; @book; @book3] as \[tevbin\] t\_[ev]{} = t\_[ff]{}for gravitationally bound system of $N$ particles, where the free fall time $t_{ff}$ for system with density $\rho$ is $t_{ff} \approx (4
\pi G \rho)^{-1/2}.$ This time scale can be shorter due to collective effects in collisionless gas [@GurSav] and be at large $N$ of the order of \[tevcol\] t\_[ev]{} \~N\^[2/3]{} t\_[ff]{}.However, since the free fall time scale for gravitationally bound systems of collisionless gas is of the order of cosmological time $t_f$ for the period, when these systems are formed, even in the latter case the particles should be very long living $\tau \gg t_f$ to form black holes in such slow evolutional process.
The evolutional time scale is much smaller for gravitationally bound systems of superheavy particles, interacting with light relativistic particles and radiation. Such systems have analogy with stars, in which evolution time scale is defined by energy loss by radiation. An example of such particles give superheavy color octet fermions of asymptotically free SU(5) model [@Kalashnikov] or magnetic monopoles of GUT models. Having decoupled from expansion, frozen out particles and antiparticles can annihilate in gravitationally bound systems, but detailed numerical simulation [@Kadnikov] has shown that annihilation can not prevent collapse of the most of mass and the timescale of collapse does not exceed the cosmological time of the period, when the systems are formed.
Spikes from phase transitions on inflationary stage
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Scale non-invariant spectrum of fluctuations, in which amplitude of small scale fluctuations is enhanced, can be another factor, increasing the probability of PBH formation. The simplest functional form of such spectrum is represented by a blue spectrum with a power law dispersion \^2(M) M\^[-k]{},with amplitude of fluctuations growing at $k>0$ to small $M$. The realistic account for existence of other scalar fields together with inflaton in the period of inflation can give rise to spectra with distinguished scales, determined by parameters of considered fields and their interaction.
In chaotic inflation scenario interaction of a Higgs field $\phi$ with inflaton $\eta$ can give rise to phase transitions on inflationary stage, if this interaction induces positive mass term $+\frac {\nu^2}{2} \eta^2 \phi^2$. When in the course of slow rolling the amplitude of inflaton decreases below a certain critical value $\eta_c = m_{\phi}/\nu$ the mass term in Higgs potential \[Higgs\] V(, )=- \^2+\^4 +\^2 \^2 changes sign and phase transition takes place. Such phase transitions on inflationary stage lead to the appearance of a characteristic spikes in the spectrum of initial density perturbations. These spike–like perturbations re-enter the horizon during the radiation or dust like era and could in principle collapse to form primordial black holes. The possibility of such spikes in chaotic inflation scenario was first pointed out in [@KofLin] and realized in [@Sakharov0] as a mechanism of of PBH formation for the model of horizontal unification [@Berezhiani1; @Berezhiani2; @Berezhiani3; @Sakharov1].
For vacuum expectation value of a Higgs field = = vand $\lambda \sim 10^{-3}$ the amplitude $\delta$ of spike in spectrum of density fluctuations, generated in phase transition on inflationary stage is given by [@Sakharov0] \[dspike\]with \[spike\] s=-, where $\kappa
\sim 1.$
If phase transition takes place at $e$–folding $N$ before the end of inflation and the spike re-enters horizon on radiation dominance (RD) stage, it forms Black hole of mass \[Mrd\] M {2 N}, where $H_0$ is the Hubble constant in the period of inflation.
If the spike re-enters horizon on matter dominance (MD) stage it should form black holes of mass \[Mmd\] M {3 N}.
First order phase transitions as a source of black holes in the early Universe {#phasetransitions}
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First order phase transition go through bubble nucleation. Remind the common example of boiling water. The simplest way to describe first order phase transitions with bubble creation in early Universe is based on a scalar field theory with two non degenerated vacuum states. Being stable at a classical level, the false vacuum state decays due to quantum effects, leading to a nucleation of bubbles of true vacuum and their subsequent expansion [@6]. The potential energy of the false vacuum is converted into a kinetic energy of bubble walls thus making them highly relativistic in a short time. The bubble expands till it collides with another one. As it was shown in [@hawking3; @5] a black hole may be created in a collision of several bubbles. The probability for collision of two bubbles is much higher. The opinion of the BH absence in such processes was based on strict conservation of the original O(2,1) symmetry. As it was shown in [@kkrs; @kkrs1; @kkrs2] there are ways to break it. Firstly, radiation of scalar waves indicates the entropy increasing and hence the permanent breaking of the symmetry during the bubble collision. Secondly, the vacuum decay due to thermal fluctuation does not possess this symmetry from the beginning. The investigations [@kkrs; @kkrs1; @kkrs2] have shown that BH can be created as well with a probability of order unity in collisions of only two bubbles. It initiates an enormous production of BH that leads to essential cosmological consequences discussed below.
In subsection \[field\] the evolution of the field configuration in the collisions of bubbles is discussed. The BH mass distribution is obtained in subsection \[Collapse\]. In subsection \[pt1\] cosmological consequences of BH production in bubble collisions at the end of inflation are considered.
Evolution of field configuration in collisions of vacuum bubbles {#field}
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Consider a theory where a probability of false vacuum decay equals $\Gamma $ and difference of energy density between the false and true vacuum outside equals $\rho_v$. Initially bubbles are produced at rest however walls of the bubbles quickly increase their velocity up to the speed of light $v=c=1$ because a conversion of the false vacuum energy into its kinetic ones is energetically favorable.
Let us discuss dynamics of collision of two true vacuum bubbles that have been nucleated in points $({\bf r}_1,t_1),({\bf r}_2,t_2)$ and which are expanding into false vacuum. Following papers [@hawking3; @7] let us assume for simplicity that the horizon size is much greater than the distance between the bubbles. Just after collision mutual penetration of the walls up to the distance comparable with its width is accompanied by a significant potential energy increase [@8]. Then the walls reflect and accelerate backwards. The space between them is filled by the field in the false vacuum state converting the kinetic energy of the wall back to the energy of the false vacuum state and slowdown the velocity of the walls. Meanwhile the outer area of the false vacuum is absorbed by the outer wall, which expands and accelerates outwards. Evidently, there is an instant when the central region of the false vacuum is separated. Let us note this false vacuum bag (FVB) does not possess spherical symmetry at the moment of its separation from outer walls but wall tension restores the symmetry during the first oscillation of FVB. As it was shown in [@7], the further evolution of FVB consists of several stages:
1\) FVB grows up to the definite size $D_M$ until the kinetic energy of its wall becomes zero;
2\) After this moment the false vacuum bag begins to shrink up to a minimal size $D^{*}$;
3\) Secondary oscillation of the false vacuum bag occurs.
The process of periodical expansions and contractions leads to energy losses of FVB in the form of quanta of scalar field. It has been shown in the [@7; @9] that only several oscillations take place. On the other hand, important note is that the secondary oscillations might occur only if the minimal size of the FVB would be larger than its gravitational radius, $D^{*}>r_g$. Then oscillating solutions of “quasilumps” can be realized [@oscilon]. The opposite case ($D^{*}<r_g$ ) leads to a BH creation with the mass about the mass of the FVB. As it was shown in [@kkrs; @kkrs1; @kkrs2] the probability of BH formation is almost unity in a wide range of parameters of theories with first order phase transitions.
Gravitational collapse of FVB and BH creation {#Collapse}
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Consider following [@kkrs; @kkrs1; @kkrs2; @book2; @book3] in more details the conditions of converting FVB into BH. The mass $M$ of FVB can be calculated in a framework of a specific theory and can be estimated in a coordinate system $K^{\prime }$ where the colliding bubbles are nucleated simultaneously. The radius of each bubble $b^{\prime}$ in this system equals to half of their initial coordinate distance at first moment of collision. Apparently the maximum size $D_M$ of the FVB is of the same order as the size of the bubble, since this is the only parameter of necessary dimension on such a scale: $D_M=2b^{\prime }C$. The parameter $C\simeq 1$ is obtained by numerical calculations in the framework of each theory, but its exact numerical value does not affect significantly conclusions.
One can find the mass of FVB that arises at the collision of two bubbles of radius:
$$\label{one}M=\frac{4\pi }3\left( Cb^{\prime }\right) ^3\rho_v$$
This mass is contained in the shrinking area of false vacuum. Suppose for estimations that the minimal size of FVB is of order of wall width $\Delta $. The BH is created if minimal size of FVB is smaller than its gravitational radius. It means that at least at the condition
$$\label{two}\Delta <r_g=2GM$$
the FVB can be converted into BH (where G is the gravitational constant).
As an example consider a simple model with Lagrangian
$$\label{three}L=\frac 12\left( \partial _\mu \Phi \right) ^2-\frac
\lambda 8\left( \Phi ^2-\Phi _0^2\right) ^2-\epsilon \Phi _0^3\left(
\Phi +\Phi _0\right) .$$
In the thin wall approximation the width of the bubble wall can be expressed as $\Delta =2\left( \sqrt{\lambda }\Phi _0\right) ^{-1}$. Using (\[two\]) one can easily derive that at least FVB with mass
$$\label{four}M>\frac 1{\sqrt{\lambda }\Phi _0G}$$
should be converted into BH of mass M. The last condition is valid only in case when FVB is completely contained within the cosmological horizon, namely $M_H>1/\sqrt{\lambda }\Phi _0G$ where the mass of the cosmological horizon at the moment of phase transition is given by $M_H\cong
m_{pl}^3/\Phi_{0}^2$. Thus for the potential (\[three\]) at the condition $\lambda >(\Phi_0/m_{pl})^2$ a BH is formed. This condition is valid for any realistic set of parameters of theory.
The mass and velocity distribution of FVBs, supposing its mass is large enough to satisfy the inequality (\[two\]), has been found in [@kkrs; @kkrs1; @kkrs2]. This distribution can be written in the terms of dimensionless mass $\mu \equiv \left( \frac \pi 3\Gamma
\right) ^{1/4}\left( \frac M{C\rho _v}\right) ^{1/3}$:
$$\label{12}
\begin{array}{c}
\frac{dP}{\Gamma ^{-3/4}Vdvd\mu }=64\pi \left( \frac \pi 3\right)
^{1/4}\mu ^3e^{\mu ^4}\gamma ^3J(\mu ,v), \\ J(\mu ,v)=\int_{\tau
_{}}^\infty d\tau e^{-\tau ^4},\tau _{-}=\mu \left[ 1+\gamma
^2\left( 1+v\right) \right] .
\end{array}$$
The numerical integration of (\[12\]) revealed that the distribution is rather narrow. For example the number of BH with mass 30 times greater than the average one is suppressed by factor $10^5$. Average value of the non dimensional mass is equal to $\mu=0.32$. It allows to relate the average mass of BH and volume containing the BH at the moment of the phase transition:
$$\label{MV}\left\langle M_{BH}\right\rangle =\frac C4\mu ^3\rho
_v\left\langle V_{BH}\right\rangle \simeq 0.012\rho _v\left\langle
V_{BH}\right\rangle .$$
First order phase transitions in the early Universe {#pt1}
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Inflation models ended by a first order phase transition hold a dignified position in the modern cosmology of early Universe (see for example [@10; @101; @102; @103; @104; @11; @111]). The interest to these models is due to, that such models are able to generate the observed large-scale voids as remnants of the primordial bubbles for which the characteristic wavelengths are several tens of Mpc. [@11; @111]. A detailed analysis of a first order phase transition in the context of extended inflation can be found in [@12]. Hereafter we will be interested only in a final stage of inflation when the phase transition is completed. Remind that a first order phase transition is considered as completed immediately after establishing of true vacuum percolation regime. Such regime is established approximately when at least one bubble per unit Hubble volume is nucleated. Accurate computation [@12] shows that first order phase transition is successful if the following condition is valid: $$\label{14}Q\equiv \frac{4\pi }9\left( \frac \Gamma {H^4}\right)
_{t_{end}}=1.$$ Here $\Gamma$ is the bubble nucleation rate. In the framework of first order inflation models the filling of all space by true vacuum takes place due to bubble collisions, nucleated at the final moment of exponential expansion. The collisions between such bubbles occur when they have comoving spatial dimension less or equal to the effective Hubble horizon $H_{end}^{-1}$ at the transition epoch. If we take $H_0=100hKm/\sec /Mpc$ in $\Omega =1$ Universe the comoving size of these bubbles is approximately $10^{-21}h^{-1}Mpc$. In the standard approach it believes that such bubbles are rapidly thermalized without leaving a trace in the distribution of matter and radiation. However, in the previous subsection it has been shown that for any realistic parameters of theory, the collision between only two bubble leads to BH creation with the probability closely to 100% . The mass of this BH is given by (see (\[MV\])) $$\label{15}M_{BH}=\gamma _1M_{bub}$$ where $\gamma _1\simeq 10^{-2}$ and $M_{bub}$ is the mass that could be contained in the bubble volume at the epoch of collision in the condition of a full thermalization of bubbles. The discovered mechanism leads to a new direct possibility of PBH creation at the epoch of reheating in first order inflation models. In standard picture PBHs are formed in the early Universe if density perturbations are sufficiently large, and the probability of PBHs formation from small post- inflation initial perturbations is suppressed (see Section \[dust\]). Completely different situation takes place at final epoch of first order inflation stage; namely collision between bubbles of Hubble size in percolation regime leads to copious PBH formation with masses
$$\label{16}M_0=\gamma _1M_{end}^{hor}= \frac{\gamma
_1}2\frac{m_{pl}^2}{H_{end}},$$
where $M_{end}^{hor}$ is the mass of Hubble horizon at the end of inflation. According to (\[MV\]) the initial mass fraction of this PBHs is given by $\beta _0\approx\gamma _1/e\approx 6\cdot 10^{-
3}$. For example, for typical value of $H_{end}\approx 4\cdot
10^{-6}m_{pl}$ the initial mass fraction $\beta $ is contained in PBHs with mass $M_0\approx 1g$.
In general the Hawking evaporation of mini BHs could give rise to a variety possible end states. It is generally assumed, that evaporation proceeds until the PBH vanishes completely [@21], but there are various arguments against this proposal (see e.g. [@22; @carr1; @222; @223]). If one supposes that BH evaporation leaves a stable relic, then it is naturally to assume that it has a mass of order $m_{rel}=km_{pl}$, where $k\simeq 1\div 10^2$. We can investigate the consequences of PBH forming at the percolation epoch after first order inflation, supposing that the stable relic is a result of its evaporation. As it follows from the above consideration the PBHs are preferentially formed with a typical mass $M_0$ at a single time $t_1$. Hence the total density $\rho$ at this time is $$\label{totdens} \rho (t_1)=\rho_{\gamma}(t_1)+\rho_{PBH}(t_1)=
\frac{3(1-\beta_0)}{32\pi t_1^2}m_{pl}^2+ \frac{3\beta_0}{32\pi
t_1^2}m_{pl}^2,$$ where $\beta_0$ denotes the fraction of the total density, corresponding to PBHs in the period of their formation $t_1$. The evaporation time scale can be written in the following form $$\label{evop} \tau_{BH}=\frac{M_0^3}{g_*m_{pl}^4}$$ where $g_*$ is the number of effective massless degrees of freedom.
Let us derive the density of PBH relics. There are two distinct possibilities to consider.
The Universe is still radiation dominated (RD) at $\tau_{BH}$. This situation will be hold if the following condition is valid $\rho_{BH}(\tau_{BH})<\rho_{\gamma}(\tau_{BH})$. It is possible to rewrite this condition in terms of Hubble constant at the end of inflation $$\label{con1} \frac{H_{end}}{m_{pl}}>\beta_0^{5/2}g_*^{-1/2}\simeq
10^{-6}$$ Taking the present radiation density fraction of the Universe to be $\Omega_{\gamma_0}=2.5\cdot 10^{-5}h^{-2}$ ($h$ being the Hubble constant in the units of $100km\cdot s^{-1}Mpc^{-1}$), and using the standard values for the present time and time when the density of matter and radiation become equal, we find the contemporary densities fraction of relics $$\label{reldens} \Omega_{rel}\approx 10^{26}h^{-2}
k\left(\frac{H_{end}}{m_{pl}}\right)^{3/2}$$ It is easily to see that relics overclose the Universe ($\Omega_{rel}>>1$) for any reasonable $k$ and $H_{end}>10^{-6}m_{pl}$.
The second case takes place if the Universe becomes PBHs dominated at period $t_1<t_2<\tau_{BH}$. This situation is realized under the condition $\rho_{BH}(t_2)>\rho_{\gamma}(t_2)$, which can be rewritten in the form $$\label{con2} \frac{H_{end}}{m_{pl}}<10^{-6}.$$ The present day relics density fraction takes the form $$\label{reldens2} \Omega_{rel}\approx 10^{28}h^{-2}
k\left(\frac{H_{end}}{m_{pl}}\right)^{3/2}$$ Thus the Universe is not overclosed by relics only if the following condition is valid $$\label{con3} \frac{H_{end}}{m_{pl}}\le 2\cdot
10^{-19}h^{4/3}k^{-2/3}.$$ This condition implies that the masses of PBHs created at the end of inflation have to be larger than $$\label{massr} M_0\ge 10^{11}g\cdot h^{-4/3}\cdot k^{2/3}.$$ From the other hand there are a number of well–known cosmological and astrophysical limits [@15; @mujana; @151; @152; @153; @154; @155] which prohibit the creation of PBHs in the mass range (\[massr\]) with initial fraction of mass density close to $\beta_0\approx 10^{-2}$.
So one have to conclude that the effect of the false vacuum bag mechanism of PBH formation makes impossible the coexistence of stable remnants of PBH evaporation with the first order phase transitions at the end of inflation.
Gravitino production by PBH evaporation and constraints on the inhomogeneity of the early Universe {#gravitino}
==================================================================================================
Presently there are no observational evidences, proving existence of PBHs. However, even the absence of PBHs provides a very sensitive theoretical tool to study physics of early Universe. PBHs represent nonrelativistic form of matter and their density decreases with scale factor $a$ as $\propto a^{-3} \propto T^{3}$, while the total density is $\propto a^{-4} \propto T^{4}$ in the period of radiation dominance (RD). Being formed within horizon, PBH of mass $M$, can be formed not earlier than at $$\label{tfRD}t(M)=\frac{M}{m_{pl}}{t_{pl}}=\frac{M}{m_{pl}^2}.$$ If they are formed on RD stage, the smaller are the masses of PBHs, the larger becomes their relative contribution to the total density on the modern MD stage. Therefore, even the modest constraint for PBHs of mass $M$ on their density \[OmPBH\]\_[PBH]{}(M)=in units of critical density $\rho_{c}=3 H^2/(8 \pi G)$ from the condition that their contribution $\alpha(M)$ into the the total density $$\label{defalpha}\alpha(M)\equiv\frac{\rho_{PBH}(M)}{\rho_{tot}}=\Omega_{PBH}(M)$$ for $\rho_{tot}=\rho_{c}$ does not exceed the density of dark matter$$\label{DMalpha}\alpha(M)=\Omega_{PBH}(M) \le
\Omega_{DM}=0.23$$ converts into a severe constraint on this contribution $$\label{defbeta}\beta \equiv
\frac{\rho_{PBH}(M,t_f)}{\rho_{tot}(t_f)}$$ in the period $t_f$ of their formation. If formed on RD stage at $t_f=t(M)$, given by (\[tfRD\]), which corresponds to the temperature $T_f=m_{pl}\sqrt{m_{pl}/M}$, PBHs contribute into the total density in the end of RD stage at $t_{eq}$, corresponding to $T_{eq}\approx 1 eV$, by factor $a(t_{eq})/a(t_f)=T_f/T_{eq}=m_{pl}/T_{eq}\sqrt{m_{pl}/M}$ larger, than in the period of their formation. The constraint on $\beta(M)$, following from Eq.(\[DMalpha\]) is then given by$$\label{DMbeta}\beta(M)=\alpha(M)\frac{T_{eq}}{m_{pl}}\sqrt{\frac{M}{m_{pl}}}
\le 0.23 \frac{T_{eq}}{m_{pl}}\sqrt{\frac{M}{m_{pl}}}.$$
The possibility of PBH evaporation, revealed by S. Hawking [@hawking4], strongly influences effects of PBHs. In the strong gravitational field near gravitational radius $r_g$ of PBH quantum effect of creation of particles with momentum $p \sim 1/r_g$ is possible. Due to this effect PBH turns to be a black body source of particles with temperature (in the units $\hbar=c=k=1$)$$\label{TPBHev}T=\frac{1}{8\pi G
M}\approx10^{13} {\rm GeV} \frac{1 {\rm g}}{M}.$$ The evaporation timescale BH is $\tau_{BH} \sim M^3/m_{pl}^4$ (see Eq.(\[evop\]) and discussion in previous section) and at $M \le
10^{14}$ g is less, than the age of the Universe. Such PBHs can not survive to the present time and the magnitude Eq.(\[DMalpha\]) for them should be re-defined and has the meaning of contribution to the total density in the moment of PBH evaporation. For PBHs formed on RD stage and evaporated on RD stage at $t<t_{eq}$ the relationship Eq.(\[DMbeta\]) between $\beta(M)$ and $\alpha(M)$ is given by [@NovikovPBH; @polnarev] $$\label{DMbetaRD}\beta(M)=\alpha(M)\frac{m_{pl}}{M}.$$ The relationship between $\beta(M)$ and $\alpha(M)$ has more complicated form, if PBHs are formed on early dust-like stages [@polnarev1; @polnarev; @khlopov6; @book], or such stages take place after PBH formation[@khlopov6; @book]. Relative contribution of PBHs to total density does not grow on dust-like stage and the relationship between $\beta(M)$ and $\alpha(M)$ depends on details of a considered model. Minimal model independent factor $\alpha(M)/\beta(M)$ follows from the account for enhancement, taking place only during RD stage between the first second of expansion and the end of RD stage at $t_{eq}$, since radiation dominance in this period is supported by observations of light element abundance and spectrum of CMB [@polnarev1; @polnarev; @khlopov6; @book].
Effects of PBH evaporation make astrophysical data much more sensitive to existence of PBHs. Constraining the abundance of primordial black holes can lead to invaluable information on cosmological processes, particularly as they are probably the only viable probe for the power spectrum on very small scales which remain far from the Cosmological Microwave Background (CMB) and Large Scale Structures (LSS) sensitivity ranges. To date, only PBHs with initial masses between $\sim 10^9$ g and $\sim 10^{16}$ g have led to stringent limits (see [*e.g.*]{} [@carr1; @carrMG; @LGreen; @polnarev]) from consideration of the entropy per baryon, the deuterium destruction, the $^4$He destruction and the cosmic-rays currently emitted by the Hawking process [@hawking4]. The existence of light PBHs should lead to important observable constraints, either through the direct effects of the evaporated particles (for initial masses between $10^{14}$ g and $10^{16}$ g) or through the indirect effects of their interaction with matter and radiation in the early Universe (for PBH masses between $10^{9}$ g and $10^{14}$ g). In these constraints, the effects taken into account are those related with known particles. However, since the evaporation products are created by the gravitational field, any quantum with a mass lower than the black hole temperature should be emitted, independently of the strength of its interaction. This could provide a copious production of superweakly interacting particles that cannot not be in equilibrium with the hot plasma of the very early Universe. It makes evaporating PBHs a unique source of all the species, which can exist in the Universe.
Following [@book; @book3; @khlopov6; @khlopov7] and [@lemoine; @green1] (but in a different framework and using more stringent constraints), limits on the mass fraction of black holes at the time of their formation ($\beta \equiv
\rho_{PBH}/\rho_{tot}$) were derived in [@KBgrain] using the production of gravitinos during the evaporation process. Depending on whether gravitinos are expected to be stable or metastable, the limits are obtained using the requirement that they do not overclose the Universe and that the formation of light nuclei by the interactions of $^4$He nuclei with nonequilibrium flux of D,T,$^3$He and $^4$He does not contradict the observations. This approach is more constraining than the usual study of photo-dissociation induced by photons-photinos pairs emitted by decaying gravitinos. It opened a new window for the upper limits on $\beta$ below $10^9$ g. The cosmological consequences of the limits, obtained in [@KBgrain], are briefly reviewed in the framework of three different scenarios: a blue power spectrum, a step in the power spectrum and first order phase transitions.
Limits on the PBH density {#evaporation}
-------------------------
Several constraints on the density of PBHs have been derived in different mass ranges assuming the evaporation of only standard model particles : for $10^9~{\rm g}<M<10^{13}~{\rm g}$ the entropy per baryon at nucleosynthesis was used [@mujana] to obtain $\beta < (10^9~{\rm g}/M)$, for $10^9~{\rm g}<M<10^{11}~{\rm g}$ the production of $n\bar{n}$ pairs at nucleosynthesis was used [@153] to obtain $\beta < 3\times 10^{-17} (10^9~{\rm
g}/M)^{1/2}$ , for $10^{10}~{\rm g}<M<10^{11}~{\rm g}$ deuterium destruction was used [@152] to obtain $\beta < 3\times 10^{-22}
(M/10^{10}~{\rm g})^{1/2}$, for $10^{11}~{\rm g}<M<10^{13}~{\rm g}$ spallation of $^4$He was used [@vainer; @khlopov6] to obtain $\beta < 3\times 10^{-21} (M/10^9~{\rm g})^{5/2}$, for $M\approx
5\times 10^{14}~{\rm g}$ the gamma-rays and cosmic-rays were used [@155; @barrau] to obtain $\beta < 10^{-28}$. Slightly more stringent limits were obtained in [@kohri], leading to $\beta <
10^{-20}$ for masses between $10^{9}~{\rm g}$ and $10^{10}~{\rm g}$ and in [@barraugamma], leading to $\beta < 10^{-28}$ for $M=5\times 10^{11}~{\rm g}$. Gamma-rays and antiprotons were also recently re-analyzed in [@barraupbar] and [@Custodio:2002jv], improving a little the previous estimates. Such constraints, related to phenomena occurring after the nucleosynthesis, apply only for black holes with initial masses above $\sim 10^9$ g. Below this value, the only limits are the very weak entropy constraint (related with the photon-to-baryon ratio) and the constraint, assuming stable remnants of black holes forming at the end of the evaporation mechanism as described in the previous Section.
To derive a limit in the initial mass range $m_{pl}<M<10^{11}$ g, gravitinos emitted by black holes were considered in [@KBgrain]. Gravitinos are expected to be present in all local supersymmetric models, which are regarded as the more natural extensions of the standard model of high energy physics (see, [*e.g.*]{}, [@olive] for an introductory review). In the framework of minimal Supergravity (mSUGRA), the gravitino mass is, by construction, expected to lie around the electroweak scale, [*i.e.*]{} in the 100 GeV range. In this case, the gravitino is [*metastable*]{} and decays after nucleosynthesis, leading to important modifications of the nucleosynthesis paradigm. Instead of using the usual photon-photino decay channel, the study of [@KBgrain] relied on the more sensitive gluon-gluino channel. Based on [@khlopovlinde; @khlopovlinde2; @khlopovlinde3; @khlopov3; @khlopov31], the antiprotons produced by the fragmentation of gluons emitted by decaying gravitinos were considered as a source of nonequilibrium light nuclei resulting from collisions of those antiprotons on equilibrium nuclei. Then, $^6$Li, $^7$Li and $^7$Be nuclei production by the interactions of the nonequilibrium nuclear flux with $^4$He equilibrium nuclei was taken into account and compared with data (this approach is supported by several recent analysis [@Karsten; @Kawasaki] which lead to similar results). The resulting Monte-Carlo estimates [@khlopov3] lead to the following constraint on the concentration of gravitinos: $n_{3/2}<
1.1\times 10^{-13}m_{3/2}^{-1/4}$, where $m_{3/2}$ is the gravitino mass in GeV. This constraint has been successfully used to derive an upper limit on the reheating temperature of the order [@khlopov3]: $T_R < 3.8\times 10^6$ GeV. The consequences of this limit on cosmic-rays emitted by PBHs was considered, [*e.g.*]{}, in [@barrauprd]. In the approach of [@KBgrain] this stringent constraint on the gravitino abundance was related to the density of PBHs through the direct gravitino emission. The usual Hawking formula [@hawking4] was used for the number of particles of type $i$ emitted per unit of time $t$ and per unit of energy $Q$. Introducing the temperature defined by Eq. (\[TPBHev\]) $
T=hc^3/(16\pi^2 k G M)\approx(10^{13}{\rm g})/{M}~{\rm GeV}, $ taking the relativistic approximation for $\Gamma_s$, and integrating over time and energy, the total number of quanta of type $i$ can be estimated as: $$\label{Niev}N_i^{TOT}=\frac{27\times 10^{24}}{64\pi ^3
\alpha_{SUGRA}}\int_{T_i}^{T_{Pl}}\frac{dT}{T^3}\int_{m/T}^x\frac{x^2dx}{e^x-(-1)^s}$$ where $T$ is in GeV, $m_{pl}\approx 10^{-5}$ g, $x\equiv Q/T$, $m$ is the particle mass and $\alpha_{SUGRA}$ accounts for the number of degrees of freedom through $M^2dM=-\alpha_{SUGRA}dt$ where $M$ is the black hole mass. Once the PBH temperature is higher than the gravitino mass, gravitinos will be emitted with a weight related with their number of degrees of freedom. Computing the number of emitted gravitinos as a function of the PBH initial mass and matching it with the limit on the gravitino density imposed by nonequilibrium nucleosynthesis of light elements leads to an upper limit on the PBH number density. If PBHs are formed during a radiation dominated stage, this limit can easily be converted into an upper limit on $\beta$ by evaluating the energy density of the radiation at the formation epoch. The resulting limit is shown on Fig. \[pot\] and leads to an important improvement over previous limits, nearly independently of the gravitino mass in the interesting range. This opens a new window on the very small scales in the early Universe.
![Constraints of [@KBgrain] on the fraction of the Universe going into PBHs (adapted from [@carr1; @carrMG; @LGreen; @polnarev]). The two curves obtained with gravitinos emission in mSUGRA correspond to $m_{3/2}$ = 100 GeV (lower curve in the high mass range) and $m_{3/2}$ = 1 TeV (upper curve in the high mass range)[]{data-label="pot"}](beta_NB.eps)
It is also possible to consider limits arising in Gauge Mediated Susy Breaking (GMSB) models [@kolda]. Those alternative scenarios, incorporating a natural suppression of the rate of flavor-changing neutral-current due to the low energy scale, predict the gravitino to be the Lightest Supersymmetric Particle (LSP). The LSP is stable if R-parity is conserved. In this case, the limit was obtained [@KBgrain] by requiring $\Omega_{3/2,0}<\Omega_{M,0}$, [*i.e.*]{} by requiring that the current gravitino density does not exceed the matter density. It can easily be derived from the previous method, by taking into account the dilution of gravitinos in the period of PBH evaporation and conservation of gravitino to specific entropy ratio, that [@KBgrain]: $$\label{BetDM}\beta \leq \frac{\Omega_{M,0}}{N_{3/2}\frac{m_{3/2}}{M}\left(
\frac{t_{eq}}{t_{f}} \right) ^{\frac{1}{2}}}$$ where $N_{3/2}$ is the total number of gravitinos emitted by a PBH with initial mass $M$, $t_{eq}$ is the end of RD stage and $t_f=\max(t_{form},t_{end})$ when a non-trivial equation of state for the period of PBH formation is considered, [*e.g.*]{} a dust-like phase which ends at $t_{end}$ [@polnarev1]. The limit (\[BetDM\]) does not imply thermal equilibrium of relativistic plasma in the period before PBH evaporation and is valid even for low reheating temperatures provided that the equation of state on the preheating stage is close to relativistic. With the present matter density $\Omega_{M,0}\approx 0.27$ [@wmap] this leads to the limit shown in Fig. \[pot2\] for $m_{3/2}=10$ GeV. Following (\[BetDM\]) this limit scales with gravitino mass as $\propto
m_{3/2}^{-1}$. Models of gravitino dark matter with $\Omega_{3/2,0}
= \Omega_{CDM,0}$, corresponding to the case of equality in the above formula, were recently considered in [@Jedamzik1; @Jedamzik11].
![Constraints of [@KBgrain] on the fraction of the Universe going into PBHs (adapted from [@carr1; @carrMG; @LGreen; @polnarev]). The curve obtained with gravitinos emission in GMSB correspond to $m_{3/2}=10$ GeV and scales with gravitino mass as $\propto m_{3/2}^{-1}$.[]{data-label="pot2"}](newplot.eps)
Cosmological consequences {#spectrum}
-------------------------
Upper limits on the fraction of the Universe in primordial black holes can be converted into cosmological constraints on models with significant power on small scales [@KBgrain].
The easiest way to illustrate the importance of such limits is to consider a blue power spectrum and to derive a related upper value on the spectral index $n$ of scalar fluctuations ($P(k)\propto
k^n$). It has recently been shown by WMAP [@wmap] that the spectrum is nearly of the Harrison-Zel’dovich type, [*i.e.*]{} scale invariant with $n\approx 1$. However this measure was obtained for scales between $10^{45}$ and $10^{60}$ times larger that those probed by PBHs and it remains very important to probe the power available on small scales. The limit on $n$ given in [@KBgrain] must therefore be understood as a way to constrain $P(k)$ at small scales rather than a way to measure its derivative at large scales : it is complementary to CMB measurements. Using the usual relations between the mass variance at the PBH formation time $\sigma_H(t_{form})$ and the same quantity today $\sigma_H(t_0)$ [@green], $$\label{Sigmh}\sigma_H(t_{form})=\sigma_H(t_0)\left(\frac{M_H(t_0)}{M_H(t_{eq})}\right)^{\frac{n-1}{6}}
\left(\frac{M_H(t_{eq})}{M_H(t_{form})}\right)^{\frac{n-1}{4}}$$ where $M_H(t)$ is the Hubble mass at time $t$ and $t_{eq}$ is the equilibrium time, it is possible to set an upper value on $\beta$ which can be expressed as $$\label{betaspect}\beta\approx \frac{\sigma_H(t_{form})}{\sqrt{2\pi}\delta_{min}}e^{-\frac{\delta_{min}^2}
{2\sigma_H^2(t_{form})}},$$ where $\delta_{min}\approx
0.3$ is the minimum density contrast required to form a PBH. The limit derived in the previous subsection leads to $n<1.20$ in the mSUGRA case whereas the usually derived limits range between 1.23 and 1.31 [@green; @Bringmann:2001yp; @kim2]. In the GMSB case, it remains at the same level for $m_{3/2}\sim10$ GeV and is slightly relaxed for smaller masses of gravitino. This improvement is due to the much more important range of masses probed by the method [@KBgrain].
In the standard cosmological paradigm of inflation, the primordial power spectrum is expected to be nearly –but not exactly– scale invariant [@liddle]. The sign of the running can, in principle, be either positive or negative. It has been recently shown that models with a positive running $\alpha_s$, defined as \[specUV\]P(k)=P(k\_0)( )\^[n\_s(k\_0)+\_s ln ( )]{},are very promising in the framework of supergravity inflation (see, [*e.g.*]{}, [@kawa]). The analysis [@KBgrain] strongly limits a positive running, setting the upper bound at a tiny value $\alpha_s<2\times 10^{-3}$. This result is more stringent than the upper limit obtained through a combined analysis of Ly$\alpha$ forest, SDSS and WMAP data [@seljak], $-0.013<\alpha_s<0.007$, as it deals with scales very far from those probed by usual cosmological observations. The order of magnitude of the running naturally expected in most models –either inflationary ones (see, [*e.g.*]{}, [@peiris]) or alternative ones (see, [*e.g.*]{}, [@khoury])– being of a few times $10^{-3}$ our upper bound should help to distinguish between different scenarios.
In the case of an early dust-like stage in the cosmological evolution [@khlopov0; @polnarev; @book; @book3], the PBH formation probability is increased to $\beta > \delta ^ {13/2}$ where $\delta$ is the density contrast for the considered small scales (see subsection \[particles\]). The associated limit on $n$ is strengthened to $n<1.19$.
Following [@green], it is also interesting to consider primordial density perturbation spectra with both a tilt and a step. Such a feature can arise from underlying physical processes [@starobinsky] and allows investigation of a wider class of inflaton potentials. If the amplitude of the step is defined so that the power on small scales is $p^{-2}$ times higher than the power on large scales, the maximum allowed value for the spectral index can be computed as a function of $p$. Figure \[pot3\], taken from [@KBgrain], shows those limits, which become extremely stringent when $p$ is small enough, for both the radiation-dominated and the dust-like cases.
![Upper limit from [@KBgrain] on the spectral index of the power spectrum as a function of the amplitude of the step.[]{data-label="pot3"}](np_NB.eps)
Another important consequence of limits [@KBgrain] concerns PBH relics dark matter (see also discussion in subsection \[pt1\]). The idea, introduced in [@macgibbon2], that relics possibly formed at the end of the evaporation process could account for the cold dark matter has been extensively studied. The amplitude of the power boost required on small scales has been derived, [*e.g.*]{}, in [@barrau2] as a function of the relic mass and of the expected density. The main point was that the “step” (or whatever structure in the power spectrum) should occur at low masses to avoid the constraints available between $10^9$ g and $10^{15}$ g. The limit on $\beta$ derived in [@KBgrain] closes this dark matter issue except within a small window below $10^3$ g.
This result can be re-formulated in a more general way. If the nature of cosmological dark matter is related with superweakly interacting particles, which can not be present in equilibrium in early Universe and for which nonequilibrium processes of production e.g. in reheating are suppressed, the early Universe should be sufficiently homogeneous on small scales to exclude copious creation of these species in miniPBH vaporation.
Finally, the limits [@KBgrain] also completely exclude the possibility of a copious PBH formation process in bubble wall collisions [@kkrs; @kkrs1; @kkrs2], considered in the previous Section. This has important consequences for the related constraints on first order phase transitions in the early Universe and on symmetry breaking pattern of particle theory.
Massive Primordial Black Holes from collapse of closed walls {#MBHwalls}
============================================================
A wide class of particle models possesses a symmetry breaking pattern, which can be effectively described by pseudo-Nambu–Goldstone (PNG) field and which corresponds to formation of unstable topological defect structure in the early Universe (see [@book2] for review and references). The Nambu–Goldstone nature in such an effective description reflects the spontaneous breaking of global U(1) symmetry, resulting in continuous degeneracy of vacua. The explicit symmetry breaking at smaller energy scale changes this continuous degeneracy by discrete vacuum degeneracy. The character of formed structures is different for phase transitions, taking place on post-inflationary and inflationary stages.
Structures from succession of U(1) phase transitions {#structures}
----------------------------------------------------
At high temperatures such a symmetry breaking pattern implies the succession of second order phase transitions. In the first transition, continuous degeneracy of vacua leads, at scales exceeding the correlation length, to the formation of topological defects in the form of a string network; in the second phase transition, continuous transitions in space between degenerated vacua form surfaces: domain walls surrounded by strings. This last structure is unstable, but, as was shown in the example of the invisible axion [@Sakharov2; @kss; @kss2], it is reflected in the large scale inhomogeneity of distribution of energy density of coherent PNG (axion) field oscillations. This energy density is proportional to the initial value of phase, which acquires dynamical meaning of amplitude of axion field, when axion mass is switched on in the result of the second phase transition.
The value of phase changes by $2 \pi$ around string. This strong nonhomogeneity of phase, leading to corresponding nonhomogeneity of energy density of coherent PNG (axion) field oscillations, is usually considered (see e.g. [@kim; @Sikivie:2006ni] and references therein) only on scales, corresponduing to mean distance between strings. This distance is small, being of the order of the scale of cosmological horizon in the period, when PNG field oscillations start. However, since the nonhomogeneity of phase follows the pattern of axion string network this argument misses large scale correlations in the distribution of oscillations’ energy density.
Indeed, numerical analysis of string network (see review in [@vs]) indicates that large string loops are strongly suppressed and the fraction of about 80% of string length, corresponding to long loops, remains virtually the same in all large scales. This property is the other side of the well known scale invariant character of string network. Therefore the correlations of energy density should persist on large scales, as it was revealed in [@Sakharov2; @kss; @kss2].
The large scale correlations in topological defects and their imprints in primordial inhomogeneities is the indirect effect of inflation, if phase transitions take place after reheating of the Universe. Inflation provides in this case equal conditions for phase transition, taking place in causally disconnected regions.
If phase transitions take place on inflational stage new forms of primordial large scale correlations appear. The value of phase after the first phase transition is inflated over the region corresponding to the period of inflation, while fluctuations of this phase change in the course of inflation its initial value within the regions of smaller size. Owing to such fluctuations, for the fixed value of $\theta_{60}$ in the period of inflation with [*e-folding*]{} $N=60$ corresponding to the part of the Universe within the modern cosmological horizon, strong deviations from this value appear at smaller scales, corresponding to later periods of inflation with $N
< 60$. If $\theta_{60} < \pi$, the fluctuations can move the value of $\theta_{N}$ to $\theta_{N} > \pi$ in some regions of the Universe. After reheating in the result of the second phase transition these regions correspond to vacuum with $\theta_{vac} =
2\pi$, being surrounded by the bulk of the volume with vacuum $\theta_{vac} = 0$. As a result massive walls are formed at the border between the two vacua. Since regions with $\theta_{vac} = 2\pi$ are confined, the domain walls are closed. After their size equals the horizon, closed walls can collapse into black holes.
This mechanism can lead to formation of primordial black holes of a whatever large mass (up to the mass of AGNs [@AGN], see for latest review [@DER]). Such black holes appear in the form of primordial black hole clusters, exhibiting fractal distribution in space [@KRS; @Khlopov:2004sc; @book2]. It can shed new light on the problem of galaxy formation [@book2; @DER1; @DER2].
Formation of closed walls in inflationary Universe {#walls}
--------------------------------------------------
To describe a mechanism for the appearance of massive walls of a size essentially greater than the horizon at the end of inflation, let us consider a complex scalar field with the potential[@AGN; @KRS; @Khlopov:2004sc; @book2] $$\label{V1} V(\varphi ) = \lambda (\left| \varphi
\right|^2 - f^2 /2)^2+\delta V(\theta ),$$ where $\varphi = re^{i\theta } $. This field coexists with an inflaton field which drives the Hubble constant $H$ during the inflational stage. The term $$\label{L1} \delta V(\theta ) = \Lambda ^4 \left(
{1 - \cos \theta } \right),$$ reflecting the contribution of instanton effects to the Lagrangian renormalization (see for example [@adams]), is negligible on the inflational stage and during some period in the FRW expansion. The omitted term (\[L1\]) becomes significant, when temperature falls down the values $T \sim \Lambda$. The mass of radial field component $r$ is assumed to be sufficiently large with respect to $H$, which means that the complex field is in the ground state even before the end of inflation. Since the term (\[L1\]) is negligible during inflation, the field has the form $\varphi \approx f/\sqrt 2 \cdot e^{i\theta }
$, the quantity $f\theta$ acquiring the meaning of a massless field.
At the same time, the well established behavior of quantum field fluctuations on the de Sitter background [@Star80] implies that the wavelength of a vacuum fluctuation of every scalar field grows exponentially, having a fixed amplitude. Namely, when the wavelength of a particular fluctuation, in the inflating Universe, becomes greater than $H^{-1}$, the average amplitude of this fluctuation freezes out at some non-zero value because of the large friction term in the equation of motion of the scalar field, whereas its wavelength grows exponentially. Such a frozen fluctuation is equivalent to the appearance of a classical field that does not vanish after averaging over macroscopic space intervals. Because the vacuum must contain fluctuations of every wavelength, inflation leads to the creation of more and more new regions containing a classical field of different amplitudes with scale greater than $H^{-1}$. In the case of an effectively massless Nambu–Goldstone field considered here, the averaged amplitude of phase fluctuations generated during each e-fold (time interval $H^{-1}$) is given by \[fluctphase\] = H/2f. Let us assume that the part of the Universe observed inside the contemporary horizon $H_0^{-1}=3000h^{-1}$Mpc was inflating, over $N_U \simeq 60$ e-folds, out of a single causally connected domain of size $H^{-1}$, which contains some average value of phase $\theta_0$ over it. When inflation begins in this region, after one e-fold, the volume of the Universe increases by a factor $e^3$ . The typical wavelength of the fluctuation $\delta\theta$ generated during every e-fold is equal to $H^{-1}$. Thus, the whole domain $H^{-1}$, containing $\theta_{0}$, after the first e-fold effectively becomes divided into $e^3$ separate, causally disconnected domains of size $H^{-1}$. Each domain contains almost homogeneous phase value $\theta_{0}\pm\delta\theta$. Thereby, more and more domains appear with time, in which the phase differs significantly from the initial value $\theta_0$. A principally important point is the appearance of domains with phase $\theta >\pi$. Appearing only after a certain period of time during which the Universe exhibited exponential expansion, these domains turn out to be surrounded by a space with phase $\theta <\pi$. The coexistence of domains with phases $\theta
<\pi$ and $\theta
>\pi$ leads, in the following, to formation of a large-scale structure of topological defects.
The potential (\[V1\]) possesses a $U(1)$ symmetry, which is spontaneously broken, at least, after some period of inflation. Note that the phase fluctuations during the first e-folds may, generally speaking, transform eventually into fluctuations of the cosmic microwave radiation, which will lead to imposing restrictions on the scaling parameter $f$. This difficulty can be avoided by taking into account the interaction of the field $\varphi$ with the inflaton field (i.e. by making parameter $f$ a variable [@book2]). This spontaneous breakdown is holding by the condition on the radial mass, $m_r=\sqrt{\lambda}f>H$. At the same time the condition \[angularmass\] m\_=\^2H on the angular mass provides the freezing out of the phase distribution until some moment of the FRW epoch. After the violation of condition (\[angularmass\]) the term (\[L1\]) contributes significantly to the potential (\[V1\]) and explicitly breaks the continuous symmetry along the angular direction. Thus, potential (\[V1\]) eventually has a number of discrete degenerate minima in the angular direction at the points $\theta_{min}=0,\ \pm
2\pi ,\ \pm 4\pi,\ ...$ .
As soon as the angular mass $m_{\theta}$ is of the order of the Hubble rate, the phase starts oscillating about the potential minimum, initial values being different in various space domains. Moreover, in the domains with the initial phase $\pi <\theta < 2\pi
$, the oscillations proceed around the potential minimum at $\theta
_{min}=2\pi$, whereas the phase in the surrounding space tends to a minimum at the point $\theta _{min}=0$. Upon ceasing of the decaying phase oscillations, the system contains domains characterized by the phase $\theta _{min}=2\pi$ surrounded by space with $\theta
_{min}=0$. Apparently, on moving in any direction from inside to outside of the domain, we will unavoidably pass through a point where $\theta =\pi$ because the phase varies continuously. This implies that a closed surface characterized by the phase $\theta
_{wall}=\pi$ must exist. The size of this surface depends on the moment of domain formation in the inflation period, while the shape of the surface may be arbitrary. The principal point for the subsequent considerations is that the surface is closed. After reheating of the Universe, the evolution of domains with the phase $\theta >\pi $ proceeds on the background of the Friedman expansion and is described by the relativistic equation of state. When the temperature falls down to $T^* \sim \Lambda$, an equilibrium state between the “vacuum” phase $\theta_{vac}=2\pi$ inside the domain and the $\theta_{vac} =0$ phase outside it is established. Since the equation of motion corresponding to potential (\[L1\]) admits a kink-like solution (see [@vs] and references therein), which interpolates between two adjacent vacua $\theta_{vac} =0$ and $\theta_{vac} =2\pi$, a closed wall corresponding to the transition region at $\theta =\pi$ is formed. The surface energy density of a wall of width $\sim 1/m\sim f/\Lambda^2$ is of the order of $\sim
f \Lambda ^2$ [^2].
Note that if the coherent phase oscillations do not decay for a long time, their energy density can play the role of CDM. This is the case, for example, in the cosmology of the invisible axion (see [@kim; @Sikivie:2006ni] and references therein).
It is clear that immediately after the end of inflation, the size of domains which contains a phase $\theta_{vac} >2\pi$ essentially exceeds the horizon size. This situation is replicated in the size distribution of vacuum walls, which appear at the temperature $T^*
\sim \Lambda$ whence the angular mass $m_{\theta}$ starts to build up. Those walls, which are larger than the cosmological horizon, still follow the general FRW expansion until the moment when they get causally connected as a whole; this happens as soon as the size of a wall becomes equal to the horizon size $R_h$. Evidently, internal stresses developed in the wall after crossing the horizon initiate processes tending to minimize the wall surface. This implies that the wall tends, first, to acquire a spherical shape and, second, to contract toward the centre. For simplicity, we will consider below the motion of closed spherical walls [^3].
The wall energy is proportional to its area at the instant of crossing the horizon. At the moment of maximum contraction, this energy is almost completely converted into kinetic energy [@Rubinwall]. Should the wall at the same moment be localized within the gravitational radius, a PBH is formed.
Detailed consideration of BH formation was performed in [@AGN]. The results of these calculations are sensitive to changes in the parameter $\Lambda$ and the initial phase $\theta _U$. As the $\Lambda$ value decreases to $\approx 1$GeV, still greater PBHs appear with masses of up to $\sim 10^{40}$ g. A change in the initial phase leads to sharp variations in the total number of black holes.As was shown above, each domain generates a family of subdomains in the close vicinity. The total mass of such a cluster is only 1.5–2 times that of the largest initial black hole in this space region. Thus, the calculations confirm the possibility of formation of clusters of massive PBHs ( $\sim 100M_{\odot}$ and above) in the pregalactic stages of the evolution of the Universe. These clusters represent stable energy density fluctuations around which increased baryonic (and cold dark matter) density may concentrate in the subsequent stages, followed by the evolution into galaxies.
It should be noted that additional energy density is supplied by closed walls of small sizes. Indeed, because the smallness of their gravitational radius, they do not collapse into BHs. After several oscillations such walls disappear, leaving coherent fluctuations of the PNG field. These fluctuations contribute to a local energy density excess, thus facilitating the formation of galaxies.
The mass range of formed BHs is constrained by fundamental parameters of the model $f$ and $\Lambda$. The maximal BH mass is determined by the condition that the wall does not dominate locally before it enters the cosmological horizon. Otherwise, local wall dominance leads to a superluminal $a \propto t^2$ expansion for the corresponding region, separating it from the other part of the Universe. This condition corresponds to the mass [@book2]\[Mmax\] M\_[max]{} = m\_[pl]{}()\^2.The minimal mass follows from the condition that the gravitational radius of BH exceeds the width of wall and it is equal to[@KRS; @book2]\[Mmin\] M\_[min]{} = f()\^2.
Closed wall collapse leads to primordial GW spectrum, peaked at \[nupeak\]\_0=310\^[11]{}(/f)[Hz]{} with energy density up to \[OmGW\]\_[GW]{} 10\^[-4]{}(f/m\_[pl]{}).At $f \sim 10^{14}$GeV this primordial gravitational wave background can reach $\Omega_{GW}\approx
10^{-9}.$ For the physically reasonable values of 1<<10\^8[GeV]{}the maximum of spectrum corresponds to 310\^[-3]{}<\_0<310\^[5]{}[Hz]{}.Another profound signature of the considered scenario are gravitational wave signals from merging of BHs in PBH cluster. These effects can provide test of the considered approach in LISA experiment.
Discussion {#Discussion}
==========
For long time scenarios with Primordial Black holes belonged dominantly to cosmological [*anti-Utopias*]{}, to “fantasies”, which provided restrictions on physics of very early Universe from contradiction of their predictions with observational data. Even this “negative” type of information makes PBHs an important theoretical tool. Being formed in the very early Universe as initially nonrelativistic form of matter, PBHs should have increased their contribution to the total density during RD stage of expansion, while effect of PBH evaporation should have strongly increased the sensitivity of astrophysical data to their presence. It links astrophysical constraints on hypothetical sources of cosmic rays or gamma background, on hypothetical factors, causing influence on light element abundance and spectrum of CMB, to restrictions on superheavy particles in early Universe and on first and second order phase transitions, thus making a sensitive astrophysical probe to particle symmetry structure and pattern of its breaking at superhigh energy scales.
Gravitational mechanism of particle creation in PBH evaporation makes evaporating PBH an unique source of any species of particles, which can exist in our space-time. At least theoretically, PBHs can be treated as source of such particles, which are strongly suppressed in any other astrophysical mechanism of particle production, either due to a very large mass of these species, or owing to their superweak interaction with ordinary matter.
By construction astrophysical constraint excludes effect, predicted to be larger, than observed. At the edge such constraint converts into an alternative mechanism for the observed phenomenon. At some fixed values of parameters, PBH spectrum can play a positive role and shed new light on the old astrophysical problems.
The common sense is to think that PBHs should have small sub-stellar mass. Formation of PBHs within cosmological horizon, which was very small in very early Universe, seem to argue for this viewpoint. However, phase transitions on inflationary stage can provide spikes in spectrum of fluctuations at any scale, or provide formation of closed massive domain walls of any size.
In the latter case primordial clouds of massive black holes around intermediate mass or supermassive black hole is possible. Such clouds have a fractal spatial distribution. A development of this approach gives ground for a principally new scenario of the galaxy formation in the model of the Big Bang Universe. Traditionally, Big Bang model assumes a homogeneous distribution of matter on all scales, whereas the appearance of observed inhomogeneities is related to the growth of small initial density perturbations. However, the analysis of the cosmological consequences of the particle theory indicates the possible existence of strongly inhomogeneous primordial structures in the distribution of both the dark matter and baryons. These primordial structures represent a new factor in galaxy formation theory. Topological defects such as the cosmological walls and filaments, primordial black holes, archioles in the models of axionic CDM, and essentially inhomogeneous baryosynthesis (leading to the formation of antimatter domains in the baryon-asymmetric Universe [@exl1; @exl2; @crg; @kolb; @we; @khl; @CSKZ; @zil; @sb; @dolgmain; @khlopgolubkov; @bgk; @FarKhl; @book; @book2; @book3]) offer by no means a complete list of possible primary inhomogeneities inferred from the existing elementary particle models.
Observational cosmology offers strong evidences favoring the existence of processes, determined by new physics, and the experimental physics approaches to their investigation. Cosmoparticle physics [@ADS; @MKH; @book; @book3], studying the physical, astrophysical and cosmological impact of new laws of Nature, explores the new forms of matter and their physical properties. Its development offers the great challenge for theoretical and experimental research. Physics of Primordial Black holes can play important role in this process.
Acknowledgement {#acknowledgement .unnumbered}
===============
I express my gratitude to J.A. de Freitas Pacheco for inviting me to write this review and I am grateful to A. Barrau, S.G.Rubin and A.S.Sakharov for discussions and help in preparation of manuscript.
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[^1]: Maxim.Khlopov@roma1.infn.it
[^2]: The existence of such domain walls in theory of the invisible axion was first pointed out in [@sikivieinvisible].
[^3]: The motion of closed vacuum walls has been driven analytically in [@tkachev; @sikivie].
|
---
abstract: 'Adversarial training is a technique for training robust machine learning models. To encourage robustness, it iteratively computes adversarial examples for the model, and then re-trains on these examples via some update rule. This work analyzes the performance of adversarial training on linearly separable data, and provides bounds on the number of iterations required for large margin. We show that when the update rule is given by an arbitrary empirical risk minimizer, adversarial training may require exponentially many iterations to obtain large margin. However, if gradient or stochastic gradient update rules are used, only polynomially many iterations are required to find a large-margin separator. By contrast, without the use of adversarial examples, gradient methods may require exponentially many iterations to achieve large margin. Our results are derived by showing that adversarial training with gradient updates minimizes a robust version of the empirical risk at a $\mathcal{O}(\ln(t)^2/t)$ rate, despite non-smoothness. We corroborate our theory empirically.'
author:
- |
Zachary Charles\
University of Wisconsin-Madison\
`zcharles@wisc.edu`\
Shashank Rajput\
University of Wisconsin-Madison\
`rajput3@wisc.edu`\
Stephen Wright\
University of Wisconsin-Madison\
`swright@cs.wisc.edu`\
Dimitris Papailiopoulos\
University of Wisconsin-Madison\
`dimitris@papail.io`\
bibliography:
- 'adv\_learning\_arxiv.bib'
title: |
Convergence and Margin of Adversarial Training\
on Separable Data
---
|
---
author:
- |
A. Karska, L. E. Kristensen, E. F. van Dishoeck, M. N. Drozdovskaya, J. C. Mottram, G. J. Herczeg, S. Bruderer, S. Cabrit, N. J. Evans II, D. Fedele, A. Gusdorf, J. K. J[ø]{}rgensen, M. J. Kaufman,\
G. J. Melnick, D. A. Neufeld, B. Nisini, G. Santangelo, M. Tafalla, S. F. Wampfler
bibliography:
- 'biblio14.bib'
date: 'Received May 9, 2014; accepted September 15, 2014'
title: |
Shockingly low water abundances in *Herschel* / PACS\
observations of low-mass protostars in Perseus
---
[Protostars interact with their surroundings through jets and winds impacting on the envelope and creating shocks, but the nature of these shocks is still poorly understood.]{} [Our aim is to survey far-infrared molecular line emission from a uniform and significant sample of deeply-embedded low-mass young stellar objects (YSOs) in order to characterize shocks and the possible role of ultraviolet radiation in the immediate protostellar environment.]{} [Herschel/PACS spectral maps of 22 objects in the Perseus molecular cloud were obtained as part of the ‘William Herschel Line Legacy’ (WILL) survey. Line emission from H$_\mathrm{2}$O, CO, and OH is tested against shock models from the literature. ]{} [Observed line ratios are remarkably similar and do not show variations with source physical parameters (luminosity, envelope mass). Most ratios are also comparable to those found at off-source outflow positions. Observations show good agreement with the shock models when line ratios of the same species are compared. Ratios of various H$_\mathrm{2}$O lines provide a particularly good diagnostic of pre-shock gas densities, $n_\mathrm{H}\sim10^{5}$ cm$^{-3}$, in agreement with typical densities obtained from observations of the post-shock gas when a compression factor of order 10 is applied (for non-dissociative shocks). The corresponding shock velocities, obtained from comparison with CO line ratios, are above 20 kms$^{-1}$. However, the observations consistently show one-to-two orders of magnitude lower H$_\mathrm{2}$O-to-CO and H$_\mathrm{2}$O-to-OH line ratios than predicted by the existing shock models.]{} [The overestimated model H$_\mathrm{2}$O fluxes are most likely caused by an overabundance of H$_\mathrm{2}$O in the models since the excitation is well-reproduced. Illumination of the shocked material by ultraviolet photons produced either in the star-disk system or, more locally, in the shock, would decrease the H$_\mathrm{2}$O abundances and reconcile the models with observations. Detections of hot H$_\mathrm{2}$O and strong OH lines support this scenario.]{}
Introduction
============
Shocks are ubiquitous phenomena where outflow-envelope interactions take place in young stellar objects (YSO). Large-scale shocks are caused by the bipolar jets and protostellar winds impacting the envelope along the ‘cavity walls’ carved by the passage of the jet [@Ar07; @Fr14]. This important interaction needs to be characterized in order to understand and quantify the feedback from protostars onto their surroundings and, ultimately, to explain the origin of the initial mass function, disk fragmentation and the binary fraction.
Theoretically, shocks are divided into two main types based on a combination of magnetic field strength, shock velocity, density, and level of ionization [@Dr80; @Dr83; @Hol89; @Hol97]. In ‘continuous’ ($C-$type) shocks, in the presence of a magnetic field and low ionization, the weak coupling between the ions and neutrals results in a continuous change in the gas parameters. Peak temperatures of a few $10^3$ K allow the molecules to survive the passage of the shock, which is therefore referred to as non-dissociative. In ‘jump’ ($J-$type) shocks, physical conditions change in a discontinuous way, leading to higher peak temperatures than in $C$ shocks of the same speed and for a given density. Depending on the shock velocity, $J$ shocks are either non-dissociative (velocities below about $\sim30$ kms$^{-1}$, peak temperatures of about a few $10^4$ K) or dissociative (peak temperatures even exceeding 10$^5$ K), but the molecules efficiently reform in the post-shock gas.
Shocks reveal their presence most prominently in the infrared (IR) domain, where the post-shock gas is efficiently cooled by numerous atomic and molecular emission lines. Cooling from H$_2$ is dominant in outflow shocks [@Ni10b; @Gi11], but its mid-IR emission is strongly affected by extinction in the dense envelopes of young protostars [@Gi01; @Ni02; @Da08; @Ma09]. In the far-IR, rotational transitions of water vapor (H$_2$O) and carbon monoxide (CO) are predicted to play an important role in the cooling process [@GL78; @ND89; @Hol97] and can serve as a diagnostic of the shock type, its velocity, and the pre-shock density of the medium [@Hol89; @KN96; @FP10; @FP12].
The first observations of the critical wavelength regime to test these models ($\lambda\sim45-200$ $\mu$m) were taken using the Long-Wavelength Spectrometer [LWS, @LWS] onboard the *Infrared Space Observatory* [ISO, @ISO]. Far-IR atomic and molecular emission lines were detected toward several low-mass deeply-embedded protostars [@Ni00; @Gi01; @EvDISO], but its origin was unclear due to the poor spatial resolution of the telescope [$\sim80''$, @Ni02; @Ce02].
The sensitivity and spectral resolution of the Photodetector Array Camera and Spectrometer [PACS, @Po10] onboard *Herschel* allowed a significant increase in the number of young protostars with far-IR line detections compared with the early ISO results and has revealed rich molecular and atomic line emission both at the protostellar [e.g. @vK10; @Go12; @He12; @Vi12; @Gr13; @Ka13; @Li14; @Ma12; @Wa13] and at pure outflow positions [@Sa12; @Sa13; @Cod12; @Lef12; @Va12; @Ni13].
The unprecedented spatial resolution of PACS allowed detailed imaging of L1157 providing firm evidence that most of far-IR H$_2$O emission originates in the outflows [@Ni10]. A mapping survey of about 20 protostars revealed similarities between the spatial extent of H$_2$O and high$-J$ CO [@Ka13]. Additional strong flux correlations between those species and similarities in the velocity-resolved profiles [@Kr10; @Kr12; @IreneCO; @Sa14] suggest that the emission from the two molecules arises from the same regions. This is further confirmed by finely-spatially sampled PACS maps in CO 16-15 and various H$_\mathrm{2}$O lines in shock positions of L1448 and L1157 [@Sa13; @Ta13]. On the other hand, the spatial extent of OH resembles the extent of \[\] and, additionally, a strong flux correlation between the two species is found [@Ka13; @Wa13]. Therefore, at least part of the OH emission most likely originates in a dissociative $J-$shock, together with \[\] [@Wa10; @Be12; @Wa13].
To date, comparisons of the far-IR observations with shock models have been limited to a single source or its outflow positions [e.g. @Ni99; @Be12; @Va12; @Sa12; @Di13; @Lee13; @FP13]. Even in these studies, separate analysis of each species (or different pairs of species) often led to different sets of shock properties that needed to be reconciled. For example, @Di13 show that CO and H$_2$ line emission in Serpens SMM3 originates from a 20 kms$^{-1}$ $J$ shock at low pre-shock densities ($\sim10^4$ cm$^{-3}$), but the H$_2$O and OH emission is better explained by a 30-40 kms$^{-1}$ $C$ shock. In contrast, @Lee13 associate emission from both CO and H$_2$O in L1448-MM with a 40 kms$^{-1}$ $C$ shock at high pre-shock densities ($\sim10^5$ cm$^{-3}$), consistent with the analysis of the L1448-R4 outflow position [@Sa13]. Analysis restricted to H$_2$O lines alone often indicates an origin in non-dissociative $J$ shocks [@Sa12; @Va12; @Bu14], while separate analysis of CO, OH, and atomic species favors dissociative $J$ shocks [@Be12; @Le12]. The question remains how to break degeneracies between these models and how typical the derived shock properties are for young protostars. Also, surveys of high-$J$ CO lines with PACS have revealed two universal temperature components in the CO ladder toward all deeply-embedded low-mass protostars [@He12; @Go12; @Gr13; @Ka13; @Kr13; @Ma12; @Lee13]. The question is how the different shock properties relate with the ‘warm’ ($T\sim300$ K) and ‘hot’ ($T\gtrsim700$ K) components seen in the CO rotational diagrams.
In this paper, far-IR spectra of 22 low-mass YSOs observed as part of the ‘William Herschel Line Legacy’ (WILL) survey (PI: E.F. van Dishoeck) are compared to the shock models from @KN96 and @FP10. All sources are confirmed deeply-embedded YSOs located in the well-studied Perseus molecular cloud spanning the Class 0 and I regime [@KS00; @En06; @Jo06; @Jo07b; @Ha07b; @Ha07a; @Da08; @En09; @Ar10]. H$_2$O, CO, and OH lines are analyzed together for this uniform sample to answer the following questions: Do far-IR line observations agree with the shock models? How much variation in observational diagnostics of shock conditions is found between different sources? Can one set of shock parameters explain all molecular species and transitions? Are there systematic differences between shock characteristics inferred using the CO lines from the ‘warm’ and ‘hot’ components? How do shock conditions vary with the distance from the powering protostar?
This paper is organized as follows. Section 2 describes our source sample, instrument with adopted observing mode, and reduction methods. §3 presents the results of the observations: line and continuum maps, and the extracted spectra. §4 shows comparison between the observations and shock models. §5 discusses results obtained in §4 and §6 presents the conclusions.
-------- --------------------------------------------- -------------------------- ------------------ ------------------------------- ------------------------------- ---------- --------------------------------------------------- -- --
Object R.A. Decl. $T_\mathrm{bol}$ $L_\mathrm{bol}$ $M_\mathrm{env}$ Region Other names
($^\mathrm{h}$ $^\mathrm{m}$ $^\mathrm{s}$) ($^\mathrm{o}$ $'$ $''$) (K) ($\mathrm{L}_\mathrm{\odot}$) ($\mathrm{M}_\mathrm{\odot}$)
Per01 03:25:22.32 +30:45:13.9 44 4.5 1.14 L1448 Per-emb 22, L1448 IRS2, IRAS03222+3034, YSO 1
Per02 03:25:36.49 +30:45:22.2 50 10.6 3.17 L1448 Per-emb 33, L1448 N(A), L1448 IRS3, YSO 2
Per03 03:25:39.12 +30:43:58.2 47 8.4 2.56 L1448 Per-emb 42, L1448 MMS, L1448 C(N), YSO 3
Per04 03:26:37.47 +30:15:28.1 61 1.2 0.29 L1451 Per-emb 25, IRAS03235+3004, YSO 4
Per05 03:28:37.09 +31:13:30.8 85 11.1 0.35 NGC1333 Per-emb 35, NGC1333 IRAS1, IRAS03255+3103, YSO 11
Per06 03:28:57.36 +31:14:15.9 85 6.9 0.30 NGC1333 Per-emb 36, NGC1333 IRAS2B, YSO 16
Per07 03:29:00.55 +31:12:00.8 37 0.7 0.32 NGC1333 Per-emb 3, HRF 65, YSO 18
Per08 03:29:01.56 +31:20:20.6 131 16.8 0.86 NGC1333 Per-emb 54, HH 12, YSO 19
Per09 03:29:07.78 +31:21:57.3 128 23.2 0.24 NGC1333 Per-emb 50
Per10 03:29:10.68 +31:18:20.6 45 6.9 1.37 NGC1333 Per-emb 21, HRF 46, YSO 23
Per11 03:29:12.06 +31:13:01.7 28 4.4 5.42 NGC1333 Per-emb 13, NGC1333 IRAS4B’, YSO 25
Per12 03:29:13.54 +31:13:58.2 31 1.1 1.30 NGC1333 Per-emb 14, NGC1333 IRAS4C, YSO 26
Per13 03:29:51.82 +31:39:06.0 40 0.7 0.51 NGC1333 Per-emb 9, IRAS03267+3128, YSO 31
Per14 03:30:15.14 +30:23:49.4 88 1.8 0.14 B1-ridge Per-emb 34, IRAS03271+3013
Per15 03:31:20.98 +30:45:30.1 35 1.7 1.29 B1-ridge Per-emb 5, IRAS03282+3035, YSO 32
Per16 03:32:17.96 +30:49:47.5 30 1.1 2.75 B1-ridge Per-emb 2, IRAS03292+3039, YSO 33
Per17 03:33:14.38 +31:07:10.9 43 0.7 1.20 B1 Per-emb 6, B1 SMM3, YSO 35
Per18 03:33:16.44 +31:06:52.5 25 1.1 1.22 B1 Per-emb 10, B1 d, YSO 36
Per19 03:33:27.29 +31:07:10.2 93 1.1 0.23 B1 Per-emb 30, B1 SMM11, YSO 40
Per20 03:43:56.52 +32:00:52.8 27 2.3 2.05 IC 348 Per-emb 1, HH 211 MMS, YSO 44
Per21 03:43:56.84 +32:03:04.7 34 2.1 1.88 IC 348 Per-emb 11, IC348 MMS, IC348 SW, YSO 43
Per22 03:44:43.96 +32:01:36.2 43 2.6 0.64 IC 348 Per-emb 8, IC348 a, IRAS03415+3152, YSO 48
-------- --------------------------------------------- -------------------------- ------------------ ------------------------------- ------------------------------- ---------- --------------------------------------------------- -- --
Observations
============
All observations presented here were obtained as part of the ‘William Herschel Line Legacy’ (WILL) OT2 program on *Herschel* (Mottram et al. in prep.). The WILL survey is a study of H$_\mathrm{2}$O lines and related species with PACS and the Heterodyne Instrument for the Far-Infrared [HIFI, @dG10] toward an unbiased flux-limited sample of low-mass protostars newly discovered in the recent [*Spitzer*]{} [c2d, @Gu09; @Gu10; @Ev09] and *Herschel* [@An10] Gould Belt imaging surveys. Its main aim is to study the physics and chemistry of star-forming regions in a statistically significant way by extending the sample of low-mass protostars observed in the ‘Water in star-forming regions with Herschel’ [WISH, @WISH] and ‘Dust, Ice, and Gas in Time’ [DIGIT, @Gr13] programs.
This paper presents the *Herschel*/PACS spectra of 22 low-mass deeply-embedded YSOs located exclusively in the Perseus molecular cloud (see Table \[catalog\]) to ensure the homogeneity of the sample (similar ages, environment, and distance). The sources were selected from the combined SCUBA and *Spitzer*/IRAC and MIPS catalog of @Jo07b and @En09, and all contain a confirmed embedded YSO [Stage 0 or I, @Ro06; @Ro07] in the center.
The WILL sources were observed using the line spectroscopy mode on PACS which offers deep integrations and finely sampled spectral resolution elements (minimum 3 samples per FWHM depending on the grating order, PACS Observer’s Manual[^1]) over short wavelength ranges (0.5–2 $\mu$m). The line selection was based on the prior experience with the PACS spectra obtained over the full far-infrared spectral range in the WISH and DIGIT programs and is summarized in Table \[lines\]. Details of the observations of the Perseus sources within the WILL survey are shown in Table \[log\].
PACS is an integral field unit with a 5$\times$5 array of spatial pixels (hereafter *spaxels*) covering a field of view of $\sim47''\times47''$. Each spaxel measured $\sim9.4''\times9.4''$, or 2$\times$10$^{-9}$ sr, and at the distance to Perseus [$d=235$ pc, @Hir08] resolves emission down to $\sim$2,300 AU. The total field of view is about 5.25$\times 10^{-8}$ sr and $\sim$11,000 AU. The properly flux-calibrated wavelength ranges include: $\sim$55–70 $\mu$m, $\sim$72–94 $\mu$m, and $\sim$105–187 $\mu$m, corresponding to the second ($<$ 100 $\mu$m) and first spectral orders ($>$ 100 $\mu$m). Their respective spectral resolving power are $R\sim$2500–4500 (velocity resolution of $\Delta\varv\sim$70-120 km s$^{-1}$), 1500–2500 ($\Delta\varv\sim$120-200 km s$^{-1}$), and 1000–1500 ($\Delta\varv\sim$200-300 km s$^{-1}$). The standard chopping-nodding mode was used with a medium (3$^\prime$) chopper throw. The telescope pointing accuracy is typically better than 2$^\prime$$^\prime$ and can be evaluated to first order using the continuum maps.
The basic data reduction presented here was performed using the Herschel Interactive Processing Environment v.10 [<span style="font-variant:small-caps;">hipe</span>, @Ot10]. The flux was normalized to the telescopic background and calibrated using observations of Neptune. Spectral flatfielding within HIPE was used to increase the signal-to-noise [for details, see @He12; @Gr13]. The overall flux calibration is accurate to $\sim 20\%$, based on the flux repeatability for multiple observations of the same target in different programs, cross-calibrations with HIFI and ISO, and continuum photometry.
Custom <span style="font-variant:small-caps;">idl</span> routines were used to extract fluxes using Gaussian fits with fixed line width [for details, see @He12]. The total, 5$\times$5 line fluxes were calculated by co-adding spaxels with detected line emission, after excluding contamination from other nearby sources except Per 2, 3, 10, and 18, where spatial separation between different components is too small (see §3.1). For sources showing extended emission, the set of spaxels providing the maximum flux was chosen for each line separately. For point-like sources, the flux is calculated at the central position and then corrected for the PSF using wavelength-dependent correction factors (see PACS Observer’s Manual). Table \[det\] shows the line detections toward each source, while the actual fluxes will be tabulated in the forthcoming paper for all the WILL sources (Karska et al. in prep.). Fits cubes containing the spectra in each spaxel will be available for download in early 2015 at http://www.strw.leidenuniv.nl/WISH/.
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Results
=======
In the following sections, PACS lines and maps of the Perseus YSOs are presented. Most sources in this sample show emission in just the central spaxel. Only a few sources show extended emission and those maps are compared to maps at other wavelengths to check for possible contamination by other sources and their outflows. In this way, the spaxels of the maps with emission originating from our objects are established and line fluxes determined over those spaxels. The emergent line spectra are then discussed.
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Spatial extent of line emission
-------------------------------
-------- ---- ----- ---- ----- --------- ---------- -------- ---------- ----------------------------------------------- -- --
Object Remarks
on off on off single multiple single multiple
Per01 X - X - X - X -
Per02 X - - X - X - Xc three sources (3), contam. by Per03 (1,2)
Per03 - X Xe - X - X a binary (1,4)
Per04 X - nd nd X - X -
Per05 X - X - X - X -
Per06 - X - X - X? - Xc contam. by NGC1333 IRAS 2A in N-W (5)
Per07 X - nd nd X - Xc - contam. in lines in N-W
Per08 - X Xe - X - X -
Per09 X - Xe - X? - X? -
Per10 - X - X - X - Xc two sources (1,2), dominated by YSO 24 (1,2)
Per11 - X Xe - - X - Xc NGC1333 IRAS4A in N-W (5)
Per12 X - nd nd X - Xc - contam. in lines by NGC1333 IRAS4A in N-W (5)
Per13 X - nd nd X? - X?
Per14 X - X - X? - X?
Per15 X - X - X - X -
Per16 X - nd nd X - X -
Per17 - X? nd nd - X - Xc? Per18 in S-E, nodded on emission
Per18 X - X - X - - Xc? B1-b outflow? (2)
Per19 X - X - X - X -
Per20 X - Xe - X - X -
Per21 X - Xe - X? - X? -
Per22 X - Xe - X? - X? -
-------- ---- ----- ---- ----- --------- ---------- -------- ---------- ----------------------------------------------- -- --
Table \[maps\] provides a summary of the patterns of H$_\mathrm{2}$O 2$_{12}$-1$_{01}$ line and continuum emission at 179 $\mu$m for all WILL sources in Perseus (maps are shown in Figs. \[specmap1\] and \[specmap2\]). The mid-infrared continuum and H$_\mathrm{2}$ line maps from @Jo06 and @Da08 [including CO 3–2 observations from @Ha07b] are used to obtain complementary information on the sources and their outflows. For a few well-known outflow sources, large-scale CO 6-5 maps from Y[i]{}ld[i]{}z et al. (subm.) are also considered.
As shown in Table \[maps\], the majority of the PACS maps toward Perseus YSOs do not show any extended line emission. The well-centered continuum and line emission originates from a single object and an associated bipolar outflow for 12 out of 22 sources. Among the sources with spatially-resolved extended emission on the maps, various reasons are identified for their origin as illustrated in Fig. \[fig:maps\]. In the map of Per 2, contribution from three nearby protostars and a strong outflow from the more distant L1448-MM source cause the extended line and continuum pattern. Emission in the Per 12 map is detected away from the continuum peak, but the emission originates from a large-scale outflow from NGC1333-IRAS4A, not the targeted source. The H$_\mathrm{2}$O emission in the Per 20 map is detected in the direction of the strong outflow and its extent is only slightly affected by the small mispointing revealed by the asymmetric continuum emission.
Extended emission beyond the well-centered continuum, as in the case of Per 20, is seen clearly only in Per 9 and Per 21-22. Additionally, the continuum peaks for Per 3, 8, and 11 are off-center, whereas the line emission peaks on-source, suggesting that some extended line emission is associated with the source itself and not only due to the mispointing. [^2] Similar continuum patterns are seen in Per 6 and Per 10, but here the line emission peaks a few spaxels away from the map center. In both cases, contribution from additional outflows / sources is the cause of the dominant off-source line emission.
To summarize, when the contamination of other sources and their outflows can be excluded, Perseus YSOs show that the H$_\mathrm{2}$O 2$_{12}$-1$_{01}$ line emission is either well-confined to the central position on the map or shows at best weak extended line emission (those are marked with e’ in Table 2). In total, 7 out of 22 sources show extended emission in the H$_\mathrm{2}$O 2$_{12}$-1$_{01}$ line associated with the targeted sources. Emission in CO, OH, and other H$_\mathrm{2}$O lines follows the same pattern (see Figure \[specmapmix\]). Similarly compact emission was seen in a sample of 30 protostars surveyed in the DIGIT program [@Gr13]. In contrast, the WISH PACS survey [@Ka13] revealed strong extended emission in about half of the 20 low-mass protostars. There, the analysis of patterns of molecular and atomic emission showed that H$_\mathrm{2}$O and CO spatially co-exist within the PACS field-of-view, while OH and \[$\ion{O}{i}$\] lines are typically less extended, but also follow each other spatially and not H$_\mathrm{2}$O and CO.
Line detections
---------------
![\[obsratio\] Flux ratios of the H$_\mathrm{2}$O 4$_{04}$-3$_{13}$ and CO 21-20 lines at $\sim$125 $\mu$m calculated using *compact* and *extended* flux extraction regions (see text) corrected for contamination from other sources / outflows. Median values for the two configurations are shown by the dashed line. The light blue rectangle shows the parameter space between the minimum and maximum values of line ratios in the more extended configuration. The error bars reflect the uncertainties in the measured fluxes of the two lines, excluding the calibration error, which is the same for those closely spaced lines.](obsratios.eps){height="6.5cm"}
[ccccccccc]{} Object & 2$_{12}$-1$_{01}$ / 16-15 & 4$_{04}$-3$_{13}$ / 24-23 & 16-15 / 24-23 & 2$_{12}$-1$_{01}$ / 4$_{04}$-3$_{13}$ & 2$_{21}$-1$_{10}$ / 4$_{04}$-3$_{13}$ & OH 84 / 79 & Ref.\
\
Perseus & 0.2–2.4 & 0.2–1.1 & 1.2–4.6 & 1.3–6.3 & 1.4–5.5 & 1.1–2.4 & This work\
\
SMM3 b & 1.0$\pm$0.1 & 0.5$\pm$ 0.2 & 3.3$\pm$0.8 & 7.1$\pm$1.0 & 3.7$\pm$1.0 & 1.4$\pm$0.7 & (1)\
c & 0.9$\pm$0.2 & 0.9$\pm$ 0.3 & 4.6$\pm$1.0 & 4.9$\pm$1.0 & 2.3$\pm$0.8 & 2.4$\pm$1.2 & (1)\
r & 1.0$\pm$0.1 & **2.0$\pm$0.7** & **9.4$\pm$2.3** & 4.7$\pm$1.0 & 3.1$\pm$0.9 & n.d. & (1)\
SMM4 r & 0.8$\pm$0.1 & 0.4$\pm$ 0.1 & 3.6$\pm$0.6 & 7.8$\pm$1.6 & 3.0$\pm$1.3 & 1.1$\pm$0.5 & (1)\
L1448-MM & 2.3$\pm$1.3 & 1.2$\pm$0.7 & 2.1$\pm$1.2 & 4.1$\pm$2.3 & 2.6$\pm$1.5 & 1.4$\pm$0.9 & (2)\
NGC1333 I4B & 1.0$\pm$0.1 & 1.0$\pm$0.1 & 1.9$\pm$0.1 & 1.9$\pm$0.1 & 2.0$\pm$0.1 & 1.2$\pm$0.2 & (3)\
\
L1157 B1 & **4.0$\pm$0.5** & – & – & 11.0 & 2.9$\pm$1.2 & n.d. & (4,5)\
B1’ & 2.1$\pm$0.2 & – & – & **9.2$\pm$2.2** & 2.8$\pm$1.0 & 0.9$\pm$0.5 & (4,5)\
B2 & **17.0$\pm$8.4** & n.d. & $>$0.3 & **$>$7.7** & n.d. & – & (6)\
R & **5.3$\pm$2.4** & $>$0.26 & $>$0.5 & **10.6$\pm$4.7** & 1.8$\pm$0.8 & – & (6)\
L1448 B2 & 2.3$\pm$0.4 & 0.4$\pm$0.3 & 2.8$\pm$0.9 & **15.0$\pm$6.5** & 3.7$\pm$1.6 & – & (6)\
R4 & **8.2$\pm$3.0** & $>$0.3 & $>$0.9 & **23.2$\pm$9.7** & 3.3$\pm$1.2 & – & (6)\
In the majority of our sources, all targeted rotational transitions of CO, H$_2$O, and OH are detected, see Fig. \[spec\] and Tables \[lines\] and \[det\]. The \[\] line at 63 $\mu$m and the \[\] line at 158 $\mu$m will be discussed separately in a forthcoming paper including all WILL sources (Karska et al. in prep.) and are not included in the figure and further analysis.
The CO 16-15 line with an upper level energy ($E_\mathrm{u}/k_\mathrm{B}$) of about 750 K is seen in 17 ($\sim$80%), the CO 24-23 ($E_\mathrm{u}/k_\mathrm{B}\sim1700$ K) in 16 ($\sim$70%), and the CO 32-31 ($E_\mathrm{u}/k_\mathrm{B}\sim3000$ K) in 8 ($\sim$30%) sources. The most commonly detected ortho-H$_2$O lines are: the 2$_{12}$-1$_{01}$ line at 179 $\mu$m ($E_\mathrm{u}/k_\mathrm{B}\sim110$ K) and the 4$_{04}$-3$_{13}$ line at 125 $\mu$m ($E_\mathrm{u}/k_\mathrm{B}\sim320$ K), seen in 15 sources ($\sim70$ %), whereas the 6$_{16}$-5$_{05}$ line at 82 $\mu$m ($E_\mathrm{u}/k_\mathrm{B}\sim640$ K) is detected in 13 sources ($\sim60$ %). The para-H$_2$O line 3$_{22}$-2$_{11}$ at 90 $\mu$m ($E_\mathrm{u}/k_\mathrm{B}\sim300$ K) is seen toward 9 sources ($\sim50$ %).
Three OH doublets targeted as part of the WILL survey, the OH ${}^{2}\Pi_{\nicefrac{1}{2}}, J = \nicefrac{1}{2} - {}^{2}\Pi_{\nicefrac{3}{2}}, J = \nicefrac{3}{2}$ at 79 $\mu$m, $^{2}\Pi_{\nicefrac{3}{2}}$ $J=\nicefrac{7}{2}-\nicefrac{5}{2}$ doublet at 84 $\mu$m ($E_\mathrm{u}/k_\mathrm{B}\sim290$ K), and $^{2}\Pi_{\nicefrac{1}{2}}$ $J=\nicefrac{3}{2}-\nicefrac{1}{2}$ doublet at 163 $\mu$m ($E_\mathrm{u}/k_\mathrm{B}\sim270$ K), are detected in 14, 15, and 13 objects, respectively ($\sim60-70$ %).
Sources without any detections of molecular lines associated with the targeted protostars are Per 4, 7, 12, and 17 (see Table \[det\]). In Per 13 only two weak H$_2$O lines at 108 and 125 $\mu$m are seen, whereas in Per 16 only a few of the lowest-$J$ CO lines are detected. A common characteristic of this weak-line group of objects, is a low bolometric temperature (all except Per 4) and a low bolometric luminosity (see Table \[catalog\]), always below 1.3 $L_{\odot}$. However, our sample also includes a few objects with similarly low values of $L_\mathrm{bol}$, that show many more molecular lines (in particular Per 3, but also Per 15 and 21), so low luminosity by itself is not a criterion for weak lines. On the other hand, low bolometric temperature and high luminosity is typically connected with strong line emission [@Kr12; @Ka13].
Observed line ratios
--------------------
Observed line ratios are calculated using the fluxes obtained from the entire 5$\times$5 PACS maps in cases of extended emission where contamination by a nearby source and / or outflows is excluded. For point sources, a wavelength-dependent PSF correction factor is applied to fluxes obtained from the central spaxel (for details, see §2). In principle, ratios using lines that are close in wavelength could be calculated using smaller flux extraction regions, and no PSF correction would be required. However, the transition wavelengths of H$_2$O are not proportional to the upper energy levels, as is the case for CO, and comparisons of lines tracing similar gas have to rely on lines that lie far apart in wavelength. We explore to what extent the size of the extraction region affects the inferred line ratios. Since our aim is to understand the influence of the extended emission associated with the source(s), the ratios of lines close in wavelength are studied to avoid the confusion due to the PSF variations.
Fluxes of the nearby H$_2$O 4$_{04}$-3$_{13}$ and CO 21-20 lines located at 124–125 $\mu$m are calculated first using only the central spaxel ([*compact region*]{}) and then using all the spaxels with detected line emission ([*extended region*]{}). Fig. \[obsratio\] illustrates that the line ratios calculated in these two regions are fully consistent, both for sources with compact and extended emission. Therefore, for the subsequent analysis the extended region is used for comparisons with models. Table \[tab:ratios\] shows the minimum and maximum values of the observed line ratios, their mean values, and the standard deviations for all sources with detections.
Even taking into account the uncertainties in flux extraction, the line ratios span remarkably narrow ranges of values, see Table \[tab:comp\] for a selection of H$_2$O, CO, and OH line ratios. The largest range is seen in the H$_2$O 2$_{12}$-1$_{01}$/CO 16-15 line ratio, which spans an order of magnitude. In all the other cases the line ratios are similar up to a factor of a few. The most similar are the OH line ratios which differ only by a factor of two, consistent with previous studies based on a large sample of low-mass YSOs in @Wa13. The observed similarities also imply that the line ratios do not depend on protostellar luminosity, bolometric temperature, or envelope mass (see Fig. \[corr125\]).
Our line ratios for Perseus sources are consistent with the previously reported values for other deeply-embedded protostars observed in the same way (‘on source’) as tabulated in Table \[tab:comp\]. Some differences are found for PACS observations of shock positions away from the protostar (‘off source’). Most notably, the ratios using the low excitation H$_2$O 2$_{12}$-1$_{01}$ line at 179 $\mu$m are up to a factor of two larger than those observed in the protostellar vicinity. Such differences are not seen when more highly-excited H$_2$O lines are compared with each other, for example H$_2$O 2$_{21}$-1$_{10}$ and 4$_{04}$-3$_{13}$ lines, or with the high$-J$ CO lines, for example CO 24–23. The ratios of two CO lines observed away from the protostar, e.g. the CO 16–15 and CO 24–23 ratios, are at the low end of the range observed toward the protostellar position.
Spectrally-resolved profiles of the H$_2$O 2$_{12}$-1$_{01}$ line observed with HIFI toward the protostar position reveal absorptions at source velocity removing about 10% of total line flux [e.g. @Kr10; @Mo14]. Our unresolved PACS observations therefore provide a lower limit to the H$_2$O emission in the 2$_{12}$-1$_{01}$ line. This effect, however, is too small to explain the differences in the line ratios at the ‘on source’ and ‘off source’ positions.
Analysis
========
The fact that multiple molecular transitions over a wide range of excitation energies are detected, points to the presence of hot, dense gas and can be used to constrain the signatures of shocks created as a result of outflow-envelope interaction. In particular, the line ratios of H$_2$O, CO, and OH are useful probes of various shock types and parameters that do not suffer from distance uncertainties.
Similarities between the spatial extent of different molecules (§3.1) coupled with similarities in the velocity-resolved line profiles among these species [@Kr10; @Yi13; @IreneCO; @Mo14] strongly suggest that all highly-excited lines of CO and H$_2$O arise from the same gas. Some differences may occur for OH, which is also associated with dissociative shocks and can be affected by radiative excitation (see §5). Modeling of absolute line fluxes requires sophisticated two-dimensional (2D) physical source models for the proper treatment of the beam filling factor [@Vi12]. Those models also show that UV heating alone is not sufficient to account for the high excitation lines. Hence, the focus in this analysis is on shocks. Since the absolute flux depends sensitively on the assumed emitting area, in the subsequent analysis only the line ratios are compared.
In the following sections, properties of shock models and the predicted line emission in various species are discussed (§4.1) and observations are compared with the models, using line ratios of the same species (§4.2), and different species (§4.3). Special focus will be given to $C-$type shocks where grids of model results are available in the literature. Observations suggest that most of the mass of hot gas is in $C-$type shocks toward the central protostellar positions, at least for H$_2$O and CO with $J$ $<$ 30 [@Kr13 Kristensen et al. in prep.]; higher-$J$ CO, OH and \[\] transitions, on the other hand, will primarily trace $J-$type shocks [e.g. @Wa13; @Kr13]. The excitation of OH and \[\] will be analyzed in a forthcoming paper; the $J>30$ CO emission is only detected toward $\sim$ 30% of all sources and so is likely unimportant for the analysis and interpretation of the data presented here. Only limited discussion of $J-$ type shocks is therefore presented below.
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Model line emission
-------------------
Models of shocks occurring in a medium with physical conditions typical for the envelopes of deeply-embedded young stellar objects provide a valuable tool for investigating shock characteristics: shock type, velocity, and the pre-shock density of (envelope) material.
Model grids have been published using a simple 1D geometry either for steady-state $C$ and $J$ type shocks [@Hol89; @KN96; @FP10] or time-dependent $C-J$ type shocks [@Gus08; @Gus11; @FP12]. The latter are non-stationary shocks, where a $J-$type front is embedded in a $C-$type shock [@Ch98; @Le04b; @Le04a]. These shocks are intermediate between pure $C-$ and $J-$type shocks and have temperatures and physical extents in between the two extremes.
$C$-$J$ shocks may be required for the youngest outflows with ages less than 10$^3$ yrs, [@FP12; @FP13 for the case of IRAS4B, Per 11]. Here the dynamical age of the outflow is taken as an upper limit of that of the shock itself, which may be caused by a more recent impact of the wind on the envelope. The age of our sources is of the order of 10$^5$ yrs [@Sad14 for Class 0 sources in Perseus] and they should have been driving winds and jets for the bulk of this period, so this timescale is long enough for any shocks close to the source position to have reached steady state. While we cannot exclude that a few individual shocks have been truncated, our primary goal is to examine trends across the sample. Invoking $C$-$J$ type shocks with a single truncation age as an additional free parameter is therefore not a proper approach for this study. The focus is therefore placed on comparing $C-$type shock results from @KN96 [KN96 from now on] and @FP10 [F+PdF10 from now on] with the observations.
All models assume the same initial atomic abundances and similarly low degrees of ionization, $x_{\rm i}$ $\sim$ 10$^{-7}$ for $C$ shocks. The pre-shock transverse magnetic field strength is parametrized as $B_{0}=b\times \sqrt{n_\mathrm{H}(\mathrm{~cm}^{-3})}$ $\mu$Gauss, where $n_\mathrm{H}$ is the pre-shock number density of atomic hydrogen and $b$ is the magnetic scaling factor, which is typically 0.1-3 in the ISM [@Dr80]. The value of $b$ in KN96 and F+PdF10 is fixed at a value of 1.
The main difference between the two shock models is the inclusion of grains in the F+PdF10 models [@FP03]. The latter models assume a standard MRN distribution of grain sizes [@Ma77] for grain radii between 0.01 and 0.3 $\mu$m and a fractional abundance of the PAH in the gas phase of $10^{-6}$ [the role of PAHs and the sizes of grains are discussed in @FP03; @FP12]. As the electrons are accelerated in the magnetic precursor, they attach themselves to grains thereby charging the grains and thus increasing the density of the ionized fluid significantly. This increase in density has the effect of enhancing the ion-neutral coupling [@Dr80], thereby effectively lowering the value of $b$ compared to the KN96 models. As a consequence, the maximum kinetic temperature is higher in the F+PdF10 models for a given shock velocity, $\varv$. The stronger coupling between the ions and neutrals results in narrower shocks [@FP10], with shock widths scaling as $\propto b^2 (x_{\rm i}
n_\mathrm{H} \varv)^{-1}$ [@Dr80]. This proportionality does not capture the ion-neutral coupling exactly, as, for example, the grain size distribution influences the coupling [@Gu07; @Gu11]. The compression in $C$ shocks also changes with the coupling since the post-shock density depends on the magnetic field, $n_\mathrm{post}\sim 0.8 \varv n_\mathrm{H} b^{-1}$ [e.g. @Ka13]. The column density of emitting molecules is a function of both shock width and compression factor, and as a zeroth-order approximation the column density is $N\sim
n_\mathrm{post}\times L \sim b x_{\rm i}^{-1}$. The ionization degree is not significantly different between the KN96 and F+PdF10 models because it is primarily set by the cosmic ray ionization rate ($\zeta=5\cdot10^{-17}$ s$^{-1}$, F+PdF10), and thus the F+PdF10 models predict lower column densities than the KN96 models for a given velocity and density.
Another important difference is that the F+PdF10 models take into account that molecules frozen out onto grain mantles can be released through sputtering when the shock velocity exceeds $\sim$ $15$ kms$^{-1}$ [@FP10; @FP12; @vL13]. Therefore, the gas-phase column densities of molecules locked up in ices increase above this threshold shock velocity with respect to the KN96 models, an effect which applies to both CO and H$_2$O. Furthermore, H$_\mathrm{2}$O forms more abundantly in the post-shock gas of F+PdF10 models, because H$_\mathrm{2}$ reformation is included, unlike in the KN96 models [@FP10].
Molecular emission is tabulated by KN96 for a wide range of shock velocities, from $\varv =5$ to 45 kms$^{-1}$ in steps of 5 kms$^{-1}$, and a wide range of pre-shock densities $n_\mathrm{H}$, from 10$^{4}$ to 10$^{6.5}$ cm$^{-3}$ in steps of 10$^{0.5}$ cm$^{-3}$. The F+PdF10 grid is more limited in size, providing line intensities for only two values of pre-shock densities, namely 10$^{4}$ and 10$^{5}$ cm$^{-3}$, and a comparable range of shock velocities, but calculated in steps of 10 kms$^{-1}$. Calculations are provided for CO transitions from $J=1-0$ to $J=60-59$ in KN96 and only up to $J=20-19$ in F+PdF10. The two sets of models use different collisional rate coefficients to calculate the CO excitation. F+PdF10 show line intensities for many more H$_\mathrm{2}$O transitions (in total $\sim$120 lines in the PACS range, see §2) than in the older KN96 grid (18 lines in the same range), which was intended for comparisons with the Submillimeter Wave Astronomy Satellite [SWAS, @SWAS] and ISO data. KN96 use collisional excitation rates for H$_\mathrm{2}$O from @Gr93 and F+PdF10 from @Fa07. Line intensities for OH are only computed by KN96, assuming only collisional excitation and using the oxygen chemical network of @Wa87. The reaction rate coefficients in that network are within a factor of 2 of the newer values by @Ba12 and tabulated in the UMIST database [www.udfa.net, @UMIST], see also a discussion in @EvD13.
We also use CO fluxes extracted from the grid of models presented by @Kr07, since high$-J$ CO lines are missing in F+PdF10. This grid is based on the shock model presented in @FP03 and covers densities from 10$^4$–10$^7$ cm$^{-3}$ and velocities from 10–50 kms$^{-1}$ (denoted as F+PdF\* from now on). The main difference compared to the results from F+PdF10 is that CO and H$_2$O level populations are not calculated explicitly through the shock; instead analytical cooling functions are used to estimate the relevant line cooling and only afterwards line fluxes are extracted [@FG09]. Models with $b$ = 1 are used. The CO line fluxes presented here are computed using the 3D non-LTE radiative transfer code LIME [@Br10], for levels up to $J$ = 80–79. The CO collisional rate coefficients from @Ya10 extended by @Ne12 are used.
In the following sections, the model fluxes of selected CO, H$_\mathrm{2}$O, and OH lines are discussed for a range of shock velocities and three values of pre-shock densities: 10$^{4}$, 10$^{5}$, and 10$^{6}$ cm$^{-3}$. Note that the post-shock densities traced by observations are related to the pre-shock densities via the compression factor dependent on the shock velocity and magnetic field. In $C$ shocks, the compression factor is about 10 [e.g. @Ka13].
Figure \[vel\] compares model fluxes of various CO, OH, and H$_\mathrm{2}$O lines from the KN96 models (panels a and d) and, for a few selected CO and H$_\mathrm{2}$O lines, compares the results with the F+PdF10 or F+PdF\* models (panels b, c, e, and f).
### CO
The KN96 model line fluxes for CO 16-15, CO 21-20, and CO 29-28 are shown in panel a of Fig. \[vel\]. The upper energy levels of these transitions lie at 750 K, 1280 K, and 2900 K, respectively, while with increasing shock velocity, the peak $C-$shock temperature increases from about 400 K to 3200 K (for 10 to 40 kms$^{-1}$) and is only weakly dependent on the assumed density (see Fig. 3 of KN96). Therefore, the CO 16-15 line is already excited at relatively low shock velocities ($\varv \sim$10 kms$^{-1}$, for $n_\mathrm{H}$=10$^{4}$ cm$^{-3}$), whereas the higher$-J$ levels become populated at higher velocities. At a given shock velocity, emission from the CO 16-15 line is the strongest due to its lower critical density, $n_\mathrm{cr}\sim9\times10^5$ cm$^{-3}$ at $T$=1000 K [@Ne12]. This situation only changes for the highest pre-shock densities, when the line becomes thermalized and the cooling in other lines dominates.
The CO 16-15 flux from the F+PdF10 $C-$type shock models is comparable to the KN96 flux for the $\varv\sim$10 kms$^{-1}$ shock, but increases less rapidly with shock velocity despite the sputtering from grain mantles (panel b of Fig. \[vel\]). Because of the lower magnetic scaling factor $b$ in the models with grains (see §4.1), it is expected that the column density and the corresponding line fluxes are lower in the F+PdF10 models. For slow shocks the higher temperatures in the latter models compensate for the smaller column density resulting in a similar CO 16-15 flux.
In $J-$type shocks, the peak temperatures of the post-shock gas are 1400, 5500, 12000, and 22000 K for the shock velocities of 10, 20, 30, and 40 kms$^{-1}$ respectively [@ND89; @KN96]. For 10-20 kms$^{-1}$ shocks, these high temperatures more easily excite the CO 16-15 line with respect to $C-$type shock emission. For shock velocities above 20 kms$^{-1}$, such high temperatures can lead to the collisional dissociation of H$_\mathrm{2}$ and subsequent destruction of CO and H$_\mathrm{2}$O molecules, resulting in the decrease of CO fluxes. This effect requires high densities and therefore the CO flux decrease is particularly strong for the pre-shock densities 10$^{5}$ cm$^{-3}$.
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### H$_\mathrm{2}$O
The H$_2$O fluxes show a strong increase with shock velocities above $\varv\sim10-15$ kms$^{-1}$ in both models, especially at low pre-shock densities (panels d-f of Fig. \[vel\]). At this velocity, the gas temperature exceeds $\sim$400 K, so the high-temperature route of H$_\mathrm{2}$O formation becomes efficient [@EJ78; @EJ78b; @Be98 KN96], which quickly transfers all gas-phase oxygen into H$_\mathrm{2}$O via reactions with H$_\mathrm{2}$ (KN96).
In contrast to CO, the upper level energies of the observed H$_\mathrm{2}$O lines are low and cover a narrow range of values, $E_\mathrm{up}\sim200-600$ K. As a result, the effect of peak gas temperature on the H$_\mathrm{2}$O excitation is less pronounced (Fig. 3 of KN96) and after the initial increase with shock velocity, the H$_\mathrm{2}$O fluxes in the KN96 models stay constant for all lines. For high pre-shock densities, the higher lying levels are more easily excited and, as a consequence, the fluxes of the H$_\mathrm{2}$O 6$_{16}$-5$_{05}$ line become larger than those of the H$_\mathrm{2}$O 2$_{12}$-1$_{01}$ line. The critical densities of these transitions are about 2 orders of magnitude higher than for the CO 16-15 line and the levels are still sub-thermally excited at densities of $10^6$-$10^7$ cm$^{-3}$ [and effectively optically thin, @Mo14].
The H$_\mathrm{2}$O 2$_{12}$-1$_{01}$ fluxes in the $C-$type F+PdF10 models are remarkably similar to those found by KN96 (panel e of Fig. \[vel\]), while the H$_\mathrm{2}$O 4$_{04}$-3$_{13}$ fluxes are lower by a factor of a few over the full range of shock velocities and pre-shock densities in the F+PdF10 models (panel f of Fig. \[vel\]). Lower H$_\mathrm{2}$O fluxes are expected due to smaller column of H$_\mathrm{2}$O in the models with grains (§4.1). For the lower$-J$ lines, the various factors (lower column density through a shock but inclusion of ice sputtering and H$_2$ reformation) apparently conspire to give similar fluxes as for KN96.
In $J-$type shocks, the fluxes of H$_\mathrm{2}$O 2$_{12}$-1$_{01}$ and 4$_{04}$-3$_{13}$ lines increase sharply for shock velocities 10–20 kms$^{-1}$ (panels e and f of Fig. 4). The increase is less steep at 30 kms$^{-1}$ shocks for $n_\mathrm{H}$=10$^{5}$ cm$^{-3}$, when the collisional dissociation of H$_\mathrm{2}$ and subsequent destruction of H$_\mathrm{2}$O molecules occurs (fluxes for larger shock velocities are not computed in F+PdF10 and hence not shown). Below 30 kms$^{-1}$, line fluxes from $J-$type shocks are comparable to those from $C-$type shock predictions except for the H$_\mathrm{2}$O 2$_{12}$-1$_{01}$ fluxes at high pre-shock densities, which are an order of magnitude higher with respect to the $C-$type shock predictions. The difference could be due to smaller opacities for the low-excitation H$_\mathrm{2}$O line in the $J$ shocks.
### OH
The fluxes of the $^{2}\Pi_{\nicefrac{3}{2}}$ $J=\nicefrac{7}{2}-\nicefrac{5}{2}$ doublet at 84 $\mu$m ($E_\mathrm{u}/k_\mathrm{B}\sim290$ K) calculated with the KN96 models are shown with the H$_2$O lines in panel d of Fig. \[vel\]. Not much variation is seen as a function of shock velocity, in particular beyond the initial increase from 10 to 15 kms$^{-1}$, needed to drive oxygen to OH by the reaction with H$_2$. At about 15 kms$^{-1}$, the temperature is high enough to start further reactions with H$_2$ leading to H$_2$O production. The trend with increasing pre-shock density is more apparent, with OH fluxes increasing by two orders of magnitude between the 10$^{4}$ cm$^{-3}$ to 10$^{6}$ cm$^{-3}$, as the density becomes closer to the critical density of the transition.
Models versus observations – line ratios of the same species
------------------------------------------------------------
Comparison of observed and modeled line ratios of different pairs of CO, H$_\mathrm{2}$O, and OH transitions is shown in Fig. \[allvel\]. The line ratios are a useful probe of molecular excitation and therefore can be used to test whether the excitation in the models is reproduced correctly, which in turn depends on density and temperature, and thus shock velocity.
### CO line ratios
Given the universal shape of the CO ladders observed toward deeply-embedded protostars (see §1 and the discussion on the origin of CO ladders in §5), three pairs of CO lines are compared with models: (i) CO 16-15 and 21-20 line ratio, corresponding to the ‘warm’, 300 K component (panel a); (ii) CO 16-15 and 29-28 line ratio, combining transitions located in the ‘warm’ and ‘hot’ ($>$ 700 K) components (panel b); (iii) CO 24-23 and 29-28, both tracing the ‘hot’ component (panel c). All ratios are consistent with the $C-$type shock models from both KN96 and F+PdF10 for pre-shock densities above $n_\mathrm{H}$=10$^4$ cm$^{-3}$. For the CO 16-15/21-20 ratio, a pre-shock density of $n_\mathrm{H}$=10$^5$ cm$^{-3}$ and shock velocities of $20$-$30$ kms$^{-1}$ best fit the observations. Shock velocities above $\sim25$ kms$^{-1}$ are needed to reproduce the observations of the other two ratios at the same pre-shock density. Alternatively, higher pre-shock densities with velocities below 30 kms$^{-1}$ are also possible.
The KN96 $C-$shock CO line ratios for lower-to-higher$-J$ transitions (panel b of Fig. \[allvel\]) decrease with velocity, due to the increase in peak temperature that allows excitation of the higher-$J$ CO transitions. The effect is strongest at low pre-shock densities (see §4.1.1) and for the sets of transitions with the largest span in $J$ numbers. The CO 16-15 / 29-28 line ratio ($\Delta J_\mathrm{up}=13$) decreases by almost three orders of magnitude between shock velocities of 10 and 40 kms$^{-1}$ over the range of pre-shock densities. In contrast, the CO 16-15 / 21-20 and CO 24-23 / 29-28 line ratios show drops of about one order of magnitude with increasing velocity (panel a and c of Fig. \[allvel\]). These model trends explain why the observed CO line ratios are good diagnostics of shock velocity.
In absolute terms, the line ratios calculated for a given velocity are inversely proportional to the pre-shock density. The largest ratios obtained for $n_\mathrm{H}$=10$^4$ cm$^{-3}$ result from the fact that the higher$-J$ levels are not yet populated at low shock-velocities, while the lower$-J$ transitions reach LTE at high shock-velocities and do not show an increase of flux with velocity. This effect is less prominent at higher pre-shock densities, where the higher$-J$ lines are more easily excited at low shock velocities.
The F+PdF\* CO line ratios, extending the @FP03 grid to higher$-J$ CO lines, are almost identical to the KN96 predictions for pre-shock densities $n_\mathrm{H}$=10$^4$ cm$^{-3}$. For higher densities, the low-velocity $C-$shock models from F+PdF\* are systematically lower than the KN96 models, up to almost an order of magnitude for 10–15 kms$^{-1}$ shocks at $n_\mathrm{H}$=10$^6$ cm$^{-3}$. Therefore, the pre-shock density is less well constrained solely by CO lines.
For densities of $10^{5}$ cm$^{-3}$, shock velocities of 20-30 kms$^{-1}$ best reproduce the ratios using only transitions from the warm’ component, while shock velocities above 25 km s$^{-1}$ match the ratios using the transitions from the hot’ component. Velocities of that order are observed in CO $J=16-15$ HIFI line profiles [@Kr13 in prep.], but higher-$J$ CO lines dominated by the hot component have not been obtained with sufficient velocity resolution.
### H$_\mathrm{2}$O line ratios
Two ratios of observed H$_\mathrm{2}$O lines are compared with the $C-$ and $J-$type shock models: (i) the ratio of the low excitation H$_\mathrm{2}$O 2$_{12}$-1$_{01}$ and moderate excitation 4$_{04}$-3$_{13}$ lines (panel d of Fig. 5) and (ii) the ratio of the highly-excited H$_\mathrm{2}$O 6$_{16}$-5$_{05}$ and 4$_{04}$-3$_{13}$ lines (panel e). Similar to the CO ratios, $C-$type shocks with pre-shock densities of 10$^5$ cm$^{-3}$ reproduce the observations well. Based on the observations of ratio (i), C shocks with a somewhat larger (F+PdF10) or smaller (KN96) pre-shock density are also possible for a broad range of shock velocities. On the other hand, no agreement with the $J-$type shocks is found for this low-excitation line ratio. Observations of ratio (ii) indicate a similar density range as ratio (i) for the KN96 models, but extend to 10$^4$ cm$^{-3}$ for the F+PdF10 models, with agreement found for both $C-$ and $J-$type.
The model trends can be understood as follows. For 10-20 kms$^{-1}$ shocks, increasing temperature in the $C$ shock models from KN96 allows excitation of high-lying H$_\mathrm{2}$O lines and causes the H$_\mathrm{2}$O 2$_{12}$-1$_{01}$/4$_{04}$-3$_{13}$ line ratio to decrease and the H$_\mathrm{2}$O 6$_{16}$-5$_{05}$/4$_{04}$-3$_{13}$ line ratio to increase. At higher shock velocities, the former ratio shows almost no dependence on shock velocity, while a gradual increase is seen in the ratio using two highly-excited lines in the KN96 models. At high pre-shock densities ($n_\mathrm{H}$=10$^6$ cm$^{-3}$), the upper level transitions are more easily excited and so the changes are even smaller.
The H$_\mathrm{2}$O 2$_{12}$-1$_{01}$/4$_{04}$-3$_{13}$ line ratios calculated using the $C-$type shock models from F+PdF10 are a factor of a few larger than the corresponding ratios from the KN96 models (see the discussion of absolute line fluxes in Sec. 4.1.2). As a result, when compared to observations, the F+PdF10 models require pre-shock densities of at least $n_\mathrm{H}$=10$^5$ cm$^{-3}$, while the KN96 models suggest a factor of few lower densities. Overall, the best fit to both the CO and H$_2$O line ratios is for pre-shock densities around $10^5$ cm$^{-3}$.
### OH line ratios
Comparison of the observed OH 84 and 79 $\mu$m line ratio with the KN96 $C-$type models (panel f of Fig. 5) indicates an order of magnitude higher pre-shock densities, $n_\mathrm{H}$=10$^6$ cm$^{-3}$, with respect to those found using the CO and H$_\mathrm{2}$O ratios. However, the KN96 models do not include any far-infrared radiation, which affects the excitation of the OH lines, in particular the 79 $\mu$m [@Wa10; @Wa13]. Additionally, part of OH most likely originates in a $J-$type shock, influencing our comparison [@Wa10; @Be12; @Ka13; @Kr13].
Similar to the absolute fluxes of the 84 $\mu$m doublet discussed in §4.1.3, not much variation in the ratio is seen with shock velocity. The ratio increases by a factor of about two between the lowest and highest pre-shock densities.
Models and observations - line ratios of different species
----------------------------------------------------------
Figure \[allvelp2\] compares observed line ratios of various H$_\mathrm{2}$O and CO transitions with the C and $J-$type shock models. The line ratios of different species are sensitive both to the molecular excitation and their relative abundances.
![\[oh\] CO to OH and H$_\mathrm{2}$O to OH line ratios as a function of shock velocities using KN96. The ratios are shown for pre-shock densities of 10$^4$ cm$^{-3}$ (red), 10$^5$ cm$^{-3}$ (blue), and 10$^6$ cm$^{-3}$ (yellow). The range of line ratios from observations is shown as filled rectangles.](ohvel2.eps){height="7cm"}
### Ratios of H$_\mathrm{2}$O and CO
Comparison of observations to the KN96 and F+PdF10 models shows that the $C-$type shocks at pre-shock density $n_{\mathrm{H}}=10^5$ cm$^{-3}$, which best reproduces the line ratios of same species, fail to reproduce the observed line ratios of different species (Fig. \[allvelp2\]). There are only a few cases where the observations seem to agree with the models at all. For $n_{\rm
H}=10^5$ cm$^{-3}$, a few H$_2$O/CO line ratios fit at low velocities ($<$ 20 kms$^{-1}$) (panels a, c and e) but this does not hold for all ratios. Moreover, such low shock velocities have been excluded in the previous section. Higher densities, $n_{\mathrm{H}}=10^6$ cm$^{-3}$, are needed to reconcile the observations of the H$_\mathrm{2}$O 2$_{12}$-1$_{01}$/CO 29-28 line ratio. Observations of all the other ratios, using more highly-excited H$_\mathrm{2}$O lines, are well below the model predictions.
The patterns seen in the panels in Fig. \[allvelp2\] can be understood as follows. The KN96 C type shock models show an initial rise in the H$_\mathrm{2}$O 2$_{12}$-1$_{01}$ and CO 16-15 line ratios from 10 to 15 kms$^{-1}$ shocks (panel a), as the temperature reaches the 400 K and enables efficient H$_\mathrm{2}$O formation. Beyond this velocity, the line ratios show no variations with velocity. The decrease in this line ratio for higher densities, from 10 at 10$^4$ cm$^{-3}$ to 0.1 at $n_\mathrm{H}$=10$^6$ cm$^{-3}$ is due to the larger increase of the column of the population in the $J_{\rm u}$=16 level with density compared to the increase in the H$_\mathrm{2}$O $2_{12}$ level (see Fig. \[vel\] above).
Line ratios of H$_\mathrm{2}$O 2$_{12}$-1$_{01}$ and higher $-J$ CO lines (e.g. 29-28, panel b) show more variation with velocity. A strong decrease by about an order of magnitude and up to two orders of magnitude are seen for the ratios with CO 24-23 and CO 29-28, respectively (the ratio with CO 24-23 is not shown here). These lines, as discussed in §4.1.1 and 4.2.1, are more sensitive than H$_\mathrm{2}$O to the increase in the maximum temperature attained in the shock that scales with shock velocities and therefore their flux is quickly rising for higher velocities (Fig. \[vel\]). The decrease is steeper for models with low pre-shock densities, since $n_\mathrm{H}\sim10^6$ cm$^{-3}$ allows excitation of high$-J$ CO lines at lower temperatures. At this density, the H$_\mathrm{2}$O/CO line ratios are the lowest and equal about unity.
Due to the lower CO 16-15 fluxes in the $C-$type shock models from F+PdF10 and similar H$_\mathrm{2}$O 2$_{12}$-1$_{01}$ fluxes (Fig. \[vel\]), the H$_\mathrm{2}$O-to-CO ratios are generally larger than in the KN96 models. The exceptions are the ratios with higher$-J$ CO which are more easily excited, especially at low shock velocities, in the hotter $C-$type shocks from F+PdF\*.
For the same reason, the increasing ratios seen in the $J$ shock models are caused by the sharp decrease in CO 16-15 flux for shock velocity $\varv$ = 30 kms$^{-1}$, rather than the change in the H$_\mathrm{2}$O lines. At such high-velocities for $J$ shocks, a significant amount of CO can be destroyed by reactions with hydrogen atoms [@FP10; @Su14]. Since the activation barrier for the reaction of H$_\mathrm{2}$O with H is about 10$^4$ K, the destruction of H$_\mathrm{2}$O does not occur until higher velocities.
Similar trends to the line ratios with H$_\mathrm{2}$O 2$_{12}$-1$_{01}$ are seen when more highly-excited H$_\mathrm{2}$O lines are used (panels c-f of Fig. \[allvelp2\]), supporting the interpretations that variations are due to differences in CO rather than H$_\mathrm{2}$O lines.
### Ratios of CO and H$_\mathrm{2}$O with OH
Fig. \[oh\] shows line ratios of CO or H$_2$O with the most commonly detected OH doublet at 84 $\mu$m. The ratios are calculated for three values of pre-shock densities (10$^4$, 10$^5$, and 10$^6$ cm$^{-3}$) using exclusively the KN96 models, because the F+PdF10 grid does not present OH fluxes.
In general, the observed CO/OH, and H$_\mathrm{2}$O/OH ratios are similar for all sources but much lower than those predicted by the models assuming that a significant fraction of the OH comes from the same shock as CO and H$_\mathrm{2}$O (see Sec. 5.1.). The only exception is the CO 24-23/OH 84 $\mu$m ratio where models and observations agree for densities 10$^{4}-10^{5}$ cm$^{-3}$ and shock velocities below 20 kms$^{-1}$. For any other set of lines discussed here, the observations do not agree with these or any other models.
As discussed in previous sections, the trends with shock velocity are determined mostly by the changes in the CO lines, rather than the OH itself, as seen in Fig. \[vel\] (panel d). At shock velocities below 20 kms$^{-1}$ the OH model flux exceeds that of CO due to the abundance effect: not all OH has been transferred to H$_\mathrm{2}$O yet at low temperatures. Due to the lower critical densities of the CO lines ($n_\mathrm{cr}\sim10^{6}-10^{7}$ cm$^{-3}$) compared with the OH line ($n_\mathrm{cr}\sim10^{9}$ cm$^{-3}$), the lines for various pre-shock densities often cross and change the order in the upper panels of Fig. \[oh\]. The corresponding trends in the H$_\mathrm{2}$O/OH line ratios are similar to those of CO/OH, except that the variations with shock velocity are smaller and the critical densities are more similar.
Discussion
==========
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Shock parameters and physical conditions
----------------------------------------
Spectrally resolved HIFI observations of the CO 10–9 and 16–15 line profiles (Kristensen et al. 2013 and in prep., Yildiz et al. 2013) as well as various H$_2$O transitions (Kristensen et al. 2012, Mottram et al. 2014) reveal at least two different kinematic shock components: non-dissociative C-type shocks in a thin layer along the cavity walls (so-called ‘cavity shocks’) and J-shocks at the base of the outflow (also called ‘spot shocks’), both caused by interaction of the wind with the envelope. Both shocks are different from the much cooler entrained outflow gas that is observed in the low-$J$ CO line profiles [@Yi13].
One possible physical explanation for our observed lack of variation is that although the outflow structure depends on the mass entrainment efficiency and the amount of mass available to entrain (the envelope mass), the wind causing the shocks does not depend on these parameters. Instead the cavity shock caused by the wind impinging on the inner envelope depends on the shock velocity and the density of the inner envelope (Kristensen et al. 2013, Mottram et al. 2014). Thus, the lack of significant variation in the line ratios suggests that the shock velocities by the oblique impact of the wind are always around 20–30 km s$^{-1}$.
In Section 4 the observed emission was compared primarily to models of $C-$type shock emission. Although $J-$type shocks play a role on small spatial scales in low-mass protostars [@Kr12; @Kr13; @Mo14] their contribution to CO emission originating in levels with $J_{\rm up} \lesssim 30$ is typically less than $\sim$ 50%. Since higher$-J$ CO emission is only detected toward 30% of the sources, the $J-$type shock component is ignored for CO. For the case of H$_2$O, spectrally resolved line profiles observed with HIFI reveal that the profiles do not change significantly with excitation up to $E_{\rm up}$ = 250 K (Mottram et al. in 2014); $J-$type shock components typically contribute $<$ 10% of the emission. It is unclear if the trend of line profiles not changing with excitation continues to higher upper-level energies, in particular all the way up to $E_{\rm up}$ = 1070 K ($J$ = 8$_{18}$—$7_{07}$ at 63.32 $\mu$m). OH and \[\], on the other hand, almost certainly trace dissociative $J-$type shocks [e.g., @vK10; @Wa13] but a full analysis of their emission will be presented in a forthcoming paper. Thus, in the following the focus remains on comparing emission to models of $C-$type shocks.
Figure \[summary\] summarizes the different line ratios as a function of pre-shock density discussed in the previous sections. General agreement is found between the observations and models when line ratios of different transitions of [*the same species*]{} are used (see top row for H$_\mathrm{2}$O, CO, and OH examples), indicating that the excitation of individual species is reproduced well by the models. The H$_\mathrm{2}$O line ratios are a sensitive tracer of the pre-shock gas density since they vary less with shock velocity than those of CO. The $C$ shock models from KN96 with pre-shock gas densities in the range of 10$^{4}$-10$^{5}$ cm$^{-3}$ are a best match to the observed ratios, consistent with values of 10$^{5}$ cm$^{-3}$ from the $C$ shock models of F+PdF10. For the considered range of shock velocities, the compression factor in those shocks, defined as the ratio of the post-shock and pre-shock gas densities, varies from about 10 to 30 [@ND89; @Dr93; @Ka13]. The resulting values of post-shock densities, traced by the observed molecules, are therefore expected to be $\geq10^{5}$-$10^{6}$ cm$^{-3}$.
The CO line ratios, on the other hand, are not only sensitive to density, but also to the shock velocities, due to their connection to the peak temperature attained in the shock. In the pre-shock density range of $\geq10^{4}$-$10^{5}$ cm$^{-3}$, indicated by the H$_\mathrm{2}$O line ratios, shocks with velocities above 20 kms$^{-1}$ best agree with the CO observations. Within this range of densities, the predictions from both the KN96 and F+PdF10 $C$ shock models show a very good agreement with each other.
The ratio of two OH lines from the KN96 models compared with the observations suggest higher pre-shock densities above $10^{5}$ cm$^{-3}$, but this ratio may be affected by infrared pumping [@Wa13]. Also, some OH emission traces (dissociative) $J-$shocks, based on its spatial connection and flux correlations to \[\] emission [@Wa10; @Wa13; @Ka13]. The single spectrally-resolved OH spectrum towards Ser SMM1 [Fig. 3, @Kr13] suggests that the contribution of the dissociative and non-dissociative shocks is comparable. Thus, observed CO/OH and H$_2$O/OH line ratios are only affected at the factor $\sim$2 level and the discrepancy in the bottom row of Fig. \[summary\] remains.
Overall, the observed CO and H$_2$O line ratios are best fit with $C-$shock models with pre-shock densities of $\sim$10$^{5}$cm$^{-3}$ and velocities $\gtrsim$20 kms$^{-1}$, with higher velocities needed for the excitation of the highest$-J$ CO lines.
The shock conditions inferred here can be compared to the temperatures and densities found from single-point non-LTE excitation and radiative-transfer models, e.g., <span style="font-variant:small-caps;">radex</span> (van der Tak et al. 2007) and from non-LTE radiative transfer analysis of line intensity ratios [@Kr13 Mottram et al., in prep.] toward various sources. Typically, densities $\gtrsim10^6$cm$^{-3}$ and temperatures of $\sim300$ K and $\gtrsim700$ K are required to account for the line emission [@He12; @Go12; @Sa12; @Va12; @Ka13; @Sa13]. Within this range of densities, the predictions from both the KN96 and F+PdF10 $C-$shock models reproduce CO observations. However, the disconnect between predicted pre-shock conditions required to reproduce H$_2$O and CO is puzzling (see below).
A small number of individual sources have been compared directly to shock models (Lee et al. 2013, Dionatos et al. 2013) and the conclusions are similar to what is reported here: pre-shock conditions of typically $10^{4}-10^{5}$ cm$^{-3}$, and emission originating in $C-$type shocks. None of the sources analyzed previously were therefore special or atypical, rather these shock conditions appear to exist toward every embedded protostar.
At shock positions away from the protostar, dissociative or non-dissociative $J-$type shocks at the same pre-shock densities are typically invoked to explain the FIR line emission [@Be12; @Sa12; @Bu14]. Differences between the protostar position and the distant shock positions are revealed primarily by our line ratios using the low-excitation H$_\mathrm{2}$O 2$_{12}$-1$_{01}$ (Table 3) and can be ascribed to the differences in the filling factors and column densities between the immediate surrounding of the protostar and the more distant shock positions (Mottram et al. 2014).
Abundances and need for UV radiation
------------------------------------
[lcrcccccc]{} log $n_\mathrm{H}$(cm $^{-3}$) & Obs. (%) &\
& &\
& & 20 & 30 & 40\
\
4 & 29 & 95.5 & 92.6 & 90.0\
4.5 & 29 & 91.6 & 86.3 & 82.8\
5 & 29 & 84.6 & 78.2 & 75.3\
5.5 & 29 & 77.3 & 71.2 & 70.4\
6 & 29 & 71.3 & 68.5 & 70.7\
6.5 & 29 & 68.3 & 69.6 & 73.3\
\
4 & 25 & 0.7 & 0.3 & 0.2\
4.5 & 25 & 0.7 & 0.3 & 0.3\
5 & 25 & 0.7 & 0.3 & 0.4\
5.5 & 25 & 0.9 & 0.4 & 0.6\
6 & 25 & 1.3 & 0.5 & 1.1\
6.5 & 25 & 1.9 & 0.8 & 2.4\
In contrast with the ratios of two H$_\mathrm{2}$O or CO lines, the ratios calculated using [*different*]{} species do not agree with the shock models (Figure \[summary\], bottom row). The ratios of H$_\mathrm{2}$O-to-CO lines are overproduced by the $C$ shock models from both the KN96 and F+PdF10 grids by at least an order of magnitude, irrespective of the assumed shock velocity. Although there are a few exceptions (e.g. the ratio of H$_\mathrm{2}$O 2$_{12}$-1$_{01}$ and CO 16-15), the majority of the investigated sets of H$_\mathrm{2}$O and CO lines follow the same trend. Observations agree only with slow, $<$20 kms$^{-1}$, $J$ shock models, but as shown above, those models do not seem to reproduce the excitation properly (Fig. \[allvel\]).
The discrepancy between the models and observations is even larger in the case of the H$_\mathrm{2}$O-to-OH line ratios, as illustrated in Fig. \[summary\]. The two orders of magnitude disagreement with the $C$ shock models cannot be accounted by any excitation effects for any realistic shock parameters. Additional comparison to $J$ shock models is not possible due to a lack of OH predictions for $J$ shocks in the F+PdF10 models.
The CO-to-OH ratios are overproduced by about an order of magnitude in the $C$ shock models, similar to the H$_\mathrm{2}$O-to-CO ratios. The agreement improves for fast ($\varv$ = 40 km s$^{-1}$) shocks in high density pre-shock medium ($\sim10^{6.5}$ cm$^{-3}$), but those parameters are not consistent with the line ratios from the same species.
An additional test of the disagreement between models and observations is provided by calculating the fraction of each species with regard to the sum of CO, H$_\mathrm{2}$O and OH emission. For that purpose, only the strongest lines observed in our program are used. As seen in Table \[tab:cool\], the observed percentage (median) of H$_\mathrm{2}$O is about 30 % and OH is about 25 %. In contrast, KN96 models predict typically 70-90 % of flux in the chosen H$_\mathrm{2}$O lines and only up to 2% in the OH lines.
The only possible way to reconcile the models with the observations, after concluding that the excitation is treated properly in the models, is to reconsider the assumed abundances. The fact that the H$_\mathrm{2}$O-to-CO and CO-to-OH ratios are simultaneously overestimated suggests a problem with the abundances of H$_\mathrm{2}$O and OH, rather than that of CO. The scenario with the overestimated H$_\mathrm{2}$O abundances and underestimated OH abundances would translate into a too large H$_\mathrm{2}$O-to-OH abundance ratio in the models. A possible and likely solution is photodissocation of H$_2$O to OH and subsequently to atomic oxygen. As noted above, some OH also comes from the dissociative shock seen in \[\].
A significant shortcoming of all these shock models lies in their inability to account for grain-grain interactions, which has been shown to significantly alter the structure of the shocks propagating in dense media [$n_{\rm H} > 10^5$ cm$^{-3}$, @Gu07; @Gui09; @Gu11]. These grain-grain interactions mostly consist of coagulation, vaporization, and shattering effects affecting the grains. Their inclusion in shock models necessitates a sophisticated treatment of the grains, especially following their charge and size distribution [@Gu07]. Most remarkably, such interactions eventually result in the creation of small grain fragments in large numbers, which increases the total dust grain surface area and thereby changes the coupling between the neutral and the charged fluids within the shock layer. The net effect is that the shock layer is significantly hotter and thinner [@Gu11], which in turn affects the chemistry and emission of molecules [@Gui09].
Unfortunately, at the moment these models are computationally expensive and are not well-suited for a grid analysis; moreover, the solutions found by Guillet et al. do not converge for preshock densities of 10$^6$ cm$^{-3}$ or higher. A recent study by @An13 shows that it is possible to approximate these effects in a computationally efficient way, and subsequently evaluated the line intensities of CO, OH and H$_2$O on a small grid of models. When including grain-grain interactions, CO lines were found to be significantly less emitting than in simpler’ models, while a smaller decrease was found for H$_2$O lines, and a small increase for OH. These trends probably still need to be systematically investigated on larger grids of models before they can be applied to our present comparison efforts.
Regardless of the effect of grain-grain interactions, @Sn05 invoked several scenarios to reconcile high absolute H$_\mathrm{2}$O fluxes with the shock models for the case of supernova remnants. These include (i) the high ratio of atomic to molecular hydrogen, which drives H$_\mathrm{2}$O back to OH and O, (ii) freeze-out of H$_\mathrm{2}$O in the post-shock gas, (iii) freeze-out of H$_\mathrm{2}$O in the pre-shock gas, and (iv) photodissociation of H$_\mathrm{2}$O in the pre- and post-shock gas. Due to the high activation barrier of the H$_\mathrm{2}$O + H $\to$ OH + H$_\mathrm{2}$ reaction ($\sim10^4$ K), the first scenario is not viable. The freeze-out in the post-shock gas (ii) is not effective in the low density regions considered in @Sn05, but can play a role in the vicinity of protostars, where densities above $\sim10^6$ cm$^{-3}$ are found [e.g. @Kr12 this work]. However, this mechanism alone would not explain the bright OH and high$-J$ H$_\mathrm{2}$O lines seen toward many deeply-embedded sources [e.g. @Ka13; @Wa13]. A similar problem is related to the freeze-out in the pre-shock gas (iii), which decreases the amount of e.g. O, OH, and H$_\mathrm{2}$O in the gas phase for shock velocities below 15 kms$^{-1}$.
Therefore, the most likely reason for the overproduction of H$_\mathrm{2}$O in the current generation of shock models, at the expense of OH, is the omission of the effects of ultraviolet irradiation (scenario iv) of the shocked material. The presence of UV radiation is directly seen in Ly-$\alpha$ emission both in the outflow-envelope shocks [@Cu95; @Wa03] and at the protostar position [@Va00; @Yan12]. Additionally, UV radiation on scales of a few 1000 AU has been inferred from the narrow profiles of $^{13}$CO 6-5 observed from the ground toward a few low-mass protostars [@Sp95; @vK09; @Yi12]. H$_\mathrm{2}$O can be photodissociated into OH over a broad range of far-UV wavelengths, including by Ly-$\alpha$, and this would provide an explanation for the disagreement between our observations and the models. Photodissociation of CO is less likely, given the fact that it cannot be dissociated by Ly-$\alpha$ and only by very hard UV photons with wavelengths $<$ 1000 Å. The lack of CO photodissociation is consistent with weak \[\] and \[\] emission observed toward low-mass YSOs [@Yi12; @Go12; @Ka13]. At the positions away from the protostars, on the other hand, the bow-shocks at the tip of the protostellar jets can produce significant emission in the \[\] [@vK09]. Therefore, it is unlikely that lower line excitation at those positions is due to the weaker UV. The differences seen in the resolved line profiles [e.g. @Sa12; @Va12 Mottram et al. in prep.] indicate that the lower column densities involved are the more likely reason for differences in the excitation.
@Vi12 proposed a scenario in which the lower-lying CO transitions observed with PACS ($14<J_\mathrm{u}<23$) originate in UV–heated gas and higher$-J$ transitions ($J_\mathrm{u}$ $>$ 24) in shocked material. All water emission would be associated with the same shocks as responsible for the higher-$J$ CO emission with less than 1% of the water emission coming from the PDR layer. Although not modeled explicitly, the UV irradiation from the star-disk boundary impinging on the shocks naturally accounts for the lower H$_\mathrm{2}$O abundance and exceeds by at least two orders of magnitude the H$_\mathrm{2}$O destruction rate by He$^{+}$ and H$_{\mathrm{3}}^{+}$, assuming a normal interstellar radiation field (G$_\mathrm{0}=1$). The authors predict that while the dynamics of the hot layers where both shocks and UV irradiation play a role will be dominated by the shocks, only the UV photons penetrate further into the envelope, where the dynamics would resemble the quiescent envelope. The lower-temperature UV-heated gas has indeed been observed to be quiescent on the spatial scales of the outflow cavity through observations of medium-$J$ $^{13}$CO lines (Y[i]{}ld[i]{}z et al. 2012, subm.).
@FP13 proposed a model where all emission originates in a non-stationary shock wave, where a $J-$type shock is embedded in a $C-$type shock. Without a detailed modeling of individual sources based on different source parameters it is not possible to rule out any of these solutions. However, the trends reported here suggest that it is possible to find a pure shock solution, in agreement with @FP13, as long as UV photons are incorporated to provide dissociation of H$_\mathrm{2}$O. Complementary observations, preferably at higher angular resolution, are required to break the solution degeneracy and determine the relative role the shocks and UV photons play on the spatial scales of the thickness of the cavity wall. Models whose results depend sensitively on a single parameter such as time, are ruled out by the fact that the observed line ratios are so similar across sources.
Conclusions
===========
We have compared the line ratios of the main molecular cooling lines detected in 22 low-mass protostars using *Herschel*/PACS with publicly available one-dimensional shock models. Our conclusions are the following:
- Line ratios of various species and transitions are remarkably similar for all observed sources. No correlation is found with source physical parameters.
- Line ratios observed toward the protostellar position are consistent with the values reported for the positions away from the protostar, except for some ratios involving the low-excitation H$_\mathrm{2}$O 2$_{12}$-1$_{01}$ line. Coupled with the larger absolute fluxes of highly-excited H$_\mathrm{2}$O and CO lines at the protostellar positions, this indicates that lines at distant off-source shock positions are less excited.
- General agreement is found between the observed line ratios of the same species (H$_\mathrm{2}$O, CO, and OH) and the $C$ shock models from Kaufman & Neufeld (1996) and Flower & Pineau des Forêts (2010). Ratios of H$_\mathrm{2}$O are particularly good tracers of the density of the ambient material and indicate pre-shock densities of order $\geq10^{5}$ cm$^{-3}$ and thus post-shock densities of order $10^6$ cm$^{-3}$. Ratios of CO lines are more sensitive to the shock velocities and, for the derived range of pre-shock densities, indicate shock velocities above 20 kms$^{-1}$.
- Ratios of CO lines located in the warm’ component of CO ladders (with $T_\mathrm{rot}\sim$300 K) are reproduced with shock velocities of 20-30 kms$^{-1}$ and pre-shock densities of $10^{5}$ cm$^{-3}$. The CO ratios using the lines from the hot’ component ($T_\mathrm{rot}\gtrsim$700 K) are better reproduced by models with shock velocities above 25 km s$^{-1}$.
- A lack of agreement is found between models and the observed line ratios of different species. The H$_\mathrm{2}$O-to-CO, H$_\mathrm{2}$O-to-OH, and CO-to-OH line ratios are all overproduced by the models by 1-2 orders of magnitude for the majority of the considered sets of transitions.
- Since the observed molecular excitation is properly reproduced in the $C$ shock models, the most likely reason for disagreement with observations is the abundances in the shock models, which are too high in case of H$_\mathrm{2}$O and too low in case of OH. Invoking UV irradiation of the shocked material, together with a dissociative $J$ shock contribution to OH and \[\], would lower the H$_2$O abundance and reconcile the models and observations.
New UV-irradiated shock models will allow us to constrain the UV field needed to reconcile the shock models with observations (M. Kaufman, priv. comm.) Those models should also account for the grain-grain processing, which affects significantly the shocks structure at densities $\sim10^6$ cm$^{-3}$ [@Gu11]. The effects of shock irradiation as a function of the distance from a protostar will help to understand the differences in the observed spectrally-resolved lines from HIFI at on source’ and distant shock-spot positions.
Herschel is an ESA space observatory with science instruments provided by European-led Principal Investigator consortia and with important participation from NASA. AK acknowledges support from the Polish National Science Center grant 2013/11/N/ST9/00400. Astrochemistry in Leiden is supported by the Netherlands Research School for Astronomy (NOVA), by a Royal Netherlands Academy of Arts and Sciences (KNAW) professor prize, by a Spinoza grant and grant 614.001.008 from the Netherlands Organisation for Scientific Research (NWO), and by the European Community’s Seventh Framework Programme FP7/2007-2013 under grant agreement 238258 (LASSIE). NJE was supported by NASA through an award issued by the Jet Propulsion Laboratory, California Institute of Technology.
Supplementary material
======================
Table \[lines\] provides molecular / atomic information about the lines observed in the WILL program.
---------- ------------------------------------------------------------------------- ---------- -------- ------------------------------- ----------------- -- --
Species Transition Wave. Freq. $E_\mathrm{u}$/$k_\mathrm{B}$ $A_\mathrm{ul}$
($\mu$m) (GHz) (K) (s$^{-1}$)
H$_2$O 2$_{21}$-2$_{12}$ 180.488 1661.0 194.1 3.1(-2)
H$_2$O 2$_{12}$-1$_{01}$ 179.527 1669.9 114.4 5.6(-2)
OH $\nicefrac{3}{2}$,$\nicefrac{1}{2}$-$\nicefrac{1}{2}$,$\nicefrac{1}{2}$ 163.398 1834.7 269.8 2.1(-2)
OH $\nicefrac{3}{2}$,$\nicefrac{1}{2}$-$\nicefrac{1}{2}$,$\nicefrac{1}{2}$ 163.131 1837.7 270.1 2.1(-2)
CO 16-15 162.812 1841.3 751.7 4.1(-4)
[\[\]]{} $^{2}P_{\nicefrac{3}{2}}-^{2}P_{\nicefrac{1}{2}}$ 157.74 2060.0 326.6 1.8(-5)
H$_2$O 4$_{04}$-3$_{13}$ 125.354 2391.6 319.5 1.7(-1)
CO 21-20 124.193 2413.9 1276.1 8.8(-4)
CO 24-23 108.763 2756.4 1656.5 1.3(-3)
H$_2$O 2$_{21}$-1$_{10}$ 108.073 2774.0 194.1 2.6(-1)
CO 29-28 90.163 3325.0 2399.8 2.1(-3)
H$_2$O 3$_{22}$-2$_{11}$ 89.988 3331.5 296.8 3.5(-1)
H$_2$O 7$_{16}$-7$_{07}$ 84.767 3536.7 1013.2 2.1(-1)
OH $\nicefrac{7}{2}$,$\nicefrac{3}{2}$-$\nicefrac{5}{2}$,$\nicefrac{3}{2}$ 84.596 3543.8 290.5 4.9(-1)
OH $\nicefrac{7}{2}$,$\nicefrac{3}{2}$-$\nicefrac{5}{2}$,$\nicefrac{3}{2}$ 84.420 3551.2 291.2 2.5(-2)
CO 31-30 84.411 3551.6 2735.3 2.5(-3)
H$_2$O 6$_{16}$-5$_{05}$ 82.032 3654.6 643.5 7.5(-1)
CO 32-31 81.806 3664.7 2911.2 2.7(-3)
CO 33-32 79.360 3777.6 3092.5 3.0(-3)
OH $\nicefrac{1}{2}$,$\nicefrac{1}{2}$-$\nicefrac{3}{2}$,$\nicefrac{3}{2}$ 79.182 3786.1 181.7 2.9(-2)
OH $\nicefrac{1}{2}$,$\nicefrac{1}{2}$-$\nicefrac{3}{2}$,$\nicefrac{3}{2}$ 79.116 3789.3 181.9 5.8(-3)
H$_2$O 6$_{15}$-5$_{24}$ 78.928 3798.3 781.1 4.6(-1)
H$_2$O 4$_{23}$-3$_{12}$ 78.742 3807.3 432.2 4.9(-1)
H$_2$O 8$_{18}$-7$_{07}$ 63.324 4734.3 1070.7 1.8
[\[\]]{} $^{3}P_{1}-^{3}P_{2}$ 63.184 4744.8 227.7 8.9(-5)
---------- ------------------------------------------------------------------------- ---------- -------- ------------------------------- ----------------- -- --
: \[lines\] Atomic and molecular data for lines observed in the WILL program
-------- ------------ ------ ------------ ------------ --------------------------------------------- ---------------------------------------------- --- -- -- -- -- -- --
Source OBSID OD Date Total time RA DEC
(s) ($^\mathrm{h}$ $^\mathrm{m}$ $^\mathrm{s}$) ($^{\mathrm{o}}$ $\mathrm{'}$ $\mathrm{''}$)
Per01 1342263508 1370 2013-02-12 851 3 25 22.32 +30 45 13.9
1342263509 1370 2013-02-12 1986 3 25 22.32 +30 45 13.9
Per02 1342263506 1370 2013-02-12 851 3 25 36.49 +30 45 22.2
1342263507 1370 2013-02-12 1986 3 25 36.49 +30 45 22.2
Per03 1342263510 1370 2013-02-12 851 3 25 39.12 +30 43 58.2
1342263511 1370 2013-02-12 1986 3 25 39.12 +30 43 58.2
Per04 1342264250 1383 2013-02-25 851 3 26 37.47 +30 15 28.1
1342264251 1383 2013-02-25 1986 3 26 37.47 +30 15 28.1
Per05 1342264248 1383 2013-02-25 851 3 28 37.09 +31 13 30.8
1342264249 1383 2013-02-25 1986 3 28 37.09 +31 13 30.8
Per06 1342264247 1383 2013-02-25 1986 3 28 57.36 +31 14 15.9
1342264246 1383 2013-02-25 851 3 28 57.36 +31 14 15.9
Per07 1342264244 1383 2013-02-25 1986 3 29 00.55 +31 12 00.8
1342264245 1383 2013-02-25 851 3 29 00.55 +31 12 00.8
Per08 1342264242 1383 2013-02-25 1986 3 29 01.56 +31 20 20.6
1342264243 1383 2013-02-25 851 3 29 01.56 +31 20 20.6
Per09 1342267611 1401 2013-03-15 1986 3 29 07.78 +31 21 57.3
1342267612 1401 2013-03-15 851 3 29 07.78 +31 21 57.3
Per10 1342267615 1401 2013-03-15 1986 3 29 10.68 +31 18 20.6
1342267616 1401 2013-03-15 851 3 29 10.68 +31 18 20.6
Per11 1342267607 1401 2013-03-15 1986 3 29 12.06 +31 13 01.7
1342267608 1401 2013-03-15 851 3 29 12.06 +31 13 01.7
Per12 1342267609 1401 2013-03-15 1986 3 29 13.54 +31 13 58.2
1342267610 1401 2013-03-15 851 3 29 13.54 +31 13 58.2
Per13 1342267613 1401 2013-03-15 1986 3 29 51.82 +31 39 06.0
1342267614 1401 2013-03-15 851 3 29 51.82 +31 39 06.0
Per14 1342263512 1370 2013-02-12 1986 3 30 15.14 +30 23 49.4
1342263513 1370 2013-02-12 851 3 30 15.14 +30 23 49.4
Per15 1342263514 1370 2013-02-12 1986 3 31 20.98 +30 45 30.1
1342263515 1370 2013-02-12 851 3 31 20.98 +30 45 30.1
Per16 1342265447 1374 2013-02-16 1986 3 32 17.96 +30 49 47.5
1342265448 1374 2013-02-16 851 3 32 17.96 +30 49 47.5
Per17 1342263486 1369 2013-02-11 1986 3 33 14.38 +31 07 10.9
1342263487 1369 2013-02-11 851 3 33 14.38 +31 07 10.9
Per18 1342265449 1374 2013-02-16 1986 3 33 16.44 +31 06 52.5
1342265450 1374 2013-02-16 851 3 33 16.44 +31 06 52.5
Per19 1342265451 1374 2013-02-16 1986 3 33 27.29 +31 07 10.2
1342265452 1374 2013-02-16 851 3 33 27.29 +31 07 10.2
Per20 1342265453 1374 2013-02-16 1986 3 43 56.52 +32 00 52.8
1342265454 1374 2013-02-16 851 3 43 56.52 +32 00 52.8
Per21 1342265455 1374 2013-02-16 1986 3 43 56.84 +32 03 04.7
1342265456 1374 2013-02-16 851 3 43 56.84 +32 03 04.7
Per22 1342265701 1381 2013-02-23 1986 3 44 43.96 +32 01 36.2
1342265702 1381 2013-02-23 851 3 44 43.96 +32 01 36.2
-------- ------------ ------ ------------ ------------ --------------------------------------------- ---------------------------------------------- --- -- -- -- -- -- --
Table \[log\] shows the observing log of PACS observations including observations identifications (OBSID), observation day (OD), date of observation, total integration time, and pointed coordinates (RA, DEC).
Table \[det\] informs about which lines are detected toward the Perseus sources. The full list of line fluxes for all WILL sources including Perseus will be tabulated in the forthcoming paper (Karska et al. in prep.).
Figures \[specmap1\] and \[specmap2\] show line and continuum maps around 179.5 $\mu$m for all the Perseus sources in the WILL program.
Figure \[specmapmix\] show maps in the H$_2$O 4$_{23}$-3$_{12}$ line at 78.74 $\mu$m, OH 84.6 $\mu$m, and CO 29-28 at 90.16 $\mu$m for Per1, Per5, Per9, and Per20, all of which show bright line emission and centrally peaked continuum. The lines are chosen to be located close in the wavelength so that the variations in the PSF does not introduce significant changes in the emission extent.
Table \[tab:ratios\] summarizes observed and modeled line ratios used in the Analysis section.
Wave ($\mu$m) Species Per 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22
--------------- ---------- -------------- -------------- -------------- -------------- -------------- -------------- -------------- -------------- -------------- -------------- -------------- -------------- -------------- -------------- -------------- -------------- -------------- -------------- -------------- -------------- -------------- -------------- -- -- -- -- -- -- -- -- -- -- -- -- --
63.184 [\[\]]{} $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ – $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$
63.324 H$_2$O $\checkmark$ $\checkmark$ $\checkmark$ – – $\checkmark$ – $\checkmark$ $\checkmark$ – $\checkmark$ – – $\checkmark$ – – – – – – – –
63.458 H$_2$O – – $\checkmark$ – – – – $\checkmark$ $\checkmark$ – $\checkmark$ – – – – – – – – – – –
78.742 H$_2$O $\checkmark$ $\checkmark$ $\checkmark$ – $\checkmark$ $\checkmark$ – $\checkmark$ $\checkmark$ – $\checkmark$ – – $\checkmark$ – – – – – $\checkmark$ $\checkmark$ $\checkmark$
78.928 H$_2$O $\checkmark$ $\checkmark$ $\checkmark$ – $\checkmark$ – – – $\checkmark$ – $\checkmark$ – – – – – – – – – – –
79.120 OH $\checkmark$ $\checkmark$ $\checkmark$ – $\checkmark$ $\checkmark$ – $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ – – $\checkmark$ – – – – $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$
79.180 OH $\checkmark$ $\checkmark$ $\checkmark$ – $\checkmark$ $\checkmark$ – $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ – – $\checkmark$ – – – – $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$
79.360 CO $\checkmark$ $\checkmark$ $\checkmark$ – $\checkmark$ – – – $\checkmark$ – $\checkmark$ – – $\checkmark$ – – – – – – – –
81.806 CO $\checkmark$ $\checkmark$ $\checkmark$ – – $\checkmark$ – $\checkmark$ $\checkmark$ – $\checkmark$ – – $\checkmark$ – – – – – – – –
82.032 H$_2$O $\checkmark$ $\checkmark$ $\checkmark$ – $\checkmark$ $\checkmark$ – $\checkmark$ $\checkmark$ – $\checkmark$ – – $\checkmark$ $\checkmark$ – – – $\checkmark$ – $\checkmark$ $\checkmark$
84.411 OH+CO $\checkmark$ $\checkmark$ $\checkmark$ – $\checkmark$ $\checkmark$ – $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ – – $\checkmark$ $\checkmark$ – – – $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$
84.600 OH $\checkmark$ $\checkmark$ $\checkmark$ – $\checkmark$ $\checkmark$ – $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ – – $\checkmark$ $\checkmark$ – – – $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$
84.767 H$_2$O – – $\checkmark$ – – – – – – – – – – – – – – – – – – –
89.988 H$_2$O $\checkmark$ $\checkmark$ $\checkmark$ – $\checkmark$ – – $\checkmark$ $\checkmark$ – $\checkmark$ – – $\checkmark$ – – – – – – $\checkmark$ –
90.163 CO $\checkmark$ $\checkmark$ $\checkmark$ – $\checkmark$ – – $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ – – $\checkmark$ – – – – – – $\checkmark$ –
108.073 H$_2$O $\checkmark$ $\checkmark$ $\checkmark$ – $\checkmark$ $\checkmark$ – $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ – $\checkmark$ $\checkmark$ $\checkmark$ – – $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$
108.763 CO $\checkmark$ $\checkmark$ $\checkmark$ – $\checkmark$ $\checkmark$ – $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ – – $\checkmark$ $\checkmark$ – – $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$
124.193 CO $\checkmark$ $\checkmark$ $\checkmark$ – $\checkmark$ $\checkmark$ – $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ – – $\checkmark$ $\checkmark$ $\checkmark$ – $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$
125.354 H$_2$O $\checkmark$ $\checkmark$ $\checkmark$ – $\checkmark$ $\checkmark$ – $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ – $\checkmark$ $\checkmark$ $\checkmark$ – – – $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$
162.812 CO $\checkmark$ $\checkmark$ $\checkmark$ – $\checkmark$ $\checkmark$ – $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ – – $\checkmark$ $\checkmark$ $\checkmark$ – $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$
163.120 OH $\checkmark$ $\checkmark$ $\checkmark$ – – $\checkmark$ – $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ – – $\checkmark$ – – – $\checkmark$ $\checkmark$ $\checkmark$ – $\checkmark$
163.400 OH $\checkmark$ $\checkmark$ $\checkmark$ – – $\checkmark$ – $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ – – $\checkmark$ $\checkmark$ – – $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ –
179.527 H$_2$O $\checkmark$ $\checkmark$ $\checkmark$ – $\checkmark$ $\checkmark$ – $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ – – $\checkmark$ $\checkmark$ – – $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$
180.488 H$_2$O $\checkmark$ $\checkmark$ $\checkmark$ – $\checkmark$ – – – $\checkmark$ – $\checkmark$ – – $\checkmark$ – – – $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$
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-------------------------- ------------------- ---- ----- ----- ------ ----- ------------ ------- ------- ------- ------- ------ ------ ------ ------
Line 1 Line 2
N Min Max Mean Std $\varv$=20 30 40 20 30 40 20 30 40
CO 16-15 21-20 17 1.2 2.5 1.7 0.4 4.4 2.9 2.4 2.7 1.4 1.0 0.8 0.5 0.5
CO 16-15 24-23 16 1.2 4.6 2.3 0.9 11.6 5.7 4.2 6.2 2.1 1.3 0.9 0.5 0.4
CO 16-15 29-28 10 1.9 8.3 3.8 1.8 72.2 31.4 2.2 19.1 5.5 0.5 11.3 2.7 0.3
CO 21-20 29-28 10 1.4 4.1 2.2 0.8 16.5 11.7 2.9 6.6 4.1 0.9 4.6 2.7 0.6
H$_2$O 2$_{12}$-1$_{01}$ 4$_{04}$-3$_{13}$ 15 1.3 6.3 3.0 1.6 6.7 5.3 4.6 1.5 1.3 1.2 0.7 0.7 0.7
H$_2$O 2$_{12}$-1$_{01}$ 6$_{16}$-5$_{05}$ 13 0.7 5.9 2.4 1.8 8.1 4.7 3.8 1.3 0.8 0.7 0.3 0.2 0.2
H$_2$O 2$_{21}$-1$_{10}$ 4$_{04}$-3$_{13}$ 16 1.4 5.5 2.8 1.1 4.2 3.6 3.3 2.3 2.0 1.8 1.5 1.3 1.3
H$_2$O 2$_{21}$-1$_{10}$ 6$_{16}$-5$_{05}$ 13 0.9 5.5 2.1 1.3 5.0 3.2 2.7 2.0 1.2 1.0 0.6 0.4 0.4
OH 84 OH 79 14 1.1 2.4 1.7 0.3 0.6 0.7 0.8 0.8 0.9 0.9 1.4 1.3 1.1
H$_2$O 2$_{12}$-1$_{01}$ CO 16-15 16 0.2 2.4 0.9 0.8 18.8 10.6 8.0 0.4 0.4 0.4 0.1 0.1 0.1
H$_2$O 2$_{21}$-1$_{10}$ CO 24-23 16 0.6 3.8 1.7 1.0 135.7 41.1 24.2 22.5 7.0 4.8 3.0 2.2 2.5
H$_2$O 3$_{22}$-2$_{11}$ CO 29-28 9 0.7 2.1 1.2 0.4 51.9 11.6 6.7 45.3 8.3 4.7 6.6 2.2 2.0
H$_2$O 4$_{04}$-3$_{13}$ CO 16-15 15 0.1 0.5 0.3 0.1 2.8 2.0 1.7 1.6 1.7 1.9 2.2 3.6 5.4
H$_2$O 4$_{04}$-3$_{13}$ CO 21-20 15 0.2 0.9 0.5 0.2 12.3 5.8 4.2 4.2 2.3 2.0 1.7 2.0 2.6
H$_2$O 6$_{16}$-5$_{05}$ CO 16-15 13 0.1 1.0 0.5 0.3 2.3 2.2 2.1 0.05 0.07 0.1 0.01 0.02 0.04
H$_2$O 6$_{16}$-5$_{05}$ CO 21-20 13 0.2 1.4 0.7 0.4 10.2 6.5 5.1 0.1 0.1 0.1 0.01 0.01 0.02
H$_2$O 6$_{16}$-5$_{05}$ CO 24-23 13 0.4 1.6 0.9 0.5 27.0 12.7 8.8 11.2 5.7 4.7 5.2 5.0 6.4
H$_2$O 6$_{16}$-5$_{05}$ CO 32-31 8 1.2 4.6 3.0 1.1 576.3 91.6 43.3 165.4 28.1 15.6 28.6 6.6 5.5
H$_2$O 2$_{12}$-1$_{01}$ OH 84 15 0.1 4.2 1.2 1.3 182.2 422.8 429.4 72.5 140.7 98.6 15.4 37.0 21.2
H$_2$O 3$_{22}$-2$_{11}$ OH 84 9 0.2 0.8 0.4 0.2 7.0 24.2 32.4 43.9 97.3 71.2 28.8 65.3 36.5
CO 16-15 OH 84 15 0.4 2.8 1.1 0.7 9.7 40.0 54.0 30.4 64.6 41.5 9.6 14.2 5.5
CO 24-23 OH 84 15 0.2 1.1 0.5 0.3 0.8 7.0 12.7 4.9 30.3 30.8 10.6 30.8 15.1
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Correlations with source parameters
===================================
Figure \[corr125\] shows selected H$_2$O-to-CO line ratios as a function of source physical parameters (bolometric luminosity, temperature, and envelope mass). Lack of correlation is seen in all cases.
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[^1]: http://herschel.esac.esa.int/Docs/PACS/html/pacs\_om.html
[^2]: On-source observations that contained Per 3 and Per 11 in the same field-of-view are discussed in separate papers by @Lee13 and @He12, respectively.
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---
abstract: 'We study the effect of globalization on the Korean market, one of the emerging markets. Some characteristics of the Korean market are different from those of the mature market according to the latest market data, and this is due to the influence of foreign markets or investors. We concentrate on the market network structures over the past two decades with knowledge of the history of the market, and determine the globalization effect and market integration as a function of time.'
author:
- 'Woo-Sung'
- Okyu
- 'Jae-Suk'
- 'Hie-Tae'
title: Effects of the globalization in the Korean financial markets
---
[^1]
Introduction
============
‘The world to Seoul, Seoul to the world.’ This was the slogan of the 1988 Seoul Olympics Games, and is also the slogan of the Korean stock market. The globalization means that foreign traders have an influence on the Korean market and its synchronization with world markets.
Interdisciplinary study has received much attention, with considerable interest in applying physics to economics and finances [@stanley; @arthur; @bouchaud; @aste; @kaizoji]. Since a financial market is a complex system, many researchers have developed network theory to analyze such systems. The concept of an asset tree constructed by a minimum spanning tree is useful in investigating market properties [@mantegna; @bonanno; @johnson]. The minimum spanning tree (MST) is derived for a unique sub-network from a fully connected network of the correlation matrix. The MST of $N$ nodes has $N-1$ links; each node represents a company or a stock and edges with the most important correlations are selected. Then clusters of companies can be identified. The clusters, a subset of the asset tree, can be extended to portfolio optimization in practice. The companies of the US stock market are clearly clustered into business sectors or industry categories [@onnela].
Nowadays, many emerging markets experience the globalization that is making rapid progress, and the influence of developed markets is becoming stronger. Most markets synchronize with the US market and globalization is leading to characteristic changes in emerging markets [@climent].
Several results have been reported on the necessity to find a model appropriate to emerging markets, because the models for mature markets cannot be applied to emerging markets universally [@india]. The Korean market is representative of emerging markets and is subject to synchronization with external markets [@jkps1; @jkps2; @jkps3; @jkps4; @jkps5]. Clustering in the Korean market differs from that in the US market and is due to foreign factors [@wsjung].
In this paper, we explore characteristics of the Korean stock market. We construct the minimum spanning tree (MST) shifting a time window of approximately two decades and analyze the time-dependent properties of the clusters in the MST that the market conditions are not stationary. Then we investigate the market with knowledge of the history of the Korean market.
Dynamics of the market
======================
The Korea Stock Exchange (KSE) opened in 1956. At that time, only 12 companies were listed on the market. As the Korean economy has developed, the stock market has undergone many changes under the influence of factors inside and outside the market.
We deal with the daily closure stock prices for companies listed on the KSE from 4 January 1980 to 30 May 2003. The stock had a total of 6648 price quotes over the period. We select 228 companies that remained in the market over this period of 23 years. Fig. \[index\] shows the index for those companies. The representative KSE index, KOSPI, is an index of the value-weighted average of current stock prices. The index of Fig. \[index\] is a price-equally-weighted index, similar to use for the Dow Jones industrial average (DJIA). Many previous studies on the stock market assumed a certain number of trading days to constitute a year. However, it is not easy to apply such an assumption to our data set, because the Korean market opening time changed in 2000. Before 20th May 2000, the market opened every day except Sunday, and from Monday to Friday after 21th May 2000. Most of data set falls into the former period, so we assume 300 trading days for one year. The x-axis values in Fig. \[index\] were calculated under this assumption.
![Index of 228 selected companies in the Korean stock market from 1980 to 2003. []{data-label="index"}](fig1){width="7cm"}
We use the logarithmic return of stock $i$, which can be written as:
$$S_i(t) = \ln Y_i(t+\Delta t)-\ln Y_i(t),$$
where $Y_i(t)$ is the price of stock $i$. The cross-correlation coefficients between stock $i$ and $j$ are defined as:
$$\lambda_{ij} =\frac{<S_{i}S_{j}>-<S_{i}><S_{j}>}
{\sqrt{(<S_{i}^{2}>-<S_{i}>^{2})(<S_{j}^{2}>-<S_{j}>^{2}) }}$$
and form a correlation matrix $\Lambda$.
![The mean, standard deviation, skewness, and kurtosis of the correlation coefficient in the Korean market as functions of time.[]{data-label="correlation"}](fig2){width="7cm"}
Category number Industry category No. of companies
----------------- ----------------------------------- ------------------
1 Fishery & Mining 1
2 Food & beverages 24
3 Tobacco 0
4 Textile 14
5 Apparel 3
6 Paper & wood 10
7 Oil 0
8 Chemicals & medical supplies 40
9 Rubber 6
10 Non-metallic minerals 12
11 Iron & metals 10
12 Manufacturing & machinery 13
13 Electrical & electronic equipment 8
14 Medical & precision machines 1
15 Transport equipment 12
16 Furniture 0
17 Electricity & gas 1
18 Construction 21
19 Distribution 17
20 Transport & storage 10
21 Banks 8
22 Insurance 11
23 Finance 4
24 Services 1
25 Movies 1
: Industry categories of the Korea Stock Exchange in our data set[]{data-label="category"}
The top panel of Fig. \[correlation\] shows the mean correlation coefficient calculated with only non-diagonal elements of $\Lambda$. The second shows the standard deviation, the third, the skewness and the last, the kurtosis. It has been reported that when the market crashes, the correlation coefficient is higher [@drozdz]. In the US market, the effect of Black Monday (19 October 1987) was clearly visible for these four coefficients, with correlations among them also apparent [@onnela]. However, crash effects on the Korean market (the late 1980s bubble crash and the 1997 Asian financial crisis) are visible, but not clear in comparison with the US market, and the Korean market coefficients do not have clear correlations.
We investigate more properties of the market through the MST that is a simple graph with $N$ nodes and $N-1$ links. The most important connection is linked when it is constructed. It is known that the US market network is centralized to a few nodes [@kim]. The hub of the US market is approximately General Electric (GE), and it is possible to make clusters (subsets of the MST) of the US market with industry categories or business sectors [@onnela]. However, the Korean market has no comparable hub for the whole market, and the clusters are constructed with the MSCI index [@wsjung]. We regard this result as the effect of globalization and market integration. Thus, we obtained the MSTs from 1980 to 2003 with time windows of width $T$ corresponding to daily data for $T$=900 days and $\delta T$=20 days. During this period there is no comparable hub, but we can form clusters with industry categories for some periods. Then we define the parameter *grouping coefficient*. The grouping coefficient of a specified industry category $C$ is defined as:
$$g_{C}=\frac{n^{C}(i_{\in C})}{n(i_{\in C})},$$
where $i_{\in C}$ represents the nodes in category $C$, $n(i)$ is the number of links that are connected to node $i$ and $n^{C}$ is the number of links from the node included in category $C$.
Table \[category\] shows 25 industry categories of the Korean stock market. In fact, there are rather more categories than 25. However, the standard for grouping is excessively detailed, and combinations of these categories are mostly used. The categories in Table \[category\] are reconstructed from Hankyoreh, a popular newspaper in Korea. Fig. \[group\]a shows the *grouping coefficient* for each category over the whole period. We observe that categories 8, 18, 21, 22 and 23 form a well-defined cluster. We focus on the maximum grouping coefficient for each industry category. For example, there are only four companies in the finance category (23) and the maximum value of the coefficient is only 0.6 (=3/5) because of the characteristics of the MST. We take the maximum value when the nodes are linked linearly. Fig. \[group\]b shows the ratio of the grouping coefficient to the maximum value for each category. Categories 18, 21, 22 and 23 are almost complete clusters in this plot.
![Plot of grouping coefficients: (a) shows the value of $g$ and (b) shows the ratio of the coefficient to the maximum value of $g$.[]{data-label="group"}](fig3){width="8cm"}
![Plot of the grouping coefficient for all categories as a function of time.[]{data-label="grouptime"}](fig4){width="7cm"}
{width="\textwidth"}
We previously investigated characteristics of the Korean stock market [@wsjung] using a data set from 2001 to 2004, and found that the market forms clusters when the Morgan Stanley Capital International (MSCI) index is exploited. However, Fig. \[group\] shows that some industry categories can be applied to form the clusters. We consider the history of the Korean market, including the globalization effect. Fig. \[grouptime\] shows the grouping coefficient $G$ for the whole market as a function of time. This coefficient is calculated with all of the nodes, and the ratio of connections between companies in the same category to the total number of links. Before the mid-1980s, the Korean market had developed according to a planned economy and had many restrictions on trading of stocks. At that time, the market was unstable because of the poor liquidity. This is one possible explanation for lower value in the early 1980s in Fig. \[grouptime\]. As the market prospered in the mid-1980s, clusters of industry categories also extensively formed.
The 1988 Seoul Olympics Games and the 1997 Asian financial crisis hastened globalization of the Korean market. In particular, globalization of the Korean market progressed to synchronization with external markets. This explains the decreasing coefficient in Fig. \[grouptime\] after 1988. The index continues to show a decreasing trend, which means that the formation of clusters in the Korean market is related to the MSCI index. The MSCI Korea index has been calculated from 1988, when the grouping coefficient in Fig. \[grouptime\] has almost a maximum value. The MSCI Korea index is a factor of the Korean market’s globalization and market integration. Because foreign traders strongly influence the Korean market, the MSCI index is a good reference for their trading [@wsjung].
Conclusions
===========
We have studied the Korean stock market network with the daily closure stock price. The analysis shows that the grouping coefficient changes with elapsing time. With globalization, the market is synchronized to external markets, and the number of clusters of industry categories decreases. Finally, the market forms clusters according to the MSCI index. We think the tendency of synchronization will be stronger and the clusters of the MSCI index or foreign factors will be firmer. Our future research will determine other properties of the globalization effect with other statistical analysis in the Korean market.
We thank G. Oh, W. C. Jun, and S. Kim for useful discussions and support.
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[^1]: Fax: +82-42-869-2510
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---
abstract: |
Learning visual features from unlabeled image data is an important yet challenging task, which is often achieved by training a model on some annotation-free information. We consider spatial contexts, for which we solve so-called jigsaw puzzles, i.e., each image is cut into grids and then disordered, and the goal is to recover the correct configuration. Existing approaches formulated it as a classification task by defining a fixed mapping from a small subset of configurations to a class set, but these approaches ignore the underlying relationship between different configurations and also limit their application to more complex scenarios.
This paper presents a novel approach which applies to jigsaw puzzles with an arbitrary grid size and dimensionality. We provide a fundamental and generalized principle, that weaker cues are easier to be learned in an unsupervised manner and also transfer better. In the context of puzzle recognition, we use an iterative manner which, instead of solving the puzzle all at once, adjusts the order of the patches in each step until convergence. In each step, we combine both unary and binary features on each patch into a cost function judging the correctness of the current configuration. Our approach, by taking similarity between puzzles into consideration, enjoys a more reasonable way of learning visual knowledge. We verify the effectiveness of our approach in two aspects. First, it is able to solve arbitrarily complex puzzles, including high-dimensional puzzles, that prior methods are difficult to handle. Second, it serves as a reliable way of network initialization, which leads to better transfer performance in a few visual recognition tasks including image classification, object detection, and semantic segmentation.
author:
- |
Chen Wei^1^, Lingxi Xie^2^, Xutong Ren^1^, Yingda Xia^2^, Chi Su^3^, Jiaying Liu^1^, Qi Tian^4^, Alan L. Yuille^2^\
Peking University^1^ The Johns Hopkins University^2^ Kingsoft^3^ Huawei Noah’s Ark Lab^4^\
[[weichen582@pku.edu.cn]{}]{}\
[[suchi@kingsoft.com]{}]{}
bibliography:
- 'egbib.bib'
title: |
Iterative Reorganization with Weak Spatial Constraints:\
Solving Arbitrary Jigsaw Puzzles for Unsupervised Representation Learning
---
Introduction {#Introduction}
============
![ We study the problem of solving jigsaw puzzles for visual recognition. Compared to the previous work [@noroozi2016unsupervised] which worked on $3\times3$ puzzles and $1\rm{,}000$ fixed configurations (left), we can solve this task in a generalized setting like 3D puzzles (right). []{data-label="Fig:JigsawPuzzles"}](Figures/teaser_v2.pdf){width="8cm"}
Deep learning especially convolutional neural networks has been boosting the performance of a wide range of applications in computer vision [@lecun2015deep]. These statistics-based approaches build hierarchical structures which contain a large number of neurons, so that visual knowledge is learned by fitting labeled training data [@krizhevsky2012imagenet]. However, annotating a large-scale dataset is often difficult and expensive. Therefore, weakly supervised or unsupervised learning has attracted a lot of research attentions [@weinberger2006unsupervised][@le2013building]. These approaches are often built on some naturally existing constraints such as temporal consistency [@wang2015unsupervised], spatial relationship [@doersch2015unsupervised] and sum-up equations [@noroozi2017representation]. Such information, though being weak, constructs loss functions without requiring annotations, and networks pre-trained in this way can either be used for weak visual feature extraction [@wang2015unsupervised] or fine-tuned in a standalone supervised learning process towards better recognition performance [@erhan2010does].
In this work, we focus on a specific way of exploiting spatial relationship, which is to solve [*jigsaw puzzles*]{} on unlabeled image data [@noroozi2016unsupervised][@noroozi2018boosting]. These approaches work by cutting an image into a grid, say, $3\times3$, of patches and then disordering them as training data, with the goal set to recover its correct spatial configuration. Examples are shown in Figure \[Fig:JigsawPuzzles\]. Thus, in order to achieve this goal, the network should have the ability to capture some semantic information, [*e.g.*]{}, learning the concept of [*car*]{} and [*ground*]{}, though not labeled, and knowing that [*car*]{} always appears above [*ground*]{}. Technically, these approaches simply assigned each configuration a unique ID, so that puzzle recognition turns into a [*plain*]{} classification problem. We point out two major drawbacks of this strategy. First, by plain classification, we assume that all configurations have the same similarity with each other, but this is often not the case, [*e.g.*]{}, two $3\times3$ configurations with only two patches swapped are often semantically closer than other two with no patches placed at the same position. Ignoring such information can bring in difficulties to representation learning. Second, the number of parameters required for plain classification increases linearly with the number of configurations, so that it is very difficult to deal with all possible configurations due to the risk of over-fitting. For example, there are ${9!}={362\rm{,}880}$ possible configurations for a $3\times3$ puzzle, but the original approach [@noroozi2016unsupervised] reached the best performance at $1\rm{,}000$ and observed over-fitting when this number continues growing. Both of these drawbacks limit us from generalizing this approach to more complex puzzles[^1] like 3D puzzles[^2]. An empirical study of this topic can be found in Section \[Experiments:TransferLearning\].
In this paper, we extend the ability of such approaches by allowing it to solve arbitrary jigsaw puzzles, [*i.e.*]{}, the puzzles are not constrained by a pre-defined set of configurations. Our major contribution is to provide a principle for unsupervised learning, that learning to recognize weak visual cues and then composing them into a complex scene is often easier and thus better in transfer. Thus, we solve jigsaw puzzles (i) in an iterative manner and (ii) using weak spatial cues, instead of determining the correct configuration all at once. To this end, we formulate puzzle recognition into an optimization problem which involves a set of unary and binary terms, with each unary term indicating whether a specified patch is located at a specified position, and each binary term measuring whether two patches should have a specified relative position. These terms are determined by a deep network backbone so that the entire system can be trained in an end-to-end manner. In both training and testing, we allow the first trial not to find the correct configuration, in which case we iterate using the configuration adjusted according to prediction until convergence. Both the above techniques, [*a.k.a.*]{}, network heads, are used to improve the quality of pre-training. They do not apply to nor introduce additional computational costs to the transfer learning stage.
We evaluate our approach in both puzzle recognition and transfer learning. The puzzle solver is trained on the ILSVRC2012 training set [@russakovsky2015imagenet] tested on the validation set, both of which do not contain class labels. Our approach solves arbitrary jigsaw puzzles with reasonable accuracy, while the prior approaches can only work on a limited set of puzzle. Then, we transfer the pre-trained model to extract features in small-scale datasets for image classification [@griffin2007caltech], as well as to be fine-tuned in the PascalVOC 2007 dataset [@everingham2010pascal] for image classification and object detection. Either learning from more complex puzzles or achieving a higher accuracy in puzzle recognition boosts transfer learning performance, which verifies our motivation. Finally, we apply our approach initialize a 3D network with unlabeled medical data, and verify its effectiveness in segmenting an abdominal organ from CT scans.
The remainder of this paper is organized as follows. Section \[RelatedWork\] briefly reviews related work, and Section \[Approach\] describes the proposed approach. After experiments are shown in Section \[Experiments\], we draw our conclusions in Section \[Conclusions\].
Related Work {#RelatedWork}
============
Deep neural networks have been playing an important role in modern computer vision systems. With the availability of large-scale datasets [@deng2009imagenet] and powerful computational device such as GPUs, researchers have designed network structures with tens [@krizhevsky2012imagenet][@simonyan2015very][@szegedy2015going] or hundreds [@he2016deep][@huang2017densely] of layers towards better recognition performance. Also, the pre-trained networks in ImageNet were transferred to other recognition tasks by either extracting visual features directly [@donahue2014decaf][@girshick2014rich][@razavian2014cnn] or being fine-tuned on a new loss function [@long2015fully][@ren2015faster]. Despite their effectiveness, these networks still strongly rely on labeled image data, but in some areas such as medical imaging, data collection and annotation can be expensive, time-consuming, or requiring expertise. Thus, there has been efforts to design unsupervised [@weinberger2006unsupervised][@le2013building] or weakly supervised [@joulin2016learning] approaches which learned visual knowledge from unlabeled data, or semi-supervised learning algorithms [@papandreou2015weakly][@qiao2018deep] which were aimed at combining a limited amount of labeled data and a large corpus of unlabeled data towards better performance. It has been verified that unsupervised pre-training helps supervised learning especially deep learning [@erhan2010does].
The key factor to learning from unlabeled data is to establish some kind of [*prior*]{}, or some weak constraints that naturally exist, [*i.e.*]{}, no annotations are required. Such prior can be either (1) embedded into the network architecture or (2) encoded as a weak supervision to optimize the network. For the first type, researchers designed clustering-based approaches to optimize visual representation so as to be beneficial to clustering [@yang2016joint][@caron2018deep], as well as generator-based approaches which assumed that all images can be represented in a low-level space and trained encoders and/or decoders to recover the image and/or representation [@radford2016unsupervised][@zhu2017unpaired]. Network architectures of these approaches are often largely modified, [*e.g.*]{}, with a set of clustering layers or encoder-decoder modules.
This paper mainly considers the second type which, in comparison to the type, is much easier in algorithmic design. Typical examples include temporal consistency which assumes that neighboring video frames contain similar visual contents [@wang2015unsupervised], spatial relationship between some pairs of unlabeled patches [@doersch2015unsupervised], learning an additive function on different regions as well as the entire image [@noroozi2017representation], [*etc*]{}. Among these priors, spatial contexts are widely believed to contain rich information which a vision system should be able to capture. Going one step beyond modeling patch relationship [@doersch2015unsupervised], researchers designed so-called jigsaw puzzles [@noroozi2016unsupervised][@noroozi2018boosting] which are more complex so that the networks are better trained in learning to solve them. Consequently, such networks perform better in transfer learning.
Researchers believed that learning from these weakly-supervised cues can help visual recognition, because many problems are indeed built on understanding and integrating this type of information. Regarding spatial contexts, a wide range of recognition tasks can benefit from understanding the relative position of two (or more) patches, such as image classification [@aghajanian2009patch], semantic segmentation [@rousseau2011supervised] and parsing [@zhang2018deepvoting], [*etc.*]{}
Our Approach {#Approach}
============
Problem and Baseline Solution {#Approach:Motivation}
-----------------------------
The problem of puzzle recognition assumes that an image is partitioned into a grid ([*e.g.*]{}, $3\times3$) of patches and then disordered, and the task is to recover the original configuration ([*i.e.*]{}, patches are ordered in the natural form). To accomplish this task, the network needs to understand what a patch contains as well as how two or more patches are related to each other ([*e.g.*]{}, in a [*car*]{} image, a [*wheel*]{} is often located to the top of the [*ground*]{}). Therefore, we expect this task to teach a network both intra-patch and inter-patch information, which we formulate as unary terms and binary terms, respectively.
We first define the terminologies used in this paper. Let $\mathbf{I}$ be an image, which is partitioned into $W\times H$ patches. Each patch, denoted $\mathbf{i}_{x,y}$ (${0}\leqslant{x}<{W}$, ${0}\leqslant{y}<{H}$), is assigned a unique ID ${a_{x,y}}\in{\left\{0,1,\ldots,WH-1\right\}}$ according to its original position, [*e.g.*]{}, the row-major policy gives ${a_{x,y}}={x+yW}$. After that, all patches are randomly disordered, and we use $c_{x,y}^\star$ to denote the ID owned by the patch that currently occupies the $\left(x,y\right)$ position. All $c_{x,y}^\star$ values compose a configuration, denoted as ${\mathbf{c}^\star}={\left(c_{x,y}^\star\right)_{x=0,y=0}^{W,H}}$. There are in total $\left(WH\right)!$ different configurations, composing the configuration set $\mathcal{C}$ that ${\left|\mathcal{C}\right|}={\left(WH\right)!}$.
Our goal is to predict the correct configuration ${\mathbf{c}^\star}\in{\mathcal{C}}$. For this purpose, a network structure with two parts was constructed [@noroozi2016unsupervised]. The network [*backbone*]{} ${\mathbb{M}^\mathrm{B}}:{\mathbf{f}_{x,y}}={\mathbf{f}\!\left(\mathbf{i}_{x,y};\boldsymbol{\theta}^\mathrm{B}\right)}$ is built upon each individual patch, and outputs a set of features for the network [*head*]{} ${\mathbb{M}^\mathrm{H}}:{\mathbf{c}}={\mathbf{g}\!\left(\mathbf{F};\boldsymbol{\theta}^\mathrm{H}\right)}$ to produce the final output ${\mathbf{c}}={\left(c_{x,y}\right)_{x=0,y=0}^{W,H}}$, where ${\mathbf{F}}={\left(\mathbf{f}_{x,y}\right)_{x=0,y=0}^{W,H}}$ is the ordered concatenation of patch features. In practice, $\mathbf{f}\!\left(\cdot;\boldsymbol{\theta}^\mathrm{B}\right)$ is often borrowed from existing network architectures [@krizhevsky2012imagenet][@simonyan2015very][@he2016deep], while $\mathbf{g}\!\left(\cdot;\boldsymbol{\theta}^\mathrm{H}\right)$ is often more interesting to investigate.
In the prior work [@noroozi2016unsupervised][@noroozi2018boosting], the network head worked by constraining the number of possible configurations, say ${K}={1\rm{,}000}$ out of $9!$, which are randomly sampled from $\mathcal{C}$ using a greedy algorithm to guarantee the Hamming distance between any two configurations is sufficiently large. Then, $\mathbf{f}\!\left(\cdot;\boldsymbol{\theta}^\mathrm{H}\right)$ was designed to be a $K$-way classifier, implemented as a fully-connected layer. The purpose of this design was mainly to control the number of parameters of the classifier (proportional to $K$) so as to prevent over-fitting[^3], but we argue that it largely limits the model from being applied more complex scenarios like 3D puzzles, while it was believed that learning from a harder task can lead to a stronger ability [@deng2010does]. This motivates us to propose a new approach in which the number of configurations can be arbitrarily large while the number of parameters remains unchanged. We will see later that the essence behind this motivation is to use weak cues with an iterative algorithm towards a more compact representation and a safer learning process.
Solving Jigsaw Puzzles with Weak Cues {#Approach:Solution}
-------------------------------------
{width="16.5cm"}
We design a network head to learn [*weak spatial constraints*]{}. By “weak” we are comparing this strategy with the aforementioned $K$-way classifier that predicts the configuration of the entire puzzle all at once. Instead, we consider an indirect cost function $S\!\left(\mathbf{I},\mathbf{c}\right)$ which outputs a cost that patch $\mathbf{i}_{x,y}$ or equivalently feature $\mathbf{f}_{x,y}$ is located at position $c_{x,y}$, and thus the most probable configuration is determined by $\arg\max_\mathbf{c}\left\{S\!\left(\mathbf{I},\mathbf{c}\right)\right\}$. $S\!\left(\mathbf{I},\mathbf{c}\right)$ is composed of two parts, namely, unary terms and binary terms. Each [*unary term*]{} provides cues for the [*absolute*]{} position of a patch, and each [*binary term*]{} provides cues for the [*relative*]{} position of two patches. Mathematically, $$\begin{aligned}
\nonumber
{S\!\left(\mathbf{I},\mathbf{c}\right)}\equiv{S\!\left(\mathbf{F},\mathbf{c}\right)}={{\sum_{\left(x,y\right)}}p_1\!\left(\mathbf{f}_{x,y},c_{x,y}\mid\mathbf{F}\right)+}\quad\quad\\
\label{Eqn:ScoreFunction}
{\quad\quad{\sum_{\left(x_1,y_1\right)\neq\left(x_2,y_2\right)}}p_2\!\left(\mathbf{f}_{x_1,y_1},\mathbf{f}_{x_2,y_2},c_{x_1,y_1},c_{x_2,y_2}\right)}.\end{aligned}$$ Here, $p_1\!\left(\mathbf{f}_{x,y},c_{x,y}\mid\mathbf{F}\right)$ is a unary term which measures how likely that patch $f_{x,y}$ is located at position $c_{x,y}$, and $p_2\!\left(\mathbf{f}_{x_1,y_1},\mathbf{f}_{x_2,y_2},c_{x_1,y_1},c_{x_2,y_2}\right)$ is a binary term measures how likely that patches $\mathbf{f}_{x_1,y_1}$ and $\mathbf{f}_{x_2,y_2}$ have the spatial relationship indicated by $c_{x_1,y_1}$ and $c_{x_2,y_2}$. Each unary term is computed based on $\mathbf{F}$, the overall variable containing feature vectors of all patches, because the position of each patch $\mathbf{f}_{x,y}$ depends on the visual messages delivered by other patches. The binary terms, on the other hand, do not have such a dependency.
In practice, the unary terms are formulated in a matrix $\mathbf{U}$ with $WH\times WH$ elements, each of which, $\left\llbracket\mathbf{U}\right\rrbracket_{a,c}$, indicates the cost obtained by putting the specified patch with ID $a$ at a specified position with ID $c$. This is implemented by a fully-connected layer between $\mathbf{F}$ and these $\left(WH\right)^2$ elements, parameterized by $\boldsymbol{\theta}^\mathrm{U}$. We perform the softmax function over all elements in each row, so that the scores corresponding to each patch sum to $1$[^4]. Then, each unary term is the log-likelihood of the score at a specified position: $$\label{Eqn:UnaryTerm}
{p_1\!\left(\mathbf{f}_{x,y},c_{x,y},\mathbf{F}\right)}={-\ln\left\llbracket\mathbf{U}\!\left(\mathbf{F};\boldsymbol{\theta}^\mathrm{U}\right)\right\rrbracket_{a_{x,y},c_{x,y}}}.$$
For each binary term involving $\mathbf{f}_{x_1,y_1}$ and $\mathbf{f}_{x_2,y_2}$, we build another mapping from these two vectors to a $9$-dimensional vector, with each index indicating the probability that the spatial relationship of $\mathbf{f}_{x_1,y_1}$ and $\mathbf{f}_{x_2,y_2}$ belongs to one of the $9$ possibilities, namely, the first patch is located to the top, bottom, left, right, top-left, top-right, bottom-left, bottom-right of the second patch or none of the above happens. Similarly, this is implemented using another fully-connected layer between $\mathbf{f}_{x_2,y_2}\oplus\mathbf{f}_{x_2,y_2}$ ($\oplus$ denotes concatenation) and a $9$-dimensional vector parameterized by $\boldsymbol{\theta}^\mathrm{V}$ followed by a softmax activation over these $9$ numbers. We denote ${r_{x_1,y_1,x_2,y_2}}\doteq{r\!\left(c_{x_1,y_1},c_{x_2,y_2}\right)}\in{\left\{0,1,\ldots,8\right\}}$ as the relative position type between $\mathbf{f}_{x_1,y_1}$ and $\mathbf{f}_{x_2,y_2}$, so that we can write the binary term as: $$\begin{aligned}
\nonumber
{p_2\!\left(\mathbf{f}_{x_1,y_1},\mathbf{f}_{x_2,y_2},c_{x_1,y_1},c_{x_2,y_2}\right)}=\quad\quad\quad\quad\quad\quad\\
\label{Eqn:BinaryTerm}
\quad\quad{-\ln\left\llbracket\mathbf{V}\!\left(\mathbf{f}_{x_1,y_1},\mathbf{f}_{x_2,y_2};\boldsymbol{\theta}^\mathrm{V}\right)\right\rrbracket_{r_{x_1,y_1,x_2,y_2}}}.\end{aligned}$$
Compared to a plain classifier assigning a class index to each puzzle, the amount of parameters required by our approach is reduced. Take a $3\times3$ puzzle as an example, and we assume that $F$ contains $D$ elements. On the one hand, the $K$-way classifier requires $KD$ parameters (a typical setting [@noroozi2016unsupervised] is ${K}={1\rm{,}000}$) which grows linearly with $K$. On the other hand, our approach requires $ \left( WH \right)^2D$ parameters for the unary terms, and $9D$ parameters for the binary terms. The total number of parameters, $\left(W^2H^2+9\right)D$ ([*e.g.*]{}, $90D$ for a $3\times3$ puzzle), is largely reduced and does not increase with $K$. Consequently, our approach is easier to be applied to the scenario with a larger set of ([*e.g.*]{}, all $9!$ possible) configurations. This advantage is verified in experiments.
Last but not least, there are many other ways of using weak spatial constraints to formulate $S\!\left(\mathbf{I},\mathbf{c}\right)$ – we just provide a practical example.
Optimization: Iterative Reorganization {#Approach:Optimization}
--------------------------------------
We aim at optimizing $S\!\left(\mathbf{F},\mathbf{c}\right)$ with respect to network parameters $\boldsymbol{\theta}^\mathrm{U}$, $\boldsymbol{\theta}^\mathrm{V}$ and configuration $\mathbf{c}$. However, note that $\mathbf{c}$ is a discrete variable which cannot be optimized by gradient descent. So we apply different strategies in training and testing.
In the training stage, we know the ground-truth configuration $\mathbf{c}^\star$, so the optimization becomes: $$\label{Eqn:OptimizationTraining}
\arg\min_{\boldsymbol{\theta}^\mathrm{U},\boldsymbol{\theta}^\mathrm{V}}S\!\left(\mathbf{F},\mathbf{c}^\star\right).$$ This is implemented by setting the supervision signal accordingly, [*i.e.*]{}, the correct cells are filled up with $1$ while others with $0$, and using stochastic gradient descent. Note that each unary term depends on the order of input patches[^5]. To sample more training data as well as adjust data distribution (explained later), we introduce iteration to the training stage. Denote the input configuration as ${\mathbf{c}^{\left(0\right)}}={\mathbf{c}^\star}$, and the corresponding feature as $\mathbf{F}^{\left(0\right)}$. In each iteration, with fixed $\boldsymbol{\theta}^\mathrm{U}$ and $\boldsymbol{\theta}^\mathrm{V}$, we maximize $S\!\left(\mathbf{F},\mathbf{c}\right)$ with respect to $\mathbf{c}$: $$\label{Eqn:OptimizationTesting}
{\mathbf{c}'}={\arg\min_{\mathbf{c}}S\!\left(\mathbf{F},\mathbf{c}^{\left(0\right)}\right)},$$ and use $\mathbf{c}'$ to find the next input $\mathbf{c}^{\left(1\right)}$, so that applying $\mathbf{c}'$ to $\mathbf{c}^{\left(1\right)}$ obtains $\mathbf{c}^{\left(0\right)}$, [*e.g.*]{}, if $\mathbf{c}'$ is perfect, then $\mathbf{c}^{\left(1\right)}$ corresponds to the original configuration that every patch is placed at the correct position. This process continues until convergence or a maximal number of iterations is reached. The losses with respect to $\boldsymbol{\theta}^\mathrm{U}$ and $\boldsymbol{\theta}^\mathrm{V}$ are accumulated, averaged, and back-propagated to update these two parameters. The same strategy, iteration, is used at the testing stage to solve jigsaw puzzles, with the only difference that no gradient back-propagation is required.
It remains a problem to solve Eqn . This is a combinatoric optimization problem, as $\mathbf{c}$ can only take $\left(WH\right)!$ discrete values which indicate the entries in $\mathbf{U}$ and $\mathbf{V}$ that are summed up. There is obviously no closed form solutions to maximize $S\!\left(\mathbf{F},\mathbf{c}\right)$, yet enumerating all $\left(WH\right)!$ possibilities is computationally intractable especially when the puzzle size becomes large. A possible solution lies in approximation, which first switches off all binary terms, so that the optimization becomes choosing $WH$ entries from a $WH\times WH$ matrix with a maximal sum, but no two entries can appear in the same row or column (this is a max-cost-max-matching problem, and the best solution $\tilde{\mathbf{c}}$ can be found using the Hungarian algorithm); then enumerates all possibilities within a limited Hamming distance from $\tilde{\mathbf{c}}$ and chooses the one with the best overall cost $S\!\left(\mathbf{F},\mathbf{c}\right)$.
Finally, we discuss strategy of introducing iteration to solve this problem. Mathematically, Eqn is a fixed-point model [@li2013fixed], [*i.e.*]{}, the output variable $\mathbf{c}$ also impacts $\mathbf{F}$ and thus $S\!\left(\mathbf{F},\mathbf{c}\right)$, so iteration is considered a regular way of optimizing it. However, the roles played by iteration are different in training and testing. In the [**training stage**]{}, after each iteration, we shall expect the configuration to be adjusted closer to the ground-truth. Therefore, if we take the input configuration fed into each round as an individual case, then the distribution of input data is changed by iteration, and the cases that are more similar to the ground-truth are more likely to be sampled. Therefore, in the [**testing stage**]{}, we can expect the iteration to improve puzzle recognition accuracy, because as the iteration continues, the input puzzle gets closer to the ground-truth by statistics, and our model sees more training data in this scenario and is stronger. We show a typical example in Figure \[Fig:PuzzleRecognition\], in which we can observe how iteration gradually predicts the correct configuration.
Experiments {#Experiments}
===========
Jigsaw Puzzle Recognition {#Experiments:JigsawPuzzles}
-------------------------
We follow [@noroozi2016unsupervised] to train and evaluate puzzle recognition on the ILSVRC2012 dataset [@russakovsky2015imagenet], a subset of the ImageNet database [@deng2009imagenet]. We train the model using all the $1.3\mathrm{M}$ training images and test it on the validation set with $50\mathrm{K}$ images, both of which do not contain class annotations.
In the training stage, we pre-process the images to prevent the model from being disturbed by pixel-level information. We first determine the size of puzzles, [*e.g.*]{}, $W\times H$, and then resize each input image into $85W\times85H$ and partition it evenly into a $W\times H$ grid. In each $85\times85$ image, we randomly crop a $64\times64$ subimage as the patch fed into the puzzle recognition network. To maximally reduce the possibility that low-level information is used, we further horizontally flip each input patch with a probability of $50\%$ and subtract mean value from each channel – we do not perform other data augmentation techniques because they are less likely to appear in real data. In practice, flip augmentation brings consistent accuracy gain to transfer learning tasks though we observe significant accuracy drop in puzzle recognition (see Table \[Tab:2DResults\]).
{width="16.5cm"}
\[Fig:PuzzleRecognition\]
The backbone of our puzzle network is borrowed from two popular architectures, namely, an $8$-layer AlexNet [@krizhevsky2012imagenet] and two deep ResNets [@he2016deep] with $18$ and $50$ layers. We do not evaluate VGGNet [@simonyan2015very] as in [@larsson2017colorization][@noroozi2018boosting] because it is more difficult to initialize and produces lower accuracy than ResNets. The outputs of the first layer with a spatial resolution of $1\times1$ ([*i.e.*]{}, [*fc6*]{} in AlexNet and [*avg-pool*]{} in ResNets) are fed into a $1\rm{,}024$-way fully-connected layer and the output is taken as $\mathbf{f}_{x,y}$, followed by our designed layers for extracting unary and binary terms for puzzle recognition. All these networks are trained from scratch. We use the SGD optimizer and a total of $250\mathrm{K}$ iterations (mini-batches) for AlexNet and $350\mathrm{K}$ for ResNets. Each batch contains $256$ puzzles. On four NVIDIA Titan-V100 GPUs, the training times on AlexNet, ResNet18 and ResNet50 are $10$, $20$ and $60$ hours, respectively.
In the testing stage, to reduce randomization factors, we switch off randomization in patch cropping and data augmentation, with each $64\times64$ patch cropped at the center of the $85\times85$ fields and not flipped. Results are summarized in Table \[Tab:2DResults\]. We first evaluate $3\times3$ puzzle recognition accuracy. For each image, there are ${9!}={362\rm{,}880}$ possible puzzles, so random guess gives a $0.0003\%$ accuracy. With only unary terms (Eqn \[Eqn:OptimizationTesting\] can be solved by the Hungarian algorithm), all network backbones achieve over $30\%$ accuracy without mirror augmentation, which shows that weak visual cues can be combined to infer global patch contexts.
On top of this baseline, we investigate the impact of other four options. [**First**]{}, adding binary terms consistently improves puzzle recognition accuracy, arguably due to the additional contextual information, which is especially useful in determining the relative position of two neighboring patches. [**Second**]{}, mirror augmentation reduces puzzle recognition accuracy dramatically in both training and testing, but as we will see later, this strategy improves the generalization ability of our pre-trained models to other recognition tasks. [**Third**]{}, compared with $2\times2$ puzzles, $3\times3$ jigsaw puzzles are naturally more difficult to solve, but they also force the model to learn more visual knowledge and thus help transfer learning, as shown in our later discussions. [**Fourth**]{}, the above phenomena remain the same as the network backbone becomes stronger, on which both puzzle recognition and transfer visual recognition becomes more accurate.
As a side comment, we point out that conventional puzzle recognition approaches with plain classification [@noroozi2016unsupervised][@noroozi2018boosting] often achieved higher puzzle recognition accuracy in a limited class set. With models trained with our approach (Line (e) in Table \[Tab:2DResults\]) we enumerate the $1\rm{,}000$ classes generated with algorithm provided by [@noroozi2016unsupervised] and find the maximal $S\!\left(\mathbf{F},\mathbf{c}\right)$, so as to mimic the behavior of plain classification. Our models with AlexNet reports a $60.2\%$ puzzle recognition accuracy which is lower than $71\%$ reported in [@noroozi2016unsupervised]. However, our approach enjoys better transfer ability, as we will see in later experiments. In addition, the performance of [@noroozi2016unsupervised] degenerates with increased puzzle size, as the fraction of explored puzzles becomes smaller, yet the weakness of ignoring underlying relationship between different configurations becomes more significant and harmful. From this perspective, the advantage of solving arbitrary puzzles becomes clearer. The same phenomenon also happens in 3D puzzles (Section \[Experiments:3DNets\]).
Some statistics for our model with ResNet50 (Line (p) in Table \[Tab:2DResults\]) as well as two typical examples are shown in Figure \[Fig:PuzzleRecognition\] (one is difficult and not solved). We can observe how the disordered patches are reorganized with weak spatial cues throughout an iterative process. As an ablation study, we experiment with fewer numbers of maximal iterations, namely $1$, $5$ and $10$ instead of $20$, but achieve lower accuracies in both puzzle recognition and transfer learning tasks. This justifies our hypothesis that iteration, together with weak spatial cues, provides a mild way of unsupervised learning, which better fits state-of-the-art deep networks.
Transfer Learning Performance {#Experiments:TransferLearning}
-----------------------------
Next, we investigate how well our models pre-trained on puzzle recognition transfer to other visual recognition tasks. Following the conventions [@noroozi2018boosting][@caron2018deep], we evaluate classification and detection tasks on the PascalVOC 2007 dataset [@everingham2010pascal]. All pre-trained networks undergo a standard fine-tuning flowchart, with a plain classifier and Fast-RCNN [@girshick2015fast] being used as network heads, respectively. We do not lock any layers in our network, because this often leads to worse transfer performance as shown in prior approaches [@noroozi2016unsupervised][@caron2018deep][@gidaris2018unsupervised].
Results are summarized in Table \[Tab:2DResults\]. We can observe some interesting phenomena. [**First**]{}, transfer recognition performance goes up with the power of network backbones, which shows the ability of our approach to tap the potential of deep networks. [**Second**]{}, both unary and binary terms contribute to transfer accuracy and they are complementary. [**Third**]{}, mirror augmentation harms puzzle recognition but improves transfer learning, because it alleviates the chance that deep networks borrow low-level pixel continuity in solving the jigsaw puzzles which falls into the category of over-fitting and helps transfer recognition very little.
Here is a side note. It was suggested in [@noroozi2016unsupervised] that forcing the network to discriminate very similar puzzles ([*e.g.*]{}, only a pair of patches are reversed) often leads to accuracy drop because the model can focus too much on local patterns. In the context of using AlexNet to solve $3\times3$ puzzles, we study different numbers of configurations, [*i.e.*]{}, $1\%$ ($3\rm{,}629$), $10\%$ ($36\rm{,}288$) and all (${9!}={362\rm{,}880}$) possible puzzles. We find that our approach reports the best transfer accuracy at the last option, while using smaller numbers of configurations leads to slightly worse performance. Hence, we make the following conjecture: it is indeed the larger number of parameters in a plain classifier, rather than solving very similar puzzles, that causes transfer performance drop.
Last, we evaluate the quality of features extracted from the pre-trained models directly (the first fully-connected layer, without being fine-tuned). We apply a linear SVM with ${C}={10}$ to the Caltech256 dataset [@griffin2007caltech] for generic object classification. Our $3\times3$ model based on AlexNet with unary terms, binary terms and mirror augmentation (Line (e) in Table \[Tab:2DResults\]) reports a $29.05\%$ accuracy, but our direct competitors [@noroozi2016unsupervised] and [@noroozi2018boosting] only reports $20.83\%$ and $23.07\%$, respectively, almost of the same quality as a randomly-initialized AlexNet ($18.73\%$).
Generalization to 3D Networks {#Experiments:3DNets}
-----------------------------
Finally, we apply our model to a 3D visual recognition task, which lies in the area of medical imaging analysis, an important prerequisite for computer-assisted diagnosis (CAD). Most medical data are volumetric ([*i.e.*]{}, appearing in a 3D form), and researchers have proposed some 3D network architectures [@cicek20163d][@milletari2016v]. Compared to 2D networks [@ronneberger2015u][@yu2018recurrent], 3D networks enjoy the benefit of seeing more contextual information, but still suffer the drawback of missing a pre-trained model. Due to the common situation that the amount of training data is limited, these 3D networks often have a relatively unstable training process and sometimes this downgrades their testing accuracy [@xia2018bridging].
Our approach provides a solution for initializing 3D networks with jigsaw puzzles. We investigate the NIH pancreas segmentation dataset [@roth2015deeporgan], which contains $82$ cases. We partition it into $4$ folds (around $20$ cases in each fold), use three of them to train a segmentation model and test it on the remaining one. To construct jigsaw puzzles, we either directly use the training samples in the NIH dataset, or refer to another public dataset named Medical Segmentation Decathlon (MSD)[^6] – the [*pancreas tumour*]{} subset with $282$ training cases. For all the data used for jigsaw puzzles, we do not use any pixel-level annotations though they are provided. We randomly crop $120\times120\times120$ volumes within each case, and cut it evenly into two puzzle sizes, namely, $2\times2\times2$ pieces with a $48\times48\times48$ subvolume cropped within each cell, or $3\times3\times3$ pieces with a $32\times32\times32$ subvolume cropped within each cell. A typical example is shown in Figure \[Fig:JigsawPuzzles\]. We randomly disorder these patches using all $8!$ or $27!$ possible configurations, and the task is to recover the original configuration. We use VNet [@milletari2016v] as the baseline (only the down-sampling layers are used in this stage), and compute the unary terms in an $8\times8$ or $27\times27$ matrix. We switch off the binary terms based on the consideration that one patch has $26$ neighbors in the 3D space which makes prediction over-complicated.
Now we recover the complete VNet structure with randomly-initialized up-sampling layers and start training on the NIH training set ($62$ cases) as well as its subsets. Results are shown in Table \[Tab:3DResults\] revealing some useful knowledge. [**First**]{}, pre-training on jigsaw-puzzles indeed helps segmentation especially in the scenarios of fewer training data. [**Second**]{}, visual knowledge learned in this manner can transfer across different datasets regardless of the different distributions in intensity (caused by the scanning device). [**Third**]{}, constructing larger and thus more difficult puzzles improves the basic ability of networks. This the value of our research – note that it is unlikely for the baseline approach to sufficiently explore the space of $3\times3\times3$ puzzles, which has ${27!}\approx{1.1\times10^{28}}$ different configurations.
Conclusions {#Conclusions}
===========
This work generalizes the framework of jigsaw puzzle recognition which was previously studied in a constrained case. To this end, we change the network head from a plain $K$-way classifier to a combinatoric optimization problem which uses both unary and binary weak spatial cues. This strategy reduces the number of learnable parameters in the model, and thus alleviates the risk of over-fitting. The increased flexibility of pre-training allows us to apply our approach to a wide range of transfer learning tasks, including directly using it for feature extraction, and generalizing it to the 3D scenarios to provide an initialization for other tasks, [*e.g.*]{}, medical imaging segmentation.
Our study reveals the ease and benefits of learning to recognize weak visual cues in unsupervised learning, in which the key problem often lies in finding a compact way of representing knowledge, [*e.g.*]{}, decomposing the entire puzzle into unary and binary terms. We point out that the exploration of unsupervised learning is still far from the end. In the future, we will also apply our method to less structured data such as graphs [@kipf2017semi] and more structured data such as videos [@karpathy2014large], and explore its ability of learning visual knowledge in an unsupervised manner.
[^1]: It was widely believed that more powerful features can be learned in more difficult vision tasks [@deng2010does], so we expect the ability of unsupervised learning to grow with the complexity of puzzles.
[^2]: This is especially useful for some areas such as medical imaging analysis, in which 3D networks [@cicek20163d][@milletari2016v] cannot easily get pre-trained weights as in 2D scenarios, yet a reasonable initialization helps a lot in training stability and testing performance.
[^3]: [@noroozi2016unsupervised] observed that setting a larger $K$ leads to performance drop in transfer experiments, and explained it as the network gets confused by very similar jigsaw puzzles. However, as shown in experiments (see Section \[Experiments:TransferLearning\]), our approach works well in the entire puzzle set $\mathcal{C}$, [*i.e.*]{}, ${K}={9!}={362\rm{,}880}$, which implies that the performance drop may due the large number of parameters.
[^4]: Ideally, the elements in each column should also sum to $1$, but it is mathematically intractable if we hope to keep the ratio between all elements. There are two arguments. First, after normalizing scores in each row, we find that there often exists one major elements in each column, and the sum of each column is close to $1$. Second, we add an additional $\ell_1$ loss term between the sum of each column and $1$, but only observe to minor changes in either puzzle recognition accuracy or transfer learning performance.
[^5]: We fully-connect $\mathbf{F}$ to the $WH\times WH$ matrix, which is an asymmetric function and thus makes the output sensitive to the order of input. We can also design a symmetric function to deal with this issue, [*e.g.*]{}, each patch $\mathbf{f}_{x,y}$ is concatenated with the average-pooled vector of other patches to form the input, but this often causes information loss and leads to lower accuracy in both puzzle recognition and transfer learning tasks.
[^6]: [http://medicaldecathlon.com/]{}
|
---
abstract: |
*Context.* Systematic Reviews (SRs) are means for collecting and synthesizing evidence from the identification and analysis of relevant studies from multiple sources. To this aim, they use a well-defined methodology meant to mitigate the risks of biases and ensure repeatability for later updates. SRs, however, involve significant effort.\
*Goal.* The goal of this paper is to introduce a novel methodology that reduces the amount of manual tedious tasks involved in SRs while taking advantage of the value provided by human expertise.\
*Method.* Starting from current methodologies for SRs, we replaced the steps of keywording and data extraction with an automatic methodology for generating a domain ontology and classifying the primary studies. This methodology has been applied in the Software Engineering sub-area of Software Architecture and evaluated by human annotators.\
*Results.* The result is a novel Expert-Driven Automatic Methodology, EDAM, for assisting researchers in performing SRs. EDAM combines ontology-learning techniques and semantic technologies with the human-in-the-loop. The first (thanks to automation) fosters scalability, objectivity, reproducibility and granularity of the studies; the second allows tailoring to the specific focus of the study at hand and knowledge reuse from domain experts. We evaluated EDAM on the field of Software Architecture against six senior researchers. As a result, we found that the performance of the senior researchers in classifying papers was not statistically significantly different from EDAM.\
*Conclusions.* Thanks to automation of the less-creative steps in SRs, our methodology allows researchers to skip the tedious tasks of keywording and manually classifying primary studies, thus freeing effort for the analysis and the discussion.
address:
- 'Knowledge Media Institute, , '
- 'DISIM Department, , '
- 'Department of Computer Science, , '
- 'Knowledge Media Institute, , '
author:
- '[^1]'
-
-
-
bibliography:
- 'bibliography.bib'
title: Reducing the Effort for Systematic Reviews in Software Engineering
---
,
,
, and
Introduction {#sec:intro}
============
Understanding the state-of-the-art in research provides the foundation for building novelty. In particular, in Software Engineering topic areas, the acquisition of knowledge for this understanding follows a clear path: started with informal reviews and surveys, it is moving towards systematic searches of the literature. @kitchenhamTR2004 clearly explains the reasons, the importance, and the advantages and disadvantages of using systematic reviews instead of informal ones. Various studies (e.g., [@Da_Silva2014; @tertiaryAli]) reveal the growing interest in systematic literature reviews and systematic mapping studies [@Wohlin2013]. A number of articles and books have been written on how to perform such systematic studies [@kitchenham2007guidelines; @wohlin2013systematic; @Wieringa2014book].
A Systematic Review (SR) is [*“a means of evaluating and interpreting all available research relevant to a particular research or topic area or phenomenon of interest”*]{} [@kitchenhamTR2004]. Given a set of research questions, and by following a systematically defined and reproducible process, a SR helps selecting primary studies that contribute to provide an answer to them. Used in combination with keywording [@mapping_se], a SR supports the systematic elicitation of an ontological classification framework [@petersen2015guidelines]. In this paper we focus specifically on the field of Software Engineering, but systematic literature reviews and mapping studies are used in several research fields, such as Biomedics [@chinapaw2011relationship], Robotics [@benitti2012exploring], Artificial Intelligence [@raza2015review], Human-Computer Interaction [@mannocci2019evolution], Psychology [@richards2012computer], an many others.
A SR can help researchers and practitioners in creating a complete, comprehensive and valid picture of the state-of-the-art about a given theme when the search-space is bounded (e.g., when the search query returns few thousands of articles to scrutinize). However, it falls short when used to investigate the state-of-the-art on an entire research area (e.g., Software Architecture) where the returned entries are hundreds of thousands - hence clearly unmanageable. As reported by @Vale2016128 while investigating the state-of-the-art of the Component-based Software Engineering area through an SR, a [*“…manual search*]{} \[restricted only to the most relevant journals and conferences related to the CBSE area\] [*was considered as the primary source, given the infeasibility of analyzing all studies collected from automatic search”*]{}. Still, they had to select, read, and thoroughly analyze 1,231 primary studies.
In contrast to manually run SRs, several state of the art automated methods allow classifying a document in a certain category or topic [@blei2003latent; @mendes2011dbpedia; @alghamdi2015survey; @schultz1999topic]. Unfortunately, most current techniques suffer from limitations that make them unsuitable for systematic reviews. The approaches which exploit keywords as proxy for research areas are unsatisfactory, as they fail to distinguish research topics from other terms that can be used to annotate papers (e.g., “user case”, “scalability”) and to take advantage of the relationships that hold between research areas (e.g., the fact that “Software Architecture” is a sub-area of “Software Engineering”). Probabilistic topic models (e.g., Latent Dirichlet Allocation [@blei2003latent]) are also unsuitable for this task since they produce cluster of terms that are not easy to map to research areas [@osborne2013exploring]. Crucially, it is often unfeasible to integrate these topic detection techniques with the needs and the knowledge of human experts. Another alternative is to apply entity linking techniques [@mendes2011dbpedia] to map papers to relevant entities in knowledge base. Unfortunately, we currently lack good granular and machine readable representation of research areas in many domains which could be used to this end.
Current techniques have complementary limitations when investigating the state-of-the-art of an entire research area: on the one hand side, SRs are [*“human-intensive”*]{}, as they require domain experts to invest a large amount of time to carry out manual tasks; on the other side, automated techniques keep the [*humans “out of the loop”*]{}, while human expertise is critical for the more conceptual analysis tasks.
This paper proposes an [*expert-driven automatic methodology*]{} (EDAM) for assisting systematic reviews that, while recognizing the essential value of human expertise, limits the amount of tedious tasks the expert has to carry out. Our methodology contributes with 1) automatically extracting an ontology of relevant topics, related to a given research area; 2) using experts to refine this knowledge base; 3) exploiting this knowledge base for classifying relevant papers that may be then further validated/analyzed by experts, and for computing research analytics. We demonstrate EDAM in the field of Software Architecture, but it can be easily applied to other research areas as well.
Naturally, the ability of domain experts to analyse the research dynamics emerging from primary studies and to distill the most important lessons and trends is still crucial. Therefore, our aim is not to fully automatize the process, but to assist domain experts by automatically generating data-driven analytics in order to free time and resources for the analysis phase.
In summary, our contributions are:
- a novel methodology for supporting ontology-driven systematic reviews, which involves both automatic techniques and human experts;
- an implementation of this methodology which exploits the Klink-2 algorithm for generating the domain ontology in the field of Software Architecture;
- an illustrative analysis of the Software Architecture trends;
- an evaluation involving six human annotators, which shows that the classification of primary studies yielded by the proposed methodology is comparable to the one produced by domain experts (p=0.77).
- an automatically generated ontology of Software Engineering, which could support further systematic reviews in the field[^2].
The rest of the paper is structured as follows. Section \[sec:rw\] introduces related works on systematic studies. Section \[sec:why\] provides an overview of some preliminary evidence of the benefits brought by using EDAM to assist a mapping study. Section \[sec:method\] then presents the EDAM methodology and its application to the research area of Software Architecture. This experiment is discussed and evaluated in Section \[sec:discussion\], which also present a comparison of several approaches for classifying research papers. Finally, in Section \[sec:conclusion\] we discuss the main implications of our study and outline future directions of research.
Related Work {#sec:rw}
============
There are many guidelines for, and reports on, carrying out systematic studies in Software Engineering. Among them, we could identify a few aimed at supporting or improving the underlying process. In our perspective, they all enable researchers to focus more on the most creative steps of a systematic study by removing what is referred to as [*manual work*]{}.
With a motivation similar to ours, i.e. to improve the search step in systematic studies in Software Engineering research, @Octaviano2015-xr propose a strategy that automates part of the primary study selection activity. @Mourao2017 present a preliminary assessment of a hybrid search strategy for systematic literature reviews that combines database search and snowballing to reduce the effort due to searches in multiple digital libraries. @Kuhrmann2017 provide recommendations specifically for the general study design, data collection, and study selection procedures. @Zhang2011, in turn, systematically select and analyze a large number of SRs. Their results have been then used to define a quasi-gold standard for future studies. In their validation, they were able to improve the rigor of the search process and provide guidelines complementing the ones already in use.
@runeson17 propose a machine learning approach that classifies papers for SRs by leveraging human experts, who iteratively validate set of publications produced by a classifier. Conversely, EDAM does not require experts to manually examine research papers, but only to review a taxonomy of research areas.
The need for guidelines in conducting empirical research has been addressed in other types of empirical studies, too. @DeMello2016 focus on opinion surveys and provide guidelines (in the form of a reference framework) aimed at improving the representativeness of samples. Also on opinion surveys, @Molleri2016 provide recommendations based on an annotated bibliography instead.
Another interesting work by @Felizardo2016 investigates how the use of forward snowballing can considerably reduce the effort in updating SRs in Software Engineering. Based on this result, complementing our method with automated forward snowballing suggests a very promising direction for future works as it could further reduce the effort for identifying relevant primary studies.
@Marshall2015 carried out an interview survey with experts in other domains (i.e. healthcare and social sciences) with the aim to identify tools that are generally used, or desirable, to ease which steps in systematic studies, and transfer the best practices to the Software Engineering domain. Among the results, data extraction and automated analysis emerge as top requirements for reducing the workload. In a similar vein, @IST-SLRtoolsupport followed by @Al-Zubidy2017-np consulted Software Engineering researchers conducting SRs to identify and prioritize the necessary SR tool features. The results identified [*search & study selection*]{} as the most desirable feature. Our work addresses the needs identified by both @Marshall2015 and @IST-SLRtoolsupport.
The idea of using ontologies for supporting SRs was discussed by few papers, but did not receive much attention. @de2007scientific introduced the Scientific Research Ontology, a resource to organize the knowledge generated from SR. This ontology offers a conceptual framework with the aim of fostering the consistency between different studies, but does not directly assist the tasks involved in SR, such as the extraction of primary studies. @sun2012towards discussed the use of ontologies for supporting key activities in SRs and presented an experiment in which they automatically classified primary studies by means of COSONT, an ontology of methods for cost estimation. Unfortunately, their approach still required to manually check hundreds of papers and the COSONT ontology was quite simplistic, being an handcrafted list of methods with no hierarchical structure. This is a common issue with manually generated ontology of research concepts, which are usually costly to produce, coarse-grained, and slow to evolve [@osborne2015klink]. Conversely, EDAM takes advantage of recent ontology learning techniques to automatically generate complex multi-level ontologies (e.g., the SE ontology presented in this paper includes 956 topics and 5,461 relationships), exploits the resulting taxonomic structure to classify the primary studies, and does not require experts to manually review a large number of papers.
An Overview of the Benefits of Automatic SRs {#sec:why}
============================================
Before entering the details of the EDAM methodology, this section provides an overview of the benefits such an automatic SR methodology can bring with respect to more traditional, manual SRs carried out according to predefined protocols.
We all agree that manual SRs based on well-defined systematic protocols help reducing (but not fully removing) subjective biases in the selection of the studies. They however are by and large unfeasible in reviewing a too large dataset (i.e. when the number of scientific publications is too large to be manually processed by the researcher).
In a similar vein, automatic SRs help reducing subjective biases (in this case by [*implementing*]{} the selection of the studies according to the predefined systematic protocol). Differently, they pose no limitation in terms of the size of the dataset of publications.
In our earlier work [@Wolfram2017-ts] we challenged these limitations and benefits by applying the automatic study selection to a manual SR carried out beforehand by other researchers [@TR-SLR-sustainab]. In this way, we could compare and contrast the results of the manual SR with the results of our automatic SR. In this earlier work, we have studied the field of software sustainability within the Software Engineering domain. While at the time the EDAM methodology was not yet fully developed, we did use the same ontology-learning algorithms and a preliminary version of the ontology for the Software Engineering domain.
![Some evidence on the Benefits of Automated SRs[]{data-label="fig:whyedam"}](img/whyEDAM.pdf)
The observations gathered during this experiment are illustrated in Fig. \[fig:whyedam\], where we represented the primary studies selected manually (see the left-hand circles) and those selected automatically (see the right-hand ovals). The experiment underwent three phases:
[**Starting point:**]{}
: The already-completed manual SR had selected 116 primary studies. Before training the algorithm and tuning the domain ontology, from the Scopus dump of scientific publications we automatically selected 950 studies. While our automatic methodology is able to handle seamlessly any size of the base of publications, the selected studies did initially include a very large number of false positives. However, it did also uncover that 12 studies selected in the manual SR where wrongly included. [*[**Observation \#1:**]{} in spite of systematic selection criteria and the involvement of multiple researchers, human errors in the manual study selection is still possible.*]{}
[**Training:**]{}
: By treating the 104 primary studies (from the manual SR) as pilot studies, we trained our domain ontology and learning algorithm to automatically select the primary studies. [*[**Observation \#2:**]{} Automatic SR is able to automatize the selection criteria of systematic reviews while handling any size of the initial dataset of scientific publications.*]{} As discussed in Section \[sec:eval1\], the domain ontology is able to classify the primary studies as correctly as the human experts do, without needing further training. As such, the domain ontology can be reused for any study in the domain of Software Engineering.
[**Final result:**]{}
: The final result of the automatic selection converged to 234 studies which included the 104 pilot studies and [*correctly*]{} identified additional 130 studies that were missing in the original manual SR. [*[**Observation \#3:**]{} By handling a much larger base of publications, automatic SRs are able to uncover primary studies that are missed by manual SRs where such scale is unfeasible.*]{}
An Expert-Driven Automatic Methodology {#sec:method}
======================================
We propose a novel expert-driven automatic methodology (EDAM) for assisting systematic reviews like systematic literature reviews and mapping studies. EDAM allows to automatize the steps that are the most time and effort consuming while requiring the least creativity, such as *selection of relevant papers*, *keywording*, and *creation of a classification schema* [@petersen2015guidelines], by exploiting ontology learning techniques and semantic technologies to foster scalability, objectivity, reproducibility, and granularity of the study (further discussed in Section \[sec:implications\]). It also supports the generation of research trends, which are typical of data synthesis in mapping studies. In this paper, we illustrate how EDAM can support mapping studies, even though it can be evidently exploited in systematic literature reviews, too.
![Steps of a systematic mappings adopting the EDAM methodology. The gray-shaded elements refer to the alternative step of reusing the previously generated ontology. []{data-label="fig:edam"}](img/EDAM_2.png)
![Classic steps of systematic mappings (inspired by [@petersen2015guidelines])[]{data-label="fig:ms"}](img/MS_1.png)
Figure \[fig:edam\] shows the steps of a mapping study using EDAM in contrast with the steps of a classic (manual) methodology - shown in Figure \[fig:ms\]. The main difference is that in the classic methodology the researchers first select and analyze each primary study (steps 2-3) and then produce a taxonomy to classify them (step 4). When assisted by EDAM, instead, the researchers first use ontology learning methods over large scholarly datasets to generate an ontology of the field (steps 2-3), then refine the ontology with the help of domain experts (step 4), and finally exploit this knowledge base to automatically select and classify the primary studies (steps 5-6).
An alternative solution for steps 2-4 (Generation of domain ontology) is the reuse of an ontology crafted by a previous study with the same scope. Indeed, in the study discussed in Section \[sec:methodologyimplementation\] we have generated an ontology of Software Engineering (SE) research topics, with the hope that it will be re-used by the research community.
In Section \[sec:generalmethodology\], we describe EDAM and discuss its advantages over a classic methodology. In Section \[sec:methodologyimplementation\], we exemplify the application of EDAM specifically aimed at identifying publication trends of the Software Architecture research area in the specific SE domain.
EDAM Description {#sec:generalmethodology}
----------------
A SR assisted by EDAM is organized along the following steps (ref. Figure \[fig:edam\]).\
**1. Research question definition.** The researchers performing the study state the research questions (RQs). These will affect the aim of the study and thus its steps. It should be noted that EDAM is applicable only to research questions that could be answered by classifying publications, authors, venues, and other entities according to the ontology for producing relevant analytics. Other research questions should be addressed according to the standard methodology [@petersen2015guidelines].\
**2. Dataset selection.** The researchers select a dataset on which to apply the chosen ontology learning technique (further elaborated in step 3) for generating the domain ontology that will be used to select and classify the primary studies. The most important characteristic of this dataset is that it must be unbiased with respect to the focus of the study. For example, if the study wants to uncover the trends in research areas (e.g., Software Architecture), the dataset should not be biased with respect to any area in the domain (e.g., Software Engineering in our case). A good strategy to select unbiased datasets is considering either a full scholarly dataset of a very high-level field (e.g., all the Computer Science papers in Microsoft Academic Search[^3] or in Scopus[^4]) or a dataset including all the papers published in the main conferences and journals of the domain under analysis. In recent years, universities, organizations, and publishing companies have released an increasing number of open datasets that could assist in this task, such as CrossRef[^5], SciGraph[^6], OpenCitations[^7], DBLP[^8], Semantic Scholar[^9], and others.\
**3. Ontology learning.** The dataset is processed by an ontology learning technique that automatically infers an ontology of the relevant concepts.
We strongly advocate the use of an ontology learning technique that generates a full domain ontology and represents it with Semantic Web standards, such as the Web Ontology Language (OWL)[^10]). The main advantage of adopting an ontology in this context is that it allows for a more comprehensive representation of the domain since it includes, in addition to hierarchical relationships, also other kinds of relationships (e.g., *sameAs*, *partOf*), which may be critical for classifying the primary studies. For example, an ontology allows to explicitly associate to each category a list of alternative labels or related terms that will be used in the classification phase. In addition, ontology learning techniques can infer very structured multi-level ontologies [@osborne2015klink], and thus describe the domain at different levels of granularity.
The task of ontology and taxonomy learning was comprehensively explored over the last 20 years. Therefore, the researcher can choose among a variety of different approaches for this step, including:
- statistical methods for deriving taxonomies from keywords [@sanderson1999deriving; @liu2012automatic];
- natural language processing approaches, e.g., FRED [@gangemi2017semantic], LODifier [@augenstein2012lodifier], Text2Onto [@cimiano2005text2onto];
- approaches based on deep learning, e.g., recurrent neural networks [@petrucci2016ontology];
- hybrid ontology learning frameworks [@wohlgenannt2012dynamic];
- specific approaches for generating research topic ontologies, e.g., Klink-2 [@osborne2015klink].
However, as discussed in the following step, researchers may also choose to skip this step and re-use a compatible ontology from a previous study.
It is useful to clarify why we suggest the adoption of an ontology learning approach, rather than the adoption of one of the currently available research taxonomies, such as the ACM computing classification system[^11], the Springer Nature classification[^12], Scopus subject areas[^13], and the Microsoft Academic Search classification. Unfortunately, these taxonomies suffer from some common issues, which make them unfeasible to support most kinds of SRs. First, they are very coarse-grained and represent wide categories of approaches, rather than the fine-grained topics addressed by researchers [@osborne2012mining]. Secondly, they are usually obsolete since they are seldom updated. For example, the 2012 version of the ACM classification was finalized fourteen years after the previous version. This is a critical point, since some interesting trends could be associated with recently emerged topics. In third instance, most ontology learning algorithms are not limited to learning research areas, but can be tailored to yield the outputs which are more apt to support a specific analysis.\
**4. Ontology refining.** The ontology resulting from the previous step is corrected and refined by domain experts. During this phase, the experts are allowed to 1) delete an existent category, 2) add a new category, 3) delete an existent relationship, 4) add a new relationship. We suggest using at least three domain experts for addressing possible disagreements.
This step is critical for two reasons. First, it may correct some errors in the automatically-generated taxonomy. Secondly, it verifies that the data-driven representation aligns with the domain experts mental model and thus the outcomes will be understandable and reusable by their research community.
Refining a very large ontology is not a trivial task, therefore if the domain comprehends a large number of topics we suggest splitting it in manageable sub branches to be addressed by different experts. Our experience suggests that a taxonomy of about 50 research areas can be reviewed in about 15-30 minutes by an expert of the field. For example, in [@osborne2015klink] three experts reviewed a Semantic Web ontology of 58 topics in about 20 minutes. In the test study for this paper, three experts took about 20 minutes to examine and produce feedback on a taxonomy of 46 topics (and 71 terms considering synonymous such as “product line”, “product-lines”, “product-line”, which were clustered automatically by the ontology learning algorithm). In both cases, we represented the ontology as tree diagram in a excel sheet[^14] and included also a list of the most popular terms in the dataset, for supporting experts in remembering all the relevant research topics.
An alternative solution is to provide experts with ontology editors that could be used to directly modify the ontology, such as Protege[^15], NeOn Toolkit[^16], TopBraid Composer[^17], Semantic Turkey[^18], or Fluent Editor[^19]. However, these tools are not always easy to learn and the adoption of a simple spreadsheet may be advisable in most cases. Indeed, the annotators who participated in the mapping study of Software Architecture described in the next section, reported that they were able to easily correct and suggest changes in the ontology using this simple solution. In particular, this task was natural to them since the same kind of spreadsheet is typically used in the analysis phase of systematic reviews (e.g., for the keywording step). We refer the reader to @sabou2018verifying for a comprehensive analysis of the verification of domain knowledge by human experts in the field of Software Engineering.
As highlighted by Figure \[fig:edam\], the aim of steps 2-4 is to generate an ontology apt to select and classify relevant papers and ultimately answer the RQs. It follows that these steps could be replaced by the adoption of an ontology previously generated and validated by a previous study with a consistent scope. For example, the ontology about Software Engineering generated for this paper’s example study (see Section \[sec:methodologyimplementation\]) can be re-used to perform many kinds of mapping studies involving other research areas in SE. Naturally, the ontology may have to be further updated to include the most recent concepts and terms. This solution allows users with no access to vast scholarly databases or no expertise in ontology learning techniques to easily implement an EDAM study.\
**5. Selection of primary studies.** The authors select a dataset of papers and define the inclusion criteria of the primary studies according to the domain ontology and other metadata of the papers (e.g., year, venue, language). The inclusion criteria are typically expressed as a search string, which uses simple logic constructs, such as AND, OR, and NOT [@aromataris2014constructing]. The search string is then used to produce the query that will be run over the dataset for selecting the primary studies. Some examples of queries include 1) “all the papers in the dataset published in a list of relevant conferences” or “all the papers in the dataset that contain a list of relevant terms from the ontology”.
In most cases this dataset will be the same or a subset of the one used for learning the domain ontology. However, the authors may want to zoom on a particular set of articles, such as the ones published in the main venues of a field, in a geographical area, or by a certain demography. It is also possible to select a different dataset altogether, since the ontology would use generic topic labels and thus be agnostic with respect to the dataset. A possible reason to do so is the availability of the full text of the studies. Many ontology learning algorithms can be run on massive metadata dataset (e.g., Scopus, Microsoft Academic Search), but some research questions may require the full text. In this case, the author may want to perform the ontology learning step on the metadata dataset, which is usually larger in size and scope, and then either select a subset composed by publications which are available online or adopt for this phase a second dataset that includes the full text of the articles, such as Core [@knoth2012core]. The growth of the Open Access movement [@wilkinson2016fair], which aims at providing free access to academic work, may alleviate this limitation in the following years.\
**6. Classification of primary studies.** The authors define a function for mapping categories to papers based on the refined ontology. This step is important to foster reproducibility since the inclusion criteria (defined in the step 5), the mapping function, and the domain ontology should contain all the information needed for replicating the classification process. The function can also be associated to an algorithmic method (e.g., a machine learning classifier), provided the method is made available and is reproducible.
The simplest way for mapping categories to papers is to associate to each category each paper that contains the label of the category or of any of its sub-categories. This simple technique for semantically characterizing documents has the advantage of being unsupervised and was applied with good results in a variety of fields, such as topic forecasting [@salatino2017topics], automatic classification of proceeding books [@osborne2016automatic], sentiment analysis [@saif2012semantic], recommender systems [@di2012linked], and many others. Alternative unsupervised methods, which we evaluate in Section \[sec:eval2\], include approaches based on TF-IDF [@ramos2003using], LDA [@blei2003latent], and word embeddings [@salatino2018csoc2].
In addition, the authors can choose to create a more complex mapping function which exploits other semantic relationships in the ontology (e.g., *relatedTerm*, *partOf*).\
**7. Data synthesis.** According to the RQs, this step may be automatic, semi-automatic or manual. Some straightforward analytics (e.g., the number of publications or citations over time) can be computed completely automatically by counting the previously classified papers or summing their number of citations. Other more complex analyses may require the use of machine learning techniques or the (manual) intervention of human experts. Starting from the groundwork formed by our research, a full analysis of the possible kinds of data synthesis and the way to automatize them will constitute interesting future works beneficial for the whole research community.\
Overall, motivated by the need to reduce the amount of manual tedious tasks involved in SRs, [**EDAM offers four main advantages over a classic methodology**]{}. [**First,**]{} human experts are not required to manually analyze and classify primary studies, but they simply have to refine the ontology, choose the inclusion criteria, and define a mapping function for associating papers to categories in the ontology. This allows researchers to carry out large scale studies that involve thousands of research papers with relative ease. [**Secondly,**]{} since the domain ontology is created with a data-driven method, it should reflect the real trends of the primary studies, rather than arbitrary human decisions about which keywords to annotate and aggregate, even if the refinement step may still introduce a degree of arbitrariness. [**Third,**]{} the use of a formal machine-readable ontology language for representing the domain taxonomy should foster the reproducibility of the study and allow authors with no expertise in data science to perform studies using previously generated ontologies. [**Fourth,**]{} this methodology allows researchers to produce and exploit complex multi-level ontologies, rather than the simple two-level classifications used by many studies [@Vale2016128].
Naturally, EDAM is suitable for research questions that can be automatized by the ontology-driven classification process previously described, or that aim at giving an overview of the state-of-the-art or state-of-practice on a topic [@wohlin2012experimentation] by analysing all of the relevant research contributions in a specific research area. We will discuss further this and other limitations in Section \[sec:limitations\].
EDAM Application {#sec:methodologyimplementation}
----------------
With the aim of presenting a reproducible pipeline and showing how EDAM can be applied, we present here an example as part of a possible systematic mapping study assisted by EDAM in the Software Architecture research area. We chose to study the research trends in this area, since trend analysis is typical of mapping studies [@wohlin2012experimentation] and it is one of the tasks that can be automatized by EDAM.
In the following, we describe how we instantiated the study example assisted by EDAM and discuss the specific technologies used to implement it. The data necessary for reproducing this study and using this same pipeline on other fields are available at <https://doi.org/10.5281/zenodo.2653924>.\
**1. Research question definition.** We wanted to focus on a task that is often addressed by mapping studies and could be completely automatized. Therefore our RQ is: “What are the trends of the main research topics of Software Architecture?”.\
**2. Dataset selection.** We selected all papers in a dump of the Scopus dataset about Computer Science in the period 2005-2013. The Scopus dataset we were given access by Elsevier BV includes papers in 1900-2013 interval, but the number of relevant articles before 2005 was too low to allow a proper trend analysis. Each paper in this dataset is described by title, abstract, keywords, venue, and author list.\
**3. Ontology learning.** We applied the Klink-2 algorithm [@osborne2015klink] on the Scopus dump for learning an ontology representing the main ’Software Architecture’ research area in SE.
Klink-2 is an algorithm that generates an ontology of research topics by processing scholarly metadata (titles, abstracts, keywords, authors, venues) and external sources (e.g., DBpedia, calls for papers, web pages). In particular, Klink-2 periodically produces the Computer Science Ontology (CSO)[^20] [@salatino2018computer] that is currently used by Springer Nature for classifying proceedings in the field of Computer Science [@osborne2016automatic], such as the well-known Lecture Notes in Computer Science series[^21]. The ontologies produced by Klink-2 use the Klink data model[^22], which is an extension of the BIBO ontology[^23] that in turn builds upon SKOS[^24]. This model includes three semantic relations: *relatedEquivalent*, which indicates that two topics can be treated as equivalent for the purpose of exploring research data; *skos:broaderGeneric*, which indicates that a topic is a subarea of another one; and *contributesTo*, which indicates that the research outputs of one topic significantly contribute to the research into another. In the following, we make use of the first two relationships for classifying studies according to their research topics.
\[alg:klink-2\]
relationships={}; keywords = FilterTopics(keywords, metadata, relationships) ontology = GenerateSemanticRelationships(relationships) return(ontology)
In Algorithm \[alg:klink-2\], we report the pseudocode of Klink-2. The algorithm takes as input a set of keywords and investigates their relationships with the set of their most co-occurring keywords. Klink-2 infers a sub-topic relationship between keyword $x$ and $y$ by means of two metrics: i) $H_R (x,y)$, which uses a semantic variation of the subsumption method; ii) $T_R (x,y)$, which uses temporal information to do the same. $H_R (x,y)$ is computed according to the following formula: $$H_R (x,y)= \left(\frac{I_R (x,y)}{I_R (x,x)} - \frac{I_R (y,x)}{I_R (y,y)}\right) c_R (x,y) ~ n(x,y)$$ where $I_R (x,y)$ is the number of elements associated with both x and y according to relation $R$ (e.g., number of co-occurrences in research papers), $\frac{I_R (x,y)}{I_R (x,x)}$ is the conditional probability that an element associated with keyword $x$ will be associated also with keyword $y$, $n_R (x,y)$ is the Levenshtein distance between the two keywords normalized by the length of the longest one, and $c_R (x,y)$ is the cosine similarity between the two vectors in which each index represents a keyword $k$ and the value is the number of time $x$ or $y$ co-occurred with $k$ in a certain context.
$T_R (x,y)$ is a temporal version of $H_R (x,y)$, which weighs more the information associated with the first years of $x$. It is useful to detect the cases in which the relationship between two terms fade because their association has become implicit (e.g., Artificial Intelligence and Machine Learning). $T_R (x,y)$ is calculated using a variation of formula (1) in which $I_R (x,y)$ is computed by weighting the intensity of the relationships in each year according to the distance from the debut of x. The weight is computed as $w(year, x)= (year - debut(x) +1)^{–\gamma}$, with $\gamma >0$ ($\gamma =2$ in the implementation used for this paper). A hierarchical relationship is inferred whenever $H_R (x,y)$ or $T_R (x,y)$ are higher then a certain threshold (0.25 in the implementation used for this study).
After inferring the hierarchical relationships, Klink-2 removes loops in the topic network (instruction $\#9$), merges similar keywords and splits ambiguous keywords associated to multiple meanings (e.g., ’Java’). The keywords produced in this step are added to the initial set of keywords to be further analysed in the next iteration and the while-loop is re-executed until there are no more keywords to be processed. Finally, Klink-2 filters the keywords considered ’too generic’ or ’not academic’ according to a set of heuristics (instruction $\#13$) and generates the triples describing the ontology.
Klink-2 was evaluated on a gold standard ontology including 88 research topics in the field of Semantic Web, which was manually generated by three senior researchers. It significantly outperformed the alternative algorithms ($p=0.0005$), yielding a precision of 86% and a recall of 85.5%. More details about Klink-2 and its evaluation can be found in @osborne2015klink.
We selected Klink-2 among the other previously discussed solutions for a number of reasons. First, it is the only approach to our knowledge that was specifically designed to generate taxonomy of research areas. Secondly, it was already integrated and evaluated on a dump of the Scopus dataset, which we adopted in this study, yielding excellent performance on the fields of artificial intelligence and semantic web [@osborne2015klink]. In third instance, it permits to define a number of pre-determinate relationships as basis for a new taxonomy. In particular, a human user can define a subsumption relation (i.e., *skos:broaderGeneric*), a *relatedEquivalent* one, or specify that two concepts should not be in any relationships. This functionality allows us to easily incorporate expert feedback in the ontology learning process. Therefore, the next iterations of the ontology will benefit from the knowledge of previous reviewers. We ran Klink-2 on the selected dataset, giving as initial seed the keyword “Software Engineering” and generated an OWL ontology of the field including 956 concepts and 5,461 relationships. We then selected the sub-branch of Software Architecture comprising 46 research areas and 71 terms (some research areas have multiple labels, such as “component based software” and “component-based software”).\
**4. Ontology refining.** We generated a spreadsheet, containing the Software Architecture (SA) ontology as a tree diagram[^25]. In this representation each concept of the ontology was illustrated by its level in the taxonomy, its labels, and the number of papers annotated with the concepts. We also included a list of the 500 more popular terms in the papers that contained the keywords “Software Architecture” and “Software Engineering”, to assist the experts in remembering other concepts or terms that the algorithm may have missed.
We sent it to three senior researchers and asked them to correct the ontology as discussed in Section \[sec:generalmethodology\]. The task took about 20 minutes and produced three revised spreadsheets. The feedback from the experts was integrated in the final ontology[^26]. In case of disagreement we went with the majority vote.
The most frequent feedback regarded: 1) the deletion sub-areas that were incorrectly classified under SA (e.g., “software evolution”), 2) the introduction of sub-areas that were neglected by Klink-2 (e.g, “architecture concerns”), and 3) the inclusion of alternative labels for some category (e.g., alternative ways to spell “component-based architecture”).\
**5. Selection of primary studies.** We then selected from the initial Scopus dump two datasets of primary studies to investigate the SA area: 1) **DSA** (Dataset SA, 3,467 publications), including all papers in the Scopus dataset that contain the terms “Software Architectures” or “Software Architecture” and include at least one of the subtopics of Software Architecture in the domain ontology, and 2) **DSA-MV** (Dataset SA - Main Venues, 1,586 publications), containing all the papers published in a list of well-known conferences and journals in the SE fields and in a particular in the SA area (see Table \[tbl:venuetable\]) and including at least one of the sub-topics of SA in the OWL ontology. We considered these two datasets since it may be interesting to analyze the discrepancy between generic SA papers and papers published in the main venues.\
**6. Classification of primary studies.** We defined the mapping function as follows. A paper was classified under a certain category (e.g., service-oriented architectures) if it contained in the title, abstract or keywords: 1) the label of the category (e.g., “service-oriented architectures”), 2) a [*relevantEquivalent*]{} of the category (e.g., “service oriented architecture”), 3) a [*skos:broaderGeneric*]{} of the category (e.g., “microservices”), or 4) a [*relevantEquivalent*]{} of any [*skos:broaderGeneric*]{} of the category (e.g., “microservice”). The advantage of this solution is that it allows us to map each category to a list of terms that can be automatically searched in the metadata of the papers. Therefore, the classification step can be handled automatically. In addition, it allows us to associate multiple categories to the same paper.
We chose this straightforward approach instead of other more complex methods based on word embeddings and string similarity [@salatino2018csoc2], since it is simple to reproduce and yields the best precision, as discussed in Section \[sec:eval2\]. There we discuss also some recent approaches [@salatino2018csoc; @salatino2018csoc2] yielding a more comprehensive set of topics and therefore a better recall, and illustrate how the choice of the method ultimately depends on the requested tradeoff between precision and recall.
In practice, we indexed titles, abstracts and keywords in an ElasticSearch[^27] instance and we ran a PHP script that imported the ontology, performed the relevant queries on the metadata, and saved the result in a MariaSQL database[^28].\
**7. Data synthesis.** Figure \[fig:study1\] shows the number of primary studies in the DSA and DSA-MV datasets. The DSA dataset follows the trend of the “Software Architecture” keyword in the Scopus dataset and decreases after 2010. Conversely, the size of DSA-MV grows steadily with the number of relevant conferences and journals.
We identified the main trends by running a script to count the number of studies about each sub-topic in each year. Since the focus of the paper is the EDAM methodology, rather than a comprehensive analysis on these research sub-areas, we will briefly discuss only the main trends associated with the more popular subtopics (in terms of number of papers). The full results of this example study, however, are available at [rexplore.kmi.open.ac.uk/data/edam](rexplore.kmi.open.ac.uk/data/edam) and on Zenodo[^29] and can be reused for supporting a more in-depth analysis of the field.
![Number of publications in DSA and DSA-MV over the years. []{data-label="fig:study1"}](img/Study_1.png)
Figure \[fig:study2\] displays the number of publications and citations associated with the most popular sub-areas of SA. The papers in DSA yield on average $4.8 \pm 2.1$ in citations versus the $13.6 \pm 7.0$ citations of those in DSA-MV. Reasonably, this tendency suggests that the papers published in the main SA venues tend to be more recognized by the research community.
![Number of publications and citations of the main topics in DSA and DSA-MV.[]{data-label="fig:study2"}](img/Study_2.png)
Figure \[fig:study3\] shows the percentage of papers published over time in the main topics within SA. We focus on the 2005-2013 period, since in this interval the number of publications is high enough to highlight the topic trends.
Software-oriented Architectures appears to have been the most prominent topic before 2009, while from 2010, Model-driven Architectures appears to be the most popular topic in this dataset. We can also appreciate the rising of Design Decisions, that seems the most significant positive trend of the last period together with Architecture Description Languages.
![Number of publications of the top ten main topics in DSA over time.[]{data-label="fig:study3"}](img/Study_4.png "fig:") ![Number of publications of the top ten main topics in DSA over time.[]{data-label="fig:study3"}](img/Study_5.png "fig:")
Interestingly, the dataset regarding the main venues (DSA-MV) exhibits some different dynamics. Figure \[fig:study5\] highlights the difference between DSA and DSA-MV by showing for each topic the ratio between its number of publications and the total publications in the ten main topics. The research areas of Design Decisions and Views appear much more prominent in the main venues, while Model-Driven Architectures and Architecture Analysis are more popular in DSA. We can further analyze these differences by considering the main trends of the DSA-MV dataset, displayed by Figure \[fig:study6\]. The trend of Design Decisions in DSA-MV mirrors the one exhibited in DSA, both growing steadily from 2010. Conversely, Service-oriented Architectures, which has a negative trend in DSA, remains stable in DSA-MV.
![Comparison DSA and DSA-MV in terms of topic distribution. The percentage value refers to the ratio between the number of publications in a topic and the total publications in the ten main topics.[]{data-label="fig:study5"}](img/Study_3.png)
![Number of publications of the top ten main topics in DSA-MV over time.[]{data-label="fig:study6"}](img/Study_6.png "fig:") ![Number of publications of the top ten main topics in DSA-MV over time.[]{data-label="fig:study6"}](img/Study_7.png "fig:")
Evaluation and Discussion {#sec:discussion}
=========================
In the following, we reflect on this preliminary application of EDAM. This section includes 1) a evaluation of our method versus six human annotators, 2) a comparison of several approaches for classifying primary studies, 3) an analysis of EDAM limitations, 4) a discussion about the implications for systematic mappings in Software Engineering, and 5) a discussion on how to reuse EDAM for other SRs.
Evaluation of the primary study classification {#sec:eval1}
----------------------------------------------
The most critical step of EDAM is the classification of primary studies. When these are correctly associated to the relevant topics, the subsequent analysis presents a realistic assessment of the landscape of the studied research field. Thus, even if working on a large number of papers can alleviate the weight of some minor misclassification mistakes, we need to be able to trust that the automatic classification process will obtain an accuracy similar to that yielded by human annotators.
We evaluated the ability of EDAM to correctly discriminate between different topics in the field of Software Architecture by (1) randomly selecting a set of 25 papers in the DSA dataset, (2) classifying them both with EDAM and with six human experts (researchers in the field of SA), and (3) comparing the results. For simplifying the task and allowing to compare the annotation algorithmically, we first selected five unambiguous categories from the main topics of SA: Design Decisions, Service-oriented Architectures, Model-driven Architectures, Architecture Description Languages, and Views. For each category, we randomly selected from the DSA dataset five primary studies that were classified by EDAM exclusively under that topic, for a total of 25 papers. These papers were described in a spreadsheet by means of their title, author list, abstract, and keywords. The human experts were given this spreadsheet and asked to classify each paper either with one of the five categories or with a “none of the above” tag. We then compared the seven annotation sets produced by the six human experts and by EDAM, considered as an additional annotator[^30].
Table \[tbl:agreement\] shows the agreement between the annotators. It was computed by calculating the ratio of papers which were tagged with the same category by both annotators. EDAM has the highest average agreement and it also yields the highest agreement with three out of six users. User5 does even better in this regards and has the highest agreement with four annotators.
The chi-square test run on the human users shows that their behaviours are significantly different ($p=0.017$). However, if we group together users $\{2,3,5,6\}$ and users $\{1,4\}$, we find no significant differences in the behaviour within each group ($p=0.81$, $p=0.38$, good intra-group agreement), while there are between the two groups ($p = 0.0007$).
Interestingly, users $\{1,4\}$ were two students at the beginning of their PhD, hence still relatively new to the domain. This could suggests the importance of considerable domain experience for this task. EDAM exhibits a behaviour consistent with the most senior group, from which it is not significantly different ($p=0.77$).
![Percentage of annotations that agree with other $n$ annotators.[]{data-label="fig:evalchart"}](img/EvalChart.png)
As anticipated, a good way to measure the performance of annotators is their agreement with the majority of other expert users. Figure \[fig:evalchart\] shows the percentage of annotations of each annotator that agree with other $n$ annotators. EDAM agrees with four out of six human annotators for 68% of the studies, it agrees with at least three of them for 80% of the studies, and it agrees with at least one of them for all the studies but one. Indeed, the categories generated by EDAM coincide with the ones suggested by the relative majority of users in 84% of the cases. Therefore, EDAM’s performance is comparable to the performance of the two annotators (User5 and User3) with the highest agreement with the user majority.
We further confirmed these findings by computing the Cohen’s kappa between each couple of annotators and between each of the annotators and EDAM. The inter-annotator agreement was $0.57$, typically indicating a moderate agreement [@landis1977measurement]. The average agreement of EDAM with the annotators was $0.58$, confirming that this method performs in line with the annotators. In addition, we note that EDAM always agrees with the majority for the studies in which no more than one annotator disagrees. It thus seems to perform well in handling simple not-ambiguous papers, that nonetheless human experts may sometimes get wrong.
In conclusion, this study suggests that the EDAM classification step generates annotations that agree with the majority of human experts and are not statistically different from the ones produced by the senior group.
Naturally, EDAM performance may change according to the quality of the ontology and the domain knowledge of the human users that refined it. EDAM is not an alternative to human experts, rather a methodology that allows humans to annotate on a larger scale, by defining a sound domain knowledge and a mapping function. However, this preliminary example application already shows very promising results.
Comparison of Classifiers for Primary Studies {#sec:eval2}
---------------------------------------------
EDAM can adopt many different approaches for automaticaly classifying primary study. The choice of the method ultimately depends on its affectiveness of the avaliable approaches on the domain under analysis and on the preferred precision-recall tradeoff. For instance, in a prevalently automatic mapping study on a large set of papers, precision is of paramount importance, and the missed topics may be compensated by the numerosity of the sample. Conversely, when the results are validated in some way by human experts (e.g., [@osborne2016automatic]) or when the goal is to detect emerging trends that may appear in few publications, a better strategy may be to sacrifice some precision for producing a more comprehensive set of topics.
In this section we compare several approaches that can be used with EDAM to automatically classify studies according to a taxonomy of research topics, and discuss their tradeoff. We focus on unsupervised approaches, since typically the authors of a systematic review do not have the resources to prepare a large gold standard to train a supervised classifier on a potentially new taxonomy.
We evaluated seven alternative approaches on a gold standard of 70 papers [@salatino2018csoc2] within the fields of Semantic Web (23 papers), Natural Language Processing (23), and Data Mining (24). These papers were selected by retrieving the most cited papers from Microsoft Academic Graph containing in the title or the abstract those relevant fields. Each paper was annotated by three domain experts (for a total of 21 different annotators) and was associated with $14.4\pm 7.0 $ topics using majority vote in case of disagreement. The inter-annotator agreement was $0.45\pm 0.18$ according to Fleiss’ Kappa, which indicates a moderate inter-rater agreement [@landis1977measurement]. The data produced in the evaluation and the Python implementation of the approaches are available at <https://cso.kmi.open.ac.uk/cso-classifier/>. A more comprehensive version of this analysis, with additional baselines that are out of scope for this paper (e.g., not producing a set of pre-defined categories), is available in @salatino2018csoc2.
All the tested classifiers analysed the title and abstract of the 70 papers and assigned them with a set of topics drawn from the Computer Science Ontology (CSO), a recently released taxonomy of research areas [@salatino2018computer]. This knowledge base covers well the three mentioned fields, including a total of 35 sub-topics for the Semantic Web, 173 for Natural Language Processing, and 396 for Data Mining. LDA100, LDA500, and LDA1000 are based on a Latent Dirichlet Allocation (LDA) model [@blei2003latent] trained over 4.6M papers in Computer Science from Microsoft Academic Search. Specifically, LDA100 used a model trained with 100 topics, LDA500 on 500 topics, and LDA100 on 1,000 topics. Similarly to [@bhatia2016automatic], these classifiers generate a set of CSO topics from the LDA topics by first producing a set of topics with a probability of at least $j$ and all their terms with a probability of at least $k$. Then they map these terms to CSO by returning all CSO topics having Levenshtein similarity higher than 0.8 with them. We performed a grid search for finding the best values of $j$ and $k$ on the gold standard and report here the best results of each classifier in term of F-measure.
TF-IDF produces a ranked list of terms using TF-IDF [@ramos2003using] (the IDF was computed on the same set adopted for LDA) and all the CSO topics having Levenshtein similarity higher than 0.8 with the first 30 terms.
Direct Mapping (DM) is the approach used for the implementation of EDAM described in step 6 of Section \[sec:methodologyimplementation\]. This same method was also used by the first version of the Smart Topic Miner (STM) [@osborne2016automatic], the system adopted by Springer Nature to classifying proceedings in the field of Computer Science. It returns all topics that explicitly appear in the papers or that are entailed by the ones appearing in the paper according to the ontology.
The CSO Classifier v.1 (CSO-C1) is an unsupervised approach presented in @salatino2018csoc that extracts a combination of n-grams (unigrams, bigrams, and trigrams) from the text and returns all the topics that have a Levenstein similarity higher than $t$ ($t=0.94$ as in the implementation reported in @salatino2018csoc). Finally, the CSO Classifier v.2 (CSO-C2) [@salatino2018csoc2] is a recent evolution of the previous classifier, which uses part-of-speech tagging to identify promising terms and then exploits word embeddings to infer semantically related topics that may not explicitly appear in the paper. This solution has also been adopted by the current version of the Smart Topic Miner [@salatino2019improving].
Table \[tbl:eval2\] reports on the resulting values of precision, recall and F-measure. The approaches based on LDA performed quite poorly. An analysis of the results revealed that the topics returned by the models are both noisy and coarse-grained, often clustering together distinct topics from CSO. Indeed, while LDA works quite well at identifying the main topics of a large collection of documents, it does not traditionally perform equally well when characterizing specific research topics, which may be associated with a relatively low number of publications (50-200), as discussed in [@osborne2012mining]. The approaches based on TF-IDF worked slightly better, yielding an F-measure of 30.1%. DM, used in our exemplary EDAM implementation, yielded the best precision of all the approaches (80.8%). Indeed, this method focuses on topics that are explicitly mentioned in the text, which tends to be very relevant. However, this solution naturally obtains a relatively low recall of 58.2%. CSO-C1, which expands the set of terms by considering string similarity, obtained a better recall (63.8%) but a lower precision (78.3%). Finally, CSO-C2 yielded the best performance in term of both recall (75.3%) and F-measure (74.1%).
DM performed significantly better ($p<10^{-7}$ with the McNemar’s test) than the first four approaches. In turn, CSO-C2 performed significantly better($p<10^{-7}$) than DM and CSO-C1.
Limitations {#sec:limitations}
-----------
In this section we discuss EDAM limitations based on the categorization given in @wohlin2012experimentation.
For [*internal validity*]{} we have identified two main threats that regard the generation of a reliable ontology, which is key to select relevant studies that directly fulfill the selection criteria (and hence correspond to the primary studies for the study at hand). In particular:
[**Ontology learning (step 3): hierarchy is important.**]{} The domain ontology, automatically inferred by the ontology learning technique, is structured hierarchically. Therefore, an area marked as [*subarea*]{} (e.g., architecture description languages) is subsumed by the previous area at the upper level of the taxonomy (e.g., Software Architecture). [*Deeper hierarchies bring finer-grained topics, and therefore a higher precision in the classification process*]{}.
During the application of ontology learning techniques to various research areas (not reported in this paper for the sake of brevity) we found that current ontology learning methods usually identify only mature (in terms of number of publications) research areas. Emerging topics may be excluded, thus reducing the granularity of recent fields’ ontologies.
To alleviate this problem, human experts may be asked to manually identify the most recent areas and to possibly adopt ontology forecasting techniques [@cano2016ontology]. Therefore, the role of experts in improving the quality and deepness of the hierarchy is indeed critical. For the sake of this study, aimed at showing the advantages of automation, the relatively small number of experts was acceptable. However, a larger and more diversified pool of experts should be involved when the research area under investigation is broader.
[**Ontology refinement (step 4): experience matters.**]{} As illustrated in Figure \[fig:edam\], EDAM requires human expertise to refine the automatically generated ontology (step 4). This task is not always straightforward, since humans can have different views on the foundational conceptual elements characterizing a certain discipline. Those differences may be related to many factors, such as the researcher’s exposure to the research area under investigation, seniority, broad vs. specialized knowledge on specific sub-disciplines. Our preliminary experiments allow us to conclude that senior domain experts, with a mature yet wide view on the research area under investigation, should be selected to minimize this threat.
The main threats for [*external validity*]{} regard the practical exploitation of EDAM. In particular:
[**Scholarly dataset: different research areas require different datasets.**]{} This paper reports on our experience with EDAM’s application to the Software Architecture research area. Since the domain of Software Engineering is well represented in the Scopus dataset, we are not facing generalizability issues. However, moving to a totally different domain would require taking into account (assuming to have access to) different scholarly datasets.
Unfortunately, finding up-to-date datasets of scholarly data covering the field under analysis is not always easy and this could be a threat to our approach. Nonetheless, the movement toward open access is helping in mitigating this issue by making available a variety of datasets containing machine-readable data about scientific publications, e.g., Microsoft Academic Graph[^31] [@sinha2015overview], CORE[^32] [@knoth2012core], OpenCitations[^33] [@peroni2015setting], DBLP[^34] [@ley2009dblp], Bio2RDF[^35] [@belleau2008bio2rdf], ScholarlyData.org[^36] [@nuzzolese2016conference], Nanopub.org[^37] [@kuhn2013broadening], Semantic Scholar[^38], and others.
[**Tool support: closed-source tools.**]{} EDAM is making use of some closed-source, proprietary tools for running some of the tasks. This may reduce the application of our approach from other research groups. In order to mitigate this threat, we are planning to release a web service accessible by other colleagues interested to carry out an EDAM study.
[**Research Questions: some may not be automatized.**]{} Many research questions that are typical of mapping studies can be answered by producing relevant analytics [@wohlin2012experimentation], e.g., by counting the number of publications, authors, and venues associated with certain topics in subsequent years. However, some more complex research questions may still require domain experts to manually analyse the relevant studies, e.g., for classifying them in categories that a state of the art classifier would be unable to detect with good accuracy. This is an inherent limitation of the methodology. Nonetheless in many of these cases a preliminary classification by an automatic system may still alleviate the expert work load, e.g., by reducing the set of publications that need to be manually analysed. In addition, the performance of entity extraction and linking tools is steadily improving [@augenstein2012lodifier; @gangemi2017semantic; @rizzo2012nerd], allowing to extract increasingly better representations of research knowledge from scientific articles. Therefore, the number of research questions that can be addressed algorithmically may increase over the following years.
Implications for Systematic Mappings {#sec:implications}
------------------------------------
There are a few implications that can potentially change the way we perform systematic mapping studies in Software Engineering. As mentioned in Section \[sec:method\], these implications regard:
[**Scalability: size does not matter anymore.**]{} EDAM can process a potentially endless set of publications. This allows e.g., mapping studies to be based on [*all*]{} relevant primary studies, previously scoped down due to the fact that humans could not manually process hundreds or thousands of papers.
[**Objectivity: the automatic classification is less biased.**]{} The automatic classification of primary studies does not suffer from the biases of specific human annotators. Nonetheless, the quality of the classification appears on par with the one produced by the human annotators.
[**Reproducibility: study duplication and extension is easy.**]{} Thanks to EDAM, replicating or extending studies, either by the same researcher or by someone else, requires simple tuning, e.g., to extend the publication period, or to select different views illustrating the publication trends of interest.
[**Granularity of the study: zooming-in and -out is simpler.**]{} Thanks to the fact that the selection and classification of primary studies is based on an domain ontology, and of course to automation, EDAM allows to tune the depth of the classification the researcher desires in a given research area. Such tuning just requires setting the level of categories and sub-categories to be included in the classification, and then re-run the methodology.
Reusing EDAM for other Systematic Reviews {#sec:selection}
-----------------------------------------
EDAM can be applied to any domain of interest and for different types of studies. The scenarios that we envisage are discussed below and illustrated in Figure \[fig:scenarios\]. They are: S1) Application of EDAM to a [*new*]{} application domain, S2) Mapping study [*replication*]{}, S3) Mapping study [*refinement*]{}, and S4) [*Systematic literature review*]{}.
![Possible EDAM applications.[]{data-label="fig:scenarios"}](img/scenarios1.png){width="\textwidth" height=".95\textheight"}
[**Application of EDAM to a new application domain (S1).**]{}
: In the basic scenario ([**S1**]{}), the ontology for the new application domain is not yet available. In this case, the complete process illustrated in Figure \[fig:edam\] (and emphasized in Figure \[fig:scenarios\].(S1)) shall be applied. This is the scenario followed in the work presented in this article. It is applicable while investigating a new domain notwithstanding its specific characteristics.
If instead a researcher wants to perform a SR in a domain for which the ontology already exists (scenario S2), such generated domain ontology can be [*reused*]{} in the following two ways, depending on the specific study goal:
[**Mapping Study Replication (same classification, S2a).**]{}
: Suppose we want to replicate a pre-existing EDAM mapping study conducted at time t$_{0}$, in order to update the list of primary studies and related analysis at time t$_{1}$ (e.g., update in year 2020 the study on Software Architecture presented in this paper). In this case, we can directly reuse the previously generated ontology (cf. Figure \[fig:scenarios\].(S2a)). The list of (updated) primary studies can be automatically re-calculated (in step 5) and used (in step 6) for classification and analysis purposes. Notice, however, that this scenario does not address the potential need to [*update*]{} the list of topics. Such a scenario is covered below.
[**Mapping Study Replication (updated classification, S2b).**]{}
: Differently from scenario S2a, we may be interested to replicate a pre-existing study [*and*]{} also include any new topics that may have emerged in the period between time t$_{0}$ and time t$_{1}$ (e.g., updating this study in year 2020 while including new topics appeared after this study). This need requires an update of the domain ontology; therefore, the process in Figure \[fig:scenarios\].(S2b) must be run from step 4 onward.
Another scenario (S3) accommodates the case in which we want to [*refine*]{} the classification and analysis conducted as a mapping study. In the current approach, as shown in the Software Architecture domain scenario, step 5 in Figure \[fig:edam\] returns a set of primary studies that can be further classified into sub-domains (e.g., Architectural Styles, being one element of our ontology, can be further refined to discover all the papers that cover selected styles). We identify two sub-scenarios in order to provide a refinement of sub-domains contents:
[**Mapping Study Refinement with classic selection criteria (S3a).**]{}
: In this scenario, one may classify the articles into sub-domains of interest by applying the inclusion and exclusion criteria [@kitchenham2007guidelines] to the primary studies selected in step 5 of EDAM. For example, knowing that Publish-Subscribe, Client-Server, and Event-driven are sub-domains of Architectural Styles, we introduce selection criteria to position Architectural Styles articles into those categories. This approach allows us to zoom into a specific sub-domain of interest and extract the articles fitting in the specific target sub-domain.
[**Mapping Study Refinement with re-generated domain ontology (S3b).**]{}
: The selected sub-domain of interest may contain hundreds of papers (for example, the Design Decisions sub-domain in our study includes 428 papers). Consequently, applying the selection criteria reported in scenario S3a may be cumbersome, requiring the manual analysis of most of those papers. Alternatively, the researcher may execute an additional round of steps 2-4 to refine the domain ontology for the specific sub-domain (cf. Figure \[fig:scenarios\].(S3b)). This scenario is similar to S1, but applied to a specific sub-domain of interest.
A fourth scenario sees the researcher is interested to run a systematic literature review (SLR) on specific research questions:
[**Systematic Literature Reviews (S4).**]{}
: In step 5 (cf. Figure \[fig:scenarios\].(S4)), given the list of primary studies generated based on the existing ontology, we may run the [*classic*]{} SLR approach [@kitchenham2013systematic] to select those papers that fit with the research questions of interest. Differently from scenario S3a, S4 adds the semantics beyond the definition of the domain, and encapsulated into the research questions and the corresponding selection criteria. E.g., given the list of all studies on Software Architecture styles, one may want to perform an SLR to analyze those approaches that are adopted in industrial settings.
Conclusions and Future Work {#sec:conclusion}
===========================
In this paper we have presented EDAM, an expert-driven automated methodology to assist systematic reviews. Its application to the Software Architecture research area shows preliminary and very promising results.
Motivated by the large amount of time and effort needed by classic methodologies to select and classify the primary studies, EDAM offers benefits that can help SE researchers to dedicate most of their time to the most cognitive-intensive tasks like e.g., interpretation of the trends and extraction of lessons and research gaps.
Additional benefits have been emphasized in Section \[sec:generalmethodology\] (after presenting EDAM) and Section \[sec:implications\] (discussing implications for systematic mappings). Among the benefits we mention the great potential for re-using EDAM and in particular domain ontologies and functions to build a shared framework helping the research community at large. Much can be done in this direction.
Our next step is to complement EDAM with automated forward snowballing to further reduce the effort for identifying relevant primary studies. With the same goal, we are planning to investigate other possible data synthesis techniques through machine learning techniques or the (manual) intervention of human experts. Last, but most important for us, we plan to reconstruct the 25 years of the Software Architecture body of knowledge by fully exploiting EDAM automation and human expertise.
Acknowledgments {#sec:acknowledgments .unnumbered}
===============
The authors would like to thank the colleagues which donated their time and expertise by contributing to this study as domain experts and/or annotators: Paris Avgeriou, Barbora Buhnova, Rafael Capilla, Jan Carlson, Ivica Crnkovic, John Grundy, Rich Hilliard, Heiko Koziolek, Anton Jansen, Ivano Malavolta, Leonardo Mariani, Marina Mongiello, Matthias Naab, Patrizio Pelliccione, Mohammad Sharaf, Damian Andrew Tamburri, Antony Tang, Jan Martijn van der Werf, Smrithi Rekha Venkatasubramanian, Rainer Weinreich, Danny Weyns, Eoin Woods, and Uwe Zdun.
We also thank Davide Falessi for reviewing an earlier version of this manuscript, and Elsevier BV for providing us with access to its large repository of scholarly data.
[^1]: Corresponding author. .
[^2]: <http://rexplore.kmi.open.ac.uk/data/edam/SE-ontology.owl>
[^3]: <http://academic.research.microsoft.com>
[^4]: <https://www.scopus.com/>
[^5]: <https://www.crossref.org/>
[^6]: <https://scigraph.springernature.com/explorer/downloads/>
[^7]: <http://opencitations.net>
[^8]: <http://dblp.uni-trier.de>
[^9]: <https://www.semanticscholar.org/>
[^10]: <https://www.w3.org/OWL/>
[^11]: <http://www.acm.org/publications/class-2012>
[^12]: <http://www.nature.com/subjects>
[^13]: <https://www.elsevier.com/solutions/scopus/content>
[^14]: See an example at <http://tinyurl.com/yal6h3wu>
[^15]: <http://protege.stanford.edu>
[^16]: <http://neon-toolkit.org/>
[^17]: <http://www.topquadrant.com/products/TB_Composer.html>
[^18]: <http://semanticturkey.uniroma2.it/>
[^19]: <http://www.cognitum.eu/Semantics/FluentEditor/>
[^20]: <http://cso.kmi.open.ac.uk/>
[^21]: <http://www.springer.com/gp/computer-science/lncs>
[^22]: <http://technologies.kmi.open.ac.uk/rexplore/ontologies/BiboExtension.owl>
[^23]: <http://purl.org/ontology/bibo/>
[^24]: <https://www.w3.org/2004/02/skos/>
[^25]: <http://tinyurl.com/yal6h3wu>
[^26]: <http://rexplore.kmi.open.ac.uk/data/edam/SE-ontology.owl>
[^27]: <https://www.elastic.co/>
[^28]: <https://mariadb.org/>
[^29]: <https://doi.org/10.5281/zenodo.2653924>
[^30]: The material and the results of the evaluation are available at <https://doi.org/10.5281/zenodo.2653924>
[^31]: <https://academic.microsoft.com/>
[^32]: <https://core.ac.uk>
[^33]: <http://opencitations.net/>
[^34]: <http://dblp.uni-trier.de/>
[^35]: [bio2rdf.org/](bio2rdf.org/)
[^36]: <http://www.scholarlydata.org/>
[^37]: <http://nanopub.org/>
[^38]: <https://www.semanticscholar.org/>
|
---
abstract: |
Here we describe the Astrophysics Source Code Library (ASCL), which takes an active approach to sharing astrophysical source code. ASCL’s editor seeks out both new and old peer-reviewed papers that describe methods or experiments that involve the development or use of source code, and adds entries for the found codes to the library. This approach ensures that source codes are added without requiring authors to actively submit them, resulting in a comprehensive listing that covers a significant number of the astrophysics source codes used in peer-reviewed studies. The ASCL now has over 340 codes in it and continues to grow.
In 2011, the ASCL[^1] has on average added 19 new codes per month. An advisory committee has been established to provide input and guide the development and expansion of the new site, and a marketing plan has been developed and is being executed. All ASCL source codes have been used to generate results published in or submitted to a refereed journal and are freely available either via a download site or from an identified source.
This paper provides the history and description of the ASCL. It lists the requirements for including codes, examines the benefits of the ASCL, and outlines some of its future plans.
author:
- 'Alice Allen$^1$, Peter Teuben$^2$, Robert J. Nemiroff$^3$, and Lior Shamir$^4$'
bibliography:
- 'P003.bib'
title: 'Practices in Code Discoverability: Astrophysics Source Code Library'
---
Note: we don’t seem to have a reference [@O27_adassxxi] for our other paper.
History of the ASCL
===================
In 1999, Robert Nemiroff and John F. Wallin founded the online Astrophysics Source Code Library (ASCL) to house codes of use to the community.[@1999AASÉ194.4408N] This was a volunteer, spare-time endeavor and resulted in a library of 37 codes which had been described in the literature and used to produce research published in or submitted to refereed journals. Twenty-seven of the codes were added to the library in 1999; the last code was added in 2002. The ASCL site also linked to other code libraries, most of which no longer exist or have not been updated in years. In 2003, a search for a new editor for the ASCL was unsuccessful. As other code libraries appeared to be under development at that time, the ASCL was no longer being updated though remained available.
In 2010, Nemiroff decided to move the information on the old ASCL site to Starship Asterisk, the discussion forum for APOD. [^2] He enlisted volunteer help to move entries to the new site and expand the library (Nemiroff, 2010). In 2011, an advisory committee was established to provide guidance for the development and expansion of the new site.
Description of the ASCL and code entry requirements
===================================================
Starship Asterisk runs on the widely-used open source bulletin board software phpBB. Its index page has two main sections, one of which is [*Learning & Resources.*]{} The ASCL is housed in this section in a separate forum called [*The Engineering Deck: Astrophysics Source Code Library*]{}.
The first three threads of the ASCL are informational threads rather than code threads and include how to add a code, which codes have recently been added, and papers and other resources which may be of interest to astrophysicists and astronomers. Each code listed in the ASCL has its own thread; the first post of a code thread contains the following information:
- Name of code
- Abstract or description of code
- Person(s) credited with writing the code
- Link to the source code site
- Link to a paper which discusses or uses the code
- Unique number for the code
Figure \[STIFF\] shows a code entry, annotated for the ADASS XXI poster presentation on the ASCL.
Questions about and discussion of the code can be posted to the thread by clicking the POST REPLY button (not shown in Fig. \[STIFF\]) at the top or bottom of the post. It is not necessary to register for the Starship Asterisk forum to read and post on the ASCL site, however, there is an advantage to doing so: registered users can subscribe to the ASCL forum and/or a particular thread on the forum; subscribing alerts a user via email when the thread or forum has been updated.
Though most entries currently do not house the codes themselves, it is possible to attach an archive file (i.e., .zip or .gz) to the code entry for downloading.
Codes are listed alphabetically by name, 100 threads to a page. A full-text search capability is available, and searches can be refined by iterative searching on the results. The ability to search will become increasingly important as the library grows.
Benefits of the ASCL
====================
Each of the various efforts to aid communication and share knowledge of codes useful for astrophysics has offered valuable information to the community; the difficulty lies in informing and reminding the community that the resource exists. According to Nemiroff, the original ASCL was not successful because [*most people just didn’t know about it and had few ways to find it.*]{}
He came to realize that consistent exposure is needed for a resource to become known and used. With APOD as an entry point to the ASCL, APOD can create the exposure needed to inform and remind astrophysicists of this resource. This is done by posting a link periodically from APOD to the ASCL.
Additional notice about the ASCL is provided by an ongoing email campaign to inform coders of the ASCL and requests to sites which link to the old ASCL site to link to the new one though the old site redirects to the new.
Because the editor seeks codes from peer-reviewed papers and adds entries for them to the library, the ASCL currently houses the largest collection of codes known to the authors. The platform on which the ASCL is housed is familiar and easy to use; it allows for discussion of each code on its own thread, attachment of archive files for those codes which do not have download sites of their own, and consistent updating and expansion. The alphabetical listing of the code entries and full-text and iterative search capabilities allow users to find codes of interest quickly.
Growth, usage, and future plans
===============================
The ASCL has been expanded greatly from last year to this, and as of this writing, has 340 codes in it. The ASCL has seen 808% growth in the number of codes from the 3rd quarter of 2010 to the 3rd quarter of 2011, as Figure \[Graphs\] shows. The expansion will continue; we currently have nearly 200 codes in the queue to be added, and actively seek newly-released codes to include. We invite the astrophysics community to suggest codes that are missing; codes that the community requests be added move to the front of the queue for inclusion.
As the resource has received exposure through APOD, the email campaign, and posts on blogs such as AstroBetter[^3] and Astronomy Computing Today,[^4] visits to the ASCL have increased; this is demonstrated by the bar graph, of two 30-day periods in 2011, in Figure \[Graphs\].
We are exploring ways to make the ASCL citable; we believe papers which use codes should cite them, and are working to provide an easy method for doing so.
Conclusion
==========
Because of the depth and breadth of the ASCL, the ongoing work to expand it, the exposure provided through APOD, the guidance of advisory committee members who know the astronomical coding community well, and the ease of using the phpbb platform, we feel the newly revised ASCL will become a valuable resource for astronomers and astrophysicists.
[**NOTE ADDED IN PROOF:**]{} ASCL codes are now incoorporated into ADS.
[^1]: <http://asterisk.apod.com/viewforum.php?f=35>
[^2]: Starship Asterisk: <http://asterisk.apod.com/viewtopic.php?f=29&t=23735>
[^3]: AstroBetter: <http://www.astrobetter.com/>
[^4]: Astronomy Computing Today: <http://astrocompute.wordpress.com/>
|
[**Noncommutative Extension of Minkowski Spacetime and Its Primary Application**]{}
Yan-Gang Miao[^1]
[**[Abstract]{}**]{}
We propose a noncommutative extension of the Minkowski spacetime by introducing a well-defined proper time from the $\kappa$-deformed Minkowski spacetime related to the standard basis. The extended Minkowski spacetime is commutative, [*i.e.*]{} it is based on the standard Heisenberg commutation relations, but some information of noncommutativity is encoded through the proper time to it. Within this framework, by simply considering the Lorentz invariance we can construct field theory models that comprise noncommutative effects naturally. In particular, we find a kind of temporal fuzziness related to noncommutativity in the noncommutative extension of the Minkowski spacetime. As a primary application, we investigate three types of formulations of chiral bosons, deduce the lagrangian theories of noncommutative chiral bosons and quantize them consistently in accordance with Dirac’s method, and further analyze the self-duality of the lagrangian theories in terms of the parent action approach.
PACS Number(s): 02.40.Gh; 11.10.Nx; 11.10.Kk
Keywords: extended Minkowski spacetime, $\kappa$-Minkowski spacetime, chiral boson
Introduction
============
In history, Snyder [@s1] published the first work on a noncommutative spacetime although the idea might be traced back earlier to W. Heisenberg, R.E. Peierls, W. Pauli, and J.R. Oppenheimer. The quantized spacetime was introduced in order to remove the divergence trouble caused by point interactions between matter and fields. In modern quantum field theory, instead of a discrete spacetime, the well-developed renormalization is utilized to overcome this difficulty in the spacetime with the scale larger than Planck’s. However, the idea of noncommutative spacetimes has revived due to the intimate relationship between the noncommutative field theory (NCFT) and string theory [@s2] and between the NCFT and quantum Hall effect [@s3], respectively. In the former relationship, some low-energy effective theory of open strings with a nontrivial background can be described by the NCFT and thus some relative features of the string theory may be clarified through the NCFT within a framework of the quantum field theory, and in the latter the quantum Hall effect can be deduced from the abelian noncommutative Chern-Simons theory at level $n$ shown to be exactly equivalent to the Laughlin theory at filling fraction $1/n$. Recently it is commonly acceptable that the noncommutativity would occur in the spacetime with the Planck scale and the NCFT [@s4] would play an important role in describing phenomena at planckian regimes.
The mathematical background for the NCFT is the noncommutative geometry [@s5]. In general, the spacetime noncommutativity can be distinguished in accordance with the Hopf-algebraic classification by the three types, [*i.e.*]{} the canonical, Lie-algebraic and quadratic noncommutativity, respectively. Among them, the ${\kappa}$-deformed Minkowski spacetime [@s6; @s7] as a specific case of the Lie-algebraic type has recently been paid much attention because it is a natural candidate for the spacetime based on which the Doubly Special Relativity [@s8] has been established. The ${\kappa}$-Minkowski spacetime is defined by the following commutation relations[^2] of the Lie-algebraic type, $$[{\hat x}^0,{\hat x}^j]=\frac{i}{\kappa}{\hat x}^j, \qquad [{\hat x}^i,{\hat x}^j]=0, \qquad i,j=1,2,3.$$ In addition, the vanishing momentum commutation relations taken from the ${\kappa}$-deformed Poincar$\acute{\rm e}$ algebra should be supplemented, $$[{\hat p}_{\mu},{\hat p}_{\nu}]=0, \qquad {\mu},{\nu}=0,1,2,3.$$ Therefore eqs. (1) and (2), together with the cross relations between ${\hat x}^{\mu}$ and ${\hat p}_{\nu}$ that should coincide with the Jacobi identity, constitute a complete algebra of the noncommutative phase space.
Although we focus in this paper on an alternative scheme (see the next section for details) on dealing with noncommutativity of spacetimes,[^3] here we briefly recapitulate the traditional scheme for the sake of presentation of a big difference between the two schemes. In the latter the noncommutativity can be described by the way of Weyl operators or for the sake of practical applications by the way of normal functions with a suitable definition of star-products. The relationship between the two ways is, as was stated in ref. [@s4], that the noncommutativity of spacetimes may be encoded through ordinary products in the noncommutative $C^{\ast}$-algebra of Weyl operators, or equivalently through the deformation of the product of the commutative $C^{\ast}$-algebra of functions to a noncommutative star-product. For instance, in the canonical noncommutative spacetime the star-product is just the Moyal-product [@s10], while in the ${\kappa}$-deformed Minkowski spacetime the star-product requires a more complicated formula [@s11]. Some recent progress on star-products shows [@s12] that a [*local*]{} field theory in the ${\kappa}$-deformed Minkowski spacetime can be described by a [*nonlocal*]{} relativistic field theory in the Minkowski spacetime.
The arrangement of this paper is as follows. In the next section, a new approach quite different from the traditional one mentioned above is proposed for disposing the noncommutativity of the ${\kappa}$-Minkowski spacetime, that is, a noncommutative extension of the Minkowski spacetime is introduced. As a byproduct, the approach makes it possible that a [*local*]{} field theory in the ${\kappa}$-deformed Minkowski spacetime can be reduced under some postulation of operator linearization to a [*still local*]{} relativistic field theory in the noncommutative extension of the Minkowski spacetime. In section 3, the idea of the extended Minkowski spacetime is applied as an example to chiral bosons, and the lagrangian theories of noncommutative chiral bosons are established in such a framework of the Minkowski spacetime and then quantized consistently in accordance with Dirac’s method. Furthermore, in section 4 the self-duality of the lagrangian theories is analyzed in terms of the parent action method. Finally section 5 is devoted to the conclusion and perspective, in which a kind of temporal fuzziness related to noncommutativity is discussed in detail.
Noncommutative extension of Minkowski spacetime
===============================================
In general the so-called noncommutativity is closely related to a noncommutative spacetime like the ${\kappa}$-deformed Minkowski spacetime, and it has nothing to do with a commutative spacetime, such as the Minkowski spacetime. In this paper we encode enough information of noncommutativity of the ${\kappa}$-Minkowski spacetime to a commutative spacetime, and propose a noncommutative extension of the Minkowski spacetime. This extended Minkowski spacetime is as commutative as the Minkowski spacetime, however, it contains noncommutativity in fact. Our motive originates from a new point of view of disposing noncommutativity, that is, whether the ${\kappa}$-deformed Minkowski spacetime can be dealt with in some sense in the same way as the Minkowski spacetime. This idea may be realized by connecting a commutative spacetime with the ${\kappa}$-Minkowski spacetime and by introducing a well-defined proper time. As a result, the noncommutative field theories defined in the ${\kappa}$-deformed Minkowski spacetime may be investigated somehow by the way of the ordinary (commutative) field theories in the noncommutative extension of the Minkowski spacetime and thus the noncommutativity may be depicted within the framework of this commutative spacetime. That is, we may provide from the point of view of noncommutativity a simplified treatment to the noncommutativity of the ${\kappa}$-Minkowski spacetime, which indeed unveils certain interesting features, such as fuzziness in the temporal dimension.
We start with an algebra of coordinate and momentum operators of a commutative spacetime, which is nothing but the standard Heisenberg commutation relations,[^4] $$[{\hat{\cal X}}^{\mu},{\hat{\cal X}}^{\nu}]=0, \qquad
[{\hat{\cal X}}^{\mu},{\hat{\cal P}}_{\nu}]=i{\delta}^{\mu}_{\nu}, \qquad
[{\hat{\cal P}}_{\mu},{\hat{\cal P}}_{\nu}]=0.$$ The connection between the commutative spacetime and the ${\kappa}$-deformed Minkowski spacetime should be found because it is the basis for us to give the noncommutative extension of the Minkowski spacetime. In accordance with ref. [@s13] the following relations may be chosen, $$\begin{aligned}
{\hat x}^0 & = & {\hat{\cal X}}^0-\frac{1}{\kappa}[{\hat{\cal X}}^j,{\hat{\cal P}}_j]_{+}, \\
{\hat x}^i & = & {\hat{\cal X}}^i+A{\eta}^{ij}{\hat{\cal P}}_j\exp\left(\frac{2}{\kappa}{\hat{\cal P}}_0\right),\end{aligned}$$ where $[{\hat{\cal O}}_1,{\hat{\cal O}}_2]_{+} \equiv \frac{1}{2}\left({\hat{\cal O}}_1{\hat{\cal O}}_2
+{\hat{\cal O}}_2{\hat{\cal O}}_1\right)$, ${\eta}^{{\mu}{\nu}}\equiv{\rm diag}(1,-1,-1,-1)$, and $A$ is an arbitrary constant. This choice is, though not unique, rational because we are able to guarantee eq. (1) in terms of eqs. (3), (4) and (5).
According to the Casimir operator of the $\kappa$-deformed Poincar[$\acute{\rm e}$]{} algebra on the standard basis [@s6], $$\hat{{\mathcal{C}}_{1}} = \left(2\kappa\sinh\frac{\hat{p}_0}{2\kappa}\right)^2-{\hat{p}_i}^2,$$ we supplement the relations of momentum operators between the commutative spacetime and the ${\kappa}$-Minkowski spacetime as follows: $$\begin{aligned}
\hat{p}_0 & = & 2\kappa\,{\sinh}^{-1}\frac{{\hat\mathcal{P}}_0}{{2\kappa}}, \\
{\hat{p}_i} & = & {\hat\mathcal{P}}_i,\end{aligned}$$ which obviously conform to eq. (2). Now eqs. (3), (4), (5), (7) and (8) can solely determine the cross commutators between ${\hat x}^{\mu}$ and ${\hat p}_{\nu}$, $$[{\hat x}^i,{\hat p}_j]=i{\delta}^{i}_{j}, \qquad
[{\hat x}^0,{\hat p}_0]=i\left(\cosh\frac{\hat{p}_0}{{2\kappa}}\right)^{-1},\qquad
[{\hat x}^0,{\hat p}_i]=-\frac{i}{\kappa}{\hat p}_i,\qquad
[{\hat x}^i,{\hat p}_0]=0.$$ Fortunately, the complete algebra of the noncommutative phase space composed of eqs. (1), (2) and (9) indeed satisfies the Jacobi identity, which is, from the point of view of consistency, crucial to the relationship, [*i.e.*]{} eqs. (4), (5), (7) and (8) established between the commutative spacetime and the ${\kappa}$-Minkowski spacetime. Moreover, such a relationship makes the Casimir operator have the usual formula as expected, $$\hat{{\mathcal{C}}_{1}} = {{\hat\mathcal{P}}_0}^2-{{\hat\mathcal{P}}_i}^2,$$ which is consistent with the standard Heisenberg commutation relations (eq. (3)).
When $\hat{p}_{\mu}$ take the usual forms, $$\begin{aligned}
\hat{p}_0 &=& -i\frac{\partial}{{\partial}t},\\
\hat{p}_i &=& -i\frac{\partial}{{\partial}x^i},\end{aligned}$$ which coincide with eq. (2), the operator ${\hat\mathcal{P}}_0$ then reads $${\hat\mathcal{P}}_0=-i{2\kappa}\left(\sin\frac{1}{2\kappa}\frac{{\partial}}{{\partial}t}\right).$$ Next we demonstrate the meaning of $t$ and $x^i$ defined by eqs. (11) and (12). For simplicity and without contradiction to eqs. (11) and (12), we set $A=0$ in eq. (5). Therefore, the canonical coordinates $x^i$ that are the eigenvalues of operators ${\hat{\cal X}}^{i}$ may be regarded as the eigenvalues of operators ${\hat x}^i$, while the eigenvalue of operator ${\hat x}^0$ does not match the ordinary time variable[^5] because operator ${\hat x}^0$, different from operator ${\hat{\cal X}}^0$, does not satisfy the standard Heisenberg commutation relations. As a result, for the ${\kappa}$-deformed Minkowski spacetime we have a certain degree of freedom to introduce a parameter which is of the temporal dimension. This parameter that is required to tend to the ordinary time variable in the limit $\kappa \rightarrow +\infty$ is used to describe the evolution of systems. Here $t$, introduced through eq. (11), is dealt with for the sake of simplicity as such a parameter, rather than the eigenvalue of operator ${\hat x}^0$, to describe the dynamical evolution of fields. It obviously tends to the ordinary time variable in the limit $\kappa \rightarrow +\infty$ as well as the eigenvalue of operator ${\hat x}^0$, which guarantees the consistency of the choice of the parameter.
Now we introduce a proper time[^6] $\tau$ by defining the operator $${\hat\mathcal{P}}_0 \equiv -i\frac{{\partial}}{{\partial}\tau}.$$ $\tau$ may be treated as the eigenvalue of operator ${\hat{\cal X}}^0$, which, together with eq. (14), is in agreement with the standard Heisenberg commutation relations. At the present stage, no connection between $\tau$ and $t$ can be determined. Here a linear realization or representation of operator ${\hat\mathcal{P}}_0$ is postulated, which gives rise to a well-defined proper time[^7] that satisfies the following differential equation in accordance with eqs. (13) and (14), $$2\kappa\left(\sin\frac{1}{2\kappa}\frac{d}{dt}\right)\tau=1.$$ Fortunately, this equation has an exact solution, $\tau=t+\sum_{n=-\infty}^{+\infty}c_{n}\exp(2{\kappa}n{\pi}t)$, where $n$ is an integer and coefficients $c_{n}$ are arbitrary real constants. The consistency requires that $\tau$ should be convergent and tend to parameter $t$ in the limit $\kappa \rightarrow +\infty$, which guarantees that both $\tau$ and $t$ can regress to the ordinary time variable. This requirement nevertheless adds[^8] the constraints, $c_{n}=0$ for $n \geq 1$. Therefore, the final form of the solution reads[^9] $$\tau=t+\sum_{n=0}^{+\infty}c_{-n}\exp(-2{\kappa}n{\pi}t).$$
The noncommutative extension of the Minkowski spacetime spanned by $(\tau, x^i)$ coordinates is thus given, in which the Casimir operator and the line element have the same forms as that in the Minkowski spacetime. However, some information of noncommutativity has been encoded through the proper time to the framework, which can be seen clearly when this kind of extended spacetimes is transformed into $(t, x^i)$ coordinates. This is one thing with two sides. Eq. (16) plays a crucial role in connecting the two coordinate systems to each other. The connection means that a noncommutative spacetime, [*i.e.*]{} the $\kappa$-Minkowski spacetime may be represented under the postulation of the operator linearization (eq. (15)) by a commutative spacetime, [*i.e.*]{} the extended Minkowski spacetime, but the price paid for such a simplified treatment is the appearance of infinitely many unfixed coefficients in the commutative spacetime. This would be understandable because it is just these unfixed coefficients that reflect the information of noncommutativity in the commutative spacetime. See the last section of this paper for a detailed discussion. By making use of eq. (16) we then reduce the Casimir operator to be $$\begin{aligned}
\hat{{\mathcal C}^{\prime}_1} &=&-\frac{{\partial}^2}{{\partial}{\tau}^2}+\frac{{\partial}}{{\partial}x^i}
\frac{{\partial}}{{\partial}x^i} \nonumber \\
&=&-\frac{1}{{\dot \tau}}\frac{{\partial}}{{\partial}t}\left(\frac{1}{{\dot \tau}}\frac{{\partial}}{{\partial}t}\right)
+\frac{{\partial}}{{\partial}x^i}\frac{{\partial}}{{\partial}x^i},\end{aligned}$$ and give the line element $$\begin{aligned}
ds^2&=&d{\tau}^2-(dx^i)^2 \nonumber \\
&=&{\dot \tau}^2dt^2-(dx^i)^2,\end{aligned}$$ where ${\dot \tau}$ means ${d\tau}/{dt}$, $${\dot \tau}=1-2{\kappa}{\pi}\sum_{n=0}^{+\infty}nc_{-n}\exp(-2{\kappa}n{\pi}t).$$
The extended Minkowski spacetime is, as expected, commutative, which is a merit for us to construct field theory models in this framework. That is, we simply consider the Lorentz invariance in the extended Minkowski spacetime, and the constructed models contain noncommutative effects naturally. In mathematics, there exists a specific map (see eqs. (4), (5), (7) and (8)) from the ${\kappa}$-Minkowski spacetime to the noncommutative extension of the Minkowski spacetime. The Lorentz invariance in the extended Minkowski spacetime reflects in fact some invariance in the $\kappa$-Minkowski spacetime that seems an unknown symmetry up to now.[^10] In this way we might have circumvented a relatively complicated procedure for searching for models that should possess such an unknown symmetry.
The line element gives the fact that the noncommutative extension of the Minkowski spacetime connects with a special flat spacetime corresponding to a twisted $t$-parameter.[^11] In consequence we may say that the ${\kappa}$-Minkowski spacetime, the source of the extended Minkowski spacetime, is somehow reduced to the flat spacetime whose metric is given by $$g_{00}={\dot \tau}^2=\left[1-2{\kappa}{\pi}\sum_{n=0}^{+\infty}nc_{-n}\exp(-2{\kappa}n{\pi}t)\right]^2,
\qquad g_{11}=g_{22}=g_{33}=-1.$$
Before the end of this section we emphasize that the Casimir operator defined by eqs. (6), (11) and (12) with infinite order in $t$-parameter derivative has now been reduced[^12] to the formula defined by eq. (17) with second order in $t$-parameter derivative under the postulation of the operator linearization (eq. (15)). This realization or representation of the Casimir operator (eq. (17)) fulfills in a way the aim that some information of noncommutativity can be encoded to a commutative spacetime. The feature of the extended Minkowski spacetime spanned by $(t, x^i)$ is that the evolution parameter is twisted while the spaces are still flat (see, [*e.g.*]{} eq. (18)), which coincides with [@s6] the characters of the ${\kappa}$-Minkowski spacetime, [*i.e.*]{} with a “quantum time” and a three-dimensional euclidean space. This implies that the extended Minkowski spacetime [*does*]{} contain enough information of noncommutativity that is able to reflect the characteristic of the ${\kappa}$-Minkowski spacetime although it does not recover the whole information of noncommutativity due to the postulation of the operator linearization.
Noncommutative chiral bosons
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We deviate from the discussion of noncommutativity temporarily and give a brief introduction of chiral bosons in the Minkowski spacetime. The main reason that chiral bosons[^13] have received much attention is that they appear in various theoretical models that relate to superstring theories, and reflect especially the existence of a variety of important dualities that connect these theories among one another. One has to envisage two basic problems in a lagrangian description of chiral bosons: the first is the consistent quantization and the second is the harmonic combination of the manifest duality and Lorentz covariance, since the equation of motion of a chiral boson, [*i.e.*]{} the self-duality condition is first order with respect to the derivatives of space and time. In order to solve these problems, various types of formulations of chiral bosons, each of which possesses its own advantages, have been proposed [@s15; @s16; @s17]. It is remarkable that these models of chiral bosons have close relationships among one another, especially various dualities that have been demonstrated in detail from the points of view of both the configuration [@s18; @s19] and momentum [@s20] spaces.
In this section we mainly propose noncommutative chiral bosons[^14] in the noncommutative extension of the Minkowski spacetime. The method is as follows: a lagrangian of noncommutative chiral bosons is given simply by the requirement of the Lorentz invariance in the extended Minkowski spacetime spanned by the coordinates $(\tau, x)$, and through the coordinate transformation eq. (16), it is then converted into its $(t, x)$-coordinate formulation with explicit noncommutativity. As a result, we establish the lagrangian theory of noncommutative chiral bosons in the extended framework of the Minkowski spacetime. Alternatively, we can also construct a lagrangian theory directly in the $(t, x)$-coordinate framework in terms of the metric eq. (20). It should be noted that for a certain formulation of chiral bosons both procedures give rise to the noncommutative generalizations that have the same physical spectrum.
Let us begin from the light-cone coordinates and their derivatives defined in the $(1+1)$-dimensional noncommutative extension of the Minkowski spacetime, respectively, as follows: $$\begin{aligned}
X^{\pm} & \equiv & \frac{1}{\sqrt 2}\left(\pm{\tau}+x\right),\\
D_{\pm} & \equiv & \frac{1}{\sqrt 2}\left(\pm\frac{{\partial}}{{\partial}\tau}+\frac{{\partial}}{{\partial}x}\right).\end{aligned}$$ It is obvious that they satisfy $D_{\pm}X^{\pm}=1$ and $D_{\pm}X^{\mp}=0$. In the spacetime spanned by the coordinates $(\tau, x)$ the equation of motion for a noncommutative chiral boson takes the usual form $$D_{\mp}{\phi}=0,$$ and its solution thus reads $${\phi}={\phi}(X^{\pm}),$$ where the upper sign corresponds to the left-moving while the lower the right-moving. Through the coordinate transformation we convert the equation of motion eq. (23) into its corresponding formulation in the coordinate system spanned by $(t, x)$, $$\mp\frac{1}{{\dot \tau}}\frac{{\partial}\phi}{{\partial}t}+\frac{{\partial}\phi}{{\partial}x}=0.$$ The solution takes the same form as eq. (24) and can easily be expressed by the $(t,x)$ coordinates through the well-defined $X^{\pm}$, $${\phi}={\phi}\left(\frac{1}{\sqrt 2}\left[\pm t+x \pm \sum_{n=0}^{+\infty}c_{-n}
\exp\left(-2{\kappa}n{\pi}t\right)\right]\right).$$ Consequently, the equation of motion and its solution comprise in a natural way the noncommutative effects related to the finite noncommutative parameter ${\kappa}$. Incidentally, they become their ordinary forms correspondent to the Minkowski spacetime in the limit $\kappa \rightarrow +\infty$.
The following task is straightforward, that is, to construct in the extended framework of the Minkowski spacetime such a lagrangian that yields the equation of motion (eq. (25)) for noncommutative chiral bosons by the method mentioned in the second paragraph of this section. Three typical formulations of chiral bosons are investigated as a primary application in this section, [*i.e.*]{} the non-manifestly Lorentz covariant version [@s15] and manifestly Lorentz covariant versions with the linear self-duality constraint [@s16] and with the quadratic one [@s17], respectively.
The non-manifestly Lorentz covariant formulation
------------------------------------------------
It is non-manifestly Lorentz covariant but indeed Lorentz invariant [@s15]. Simply considering the Lorentz invariance in the extended Minkowski spacetime $(\tau, x)$, we give the action $$S_{1}=\int d{\tau}dx\left[\frac{{\partial}\phi}{{\partial}\tau}\frac{{\partial}\phi}{{\partial}x}
-\left(\frac{{\partial}\phi}{{\partial}x}\right)^2\right],$$ which is nothing but the formulation of Floreanini and Jackiw’s left-moving chiral bosons if $\tau$ is replaced by the ordinary time. After making the coordinate transformation, we obtain the action written in terms of the coordinates $(t,x)$, $$S_{1}=\int dtdx\sqrt{-g}\left[\frac{1}{{\dot \tau}}\frac{{\partial}\phi}{{\partial}t}
\frac{{\partial}\phi}{{\partial}x}-\left(\frac{{\partial}\phi}{{\partial}x}\right)^2\right],$$ where $\sqrt{-g}$ is the Jacobian and also the nontrivial measure of the flat spacetime eq. (20) connected with the ${\kappa}$-Minkowski spacetime, $$\sqrt{-g}=\vert{\dot \tau}\vert.$$ Therefore the lagrangian takes the form $${\cal L}_{1}=\pm{\dot \phi}{\phi}^{\prime} -\sqrt{-g}{{\phi}^{\prime}}^2,$$ where a dot and a prime stand for derivatives with respect to time[^15] $t$ and space $x$, respectively. A plus and minus sign appears in front of the first term due to the ratio $\sqrt{-g}/{\dot \tau}$, and the choice depends on either ${\dot \tau} > 0$ or ${\dot \tau} < 0$. However, this does not cause any ambiguity.[^16] By making use of Dirac’s quantization [@s21] we can prove that the lagrangian only describes a left-moving noncommutative chiral boson in the extended framework of the Minkowski spacetime, which is independent of whether ${\dot \tau} > 0$ or ${\dot \tau} < 0$, and that the similar case also happens in the linear and quadratic self-duality constraint formulations of chiral bosons. Incidentally, this feature does not exist in the case of ordinary (commutative) chiral bosons due to the triviality $\sqrt{-g}={\dot \tau}=1$ in the limit $\kappa \rightarrow +\infty$.
In terms of Dirac’s quantization [@s21] we can verify that the lagrangian ${\cal L}_{1}$ indeed describes a noncommutative chiral boson which satisfies the equation of motion eq. (25) with the choice of the upper sign correspondent to the left-handed chirality. To this end, at first define the momentum conjugate to $\phi$, $${\pi}_{\phi} \equiv \frac{{\partial}{\cal L}_{1}}{{\partial}{\dot \phi}}=\pm{\phi}^{\prime},$$ and then give the hamiltoian through the Legendre transformation, $${\cal H}_{1}={\pi}_{\phi}{\dot \phi}-{\cal L}_{1}=\sqrt{-g}{{\phi}^{\prime}}^2.$$ Note that this hamiltoian explicitly contains time[^17] and that it is positive definite as its counterpart in the Minkowski spacetime. The definition of momenta actually yields one primary constraint $${\Omega}(x)\equiv {\pi}_{\phi} \mp {\phi}^{\prime} \approx 0,$$ where “$\approx$” stands for Dirac’s weak equality. Because of no further constraints and of the non-vanishing equal-time Poisson bracket, $$\{{\Omega}(x),{\Omega}(y)\}_{PB}=\mp 2{\partial}_{x}{\delta}(x-y),$$ this constraint itself constitutes a second-class set. With the inverse of the Poisson bracket, $$\{{\Omega}(x),{\Omega}(y)\}^{-1}_{PB}=\mp \frac{1}{2}{\varepsilon}(x-y),$$ where ${\varepsilon}(x)$ is the step function with the property $d{\varepsilon}(x)/dx=\delta(x)$, we calculate the equal-time Dirac brackets: $$\begin{aligned}
\{{\phi}(x),{\phi}(y)\}_{DB}&=&\mp \frac{1}{2}{\varepsilon}(x-y),\nonumber \\
\{{\phi}(x),{\pi}_{\phi}(y)\}_{DB}&=&\frac{1}{2}\delta(x-y),\nonumber \\
\{{\pi}_{\phi}(x),{\pi}_{\phi}(y)\}_{DB}&=&\pm \frac{1}{2}{\partial}_{x}\delta(x-y).\end{aligned}$$ In the sense of Dirac brackets weak constraints become strong conditions. As a consequence, we write the reduced hamiltonian $${\cal H}^{r}_{1}=\sqrt{-g}{{\phi}^{\prime}}^2=\sqrt{-g}{\pi}_{\phi}^2
=\frac{1}{2}\sqrt{-g}\left({{\phi}^{\prime}}^2+{\pi}_{\phi}^2\right),$$ and derive from the canonical hamiltonian equation $${\dot \phi}=\int dy \{{\phi}(x),{\cal H}^{r}_{1}(y)\}_{DB},$$ the equation of motion for the noncommutative chiral boson, $${\dot \phi}=\pm \sqrt{-g}{\phi}^{\prime}={\dot \tau}{\phi}^{\prime},$$ which is nothing but eq. (25) with the upper sign corresponding to the left-handed chirality.
The manifestly Lorentz covariant formulation with linear self-duality constraint
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In this formulation the self-duality constraint is imposed upon a massless real scalar field [@s16]. Although it has some defects [@s23], the linear formulation strictly describes a chiral boson from the point of view of equations of motion at both the classical and quantum levels. Its generalization in the canonical noncommutative spacetime has been studied in detail, and in particular a kind of fuzziness on the left- and right-handed chiralities in the spatial dimension has been noticed [@s22]. Under the requirement of the Lorentz invariance in the extended Minkowski spacetime, we can write the action in a straightforward way, $$S_{2}=\int d{\tau}dx\left(-D_{+}{\phi}D_{-}{\phi}-\sqrt{2}{\lambda}_{+}D_{-}{\phi}\right),$$ where ${\lambda}_{+} \equiv \frac{1}{\sqrt{2}}({\lambda}_{0}+{\lambda}_{1})$, a light-cone component of the vector field ${\lambda}_{\mu}$, ${\mu}=0,1$, introduced as a Lagrange multiplier. In terms of the method adopted in subsection 3.1, we then convert it into its formulation in the framework spanned by the coordinates $(t, x)$, $$S_{2}=\int dtdx\sqrt{-g}\left[\frac{1}{2{\dot \tau}^2}\left(\frac{{\partial}\phi}{{\partial}t}\right)^2-\frac{1}{2}
\left(\frac{{\partial}\phi}{{\partial}x}\right)^2+{\lambda}_{+}
\left(\frac{1}{{\dot \tau}}\frac{{\partial}\phi}{{\partial}t}-\frac{{\partial}\phi}{{\partial}x}\right)\right],$$ from which the lagrangian reads $${\cal L}_{2}=\frac{1}{2\sqrt{-g}}\left[{\dot \phi}^2-\left(\sqrt{-g}{\phi}^{\prime}\right)^2\right]
+{\lambda}_{+}\left(\pm{\dot \phi}-\sqrt{-g}{\phi}^{\prime}\right),$$ where a plus and minus sign also exists in front of ${\dot \phi}$ as explained in subsection 3.1.
As was done in the above subsection, we make the hamiltonian analysis by using Dirac’s method and prove that ${\cal L}_{2}$ describes, as expected, a noncommutative chiral boson with the left-handed chirality. Let us define momenta conjugate to ${\phi}$ and ${\lambda}_{+}$, respectively, $$\begin{aligned}
{\pi}_{\phi} & \equiv & \frac{{\partial}{\cal L}_{2}}{{\partial}{\dot \phi}}
= \frac{1}{\sqrt{-g}}{\dot \phi} \pm {\lambda}_{+},
\nonumber \\
{\pi}_{{\lambda}_{+}} & \equiv & \frac{{\partial}{\cal L}_{2}}{{\partial}{\dot {\lambda}_{+}}} \approx 0.\nonumber\end{aligned}$$ The latter gives in fact one primary constraint $${\Omega}_{1}(x)\equiv {\pi}_{{\lambda}_{+}} \approx 0.$$ Making the Legendre transformation, we get the canonical hamiltonian $$\begin{aligned}
{\cal H}_{2} & =& {\pi}_{\phi}{\dot \phi}+{\pi}_{{\lambda}_{+}}{\dot {\lambda}_{+}}-{\cal L}_{2}\nonumber \\
& =& \sqrt{-g}\left[\frac{1}{2}\left({\pi}_{\phi}^2
+{{\phi}^{\prime}}^2\right)+{\lambda}_{+}\left(\mp{\pi}_{\phi}+{\phi}^{\prime}\right)
+\frac{1}{2}{\lambda}_{+}^2\right].\end{aligned}$$ As a basic consistency requirement in dynamics of constrained systems, ${\Omega}_{1}(x)$ should be preserved in time, which yields one secondary constraint $${\Omega}_{2}(x)\equiv \pm {\pi}_{\phi}-{\phi}^{\prime}-{{\lambda}_{+}} \approx 0.$$ Because the preservation of ${\Omega}_{2}(x)$ does not give further constraints, the constraint set consists of ${\Omega}_{1}(x)$ and ${\Omega}_{2}(x)$. With the matrix elements of equal-time Poisson brackets of the constraints, $$\begin{aligned}
& & C_{11}(x,y)=0,\nonumber \\
& & C_{12}(x,y)=-C_{21}(x,y)=\delta(x-y),\nonumber \\
& & C_{22}(x,y)=\mp 2{\partial}_{x}{\delta}(x-y),\nonumber\end{aligned}$$ we can easily deduce their inverse elements $$\begin{aligned}
& & C^{-1}_{11}(x,y)=\mp 2{\partial}_{x}{\delta}(x-y),\nonumber \\
& & C^{-1}_{12}(x,y)=-C^{-1}_{21}(x,y)=-\delta(x-y),\nonumber \\
& & C^{-1}_{22}(x,y)=0,\nonumber\end{aligned}$$ and then compute the non-vanishing equal-time Dirac brackets $$\begin{aligned}
\{{\phi}(x),{\pi}_{\phi}(y)\}_{DB} &=& \delta(x-y),\nonumber \\
\{{\phi}(x),{\lambda}_{+}(y)\}_{DB} &=& \pm \delta(x-y),\nonumber \\
\{{\pi}_{\phi}(x),{\lambda}_{+}(y)\}_{DB} &=& -{\partial}_{x}{\delta}(x-y),\nonumber \\
\{{\lambda}_{+}(x),{\lambda}_{+}(y)\}_{DB} &=& \mp 2{\partial}_{x}{\delta}(x-y).\nonumber\end{aligned}$$ After making Dirac’s weak constraints be strong conditions, we obtain the reduced hamiltonian in terms of independent phase space variables, $${\cal H}^{r}_{2}=\pm \sqrt{-g}{\pi}_{\phi}{{\phi}^{\prime}}.$$ Note that the linear self-duality model of chiral bosons in the ordinary spacetime [@s16] is intrinsically non-positive definite, the non-positive definition here should not be induced by its generalization in the extended Minkowski spacetime but emerges from the original (commutative) formulation. From the canonical hamiltonian equation, $${\dot \phi}=\int dy \{{\phi}(x),{\cal H}^{r}_{2}(y)\}_{DB},$$ we thus arrive at the expected equation of motion $${\dot \phi}=\pm \sqrt{-g}{\phi}^{\prime}={\dot \tau}{\phi}^{\prime}.$$
The manifestly Lorentz covariant formulation with quadratic self-duality constraint
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In this formulation the square of the self-duality constraint, instead of the self-duality itself, is imposed upon a massless real scalar field [@s17]. In accordance with the Lorentz invariance we firstly write the following action in the extended Minkowski spacetime related to the coordinates $(\tau, x)$, $$S_{3}=\int d{\tau}dx\left[-D_{+}{\phi}D_{-}{\phi}-{\lambda}_{++}\left(D_{-}{\phi}\right)^2\right],$$ where ${\lambda}_{++} \equiv \frac{1}{2}({\lambda}_{00}+{\lambda}_{01}+{\lambda}_{10}+{\lambda}_{11})$, a light-cone component of the tensor field ${\lambda}_{{\mu}{\nu}}$, ${\mu}, {\nu}=0,1$, introduced as a Lagrange multiplier, and then rewrite it in the $(t, x)$-coordinate framework in terms of eq. (16), $$S_{3}=\int dtdx\sqrt{-g}\left[\frac{1}{2{\dot \tau}^2}\left(\frac{{\partial}\phi}{{\partial}t}\right)^2-\frac{1}{2}
\left(\frac{{\partial}\phi}{{\partial}x}\right)^2-\frac{1}{2}{\lambda}_{++}
\left(\frac{1}{{\dot \tau}}\frac{{\partial}\phi}{{\partial}t}-\frac{{\partial}\phi}{{\partial}x}\right)^2\right],$$ which yields at last the lagrangian $${\cal L}_{3}=\frac{1}{2\sqrt{-g}}\left[{\dot \phi}^2-\left(\sqrt{-g}{\phi}^{\prime}\right)^2\right]
-\frac{1}{2\sqrt{-g}}{\lambda}_{++}\left(\pm{\dot \phi}-\sqrt{-g}{\phi}^{\prime}\right)^2,$$ where a plus and minus sign emerges in front of ${\dot \phi}$ once again as occurred in subsections 3.1 and 3.2.
Briefly repeating the procedure gone through in the above two subsections, we can make the conclusion that ${\cal L}_{3}$ also describes a noncommutative chiral boson with the left-handed chirality in the extended Minkowski spacetime. At first, through introducing canonical momenta conjugate to ${\phi}$ and ${\lambda}_{++}$, respectively, $$\begin{aligned}
{\pi}_{\phi} & \equiv & \frac{{\partial}{\cal L}_{3}}{{\partial}{\dot \phi}}=\frac{1}{\sqrt{-g}}
\left(1-{\lambda}_{++}\right){\dot\phi} \pm {\lambda}_{++}{\phi}^{\prime},\nonumber \\
{\pi}_{{\lambda}_{++}} & \equiv & \frac{{\partial}{\cal L}_{3}}{{\partial}{\dot {\lambda}_{++}}} \approx 0,\nonumber\end{aligned}$$ we get a primary constraint $${\Omega}_{1}(x)\equiv {\pi}_{{\lambda}_{++}} \approx 0,$$ and derive the canonical hamiltonian in terms of the Legendre transformation, $${\cal H}_{3}={\pi}_{\phi}{\dot \phi}+{\pi}_{{\lambda}_{++}}{\dot{\lambda}_{++}}-{\cal L}_{3}
=\frac{\sqrt{-g}}{1-{\lambda}_{++}}\left(\frac{1}{2}{\pi}_{\phi}^2 \mp
{\lambda}_{++}{\pi}_{\phi}{\phi}^{\prime}+\frac{1}{2}{{\phi}^{\prime}}^2\right).$$ The consistency of time evolution of ${\Omega}_{1}(x)$ then gives rise to one secondary constraint $${\tilde{\Omega}}_{2}(x) \equiv \left({\pi}_{\phi} \mp {\phi}^{\prime}\right)^2 \approx 0,$$ which is first-class and no longer induces further constraints. At this stage, we may replace ${\tilde{\Omega}}_{2}(x)$ by its linearized version, $${\Omega}_{2}(x) \equiv {\pi}_{\phi} \mp {\phi}^{\prime} \approx 0,$$ which is second-class, as was dealt with [@s24] to the ordinary quadratic self-duality constraint formulation in the Minkowski spacetime under the consideration of the classical equivalence between the two constraints. According to Dirac’s method, a gauge fixing condition $${\chi}(x)\equiv {\lambda}_{++}(x)-F(x)\approx 0$$ with $F(x)$ an arbitrary function in the extended framework of the Minkowski spacetime, should be added, and thus the constraint set, $({\chi}(x),
{\Omega}_{1}(x), {\Omega}_{2}(x))$, becomes second-class. The remainder of the canonical analysis can be followed straightforwardly and the results, such as the non-vanishing equal-time Dirac brackets, the reduced hamiltonian, and the equation of motion, are exactly same as that obtained in subsection 3.1. As a consequence, we verify that the quadratic formulation can be reduced to the noncommutative generalization of Floreanini and Jackiw’s chiral bosons, which reveals the connection between the two formulations in the noncommutative extension of the Minkowski spacetime.
Self-duality of noncommutative chiral bosons
============================================
Based on the important role played by the duality and/or self-duality in the ordinary chiral $p$-forms and the related theories [@s18; @s19; @s20], it is curious to argue whether such a symmetry maintains or not in their various noncommutative generalizations. It was revealed [@s22] that the self-duality is not preserved when the chiral boson action with the linear chirality constraint [@s16] is generalized from the Minkowski spacetime to the canonical noncommutative spacetime. However, it will be verified in the following context that this kind of symmetries still exists in the generalizations performed in the noncommutative extension of the Minkowski spacetime. The situation implies somehow a complex relationship between duality/self-duality and noncommutativity, which may be worth notice.
In this section a systematic approach, known as the parent action method [@s25], is used to investigate the self-duality of the actions eqs. (28), (32) and (35). The approach, from the Legendre transformation and for the foundation of equivalence among theories at the level of actions instead of equations of motion, can be summarized[^18] briefly as follows: (i) to introduce auxiliary fields and then to construct a parent or master action based on a source action, and (ii) to make variation of the parent action with respect to each auxiliary field, to solve one auxiliary field in terms of other fields and then to substitute the solution into the parent action. Through making variations with respect to different auxiliary fields, we can obtain different forms of an action. These forms are, of course, equivalent classically, and the relation between them is usually referred to duality. If the resulting forms are same, their relation is called self-duality.
Model in subsection 3.1
-----------------------
According to the parent action approach [@s25], we introduce two auxiliary vector fields $G^{\mu}$ and $F_{\mu}$, and write down[^19] the parent action correspondent to $S_{1}$, [*i.e.*]{} eq. (28), $$S_{1}^{{\rm p}}=\int dt dx\left[F_0F_1-\sqrt{-g}{F_1}^2+G^{\mu}\left(F_{\mu}-{\partial}_{\mu}\phi\right)\right].$$ Now varying eq. (37) with respect to $G^{\mu}$ gives $F_{\mu}={\partial}_{\mu}\phi$, together with which eq. (37) regresses to the action eq. (28). This shows the classical equivalence between the parent action $S_{1}^{{\rm p}}$ and its source action $S_{1}$. However, varying eq. (37) with respect to $F_{\mu}$ leads to the expression of $F_{\mu}$ in terms of $G^{\mu}$: $$\begin{aligned}
F_0 &=& -2\sqrt{-g}G^0-G^1,\nonumber \\
F_1 &=& -G^0.\end{aligned}$$ Substituting eq. (38) into eq. (37), we obtain a kind of dual versions for the action $S_{1}$, $${\tilde S}_{1}^{{\rm dual}}=\int dt dx\left[-G^0G^1-\sqrt{-g}\left({G^0}\right)^2+\phi{\partial}_{\mu}G^{\mu}\right],$$ where $\phi$ is dealt with at present as a Lagrangian multiplier. Further varying eq. (39) with respect to $\phi$ gives ${\partial}_{\mu}G^{\mu}=0$, whose solution takes the form $$G^{\mu}={\epsilon}^{{\mu}{\nu}}{\partial}_{\nu}\varphi,$$ where ${\epsilon}^{00}={\epsilon}^{11}=0$, ${\epsilon}^{01}=-{\epsilon}^{10}=+1$, and $\varphi$ is a scalar filed that is in general different from $\phi$. Therefore the dual version eq. (39) can be reduced to its simpler formula described only by $\varphi$, $$S_{1}^{{\rm dual}}=\int dt dx\left[{\dot \varphi}{\varphi}^{\prime} -\sqrt{-g}{{\varphi}^{\prime}}^2\right].$$ It has the same form as eq. (28) just with the replacement of $\phi$ by $\varphi$, that is, eq. (28) or eq. (41) is self-dual with respect to the chiral boson field.
In accordance with the illustration utilized for duality in ref. [@s25], the self-duality of $S_{1}$ and $S_{1}^{{\rm dual}}$ may be illustrated by Fig. 1.
(100,50)(0,10) (46,60)[$S_{1}^{{\rm p}}$]{} (46,57)[(-2,-3)[23]{}]{}(54,57)[(2,-3)[23]{}]{} (22,40)[${\delta G^{\mu}}$]{}(68,40)[$\delta F_{\mu}+\delta
\phi$]{} (18,17)[${S_{1}}$]{}(78,17)[${S_{1}^{{\rm dual}}}$]{} (20,10)[(0,1)[5]{}]{}(80,10)[(0,1)[5]{}]{} (20,10)[(1,0)[16]{}]{}(64,10)[(1,0)[16]{}]{} (38,9)[**[self-dual]{}**]{}
Model in subsection 3.2
-----------------------
Following the procedure gone through in the above subsection, we can verify that $S_{2}$ (eq. (32)) is self-dual with respect to the chiral boson field. To this end, let us at first regard $S_{2}$ as a source action and give its corresponding parent action $$S_{2}^{{\rm p}}=\int dt dx\left\{\frac{1}{2\sqrt{-g}}\left[{F_0}^2-\left(\sqrt{-g}F_1\right)^2\right]
+{\lambda}_{+}\left({F_0}-\sqrt{-g}F_1\right)+G^{\mu}\left(F_{\mu}-{\partial}_{\mu}\phi\right)\right\},$$ where $G^{\mu}$ and $F_{\mu}$ stand for auxiliary vector fields as before. Next, through making the variation of eq. (42) with respect to $G^{\mu}$ we simply obtain $F_{\mu}={\partial}_{\mu}\phi$, with which eq. (42) becomes the source action eq. (32). This implies that $S_{2}^{{\rm p}}$ is classically equivalent to $S_{2}$. Thirdly, by making the variation of eq. (42) with respect to $F_{\mu}$ we have $$\begin{aligned}
F_0 &=& -\sqrt{-g}\left(G^0+{\lambda}_{+}\right),\nonumber \\
F_1 &=& \frac{1}{\sqrt{-g}}\left(G^1-\sqrt{-g}{\lambda}_{+}\right).\end{aligned}$$ After substituting eq. (43) into eq. (42) we then deduce a dual version of $S_{2}$, $${\tilde S}_{2}^{{\rm dual}}=\int dt dx\left\{-\frac{1}{2{\sqrt{-g}}}\left[\left({\sqrt{-g}}G^0\right)^2-
\left(G^1\right)^2\right]-{\lambda}_{+}\left(\sqrt{-g}{G^0}+G^1\right)+\phi{\partial}_{\mu}G^{\mu}\right\}.$$ At last, varying eq. (44) with respect to $\phi$ which is now dealt with as a Lagrangian multiplier, we derive eq. (40) again and therefore simplify the above dual action to be $$S_{2}^{{\rm dual}}=\int dt dx\left\{\frac{1}{2\sqrt{-g}}\left[{\dot \varphi}^2
-\left(\sqrt{-g}{\varphi}^{\prime}\right)^2\right]
+{\lambda}_{+}\left({\dot \varphi}-\sqrt{-g}{\varphi}^{\prime}\right)\right\},$$ which is nothing but eq. (32) if $\varphi$ is replaced by $\phi$. The illustration of self-duality of $S_{2}$ and $S_{2}^{{\rm dual}}$ can be seen in Fig. 2.
(100,50)(0,10) (46,60)[$S_{2}^{{\rm p}}$]{} (46,57)[(-2,-3)[23]{}]{}(54,57)[(2,-3)[23]{}]{} (22,40)[${\delta G^{\mu}}$]{}(68,40)[$\delta F_{\mu}+\delta
\phi$]{} (18,17)[${S_{2}}$]{}(78,17)[${S_{2}^{{\rm dual}}}$]{} (20,10)[(0,1)[5]{}]{}(80,10)[(0,1)[5]{}]{} (20,10)[(1,0)[16]{}]{}(64,10)[(1,0)[16]{}]{} (38,9)[**[self-dual]{}**]{}
Model in subsection 3.3
-----------------------
Simply repeating the procedure followed in the above two subsections, we can easily show the self-duality of $S_{3}$ (eq. (35)) with respect to the chiral boson field $\phi$. The parent action, correspondent to $S_{3}$ as a source, takes the form $$S_{3}^{{\rm p}}=\int dt dx\left\{
\frac{1}{2\sqrt{-g}}\left[{F_0}^2-\left(\sqrt{-g}F_1\right)^2\right]
-\frac{1}{2\sqrt{-g}}{\lambda}_{++}\left({F_0}-\sqrt{-g}F_1\right)^2 +G^{\mu}\left(F_{\mu}-{\partial}_{\mu}\phi\right)\right\},$$ where $G^{\mu}$ and $F_{\mu}$ are auxiliary vector fields introduced. Varying the parent action with respect to $G^{\mu}$ gives $F_{\mu}={\partial}_{\mu}\phi$, which provides nothing new but just the classical equivalence between $S_{3}^{{\rm p}}$ and $S_{3}$. However, varying it with respect to $F_{\mu}$ instead leads to the useful relations between the two auxiliary fields, $$\begin{aligned}
F_0 &=& -\sqrt{-g}\left(1+{\lambda}_{++}\right)G^0-{\lambda}_{++}G^1,\nonumber \\
F_1 &=& -{\lambda}_{++}G^0+\frac{1}{\sqrt{-g}}\left(1-{\lambda}_{++}\right)G^1.\end{aligned}$$ Substituting the above equation into the parent action, we derive a dual action described by $G^{\mu}$ and $\phi$, $${\tilde S}_{3}^{{\rm dual}}=\int dt dx\left\{-\frac{1}{2{\sqrt{-g}}}\left[\left({\sqrt{-g}}G^0\right)^2-
\left(G^1\right)^2\right]-\frac{1}{2{\sqrt{-g}}}{\lambda}_{++}\left(\sqrt{-g}{G^0}+G^1\right)^2
+\phi{\partial}_{\mu}G^{\mu}\right\}.$$ Now treating $\phi$ as a Lagrangian multiplier, we arrive at eq. (40) once more and thus deduce the formula of the dual action with the obvious self-duality, $$S_{3}^{{\rm dual}}=\int dt dx\left\{\frac{1}{2\sqrt{-g}}\left[{\dot \varphi}^2
-\left(\sqrt{-g}{\varphi}^{\prime}\right)^2\right]
-\frac{1}{2{\sqrt{-g}}}{\lambda}_{++}\left({\dot \varphi}-\sqrt{-g}{\varphi}^{\prime}\right)^2\right\}.$$ As done in the above two subsections, the self-duality of $S_{3}$ and $S_{3}^{{\rm dual}}$ may be illustrated by Fig. 3.
(100,50)(0,10) (46,60)[$S_{3}^{{\rm p}}$]{} (46,57)[(-2,-3)[23]{}]{}(54,57)[(2,-3)[23]{}]{} (22,40)[${\delta G^{\mu}}$]{}(68,40)[$\delta F_{\mu}+\delta
\phi$]{} (18,17)[${S_{3}}$]{}(78,17)[${S_{3}^{{\rm dual}}}$]{} (20,10)[(0,1)[5]{}]{}(80,10)[(0,1)[5]{}]{} (20,10)[(1,0)[16]{}]{}(64,10)[(1,0)[16]{}]{} (38,9)[**[self-dual]{}**]{}
Conclusion and perspective
==========================
In conclusion we emphasize the key point of this paper, that is, the proposal of the noncommutative extension of the Minkowski spacetime. This newly proposed spacetime is founded by introducing a proper time from the ${\kappa}$-deformed Minkowski spacetime related to the standard basis. It is a commutative spacetime, but contains some information of noncommutativity encoded from the ${\kappa}$-deformed Minkowski spacetime. Our performance thus gives the possibility to deal with noncommutativity in a simplified way within a commutative framework. As a byproduct, a [*local*]{} field theory in the ${\kappa}$-deformed Minkowski spacetime can be reduced to a [*still local*]{} relativistic field theory in the noncommutative extension of the Minkowski spacetime. Due to the postulation of the operator linearization (eq. (15)) the new spacetime may be regarded as a minimal extension of the Minkowski spacetime to which the noncommutativity is encoded. Mathematically, we give a specific map (see eqs. (4), (5), (7) and (8)): $({\hat x}^{\mu}, {\hat p}_{\nu}) \rightarrow ({\hat{\cal X}}^{\mu}, {\hat{\cal P}}_{\nu})$, [*i.e.*]{} from the ${\kappa}$-Minkowski spacetime to the extended Minkowski spacetime, and nevertheless the latter spacetime contains intrinsically the primordial information of the former, [*i.e.*]{} the noncommutativity expressed by the finite noncommutative parameter ${\kappa}$. In applications one keeps a model Lorentz invariant in the extended Minkowski spacetime, which actually reflects it some invariance in the ${\kappa}$-Minkowski spacetime related to the standard basis. This is the method that makes the model comprise noncommutative effects naturally. Moreover, we notice the connection between the ${\kappa}$-Minkowski spacetime and the flat spacetime with the nontrivial metric eq. (20), which provides an alternative way to fulfil noncommutative generalizations for field theory models. As a primary application of this extended Minkowski spacetime, three types of formulations of chiral bosons are generalized in terms of this method to the corresponding noncommutative versions and the lagrangian theories of noncommutative chiral bosons are acquired and then quantized consistently by the use of Dirac’s method. In addition, the self-duality of the lagrangian theories of noncommutative chiral bosons is verified in terms of the parent action method. Here we should mention that the self-duality is not preserved [@s22] when the chiral boson with the linear chirality constraint is generalized to the canonical noncommutative spacetime. This implies that the noncommutativity of spacetimes has a complex relationship with the self-duality of lagrangian theories as has been stated in ref. [@s22].
In particular, we note that the proper time $\tau$ (eq. (16)) contains infinitely many real coefficients, $c_{-n}$ for $n \geq 0$, which can not be fixed within the framework of the noncommutative extension of the Minkowski spacetime. An interesting feature caused by the indefinite coefficients is that the initial value of the proper time, $$\left.{\tau}\right|_{t=0}=\sum_{n=0}^{+\infty}c_{-n},$$ is uncertain even if the convergence requirement of the series of constant terms is added, and the uncertainty can further extend to any value of the proper time. This may be interpreted to be a kind of temporal fuzziness that exists in the extended Minkowski spacetime. Such a fuzziness is compatible with the ${\kappa}$-Minkowski spacetime in which a time and a space operators commute to the space operator, that is, it originates from the special noncommutativity described in mathematics by the ${\kappa}$-deformed Poincar[$\acute{\rm e}$]{} algebra. For noncommutative chiral bosons, nevertheless, we point out that the fuzziness is different from that discovered [@s22] in the canonical noncommutativity where the similar phenomenon presents ambiguous left- and right-handed chiralities in the spatial dimension. In spite of the distinction mentioned, the existence of fuzziness that is closely related to noncommutative spacetimes might be inevitable although it is not clear how the noncommutativity brings about fuzziness in the temporal or spatial dimension, or even probably in multiple temporal and spatial dimensions.
Further considerations focus on the following two aspects. The first is whether we can give a map from the ${\kappa}$-deformed Minkowski spacetime related to the bicrossproduct basis [@s7] to a commutative spacetime. This is worth noticing because the ${\kappa}$-Poincar[$\acute{\rm e}$]{} algebra in the bicrossproduct basis contains the undeformed (classical) Lorentz subalgebra. Different from the case of the standard basis, now the difficulty lies in the appearance of entangled terms of ${\hat{p}_0}$ and ${\hat{p}_i}$ in the Casimir operator related to the bicrossproduct basis. Namely, we try to find, by following the way adopted for the standard basis or by setting up some new way, such a proper time and such “proper” spatial coordinates as well that the entanglement could be removed. The second aspect is to enlarge the application of the noncommutative extension of the Minkowski spacetime by considering any models that are interesting in the ordinary (commutative) spacetime, in particular, the models whose noncommutative generalizations connect with phenomena probably tested in experiments, such as in ref. [@s26] where the canonical noncommutativity is involved. In this way we may have opportunities to test the extended Minkowski spacetime and/or to determine the value of the noncommutative parameter by comparing theoretical results with available experimental data. A quick and direct consideration is to investigate chiral $p$-forms [@s18; @s19] and the interacting theory of chiral bosons and gauge fields [@s27] in the noncommutative extension of the Minkowski spacetime, and results will be given separately.
[**Acknowledgments**]{}
The author would like to thank H.J.W. Müller-Kirsten of the University of Kaiserslautern for helpful discussions and H.P. Nilles of the University of Bonn for warm hospitality. This work was supported in part by the DFG (Deutsche Forschungsgemeinschaft), by the National Natural Science Foundation of China under grant No.10675061, and by the Ministry of Education of China under grant No.20060055006.
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[^1]: e-mail address: miaoyg@nankai.edu.cn
[^2]: Here the noncommutative parameter ${\kappa}$ with the mass dimension is considered to be real and positive.
[^3]: The concept of noncommutative fields [@s9] is different from that of the noncommutative spacetime and is not involved in this paper.
[^4]: The speed of light $c$ and the Planck constant $\hbar$ are set to unit throughout this paper.
[^5]: The eigenvalue of operator ${\hat x}^0$ does not match the $t$-parameter either, but takes a more complicated formula that should coincide with the algebra of the noncommutative phase space defined by eqs. (1), (2) and (9).
[^6]: This terminology is adopted here only because of its temporal dimension. It has nothing to do with that of the special relativity.
[^7]: This form is a natural choice that corresponds to a kind of minimal extensions of the Minkowski spacetime from the point of view of noncommutativity. The idea of minimal extensions is basic and usually adopted in physics, in particular at the beginning stage to establish a theory, such as the minimal supersymmetry. When we introduce light-cone components in the coordinate system related to $\tau$, this form indeed induces models of noncommutative chiral bosons with interesting physical properties. For the details, see further discussions.
[^8]: For the usual notation of $t \geq 0$. When $t < 0$, the solution takes the form, $\tau=t+\sum_{n=1}^{+\infty}c^{\prime}_{n}\exp(2{\kappa}n{\pi}t)$, where $c^{\prime}_{n}$ are constants and the convergence requires $c^{\prime}_{n}=0$ for ${n} < 0$.
[^9]: An additional constraint should be imposed upon the non-vanishing coefficients, $c_{-n}$ for $n \geq 0$, if the initial value of $\tau$ is required to be finite, which is related to the so-called temporal fuzziness that will be analyzed in detail in the last section of this paper.
[^10]: See, [*e.g.*]{} ref. [@s14] and the references therein.
[^11]: The curvature of this spacetime equals to zero. The author would like to thank the anonymous referee for pointing it out.
[^12]: Here it only means that $\hat{{\mathcal C}_1}$ is reduced to $\hat{{\mathcal C}^{\prime}_1}$, while the inverse procedure is never implied in this paper.
[^13]: In general, one should mention chiral [*p*]{}-forms that include chiral bosons as the $p=0$ case. A chiral 0-form in the (1+1)-dimensional Minkowski spacetime is usually called a chiral boson which describes a left- or right-moving boson in one spatial dimension.
[^14]: We can also consider chiral [*p*]{}-forms ($p\geq 1$) and their noncommutative generalizations. This is one of the further topics that will probably be reported elsewhere soon.
[^15]: For the sake of convenience in description, [*time*]{}, here different from the ordinary time variable, stands only for [*t-parameter*]{} in the three subsections of section 3.
[^16]: The same result can be obtained if we start with the action for right-moving chiral bosons, which is also available in subsections 3.2 and 3.3.
[^17]: The Dirac quantization is shown to be available in constrained systems whose hamiltoians contain time explicitly. For instance, see ref. [@s22].
[^18]: As to the original proposal and recent development of the parent action method, see, for instance, the references cited by ref. [@s25].
[^19]: For simplicity, we only consider case ${\dot \tau} > 0$ in the discussion of self-duality in this section. As to case ${\dot \tau} < 0$, the same result will be deduced.
|
---
abstract: 'In this paper, we introduce a new class of nonsmooth convex functions called SOS-convex semialgebraic functions extending the recently proposed notion of SOS-convex polynomials. This class of nonsmooth convex functions covers many common nonsmooth functions arising in the applications such as the Euclidean norm, the maximum eigenvalue function and the least squares functions with $\ell_1$-regularization or elastic net regularization used in statistics and compressed sensing. We show that, under commonly used strict feasibility conditions, the optimal value and an optimal solution of SOS-convex semi-algebraic programs can be found by solving a single semi-definite programming problem (SDP). We achieve the results by using tools from semi-algebraic geometry, convex-concave minimax theorem and a recently established Jensen inequality type result for SOS-convex polynomials. As an application, we outline how the derived results can be applied to show that robust SOS-convex optimization problems under restricted spectrahedron data uncertainty enjoy exact SDP relaxations. This extends the existing exact SDP relaxation result for restricted ellipsoidal data uncertainty and answers the open questions left in [@JLV15] on how to recover a robust solution from the semi-definite programming relaxation in this broader setting.'
author:
- 'N. H. Chieu[^1], J.W. Feng [^2], W. Gao [^3], G. Li[^4] and D. Wu [^5]'
date: '[**Dedicated to the memory of Jon Borwein who was of great inspiration to us**]{}\'
title: '**SOS-convex Semi-algebraic Programs and its Applications to Robust Optimization: A Tractable Class of Nonsmooth Convex Optimization [^6]** '
---
[**Keywords:**]{} Nonsmooth optimization, Convex optimization, SOS-convex polynomial, semi-definite program, robust optimization.
Introduction
============
Convex optimization is ubiquitous across science and engineering [@BN01; @Jon]. It has found applications in a wide range of disciplines, such as automatic control systems, signal processing, electronic circuit design, data analysis, statistics (optimal design), and finance (see [@BN01; @Boyd] and the references therein). The key to the success in solving convex optimization problems is that convex functions exhibit a local to global phenomenon: every local minimizer is a global minimizer. Despite the great success of theoretical and algorithmic development and its wide application, we note that a convex optimization problem is, in general, NP-hard from the complexity point of view.
Recently, for convex polynomials, a new notion of sums-of-squares-convexity (SOS-convexity) [@Parrilo; @HN10] has been proposed as a tractable sufficient condition for convexity based on semidefinite programming. The SOS-convex polynomials cover many commonly used convex polynomials such as convex quadratic functions and convex separable polynomials. An appealing feature of an SOS-convex polynomial is that deciding whether a polynomial is SOS-convex or not can be equivalently rewritten as a feasibility problem of a semi-definite programming problem (SDP) which can be validated efficiently. It has also been recently shown that for an SOS-convex optimization problems, its optimal value and optimal solution can be found by solving a single semi-definite programming problem [@Lasserre1] (see also [@jl1; @jl_ORL]). On the other hand, many modern applications of optimization to the area of statistics, machine learning, signal processing and image processing often result in structured nonsmooth convex optimization problems [@Boyd]. These optimization problems often take the following generic form $\min_{x \in \mathbb{R}^{n}} \{g(x)+h(x)\}$, where $g:\mathbb{R}^{n} \rightarrow \mathbb{R}$ is a convex quadratic function and $h:\mathbb{R}^{n} \rightarrow \mathbb{R}$ is a nonsmooth function. For example, in many signal processing applications $g$ represents the quality of the recovered signal while $h$ serves as a regularization which enforces prior knowledge of the form of the signal, such as simplicity/sparsity (in the sense that the solution has fewest nonzero entries). Some typical choices of the regularization function promoting the sparsity of the solution are the so-called $\ell_1$-norm and the weighted sum of $\ell_1$-norm and $\ell_2$-norm (referred as the elastic net regularization [@Zou]), and is therefore nonsmooth. With these applications in mind, this then motivates the following natural and important question:
[*Is it possible to extend the SOS-convex polynomials and SOS-convex optimization problems to the nonsmooth setting which not only covers broad nonsmooth problems arising in common applications but also maintains the appealing feature of tractability (in terms of semidefinite programming)?* ]{}
The purpose of this paper is to provide an affirmative answer for the above question. In particular, in this paper, we make the following contributions:
- In Section 3, we identify a new class of nonsmooth convex functions which we refer as SOS-convex semi-algebraic functions (Definition 3.1). This class of nonsmooth convex functions covers not only convex functions which can be expressed as the maximum of finitely many SOS-convex polynomials (in particular, SOS-convex polynomials) but also many common nonsmooth functions arising in the applications such as the Euclidean norm, the maximum eigenvalue function (by identifying the symmetric matrix spaces $S^n$ as an Euclidean space with dimension $n(n+1)/2$) and the least squares functions with $\ell_1$-regularizer or elastic net regularizer used in compressed sensing.
- In Section 4, we show that, under a commonly used strict feasibility condition, the optimal value and an optimal solution of SOS-convex semi-algebraic optimization problems can be found by solving a single semi-definite programming problem which extends the previous known result of SOS-convex polynomial optimization problems (Theorem \[th:exact\_SDP\] and Theorem \[th:3.2\]). We achieve this by exploiting tools from semi-algebraic geometry, convex-concave minimax theorem and a recently established Jensen inequality type result for SOS-convex polynomials.
- In Section 5, we briefly outline how our results can be applied to show that robust SOS-convex optimization problems under restricted spectrahedron data uncertainty enjoy exact semi-definite programming relaxations. This extends the existing result for restricted ellipsoidal data uncertainty established in [@jl1] and answers the open questions left in [@jl1] on how to recover a robust solution from the semi-definite programming relaxation in this broader setting.
Preliminaries
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First of all, let us recall some notations and basic facts on sums-of-squares polynomial and semi-definite programming problems. Recall that $S_{n}$ denotes the space of symmetric $(n\times n)$ matrices with the trace inner product and $\succeq $ denotes the Löwner partial order of $
S^{n}$, that is, for $M,N\in S^{n},$ $M\succeq N$ if and only if $(M-N)$ is positive semidefinite. Let $S^{n}_+:=\{M\in S_{n}\mid M\succeq 0\}$ be the closed convex cone of positive semidefinite symmetric $(n\times n)$ matrices. Note that for $M,N\in S^{n}_+$, the inner product, $(M,N):=\mathrm{Tr\ }[MN]$, where $\mathrm{Tr\ }[.]$ refers to the trace operation. Note also that $ M\succ 0$ means that $M$ is positive definite. In the sequel, unless otherwise stated, the space $\R^n$ is equipped with the Euclidean norm, that is, $\|x\|:=(\sum\limits_{i=1}^n |x_i|^2)^{1/2}$ for all $x=(x_1,x_2,...,x_n)\in \R^n$. Consider a polynomial $f$ with degree at most $d$ where $d$ is an even number. Let $\mathbb{R}_d[x_1,\ldots,x_n]$ be the space consisting of all real polynomials on $\mathbb{R}^n$ with degree at most $d$ and let $s(d,n)$ be the dimension of $\mathbb{R}_d[x_1,\ldots,x_n]$. Write the canonical basis of $\mathbb{R}_d[x_1,\ldots,x_n]$ by $$x^{(d)}:=(1,x_1,x_2,\ldots,x_n,x_1^2,x_1x_2,\ldots,x_2^2,\ldots,x_n^2,\ldots,x_1^{d},\ldots,x_n^d)^T$$ and let $x^{(d)}_{\alpha}$ be the $\alpha$-th coordinate of $x^{(d)}$, $1 \le \alpha \le s(d,n)$. Then, we can write $f(x)= \sum_{\alpha=1}^{s(d,n)} f_{\alpha}x^{(d)}_{\alpha}$.
We say that a real polynomial $f$ is sums-of-squares (cf. [@Lasserre]) if there exist real polynomials $f_j$, $j=1,\ldots,r$, such that $f=\sum_{j=1}^rf_j^2$. The set consisting of all sum of squares real polynomials in the variable $x$ is denoted by $\Sigma^2[x]$. Moreover, the set consisting of all sum of squares real polynomials with degree at most $d$ is denoted by $\Sigma^2_d[x]$. For a polynomial $f$, we use ${\rm deg}f$ to denote the degree of $f$. Let $l=d/2$. Then, $f$ is a sum-of-squares polynomial if and only if there exists a positive semi-definite symmetric matrix $W \in S_+^{s(l,n)}$ such that $$\label{eq:useful}
f(x)=(x^{(l)})^TW x^{(l)},$$ where $x^{(l)}=(1,x_1,x_2,\ldots,x_n,x_1^2,x_1x_2,\ldots,x_2^2,\ldots,x_n^2,\ldots,x_1^{l},\ldots,x_n^l)^T$. For each $1 \le \alpha \le s(d,n)$, we denote $i(\alpha)=(i_1(\alpha),\ldots,i_n(\alpha)) \in (\mathbb{N} \cup \{0\})^n$ to be the multi-index such that $$x^{(d)}_{\alpha}=x^{i(\alpha)}:=x_1^{i_1(\alpha)}\ldots x_n^{i_n(\alpha)}.$$ Then, by comparing the coefficients in (\[eq:useful\]), we have the following linear matrix inequality characterization of a sum-of-squares polynomial.
\[th:sos\] Let $d$ be an even number. For a polynomial $f$ on $\mathbb{R}^n$ with degree at most $d$, $f$ is a sum-of-squares polynomial if and only if the following linear matrix inequality problem has a solution $$\begin{aligned}
\left\{ \begin{array}{l}
W \in S_+^{s(l,n)} \\
\displaystyle f_{\alpha}=\sum_{1 \le \beta,\gamma \le s(l,n), i(\beta)+i(\gamma)=i(\alpha)} W_{\beta,\gamma}, \ 1 \le \alpha \le s(d,n),\ l=d/2.
\end{array}\right.\end{aligned}$$
We now recall the definition of SOS-convex polynomial. The notion of SOS-convex polynomial was first proposed in [@HN10] and further developed in [@Parrilo]. Here, for convenience of our discussion, we follow the definition used in [@Parrilo].
A real polynomial $f$ on $\mathbb{R}^n$ is called *SOS-convex* if the polynomial $F: (x,y) \mapsto f(x)-f(y)-\nabla f(y)^T(x-y)$ is a sums-of-squares polynomial on $\mathbb{R}^n \times \mathbb{R}^n$.
The significance of the class of SOS-convex polynomials is that checking whether a polynomial is SOS-convex is equivalent to solving a semi-definite programming problem (SDP) which can be done in polynomial time; while checking a polynomial is convex or not is, in general, an NP-hard problem [@HN10; @Parrilo]. Moreover, another important fact is that, for SOS-convex polynomial program, an exact SDP relaxation holds under the usual strict feasibility condition. In contrast, solving a convex polynomial program, is again, in general, an NP hard problem [@Parrilo].
Clearly, a SOS-convex polynomial is convex. However, the converse is not true, that is, there exists a convex polynomial which is not SOS-convex [@Parrilo]. The sum of two SOS-convex polynomials and nonnegative scalar multiplication of an SOS-convex polynomial are still SOS convex polynomials. It is known that any convex quadratic function and any convex separable polynomial is an SOS-convex polynomial [@jl1]. Moreover, an SOS-convex polynomial can be non-quadratic and non-separable. For instance, $f(x)=x_1^8+x_1^2+x_1x_2+x_2^2$ is a SOS-convex polynomial which is non-quadratic and non-separable.
The following existence result for solutions of a convex polynomial optimization problem will also be useful for our later analysis.
\[minattain\] Let $f_0,f_1,\ldots,f_m$ be convex polynomials on $\mathbb{R}^n$ and let $C:=\left\{x \in \mathbb{R}^n : f_i(x) \leq 0, i=1,\ldots,m\right\}$ be nonempty. If $\inf\limits_{x\in C}f_0(x)>-\infty$ then $\operatorname{argmin}\limits_{x\in C}f_0(x) \neq \emptyset$.
SOS-convex semi-algebraic functions
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We begin this section with introducing the notion of SOS-convex semi-algebraic functions. The class of SOS-convex semi-algebraic functions is a subclass of the class of locally Lipschitz nonsmooth convex functions, and includes SOS-convex polynomials.
[**(SOS-convex semi-algebraic functions)** ]{} We say $f:\R^n\rightarrow \R$ is an SOS-convex semi-algebraic function on $\mathbb{R}^n$ if it admits a representation $$\label{rep} \displaystyle f(x)=\sup_{y \in \Omega} \{h_0(x)+\sum_{j=1}^m y_j h_j(x)\},\, m \in \mathbb{N},$$ where
- each $h_j$, $j=0,1,\ldots,m$, is a polynomial and for each $y \in \Omega$, $\displaystyle h_0+\sum_{j=1}^m y_j h_j$ is a SOS-convex polynomial on $\mathbb{R}^n$;
- $\Omega$ is a nonempty compact semi-definite program representable set given by $$\label{eq:1}
\Omega=\{y \in \mathbb{R}^m: \exists \, z \in \mathbb{R}^p \mbox{ s.t. } A_0+\sum_{j=1}^m y_j A_j+ \sum_{l=1}^p z_l B_l \succeq 0\},$$
for some $p \in \mathbb{N},$ $A_j$ and $B_l,$ $j=0,1,...,m,$ $l=1,...,p,$ being $(t\times t)$-symmetric matrices with some $t\in \mathbb N.$
Moreover, the maximum of the degree of the polynomial $h_j$, $j=1,\ldots,m,$ is said to be the degree of the SOS-convex semi-algebraic function $f$ with respect to the representation .
The class of SOS-convex semi-algebraic functions contains many common nonsmooth convex functions. Below, we provide some typical examples.
[**(Examples of SOS-convex semi-algebraic functions)**]{}\[ex3.1\]
- Let $f(x)=\max_{1 \le i \le m}f_i(x)$ where each $f_i$, $i=1,\ldots,m$, is an SOS-convex polynomial. Note that $f(x)=\sup_{y \in \Delta} g(x,y)$ where $\Delta$ is the simplex in $\mathbb{R}^m$ given by $\Delta=\{y: y_i \ge 0, \sum_{i=1}^m y_i=1\}$ and $g(x,y)=\sum_{i=1}^m y_i f_i(x)$. Then, we see that $f$ is an SOS-convex semi-algebraic function.
- Let $f(x)=\|x\|$. Then, $f$ is an SOS-convex semi-algebraic function. To see this, we only need to note that $$\|x\|=\sup_{\|(y_1,\ldots,y_n)\| \le 1} \sum_{i=1}^n x_iy_i,$$ and the unit ball defined by $\|\cdot\|$ is a compact semi-definite program representable set. More generally, $f(x)=\|x\|_p:=\big(\sum_{i=1}^n |x_i|^p\big)^{\frac{1}{p}}$ with $p=\frac{s}{s-1}$ and $s$ being an even positive integer, is an SOS-convex semi-algebraic function. To see this, we only need to note that $$\|x\|_p=\sup_{\|(y_1,\ldots,y_n)\|_s \le 1} \sum_{i=1}^n x_iy_i.$$ and the set $\{y \in \mathbb{R}^n: \|y\|_s \le 1\}=\{y \in \mathbb{R}^n:
\sum_{i=1}^n y_i^s \le 1\}$ is described by an SOS-convex polynomial inequality (as $s$ is even) and so, is a compact semi-definite program representable set [@HN10].
- Identify the $(n \times n)$ symmetric matrices space $S^n$ with the trace inner product ${\rm Tr}(AB)=\sum_{ij} A_{ij}B_{ij}$ as $\mathbb{R}^{n(n+1)/2}$ with the usual inner product. Let $f:S^n \rightarrow \mathbb{R}$ be defined by $f(X)=\lambda_{\max}(X)$ where $\lambda_{\max}$ is the maximum eigenvalue. Then, $f$ is an SOS-convex semi-algebraic function on $S^n$. To see this, we only need to notice that $$\lambda_{\max}(X)=\sup\{{\rm Tr}(XY): Y \in S^n, {\rm Tr}(Y)=1, Y \succeq 0\}$$ and the set $\{Y \in S^n: {\rm Tr}(Y)=1, Y \succeq 0\}$ is a compact semi-definite program representable set.
Next, we see that SOS-convex semi-algebraic functions cover many least squares functions with regularization. To see this, we need the following simple lemma which shows that finite addition preserves SOS-convex semi-algebracity.
\[prop3.1\] Let $f_i$ be SOS-convex semi-algebraic functions on $\mathbb{R}^n$, $i=1,\ldots,q$. Then, $\sum_{i=1}^q f_i$ is an SOS-convex semi-algebraic function on $\mathbb{R}^n$.
To see the conclusion, it suffices to show the case where $q=2$. We first show that $f_1+f_2$ is an SOS-convex semi-algebraic function. As $f_i$, $i=1,2$ are SOS-convex semi-algebraic functions, $f_i(x)=\sup_{y^i \in \Omega_i}\{h_0^i(x)+\sum_{j=1}^{m_i} y_j^i h_j^i(x)\}$, where $m_i \in \mathbb{N}$, $h_l^i$, $l=0,1,\ldots,m$ are SOS-convex polynomials and $\Omega_i$ is a compact semi-definite program representable sets given by $$\Omega_i=\{y^i \in \mathbb{R}^{m_i}: \exists \, z^i \in \mathbb{R}^{p_i} \mbox{ s.t. } A_0^i+\sum_{j=1}^{m_i} y_j^i A_j^i+ \sum_{l=1}^{p_i} z_l^i B_l^i \succeq 0\}.$$ Then, $$f_1(x)+f_2(x)=\sup_{(y^1,y^2) \in \Omega_1\times \Omega_2}\left\{h_0^1(x)+h_0^2(x)+\sum_{j=1}^{m_1} y_j^1 h_j^1(x)+\sum_{j=1}^{m_2} y_j^2 h_j^2(x)\right\}.$$ Note that $\Omega_1 \times \Omega_2$ is also a compact semi-definite program representable set. Thus, $f_1+f_2$ is also an SOS-convex semi-algebraic function.
[**(Further examples: least squares problems with regularization)**]{} Let $A \in \mathbb{R}^{m \times n}$ and $b \in \mathbb{R}^m$. From the preceding proposition, we see that the following functions which arises in sparse optimization are SOS-convex semi-algebraic:
- The least squares function with $\ell_1$-regularization $f(x)=\|Ax-b\|^2+ \mu \|x\|_1$ where $\mu >0$. Note that since $\|x\|_1:=|x_1|+|x_2|+...+|x_n|$ and $|x_i|=\max\{x_i,-x_i\},$ $i=1,2,...,n,$ it follows from Example \[ex3.1\] and Proposition \[prop3.1\] that $\|\cdot\|_1$ is an SOS-convex semi-algebraic function, while the function $x\mapsto\|Ax-b\|^2$ is a convex quadratic function and thus is SOS-convex semi-algebraic.
- The least squares function with elastic net regularization [@Zou] $f(x)=\|Ax-b\|^2+ \mu_1 \|x\|_1+ \mu_2 \|x\|^2$ where $\mu_1,\mu_2>0$.
Exact SDP relaxation for SOS-convex semi-algebraic programs
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In this section, we show that an SOS-convex semi-algebraic program admits an exact SDP relaxation in the sense that the optimal value of the SDP relaxation problem equals the optimal value of the underlying SOS-convex semi-algebraic program. Moreover, a solution for the SOS-convex semi-algebraic program can be recovered from its SDP relaxation, under strict feasibility assumptions.
Consider the following SOS-convex semi-algebraic program: $$\begin{aligned}
(P) & \min & f_0(x) \\
& \mbox{ subject to } & f_i(x) \le 0, \ i=1,\ldots,s,\end{aligned}$$ where each $f_i$, $i=0,1,\ldots,s$, is an SOS-convex semi-algebraic function in the form $$\displaystyle f_i(x)=\sup_{(y_1^i,\ldots,y_m^i) \in \Omega_i} \{h_0^i(x)+\sum_{j=1}^m y_j^i h_j^i(x)\}, \ m \in \mathbb{N},$$ such that
- each $h_j^i$ is a polynomial with degree at most $d$, and for each $y^i=(y^i_1,...,y^i_m)\in \Omega_i$, the function $\displaystyle h_0^i+\sum_{j=1}^m y^i_j h_j^i$ is an SOS-convex polynomial on $\mathbb{R}^n$;
- $\Omega_i$, $i=0,1,\ldots,s$, is a nonempty compact semi-definite program representable set given by $$\Omega_i=\big\{(y_1^i,\ldots,y_m^i) \in \mathbb{R}^m: \exists z^i=(z^i_1,...,z^i_{p_i}) \in \mathbb{R}^{p_i} \mbox{ s.t. } A_0^i+\sum_{j=1}^m y_j^i A_j^i+ \sum_{l=1}^{p_i} z_l^i B_l^i \succeq 0\big\},$$ for some $p_i \in \mathbb{N}.$
[*Without loss of generality, throughout this paper, we assume that $d$ is an even number.* ]{}
We now introduce a relaxation problem for problem (P) as follows $$\begin{aligned}
({SDP}) \ \ \ \sup_{\substack{\lambda_0^i \ge 0, (\lambda_1^i,...,\lambda_m^i) \in \mathbb{R}^m \\
z_l^i \in \mathbb{R}, \mu \in \mathbb{R}}} \big\{\mu&:& h_0^0+\sum_{j=1}^m \lambda_j^0 h_j^0+\sum_{i=1}^s \left(\lambda_0^i h_0^i + \sum_{j=1}^m \lambda_j^i h_j^i\right) -\mu \in \Sigma^2_d[x], \\
& & \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ A_0^0+\sum_{j=1}^m\lambda_j^0 A_j^0+\sum_{l=1}^{p_0} z_l^0 B_l^0 \succeq 0,\\
& & \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \lambda_0^i A_0^i+\sum_{j=1}^m\lambda_j^i A_j^i+\sum_{l=1}^{p_i} z_l^i B_l^i \succeq 0, \, i=1,\ldots,s\big\}. \end{aligned}$$ We note (from Lemma \[th:sos\]) that $(SDP)$ can be equivalently rewritten as the following semi-definite programming problem: [$$\begin{aligned}
& & \sup_{\substack{\lambda_0^i \ge 0, (\lambda_1^i,...,\lambda_m^i) \in \mathbb{R}^m\\
z_l^i \in \mathbb{R}, \mu \in \mathbb{R},
W \in S^{s(d/2,n)}}} \{\mu: (h_0^0)_1+\sum_{j=1}^m \lambda_j^0 (h_j^0)_1+\sum\limits_{i=1}^{s}\left(\lambda_0^i (h_0^i)_1 + \sum_{j=1}^m \lambda_j^i (h_j^i)_1\right)-\mu =W_{1,1}, \nonumber \\
& & \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (h_0^0)_\alpha+\sum_{j=1}^m \lambda_j^0 (h_j^0)_\alpha+\sum\limits_{i=1}^{s}\left(\lambda_0^i (h_0^i)_\alpha + \sum_{j=1}^m \lambda_j^i (h_j^i)_\alpha\right)=\sum_{\substack{ 1\le \beta,\gamma \le s(d/2,n) \\i(\beta)+i(\gamma)=i(\alpha)}}W_{\beta,\gamma} \, ,\ 2 \le \alpha \le s(d,n)\nonumber \\
& & \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ W \succeq 0, \quad \, A_0^0+\sum_{j=1}^m\lambda_j^0 A_j^0+\sum_{l=1}^{p_0} z_l^0 B_l^0 \succeq 0,\\
& & \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \lambda_0^i A_0^i+\sum_{j=1}^m\lambda_j^i A_j^i+\sum_{l=1}^{p_i} z_l^i B_l^i \succeq 0, \, i= 1,\ldots,s\}. \nonumber\end{aligned}$$]{}
Next, we show that an exact SDP relaxation holds between (P) and $(SDP)$ in the sense that their optimal values are the same. We start with a simple property for a bounded set which describes by linear matrix inequalities.
\[lemma:1\] Let $\mathcal{U}$ be a nonempty compact set with the form $\mathcal{U}=\{(u_1,\ldots,u_m) \in \mathbb{R}^m: \exists z \in \mathbb{R}^p \mbox{
such that } A_0+\sum_{j=1}^mu_j A_j+ \sum_{l=1}^p z_l B_l \succeq 0\}$ where $A_j,B_l \in S^{q}$. Let $(\lambda_0,\ldots,\lambda_m) \in \mathbb{R}^{m+1}$ and $\lambda_0 A_0+\sum_{j=1}^m\lambda_j A_j+ \sum_{l=1}^p v_l B_l \succeq 0$ for some $(v_1,\ldots,v_p) \in \mathbb{R}^p$. Then, the following implication holds: $$\label{eq:trick}
\lambda_0=0 \ \Rightarrow \ \lambda_j=0 \ \mbox{ for all } j=1,\ldots,m.$$
Let $(\lambda_0,\ldots,\lambda_m) \in \mathbb{R}^{m+1}$ and $\lambda_0 A_0+\sum_{j=1}^m\lambda_j A_j+ \sum_{l=1}^p v_l B_l \succeq 0$ for some $(v_1,\ldots,v_p) \in \mathbb{R}^p$. We proceed by the method of contradiction. Suppose that $\lambda_0=0$ and there exists $j_0 \in \{1,\ldots,m\}$ with $\lambda_{j_0} \neq 0.$ This means that $$\sum_{j=1}^m\lambda_j A_j+ \sum_{l=1}^p v_l B_l \succeq 0 \mbox{ and } (\lambda_1,\ldots,\lambda_m) \neq 0_{\mathbb{R}^m}.$$ Now take $\hat{u}=(\hat u_1,\ldots,\hat u_m) \in \mathcal{U}$. Then, we have $A_0+\sum_{j=1}^m\hat{u}_j A_j+ \sum_{l=1}^p \hat{v}_l B_l \succeq 0$ for some $(\hat{v}_1,\ldots,\hat{v}_p)\in \R^p$, and so, $$A_0+\sum_{j=1}^m (\hat{u}_j+ t \lambda_j) A_j + \sum_{l=1}^p (\hat{v}_l+t v_l) B_l\succeq 0\ \mbox{ for all } t \ge 0.$$ The latter implies that $$(\hat u_1,\ldots,\hat u_m)+t(\lambda_1,\ldots,\lambda_m) \in \mathcal{U}\ \mbox{ for all } t \ge 0,$$ which contradicts the boundedness of $\mathcal{U}$. Thus, the conclusion follows.
We are now ready to state and prove the first main result of this section, showing the exactness of the SDP relaxation for SOS-convex semi-algebraic programs under a strict feasibility condition.
[**(Exact SDP Relaxation for SOS-convex Semi-algebraic Programs)**]{}\[th:exact\_SDP\] For problem $(P),$ suppose the following strict feasibility condition holds: there exists $x_0 \in \mathbb{R}^n$ such that $f_i(x_0)<0$, $i=1,\ldots,s$. Then, we have $$\begin{aligned}
{\rm val}(P)&=& {\rm val}(SDP),\end{aligned}$$ where ${\rm val}(P)$ and ${\rm val}(SDP)$ are the optimal values of problems $(P)$ and $(SDP),$ respectively.
We first justify that ${\rm val}(P) \ge {\rm val}(SDP).$ Let $\lambda_0^i \ge 0,$ $(\lambda_1^i,...,\lambda_m^i) \in \mathbb{R}^m,$ $z_l^i \in \mathbb{R},$ and $\mu \in \mathbb{R},$ be feasible for $(SDP)$. Then, we have $$\begin{aligned}
& & h_0^0+\sum_{j=1}^m \lambda_j^0 h_j^0+\sum_{i=1}^s \left(\lambda_0^i h_0^i + \sum_{j=1}^m \lambda_j^i h_j^i\right) -\mu \in \Sigma^2_d[x], \\
& & A_0^0+\sum_{j=1}^m\lambda_j^0 A_j^0+\sum_{l=1}^{p_0} z_l^0 B_l^0 \succeq 0,\\
& & \lambda_0^i A_0^i+\sum_{j=1}^m\lambda_j^i A_j^i+\sum_{l=1}^{p_i} z_l^i B_l^i \succeq 0, \, i=1,\ldots,s .\end{aligned}$$ Take any $x \in \mathbb{R}^n$ with $f_i(x) \le 0,$ $i=1,...,s$. We want to show $f_0(x) \ge \mu$. For each $i=1,\ldots,s,$ pick $\bar y_i=(\bar y_1^i,\ldots,\bar y_m^i) \in \Omega_i.$ By the definition of $\Omega_i,$ there exist $\bar z^i \in \mathbb{R}^{p_i}$ such that $A_0^i+\sum_{j=1}^m \bar y_j^i A_j^i+ \sum_{l=1}^{p_i} \bar z_l^i B_l^i \succeq 0$. For each $j=1,\ldots,m$ and $i=1,\ldots,s,$ put $$\widetilde{y}_j^i:= \left\{\begin{array}{cc}
\frac{\lambda_j^i}{\lambda_0^i} &\mbox{if}\ \lambda_0^i \neq 0, \\
\bar y_j^i &\mbox{if}\ \lambda_0^i = 0,
\end{array}
\right.$$ and $$\widetilde{z}_l^i:= \left\{\begin{array}{cc}
\frac{z_l^i}{\lambda_0^i}&\mbox{if}\ \lambda_0^i \neq 0, \\
\bar z_l^i &\mbox{if}\ \lambda_0^i = 0.
\end{array}
\right.$$ Then, for each $i=1,\ldots,s$, we have $$A_i^0+\sum_{j=1}^m \widetilde{y}_j^i A_j^i+ \sum_{l=1}^{p_i} \tilde{z}_l^i B_l^i = \left\{\begin{array}{cc}
\frac{1}{\lambda_0^i} \big(\lambda_0^i A_0^i+\sum_{j=1}^m \lambda_j^i A_j^i+ \sum_{l=1}^{p_i} z_l^i B_l^i\big)& \mbox{if}\ \lambda_0^i \neq 0, \\
A_0^i+\sum_{j=1}^m \bar y_j^i A_j^i+ \sum_{l=1}^{p_i} \bar z_l^i B_l^i &\mbox{if}\ \lambda_0^i = 0,
\end{array}
\right.$$ which is always a positive semidefinite symmetric matrix. So, $\widetilde{y}^i:=(\widetilde{y}^i_1,...,\widetilde{y}^i_m) \in \Omega_i$ and hence $$\label{tv1}h_0^i(x)+ \sum_{j=1}^m \widetilde{y}_j^i h_j^i(x) \le f_i(x)\leq 0\quad\mbox{for all}\ i=1,...,s.$$ Moreover, as the sets $\mathcal{U}_i$ are bounded, according to Lemma \[lemma:1\], for each $i=1,\ldots,s$, if $\lambda_0^i=0$, then $\lambda_j^i=0$ for all $j=1,\ldots,m$. This implies that $$\begin{aligned}
h_0^0+\sum_{j=1}^m \lambda^0_j h_j^0+\sum_{i=1}^s \lambda_0^i \left( h_0^i + \sum_{j=1}^m \widetilde{y}_j^i h_j^i\right)-\mu
&=& h_0^0+\sum_{j=1}^m \lambda^0_j h_j^0+\sum_{i=1}^s \left( \lambda_0^i h_0^i + \sum_{j=1}^m (\lambda_0^i \widetilde{y}_j^i) h_j^i\right)-\mu \\
&= & h_0^0+\sum_{j=1}^m \lambda^0_j h_j^0+\sum_{i=1}^s \left( \lambda_0^i h_0^i + \sum_{j=1}^m \lambda_j^i h_j^i\right)-\mu \in \Sigma^2_d[x].\end{aligned}$$ So, noting that $(\lambda^0_1,\ldots,\lambda^0_m) \in \Omega_0$ and $\lambda_0^i\geq 0$ for all $i=1,...,s$, by it holds that $$\begin{array}{rl}
f_0(x) &\ge h_0^0(x)+\sum\limits_{j=1}^m \lambda^0_jh_j^0(x) \\ \cr
& \ge h_0^0(x)+\sum\limits_{j=1}^m \lambda^0_j h_j^0(x)+\sum\limits_{i=1}^s \lambda_0^i \left( h_0^i(x) + \sum\limits_{j=1}^m \widetilde{y}_j^i h_j^i(x)\right) \ge \mu.\end{array}$$ Therefore, ${\rm val}(P) \ge {\rm val}(SDP)$.
Next, we will justify that ${\rm val}(P) \le {\rm val}(SDP).$ As ${\rm val}(P) \ge {\rm val}(SDP)$ always holds, it suffices to consider the case ${\rm val}(P)>-\infty$. Noting that the feasible set of $(P)$ is nonempty, we may assume that $r:={\rm val}(P) \in \mathbb{R}$. Our assumptions guarantee that there exists $x_0$ such that $f_i(x_0)<0$, $i=1,\ldots,s,$ and each $f_i$ is a continuous convex function. So, the standard Lagrangian duality for convex programming problem shows that $$\label{tv2}\begin{array}{rl}
r &:= \inf\limits_{x \in \mathbb{R}^n}\big\{f_0(x): f_i(x) \le 0,\ i=1,...,s\big\} \\ \cr
& = \max\limits_{\lambda\in \R^s_+} \inf\limits_{x \in \mathbb{R}^n}\big\{f_0(x)+\sum\limits_{i=1}^s \lambda_i f_i(x)\big\}\\ \cr
&= \max\limits_{\lambda\in \R^s_+} \inf\limits_{x \in \mathbb{R}^n}\max\limits_{y\in \prod\limits_{i=0}^s \Omega_i}h_\lambda(x,y), \end{array}$$ where $\lambda:=(\lambda_1,...,\lambda_s)\in \R^s,$ $y:=(y_1^0,\ldots,y_m^0,...,y_1^s,\ldots,y_m^s)\in \R^{m(s+1)},$ and $$h_\lambda(x,y):=h_0^0(x)+\sum_{j=1}^m y_j^0 h_j^0(x)+\sum_{i=1}^s \lambda_i\big(h_0^i(x)+\sum_{j=1}^m y_j^i h_j^i(x)\big).$$ Note that $ \prod\limits_{i=0}^s \Omega_i$ is a convex compact set, and for any $\lambda\in \R^s_+$ the function $h_\lambda(x,y)$ is convex in $x$ for each fixed $y$ and is concave in $y$ for each fixed $x.$ Thus, for each $\lambda\in \R^s_+,$ by the convex-concave minimax theorem we have $$\begin{aligned}
\inf_{x \in \mathbb{R}^n}\max_{y\in \prod\limits_{i=0}^s \Omega_i}h_\lambda(x,y)= \max_{y\in \prod\limits_{i=0}^s \Omega_i}\inf_{x \in \mathbb{R}^n}h_\lambda(x,y).\end{aligned}$$ This together with yields $$\begin{array}{rl}
r &= \max\limits_{\lambda\in \R^s_+} \max\limits_{y\in \prod\limits_{i=0}^s \Omega_i}\inf\limits_{x \in \mathbb{R}^n}h_\lambda(x,y)\\ \cr
&= \max\limits_{\substack{(y_1^i,\ldots,y_m^i) \in \Omega_i, 0 \le i\le s \\
\lambda_1 \ge 0,...,\lambda_s\geq0}}\inf\limits_{x \in \mathbb{R}^n}\big\{h_0^0(x)+\sum\limits_{j=1}^m y_j^0 h_j^0(x)+\sum\limits_{i=1}^s \lambda_i\big(h_0^i(x)+\sum\limits_{j=1}^m y_j^i h_j^i(x)\big)\big\}.\end{array}$$ In particular, the latter shows that there exist $(\tilde{y}_1^i,\ldots,\tilde{y}_m^i) \in \Omega_i,$ $0 \le i\le s,$ and $\tilde{\lambda}_1 \ge 0,...,\tilde{\lambda}_s \ge 0,$ such that $$\inf_{x \in \mathbb{R}^n}\big\{h_0^0(x)+\sum_{j=1}^m \tilde{y}_j^0 h_j^0(x)+\sum_{i=1}^s \tilde{\lambda}_i\big(h_0^i(x)+\sum_{j=1}^m \tilde{y}_j^i h_j^i(x)\big)\big\} =r.$$ Denote $G(x)=h_0^0(x)+\sum_{j=1}^m \tilde{y}_j^0 h_j^0(x)+\sum_{i=1}^s \tilde{\lambda}_i\big(h_0^i(x)+\sum_{j=1}^m \tilde{y}_j^i h_j^i(x)\big)-r$. By Lemma \[minattain\], there exists $a \in \mathbb{R}^n$ such that $G(a)=\inf_{x \in \mathbb{R}^n} G(x)=0$ (and so, $\nabla G(a)=0$). As $G$ is an SOS-convex polynomial, $H(x,y):=G(x)-G(y)-\nabla G(y)^T(x-y)$ is a sums-of-squares polynomial. Letting $y=a$, it follows that $G(x)=H(x,a)$ is also a sums-of-squares polynomial, that is, $$\label{eq:pp1}
h_0^0+\sum_{j=1}^m \tilde{y}_j^0 h_j^0+\sum_{i=1}^s \tilde{\lambda}_i\big(h_0^i+\sum_{j=1}^m \tilde{y}_j^i h_j^i\big)-r \in \Sigma^2_d[x].$$ On the other hand, for each $i=0,...,s,$ since $(\tilde{y}_1^i,\ldots,\tilde{y}_m^i) \in \Omega_i,$ there exists $ \tilde z^i=(\tilde z^i_1,...,\tilde z^i_{p_i})\in \R^{p_i}$ such that $$\label{eq:pp}
A_0^i+\sum_{j=1}^m \tilde y_j^i A_j^i+ \sum_{l=1}^{p_i} \tilde z_l^i B_l^i \succeq 0.$$ Now, let $\lambda^0_j:=\tilde y_j^0$, $j=1,\ldots,m$, $z^0_l:=\tilde z^0_l,$ $l=1,...,p_0,$ $\lambda_0^i:=\tilde{\lambda}_i$ and $\lambda_j^i:=\tilde{\lambda}_i \tilde{y}_j^i$ and $z_l^i:=\tilde{\lambda}_i \tilde{z}_l^i$ for each $j=1,\ldots,m,$ $i=1,\ldots,s,$ $l=1,...,p_i$. From , and $\tilde{\lambda}_i \ge 0$, we see that $$h_0^0+\sum_{j=1}^m \lambda^0_j h_j^0+\sum_{i=1}^s \left(\lambda_0^i h_0^i + \sum_{j=1}^m \lambda_j^i h_j^i\right) -r \in \Sigma^2_d[x],$$$$A_0^0+\sum_{j=1}^m\lambda_j^0 A_j^0+\sum_{l=1}^{p_0} z_l^0 B_l^0 \succeq 0,$$ and $$\lambda_0^i A_0^i+\sum_{j=1}^m\lambda_j^i A_j^i+\sum_{l=1}^{p_i} z_l^i B_l^i \succeq 0, \, i=1,\ldots,s.$$ This says that $ \lambda_0^i \ge 0, (\lambda_1^i,....,\lambda_m^i) \in \mathbb{R}^m, z_l^i \in \mathbb{R}, r \in \mathbb{R}$ is feasible for $(SDP).$ Thus, $${\rm val}(P) =r\le {\rm val}(SDP),$$ and hence ${\rm val}(P) = {\rm val}(SDP)$. The proof is complete.
[**(Special Cases: Min-max programs involving SOS-convex polynomials)**]{} In the case where the objective function $f_0$ can be expressed as a finite maximum of SOS-convex polynomials and the constraint functions $f_i$, $i=1,\ldots,s$, are SOS-convex polynomials, Theorem \[th:exact\_SDP\] has established in [@minmax Theorem 3.1] for min-max programs.
In the preceding theorem, we see that the optimal value of a SOS-convex semi-algebraic optimization problem (P) can be found by solving a single semi-definite programming problem, that is, its SDP relaxation problem (SDP). Next, we examine the important question that: how to recover an optimal solution of (P) from its SDP relaxation problem?
For a given $z=(z_{\alpha}) \in \mathbb{R}^{s(r,n)}$, we define a linear function $L_z: \mathbb{R}_r[x_1,\ldots,x_n] \rightarrow \mathbb{R}$ by $$\label{Reisz}
L_z(u)=\sum_{\alpha=1}^{s(r,n)} u_{\alpha} z_{\alpha} \mbox{ with } u(x)=\sum_{\alpha=1}^{s(r,n)} u_\alpha x^{(r)}_{\alpha}.$$ For each $\alpha=1,\ldots,s(2r,n)$, define $M_{\alpha}$ to be the $(s(r,n) \times s(r,n))$ symmetric matrix such that $${\rm Tr} (M_{\alpha} W)= \sum_{\substack{1 \le \beta,\gamma \le s(r,n) \\
i(\beta)+i(\gamma)=i(\alpha)}} W_{\beta,\gamma} \mbox{ for all } W \in S^{s(r,n)}.$$ Then, for $z=(z_{\alpha}) \in \mathbb{R}^{s(2r,n)}$, the moment matrix with respect to the sequence $z=(z_{\alpha})$ with degree $r$ is denoted by ${\bf M}_r(z)$, and is defined by $${\bf M}_r(z)=\sum_{1 \le \alpha \le s(2r,n)} z_{\alpha}M_{\alpha}.$$
As a simple illustration, let $r=4$ and $n=1$, for $z=(z_1,\ldots,z_5)^T \in \mathbb{R}^{s(4,1)}=\mathbb{R}^5$, $$L_z(u)= \sum_{i=1}^{5} \alpha_iz_i, \mbox{ for all } u(x)=\alpha_1+\alpha_2 x+\alpha_3 x^2+\alpha_4x^3+ \alpha_5 x^4.$$ Moreover, for $r=1$, $n=2$ and $z \in \mathbb{R}^{s(2,2)}=\mathbb{R}^{6}$ $${\bf M}_1(z)= \left(\begin{array}{ccc}
z_1 & z_2 & z_3 \\
z_2 & z_4 & z_5 \\
z_3 & z_5 & z_6
\end{array}
\right).$$ Recall that $(SDP)$ can be equivalently rewritten as a semi-definite programming problem. [$$\begin{aligned}
& & \sup_{\substack{\lambda_0^i \ge 0, (\lambda_1^i,...,\lambda_m^i) \in \mathbb{R}^m\\
z_l^i \in \mathbb{R}, \mu \in \mathbb{R},
W \in S^{s(d/2,n)}}} \{\mu: (h_0^0)_1+\sum_{j=1}^m \lambda_j^0 (h_j^0)_1+\sum\limits_{i=1}^{s}\left(\lambda_0^i (h_0^i)_1 + \sum_{j=1}^m \lambda^i_j (h_j^i)_1\right)-\mu =W_{1,1}, \nonumber \\
& & \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (h_0^0)_\alpha+\sum_{j=1}^m \lambda_j^0 (h_j^0)_\alpha+\sum\limits_{i=1}^{s}\left(\lambda_0^i (h_0^i)_\alpha + \sum_{j=1}^m \lambda_j^i (h_j^i)_\alpha\right)=\sum_{\substack{ 1\le \beta,\gamma \le s(d/2,n) \\i(\beta)+i(\gamma)=i(\alpha)}}W_{\beta,\gamma} \, , \ 2 \le \alpha \le s(d,n)\nonumber \\
& & \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ W \succeq 0, \quad A_0^0+\sum_{j=1}^m\lambda_j^0 A_j^0+\sum_{l=1}^{p_0} z_l^0 B_l^0 \succeq 0,\\
& & \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \lambda_0^i A_0^i+\sum_{j=1}^m\lambda_j^i A_j^i+\sum_{l=1}^{p_i} z_l^i B_l^i \succeq 0, \, i= 1,\ldots,s\}. \nonumber\end{aligned}$$]{} The Lagrangian dual of the above semi-definite programming reformulation of $(SDP)$ can be stated as follows: $${\small \begin{array}{rl}
&\inf\limits_{\substack{y=(y_{\alpha}) \in \mathbb{R}^{s(d,n)} \\ Z_i \succeq 0}}\, \sup\limits_{\substack{\lambda_0^i \ge 0, (\lambda_1^i,...,\lambda_m^i) \in \mathbb{R}^m\\
z_l^i \in \mathbb{R}, \mu \in \mathbb{R},
W\succeq0}}\Big\{\mu + y_1\left( (h_0^0)_1+\sum\limits_{j=1}^m \lambda_j^0 (h_j^0)_1+\sum\limits_{i=1}^{s}\left(\lambda_0^i (h_0^i)_1 + \sum\limits_{j=1}^m \lambda^i_j (h_j^i)_1\right)-\mu -W_{1,1}\right) \\
& +\sum\limits_{2 \le \alpha \le s(d,n)} y_{\alpha}\left((h_0^0)_\alpha+\sum\limits_{j=1}^m \lambda_j^0 (h_j^0)_\alpha+\sum\limits_{i=1}^{s}\left(\lambda_0^i (h_0^i)_\alpha + \sum\limits_{j=1}^m \lambda_j^i (h_j^i)_\alpha\right)-\sum\limits_{\substack{ 1\le \beta,\gamma \le s(d/2,n) \\i(\beta)+i(\gamma)=i(\alpha)}}W_{\beta,\gamma}\right) \\
&+{\rm Tr}\big(Z_0 (A_0^0+\sum\limits_{j=1}^m\lambda_j^0 A_j^0+\sum\limits_{l=1}^{p_0} z_l^0 B_l^0)\Big) +\sum\limits_{i=1}^s{\rm Tr}\big(Z_i (\lambda_0^i A_0^i+\sum\limits_{j=1}^m\lambda_j^i A_j^i+\sum\limits_{l=1}^{p_i} z_l^i B_l^i\Big)\Big\},
\end{array}}$$ which can be further simplified as $$\begin{aligned}
(SDP^*) & \displaystyle \inf_{y=(y_{\alpha}) \in \mathbb{R}^{s(d,n)}, Z_i \succeq 0} & \sum_{1 \le \alpha \le s(d,n)} (h^0_0)_{\alpha} y_{\alpha}+{\rm Tr}\big(Z_0 A_0^0\big) \\
& \mbox{ s.t. } & \sum_{1 \le \alpha \le s(d,n)} (h^i_0)_{\alpha} y_{\alpha}+{\rm Tr} \big( Z_i A^i_0\big) \le 0, \, i= 1,\ldots,s, \\
& & \sum_{1 \le \alpha \le s(d,n)} (h_j^i)_{\alpha} y_{\alpha}+{\rm Tr} \big( Z_i A_j^i\big) = 0, \, i=0, 1,\ldots,s, j=1,\ldots,m, \\
& &{\rm Tr} \big( Z_i B_l^i\big) = 0,\, i=0, 1,...,s,\, l=1,...,p_i,\\
& & {\bf M}_{\frac{d}{2}}(y)=\sum_{1 \le \alpha \le s(d,n)} y_{\alpha} M_{\alpha} \succeq 0,\\
& & y_1=1.\end{aligned}$$ We note that the problem $(SDP^*)$ is also a semi-definite programming problem, and hence can be efficiently solved as well.
Next, we recall the following Jensen’s inequality for SOS-convex polynomial (cf [@Lasserre]) which will play an important role in our later analysis.
[**(Jensen’s inequality for SOS-convex polynomial [@Lasserre Theorem 5.13])**]{} \[lemma:Jensen\] Let $f$ be an SOS-convex polynomial on $\mathbb{R}^n$ with degree $2r$. Let $y \in \mathbb{R}^{s(2r,n)}$ with $y_1=1$ and ${\bf M}_{r}(y) \succeq 0$. Then, we have $$L_{y}(f) \ge f(L_{y}(X_1),\ldots,L_{y}(X_n)),$$ where $L_y$ is given as in (\[Reisz\]) and $X_i$ denotes the polynomial which maps a vector in $\R^n$ to its $i$th coordinate.
The next main result of this section is the following theorem, providing the way to recover a solution to problem (P) from a solution to its SDP relaxation.
[**(Recovery of the solution)**]{}\[th:3.2\] For problem $(P)$, suppose that the following strict feasibility conditions hold:
- there exists $\bar x \in \mathbb{R}^n$ such that $f_i(\bar x)<0 $ for all $i=1,...,s$;
- for each $i=0,1,\ldots,s$, there exist $\bar y^i \in \mathbb{R}^m$ and $\bar z^i \in \mathbb{R}^{p_i}$ such that $A_0^i+\sum_{j=1}^m \bar y_j^i A_j^i+ \sum_{l=1}^{p_i} \bar z_l^i B_l^i \succ 0$.
Let $(y^*,Z_0^*,Z_1^*,...,Z_s^*)$ be an optimal solution for $(SDP^*)$ and let $x^*:=(L_{y^*}(X_1),\ldots,L_{y^*}(X_n))^T \in \mathbb{R}^n$ where $X_i$ denotes the polynomial which maps a vector $x\in \R^n$ to its $i$th coordinate. Then, $x^*$ is an optimal solution for $(P)$.
From condition [(i)]{}, the exact SDP relaxation result (Theorem \[th:exact\_SDP\]) gives us that ${\rm val}(P)={\rm val}(SDP)$. Note that $(SDP)$ and $(SDP^*)$ are dual problems to each other. The usual weak duality for semi-definite programming implies that ${\rm val}(SDP^*) \ge {\rm val}(SDP)={\rm val}(P)$. Next, we establish that ${\rm val}(SDP^*)={\rm val}(P)$, where ${\rm val}(SDP^*)$ is the optimal value of problem $(SDP^*)$. To see this, let $x$ be a feasible point of $(P)$ and let $r=f_0(x)$. Then $$f_0(x)=\sup_{(y_1^0,\ldots,y_m^0) \in \Omega_0} \{h_0^0(x)+\sum_{j=1}^m y_j^0 h_j^0(x)\}=r$$ and $$f_i(x)=\sup_{(y_1^i,\ldots,y_m^i) \in \Omega_i} \{h_0^i(x)+\sum_{j=1}^m y_j^i h_j^i(x)\} \le 0, \, i=1,\ldots,s,$$ where $\Omega_i$, $i=0,1,\ldots,s$ are compact sets given by $\Omega_i=\big\{(y_1^i,\ldots,y_m^i) \in \mathbb{R}^m: \exists z^i=(z^i_1,...,z^i_{p_i}) \in \mathbb{R}^{p_i} \mbox{ s.t. } A_0^i+\sum_{j=1}^m y_j^i A_j^i+ \sum_{l=1}^{p_i} z_l^i B_l^i \succeq 0\big\}$. This shows that $$(y^0,z^0) \in \mathbb{R}^m \times \mathbb{R}^{p_0}, \ A_0^0+\sum_{j=1}^m y_j^0 A_j^0+ \sum_{l=1}^{p_0} z_l^0 B_l^0 \succeq 0 \, \Rightarrow \, h_0^0(x)+\sum_{j=1}^m y_j^0 h_j^0(x) \le r,$$ and $$(y^i,z^i) \in \mathbb{R}^m \times \mathbb{R}^{p_i}, \ A_0^i+\sum_{j=1}^m y_j^i A_j^i+ \sum_{l=1}^{p_i} z_l^i B_l^i \succeq 0 \, \Rightarrow \, h_0^i(x)+\sum_{j=1}^m y_j^i h_j^i(x) \le 0, \ i=1,\ldots,s.$$ It then follows from condition [(ii)]{} and the strong duality theorem for semi-definite programming that there exist $Z_i \succeq 0$, $i=0,1,\ldots,s$ such that $$\left\{ \begin{array}{cc}
& h_0^0 (x)+{\rm Tr} \big( Z_0 A_0^0\big) \le r, \\
& h^i_0(x)+{\rm Tr}\big(Z_iA^i_0\big)\le 0,\, i=1,...,s,\\
& h_j^i(x)+{\rm Tr} \big( Z_i A_j^i\big) = 0, \, i= 0, 1,\ldots,s,\, j=1,\ldots,m, \\
& {\rm Tr} \big( Z_i B_l^i\big) = 0,\, i=0, 1,...,s,\, l=1,...,p_i,
\end{array} \right.$$ Let $x^{(d)}=(1,x_1,x_2,\ldots,x_n,x_1^2,x_1x_2,\ldots,x_2^2,\ldots,x_n^2,\ldots,x_1^{d},\ldots,x_n^d)^T$. Then, $(x^{(d)},Z_0,Z_1,\ldots,Z_s)$ is feasible for $(SDP^*)$ and $$f_0(x)=r \ge h_0^0 (x)+{\rm Tr} \big( Z_0 A_0^0\big)= \sum\limits_{1 \le \alpha \le s(d,n)} (h_0^0)_{\alpha} x^{(d)}_{\alpha}+{\rm Tr} \big( Z_0 A_0^0\big).$$ This shows that ${\rm val}(P) \ge {\rm val}(SDP^*)$, and hence ${\rm val}(P)={\rm val}(SDP^*)$.
Now, let $(y^*,Z_0^*,Z_1^*,...,Z_s^*)$ be an optimal solution for $(SDP^*)$. Then, $Z_i^* \succeq 0$, $i=0,1,...,s$, and $$\begin{array}{rl} & \sum\limits_{1 \le \alpha \le s(d,n)} (h_0^i)_{\alpha} y_{\alpha}^*+{\rm Tr} \big( Z_i^* A_0^i\big) \le 0, \, i=1,\ldots,s, \\
& \sum\limits_{1 \le \alpha \le s(d,n)} (h^i_j)_{\alpha} y^*_{\alpha}+{\rm Tr} \big( Z_i^* A_j^i\big) = 0, \, i=0, 1,\ldots,s, j=1,\ldots,m, \\
&{\rm Tr} \big( Z_i^* B_l^i\big) = 0,\, i=0, 1,...,s,\, l=1,...,p_i,\\
& {\bf M}_{\frac{d}{2}}(y^*)=\sum\limits_{1 \le \alpha \le s(d,n)} y^*_{\alpha} M_{\alpha} \succeq 0,\\
& y_1^*=1.\end{array}$$ Note that for each $(y^0_1,...,y^0_m)\in \Omega_0,$ one can find $z^0=(z^0_1,...,z^0_{p_0}) \in \mathbb{R}^{p_0}$ such that $$A^0_0+\sum\limits_{j=1}^my^0_j A_j^0+\sum\limits_{l=1}^{p_0}z^0_lB_l^0\succeq0.$$ So, for each $(y^0_1,...,y^0_m)\in \Omega_0,$ it holds that $$\label{eq1}\begin{array}{rl} \sum\limits_{1 \le \alpha \le s(d,n)} (h_0^0)_{\alpha} y_{\alpha}^*+{\rm Tr}\big( Z_0^* A_0^0\big)&\geq \sum\limits_{1 \le \alpha \le s(d,n)} (h_0^0)_{\alpha} y_{\alpha}^* -{\rm Tr} \big( Z_0^*(\sum\limits_{j=1}^my^0_j A_j^0+\sum\limits_{l=1}^{p_0}z^0_lB_l^0)\big)\\
&=\sum\limits_{1 \le \alpha \le s(d,n)} (h_0^0)_{\alpha} y_{\alpha}^*-\sum\limits_{j=1}^my^0_j{\rm Tr} \big( Z_0^*A_j^0\big)\\
&=\sum\limits_{1 \le \alpha \le s(d,n)} (h_0^0)_{\alpha} y_{\alpha}^*+\sum\limits_{j=1}^my^0_j \sum\limits_{1 \le \alpha \le s(d,n)}(h^0_j)_{\alpha} y^*_{\alpha}\\
& =L_{y^*}\big(h_0^0+\sum\limits_{j=1}^my^0_jh^0_j\big).
\end{array}$$ Since $h_0^0+\sum\limits_{j=1}^my^0_jh^0_j$ is SOS-convex, ${\bf M}_{\frac{d}{2}}(y^*)\geq 0,$ and $y_1^*=1,$ by Lemma \[lemma:Jensen\], we have $$\label{eq2}L_{y^*}\big(h_0^0+\sum\limits_{j=1}^my^0_jh^0_j\big)\geq \big(h_0^0+\sum\limits_{j=1}^my^0_jh^0_j\big)(L_{y^*}(X_1),\ldots,L_{y^*}(X_n))=h_0^0(x^*)+\sum\limits_{j=1}^my^0_jh^0_j(x^*)$$ for every $(y^0_1,...,y^0_m)\in \Omega_0.$ Taking supremum over all $(y^0_1,...,y^0_m)\in \Omega_0$ in and using , it follows that $$f_0(x^*)\leq \sum\limits_{1 \le \alpha \le s(d,n)} (h_0^0)_{\alpha} y_{\alpha}^*+{\rm Tr}\big( Z_0^* A_0^0\big).$$ Taking into account that $(y^*,Z_0^*,Z_1^*,...,Z_s^*)$ is an optimal solution for $(SDP^*),$ we get $${\rm val}(SDP^*)=\sum_{1 \le \alpha \le s(d,n)} (h_0^0)_{\alpha} y_{\alpha}^*+{\rm Tr} \big( Z_0^* A_0^0\big) \ge f_0(x^*).$$ We claim that $x^*$ is feasible for (P). Granting this, we have $${\rm val}(SDP^*)\geq f_0(x^*) \ge {\rm val}(P)={\rm val}(SDP^*).$$ This forces that $f_0(x^*)={\rm val}(P)$, and so, $x^*$ is an optimal solution for (P).
We now verify our claim. Take any $i=1,...,s$ and $(y^i_1,...,y^i_m)\in \Omega_i.$ Then one can find $z^i=(z^i_1,...,z^i_{p_i}) \in \mathbb{R}^{p_i}$ such that $$A^i_0+\sum\limits_{j=1}^my^i_j A_j^i+\sum\limits_{l=1}^{p_i}z^i_lB_l^i\succeq0.$$ Arguing as before, we arrive at $$f_i(x^*)\leq \sum\limits_{1 \le \alpha \le s(d,n)} (h_0^i)_{\alpha} y_{\alpha}^*+{\rm Tr} \big( Z_i^* A_0^i\big)\leq 0.$$ This shows that $x^*$ is feasible for (P). So, the conclusion follows.
Finally, we illustrate how to find the optimal value and an optimal solution for an SOS-convex semi-algebraic program by solving a single semi-definite programming problem.
[**(Illustrative example)**]{}
Consider the following simple 2-dimensional nonsmooth convex optimization problem: $$\begin{aligned}
(EP) & \min & x_1^4-x_2 \\
& \mbox{\rm s.t. } & x_1^2+ x_2^2+ 2\|(x_1,x_2)\| -1\le 0. $$ Let $$\Omega_1=\{(y_1^1,y_2^1): (y_1^1)^2+(y^1_2)^2 \le 1\}= \{(y_1^1,y^1_2): \left(\begin{array}{ccc}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1
\end{array}
\right)+ y_1^1 \left(\begin{array}{ccc}
0 & 0 & 1 \\
0 & 0 & 0 \\
1 & 0 & 0
\end{array}
\right)+y^1_2\left(\begin{array}{ccc}
0 & 0 & 0 \\
0 & 0 & 1 \\
0 & 1 & 0
\end{array}
\right) \succeq 0\}.$$ Let $h^1_0(x)=x_1^2+ x_2^2 -1$ and $h^1_j(x)=2x_j$, $j=1,2$. We first observe that, for each $(y^1_1,y^1_2)\in \Omega_1,$ the function $h^1_0+\sum_{j=1}^2 y^1_j h^1_j$ is an SOS-convex polynomial. Denote $f_0(x)=x_1^4-x_2$ and $f_1(x)= x_1^2+ x_2^2+ 2\|(x_1,x_2)\| -1$. Then, $f_1(x)=\sup_{(y^1_1,y^1_2)\in \Omega_1}\{h^1_0(x)+y^1_1h^1_1(x)+y^1_2h^1_2(x)\}$, and so, $f_1$ is an SOS-convex semi-algebraic function. Obviously, $f_0$ is an SOS-convex polynomial and thus is also an SOS-convex semi-algebraic function. This shows that (EP) is an SOS-convex semi-algebraic program.
Let $x_0=(0,0)$. It can be verified that $f_1(x_0)=-1<0.$ Thus, Theorem \[th:exact\_SDP\] implies that ${\rm val}(EP)={\rm val}(ESDP)$ where (ESDP) is given by $$\begin{aligned}
(ESDP) & & \sup_{\lambda^1_0 \ge 0, \lambda^1_j \in \mathbb{R}, \mu \in \mathbb{R}} \{\mu: f_0+\left(\lambda^1_0 h^1_0 + \sum_{j=1}^2\lambda^1_j h^1_j\right) -\mu \in \Sigma^2_4[x], \\
& & \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \lambda^1_0 \left(\begin{array}{ccc}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1
\end{array}
\right)+ \lambda^1_1 \left(\begin{array}{ccc}
0 & 0 & 1 \\
0 & 0 & 0 \\
1 & 0 & 0
\end{array}
\right)+\lambda^1_2 \left(\begin{array}{ccc}
0 & 0 & 0 \\
0 & 0 & 1 \\
0 & 1 & 0
\end{array}
\right) \succeq 0\}.\end{aligned}$$ Note that $$\begin{aligned}
& & f_0+\left(\lambda^1_0 h_1^0 + \sum_{j=1}^2\lambda^1_j h^1_j\right) -\mu \in \Sigma^2_4[x] \\
&\Leftrightarrow &
x_1^4-x_2+\lambda^1_0 (x_1^2+x_2^2-1)+ 2\lambda_1^1x_1+ 2\lambda^1_2x_2-\mu \\
& & =\left(1, x_1,x_2,x_1^2,x_1x_2,x_2^2\right)\left(\begin{array}{cccccc}
W_{11} & W_{12} & W_{13} & W_{14} & W_{15} & W_{16}\\
W_{12} & W_{22} & W_{23} & W_{24}& W_{25} & W_{26}\\
W_{13} & W_{23} & W_{33} & W_{34}& W_{35} & W_{36}\\
W_{14} & W_{24} & W_{34} & W_{44}& W_{45} & W_{46}\\
W_{15} & W_{25} & W_{35} & W_{45}& W_{55} & W_{56}\\
W_{16} & W_{26} & W_{36} & W_{46}& W_{56} & W_{66}
\end{array}\right)\left(\begin{array}{c}
1 \\
x_1 \\
x_2 \\
x_1^2 \\
x_1x_2 \\
x_2^2 \end{array}\right), W=(W_{ij}) \in S_+^{6}, \\
& \Leftrightarrow & W_{11}=-\lambda^1_0-\mu, W_{12}=\lambda_1^1, 2W_{14}+W_{22}=2\lambda^1_0, \\
& & W_{33}+2W_{16}=\lambda^1_0, 2W_{13}+W_{66}=-1+2\lambda^1_2, W_{44}=1, \\
& & W_{23}=W_{24}=W_{34}=0, W_{i5}=0, i=1,\ldots,5, W_{i6}=0, i=1,\ldots,6, \\
& & W=(W_{ij}) \in S_+^{6}. \end{aligned}$$ Thus, $(ESDP)$ can be equivalently rewritten as the following semidefinite programming problem: $$\begin{aligned}
\sup_{\lambda^1_0 \ge 0, \lambda^1_j \in \mathbb{R}, \mu \in \mathbb{R},W \in S^6} \{\mu&:& W_{11}=-\lambda^1_0-\mu, 2W_{12}=\lambda^1_1, 2W_{14}+W_{22}=\lambda^1_0, \\
& & W_{11}=-\lambda^1_0-\mu, W_{12}=\lambda_1^1, 2W_{14}+W_{22}=2\lambda^1_0, \\
& & W_{33}+2W_{16}=\lambda^1_0, 2W_{13}+W_{66}=-1+2\lambda^1_2, W_{44}=1, \\
& & W_{23}=W_{24}=W_{34}=0, W_{i5}=0, i=1,\ldots,5, W_{i6}=0, i=1,\ldots,6, \\
& & W=(W_{ij}) \in S_+^{6} \\
& & \lambda^1_0 \left(\begin{array}{ccc}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1
\end{array}
\right)+ \lambda_1^1 \left(\begin{array}{ccc}
0 & 0 & 1 \\
0 & 0 & 0 \\
1 & 0 & 0
\end{array}
\right)+\lambda^1_2 \left(\begin{array}{ccc}
0 & 0 & 0 \\
0 & 0 & 1 \\
0 & 1 & 0
\end{array}
\right) \succeq 0\}.\end{aligned}$$ Solving this semi-definite programming problem using CVX [@CVX1; @CVX2], we obtain the optimal value ${\rm val}(RP)={\rm val}(ESDP)=-0.414214 \approx 1-\sqrt{2}$ and the dual variable $y^*=(y_1^*,\ldots,y_{15}^*) \in \mathbb{R}^{15}=\mathbb{R}^{s(4,2)}$ with $y_1^*=1$, $y_2^*=0$ and $y_3^*=0.414214 \approx \sqrt{2}-1$. It can be verified that the conditions in Theorem \[th:3.2\] are satisfied. So, Theorem \[th:3.2\] implies that $x^*=(L_{y^*}(X_1),L_{y^*}(X_2))=(y_2^*,y_3^*)=(0,\sqrt{2}-1)$ is a solution for (EP).
Indeed, the optimality of $(0,\sqrt{2}-1)$ for (EP) can be verified independently. To see this, note that for all $(x_1,x_2)$ which is feasible for (EP), one has $$x_1^2+x_2^2+2\|(x_1,x_2)\|-1 \le 0.$$ In particular, $$|x_2|^2+2|x_2|-1=x_2^2+2|x_2|-1 \le 0,$$ which implies that $|x_2| \le \sqrt{2}-1$. Thus, for all feasible point $(x_1,x_2)$ for (EP), $x_1^4-x_2 \ge -x_2 \ge -|x_2| \ge 1-\sqrt{2}$, and so, ${\rm val}(EP) \ge 1-\sqrt{2}$. On the other hand, direct verification shows that $(0,\sqrt{2}-1)$ is feasible for (EP) with the object value $1-\sqrt{2}$. So, ${\rm val}(EP)=1-\sqrt{2}$ and $(0,\sqrt{2}-1)$ is a solution of the problem (EP).
Applications to robust optimization
===================================
In this section, we briefly outline how our results can be applied to the area of robust optimization [@BEN09] (for some recent development see [@Goberna; @Gold; @jl; @jl1]). Consider the following robust SOS-convex optimization problem $$\begin{aligned}
(RP) & \min & f(x) \\
& \mbox{ subject to } & g_i^{(0)}(x)+\sum\limits_{j=1}^{t_i}u_i^{(j)}g_i^{(j)}(x)+\sum\limits_{j=t_i+1}^su_i^{(j)}g_i^{(j)}(x) \le 0, \,\ \forall u_i\in \mathcal{U}_i,\, \ i=1,\ldots,s,\end{aligned}$$ where $f,$ $g_i^{(j)}$, $i=1,\ldots,m$, $j=0, 1,...,t_i,$ are SOS-convex polynomials, $g_i^{(j)}$, $i=1,\ldots,m$, $j=t_i+1,...,s,$ are affine functions, and $u_i$ are uncertain parameters and belong to uncertainty sets $\mathcal{U}_i$, $i=1,\ldots,s$.
In the case where $\mathcal{U}_i$ is the so-called restricted ellipsoidal uncertainty set given by $$\begin{aligned}
\mathcal{U}_i^e=\{(u_i^1,\ldots,u_i^{t_i},u_i^{t_i+1},\ldots,u_i^s)&:& \|(u_i^1,\ldots,u_i^{t_i})\|\le 1, \ u_i^j \ge 0, j=1,\ldots,t_i\} \\
& & \|(u_i^{t_i+1},\ldots,u_i^{s})\| \le 1\},\end{aligned}$$ this robust optimization problem was first examined in [@Gold] in the special case of robust convex quadratic optimization problems, and then subsequently in [@JLV15] for general robust SOS-polynomial optimization problems. In particular, [@JLV15] showed that the optimal value of (RP) with $\mathcal{U}_i=\mathcal{U}_i^e$ can be found by solving a related semi-definite programming problem (SDP) and raised an open question that how to found an optimal solution of (RP) from the corresponding related SDP.
As we will see, as a simple application of the result in Section 4, we can extend the exact semi-definite programming relaxations result in [@JLV15] to a more general setting and answer the open questions left in [@JLV15] on how to recover a robust solution from the semi-definite programming relaxation in this broader setting.
To do this, we first introduce the notion of restricted spectrahedron data uncertainty set which is a compact set given by $$\begin{aligned}
\mathcal{U}^{s}_i=\{(u_i^{(1)},\ldots,u_i^{(t_i)},u_i^{(t_i+1)},\ldots,u_i^{(s)}) \in \mathbb{R}^{s}&: & A_i^0+\sum_{j=1}^s u_i^{(j)} A_i^j \succeq 0, \\
& & (u_i^{(1)},\ldots,u_i^{(t_i)}) \in \mathbb{R}^{t_i}_+, (u_i^{(t_i+1)},\ldots,u_i^{(s)}) \in \mathbb{R}^{s-t_i} \}.\end{aligned}$$ It is not hard to see that the restricted ellipsoidal uncertainty set is a special case of the restricted spectrahedron data uncertainty set as the norm constraint can be expressed as a linear matrix inequality.
Let $f_0(x)=f(x)$, $g_i(x,u_i)=g_i^{(0)}(x)+\sum\limits_{j=1}^{t_i}u_i^{(j)}g_i^{(j)}(x)+\sum\limits_{j=t_i+1}^su_i^{(j)}g_i^{(j)}(x)$ and $f_i(x)=\sup_{u_i \in \mathcal{U}_i}\{g_i(x,u_i)\}$, $i=1,\ldots,s$. From the construction of the restricted spectrahedron data uncertainty, for each $u_i \in \mathcal{U}_i^s$, $g_i(\cdot,u_i)$ is an SOS-convex polynomial. Moreover, each uncertainty set $\mathcal{U}_i^s$ can be written as $$\begin{aligned}
\mathcal{U}^{s}_i=\{(u_i^{(1)},\ldots,u_i^{(t_i)},u_i^{(t_i+1)},\ldots,u_i^{(s)}) \in \mathbb{R}^{s}&: & \widetilde{A}_i^0+\sum_{j=1}^s u_i^{(j)} \widetilde{A}_i^j \succeq 0\},\end{aligned}$$ where $$\label{A_i}
\widetilde{A}_i^0=\left(\begin{array}{cc}
0_{t_i \times t_i} & 0\\
0 & A_i^0
\end{array}
\right), \widetilde{A}_i^j=\left(\begin{array}{cc}
{\rm diag} \, e_j & 0\\
0 & A_i^j
\end{array}
\right), j=1,\ldots,t_i, \mbox{ and } \widetilde{A}_i^j=\left(\begin{array}{cc}
0_{t_i \times t_i} & 0\\
0 & A_i^j
\end{array}
\right), j=t_i+1,\ldots,s.$$ Here, $e_j \in \mathbb{R}^{n}$ denotes the vector whose $j$th element equals to one and $0$ otherwise. Therefore, we see that the robust convex problem (RP) under the restricted spectrahedron data uncertainty (that is, $\mathcal{U}_i=\mathcal{U}_i^s$) can be regarded as a special SOS-convex semi-algebraic program. Therefore, Theorem \[th:exact\_SDP\] and Theorem \[th:3.2\] can be applied directly to obtain the desired exact SDP relaxation result and the exact solution recovery property. For brevity, we omit the details here. This extends the exact semi-definite programming relaxations result in [@JLV15] to a more general setting and answer the open questions left in [@JLV15] on how to recover a robust solution from the semi-definite programming relaxation in this broader setting.
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[^1]: School of Mathematics and Statistics, University of New South Wales, Sydney NSW 2052, Australia, and Department of Mathematics, Vinh University, Vinh, Nghe An 42118, Vietnam. Email: nhchieu@unsw.edu.au, nghuychieu@vinuni.edu.com.
[^2]: School of Civil and Environmental Engineering, University of New South Wales, Sydney NSW 2052, Australia. E-mail: jinwen.feng@unsw.edu.au
[^3]: School of Civil and Environmental Engineering, University of New South Wales, Sydney NSW 2052, Australia. E-mail: w.gao@unsw.edu.au
[^4]: School of Mathematics and Statistics, University of New South Wales, Sydney NSW 2052, Australia. E-mail: g.li@unsw.edu.au
[^5]: School of Civil and Environmental Engineering, University of New South Wales, Sydney, NSW 2052, Australia. Email: di.wu@unsw.edu.au
[^6]: Research was partially supported by a research grant from Australian Research Council.
|
---
abstract: |
In 1983, Bouchet proposed a conjecture that every flow-admissible signed graph admits a nowhere-zero $6$-flow. Bouchet himself proved that such signed graphs admit nowhere-zero $216$-flows and Zýka further proved that such signed graphs admit nowhere-zero $30$-flows. In this paper we show that every flow-admissible signed graph admits a nowhere-zero $11$-flow.
[**Keywords:**]{} [Integer flow; Modulo flow; Balanced $\mathbb Z_{2} \times \mathbb Z_{3}$-NZF; Signed graph;]{}
author:
- |
Matt DeVos\
Department of Mathematics\
Simon Fraser University, Burnaby, B.C., Canada V5A1S6\
Email: mdevos@sfu.ca\
Jiaao Li, You Lu, Rong Luo, Cun-Quan Zhang[^1], Zhang Zhang\
Department of Mathematics\
West Virginia University\
Morgantown, WV 26505\
Email: {joli,yolu1, rluo, cqzhang}@math.wvu.edu, zazhang@mix.wvu.edu
title: 'Flows on flow-admissible signed graphs '
---
Introduction {#sc: Introduction}
============
Graphs or signed graphs considered in this paper are finite and may have multiple edges or loops. For terminology and notations not defined here we follow [@BM2008; @Diestel2010; @West1996].
In 1983, Bouchet [@Bouchet83] proposed a flow conjecture that [*every flow-admissible signed graph admits a nowhere-zero $6$-flow*]{}. Bouchet [@Bouchet83] himself proved that such signed graphs admit nowhere-zero $216$-flows; Zýka [@Zyka1987] proved that such signed graphs admit nowhere-zero $30$-flows. In this paper, we prove the following result.
\[thm: 11-flow\] Every flow-admissible signed graph admits a nowhere-zero $11$-flow.
In fact, we prove a stronger and very structural result as follows, and Theorem \[thm: 11-flow\] is an immediate corollary.
\[TH: e=10\] Every flow-admissible signed graph $G$ admits a $3$-flow $f_1$ and a $5$-flow $f_2$ such that $f = 3f_1 + f_2$ is a nowhere-zero $11$-flow, $|f(e)| \not = 9$ for each edge $e$, and $|f(e)| = 10$ only if $e \in B(\operatorname{{\rm supp}}(f_1))\cap B(\operatorname{{\rm supp}}(f_2))$, where $B(\operatorname{{\rm supp}}(f_i))$ is the set of all bridges of the subgraph induced by the edges of $\operatorname{{\rm supp}}(f_i)$ $(i=1,2)$.
Theorem \[TH: e=10\] may suggest an approach to further reduce $11$-flows to $9$-flows.
The main approach to prove the $11$-flow theorem is the following result, which, we believe, will be a powerful tool in the study of integer flows of signed graphs, in particular to resolve Bouchet’s $6$-flow conjecture.
\[thm: balanced-6-flow\] Every flow-admissible signed graph admits a balanced nowhere-zero $\mathbb Z_{2} \times \mathbb Z_{3}$-flow.
A $\mathbb{Z}_2\times \mathbb{Z}_3$-flow $(f_1,f_2)$ is called [*balanced*]{} if $\operatorname{{\rm supp}}(f_1)$ contains an even number of negative edges.
The rest of the paper is organized as follows: Basic notations and definitions will be introduced in Section \[se:defintions\]. Section \[se:mod flow\] will discuss the conversion of modulo flows into integer flows. In particular a new result to convert a modulo $3$-flow to an integer $5$-flow will be introduced and its proof will be presented in Section \[se:mod3-5\]. The proofs of Theorems \[TH: e=10\] and \[thm: balanced-6-flow\] will be presented in Sections \[se:proof 11 flow\] and \[se:6-flow\], respectively.
Signed graphs, switch operations, and flows {#se:defintions}
===========================================
Let $G$ be a graph. For $U_1, U_2\subseteq V(G)$, denote by $\delta_{G}(U_1, U_2)$ the set of edges with one end in $U_1$ and the other in $U_2$. For convenience, we write $\delta_G(U_1, V(G)\setminus U_1)$ and $\delta_G(\{v\})$ for $\delta_G(U_1)$ and $\delta_G(v)$ respectively. The degree of $v$ is $d_G(v)=|\delta_G(v)|$. A $d$-vertex is a vertex with degree $d$. Let $V_d(G)$ be the set of $d$-vertices in $G$. The maximum degree of $G$ is denoted by $\Delta(G)$. We use $B(G)$ to denote the set of cut-edges of $G$.
A *signed graph* $(G,\sigma)$ is a graph $G$ together with a [*signature*]{} $\sigma: E(G) \to \{-1,1\}$. An edge $e \in E(G)$ is *positive* if $\sigma(e) =1$ and *negative* otherwise. Denote the set of all negative edges of $(G,\sigma)$ by $E_N(G,\sigma)$. For a vertex $v$ in $G$, we define a new signature $\sigma'$ by changing $\sigma'(e) = -\sigma(e)$ for each $e\in \delta_G(v)$. We say that $\sigma'$ is obtained from $\sigma$ by making a *switch* at the vertex $v$. Two signatures are said to be [*equivalent*]{} if one can be obtained from the other by making a sequence of switch operations. Define the [*negativeness*]{} of $G$ by $\epsilon (G,\sigma)=\min\{|E_N(G,\sigma')| : \mbox{ $\sigma'$ is equivalent to $\sigma$}\}$. A signed graph is balanced if its negativeness is $0$. That is it is equivalent to a graph without negative edges. For a subgraph $G'$ of $G$, denote $\sigma(G') = \prod_{e\in E(G')} \sigma(e)$.
For convenience, the signature $\sigma$ is usually omitted if no confusion arises or is written as $\sigma_G$ if it needs to emphasize $G$. If there is no confusion from the context, we simply use $E_N(G)$ for $E_N(G,\sigma)$ and use $\epsilon (G)$ for $\epsilon (G,\sigma)$.
Every edge of $G$ is composed of two half-edges $h$ and $\hat{h}$, each of which is incident with one end. Denote the set of half-edges of $G$ by $H(G)$ and the set of half-edges incident with $v$ by $H_G(v)$. For a half-edge $h \in H(G)$, we refer to $e_{h}$ as the edge containing $h$. An [*orientation*]{} of a signed graph $(G, \sigma)$ is a mapping $\tau: H(G) \to \{-1,1\}$ such that $\tau(h) \tau(\hat{h}) = -\sigma(e_{h})$ for each $h \in H(G)$. It is convenient to consider $\tau$ as an assignment of orientations on $H(G)$. Namely, if $\tau(h) =1$, $h$ is a half-edge oriented away from its end and otherwise towards its end. Such an ordered triple $(G, \sigma, \tau)$ is called a [*bidirected graph*]{}.
Assume that $G$ is a signed graph associated with an orientation $\tau$. Let $A$ be an abelian group and $f: E(G) \to A$ be a mapping. The [*boundary*]{} of $f$ at a vertex $v$ is defined as $$\partial f (v)=\sum_{h\in H_G(v)} \tau (h) f(e_h).$$ The pair $(\tau,f)$ (or simplify, $f$) is an *$A$-flow* of $G$ if $\partial f (v)=0$ for each $v\in V(G)$, and is an (integer) *$k$-flow* if it is a $\mathbb{Z}$-flow and $|f(e)|<k$ for each $e\in E(G)$.
Let $f$ be a flow of a signed graph $G$. The [support]{} of $f$, denoted by $\operatorname{{\rm supp}}(f)$, is the set of edges $e$ with $f(e)\neq 0$. The flow $f$ is [*nowhere-zero*]{} if $\operatorname{{\rm supp}}(f) = E(G)$. For convenience, we abbreviate the notions of [*nowhere-zero $A$-flow*]{} and [*nowhere-zero $k$-flow*]{} as [*$A$-NZF*]{} and [*$k$-NZF*]{}, respectively. Observe that $G$ admits an $A$-NZF (resp., a $k$-NZF) under an orientation $\tau$ if and only if it admits an $A$-NZF (resp., a $k$-NZF) under any orientation $\tau'$. A $\mathbb Z_k$-flow is also called a modulo $k$-flow. For an integer flow $f$ of $G$ and a positive integer $t$, let $E_{f=\pm t} :=\{e\in E(G) : |f(e)|=t\}$.
A signed graph $G$ is [*flow-admissible*]{} if it admits a $k$-NZF for some positive integer $k$. Bouchet [@Bouchet83] characterized all flow-admissible signed graphs as follows.
[([@Bouchet83])]{}\[flow admissible\] A connected signed graph $G$ is flow-admissible if and only if $\epsilon(G)\neq 1$ and there is no cut-edge $b$ such that $G-b$ has a balanced component.
Modulo flows on signed graphs {#se:mod flow}
=============================
Just like in the study of flows of ordinary graphs and as Theorem \[thm: balanced-6-flow\] indicates, the key to make further improvement and to eventually solve Bouchet’s $6$-flow conjecture is to further study how to convert modulo $2$-flows and modulo $3$-flows into integer flows. The following lemma converts a modulo $2$-flow into an integer $3$-flow.
\[[@CLLZ2018]\] \[TH: 2-to-3\] If a signed graph is connected and admits a $\mathbb{Z}_2$-flow $f_1$ such that $\operatorname{{\rm supp}}(f_1)$ contains an even number of negative edges, then it also admits a $3$-flow $f_2$ such that $\operatorname{{\rm supp}}(f_1)\subseteq \operatorname{{\rm supp}}(f_2)$ and $|f_2(e)| = 2$ if and only if $e \in B(\operatorname{{\rm supp}}(f_2))$.
In this paper, we will show that one can convert a $\mathbb{Z}_3$-NZF to a very special $5$-NZF.
\[lm: 1,2,4-flow\] Let $G$ be a signed graph admitting a $\mathbb Z_{3}$-NZF. Then $G$ admits a $5$-NZF $g$ such that $E_{g=\pm3}=\emptyset$ and $E_{g=\pm4}\subseteq B(G)$.
Theorem \[lm: 1,2,4-flow\] is also a key tool in the proof of the $11$-theorem and its proof will be presented in Section \[se:mod3-5\].
Theorem \[lm: 1,2,4-flow\] is sharp in the sense that there is an infinite family of signed graphs that admits a $\mathbb Z_{3}$-NZF but does not admit a $4$-NZF. For example, the signed graph obtained from a tree in which each vertex is of degree one or three by adding a negative loop at each vertex of degree one. An illustration is shown in Fig. \[FIG: no-3-flow\].
(-0.8,0.5) coordinate (u1);(u1) circle (0.08cm); (0.8,0.5) coordinate (u2);(u2) circle (0.08cm); (-2.4,0.5) coordinate (u3);(u3) circle (0.08cm); (2.4,0.5) coordinate (u4);(u4) circle (0.08cm); (-0.8,-0.5) coordinate (u5);(u5) circle (0.08cm); (0.8,-0.5) coordinate (u6);(u6) circle (0.08cm);
(u1)–(u3); (u1)–(u2); (u4)–(u2); (u1)–(u5); (u6)–(u2);
(-0.8,-1.5) coordinate (v1); (v1) arc (270:90:0.5cm); (-0.8,-1.5) coordinate (v11); (v11) arc (-90:90:0.5cm);
(0.8,-0.5) coordinate (v2); (v2) arc (90:270:0.5cm); (0.8,-0.5) coordinate (v21); (v21) arc (90:-90:0.5cm);
(2.4,0.5) coordinate (v3); (v3) arc (180:0:0.5cm); (2.4,0.5) coordinate (v31); (v31) arc (180:360:0.5cm);
(-3.4,0.5) coordinate (v4); (v4) arc (180:0:0.5cm); (-3.4,0.5) coordinate (v41); (v41) arc (180:360:0.5cm);
Proof of the $11$-flow theorem {#se:proof 11 flow}
==============================
Now we are ready to prove Theorem \[TH: e=10\].
[**Proof of Theorem \[TH: e=10\].**]{} Let $G$ be a connected flow-admissible signed graph. By Theorem \[thm: balanced-6-flow\], $G$ admits a balanced $\mathbb Z_{2} \times \mathbb Z_{3}$-NZF $(g_1, g_2)$. By Lemma \[TH: 2-to-3\], $G$ admits a $3$-flow $f_1$ such that $\operatorname{{\rm supp}}(g_1)\subseteq \operatorname{{\rm supp}}(f_1)$ and $|f_1(e)|=2$ if and only if $e \in B(\operatorname{{\rm supp}}(f_1))$.
By Theorem \[lm: 1,2,4-flow\], $G$ admits a $5$-flow $f_2$ such that $\operatorname{{\rm supp}}(f_2)=\operatorname{{\rm supp}}(g_2)$ and $$E_{f_2=\pm 3}=\emptyset.
\label{EQ: no 3}$$
Since $(g_1,g_2)$ is a $\mathbb{Z}_2\times \mathbb{Z}_3$-NZF of $G$, $${\rm supp}(f_1)\cup{\rm supp}(f_2)=\operatorname{{\rm supp}}(g_1)\cup \operatorname{{\rm supp}}(g_2) = E(G).
\label{EQ: supp}$$
We are to show that $f= 3f_1 + f_2$ is a nowhere-zero $11$-flow described in the theorem. Since $|f_1(e)|\leq 2$ and $|f_2(e)| \leq 4$, we have $$|f(e)| = |(3f_1+f_2)(e)| \leq 3|f_1(e)| + |f_2(e)| \leq 10
~~~~ \forall e\in E(G).$$ Furthermore, by applying Equations (\[EQ: no 3\]) and (\[EQ: supp\]), $$3f_1(e)+ f_2(e) \neq 0, \pm 9 ~~~~ \forall e\in E(G).$$ If $|f(e)| = 10$ for some edge $e \in E(G)$, then $|f_1(e)| = 2$ and $|f_2(e)| = 4$. Thus, by Lemmas \[TH: 2-to-3\] and \[lm: 1,2,4-flow\] again, the edge $e \in B(\operatorname{{\rm supp}}(f_1))\cap B(\operatorname{{\rm supp}}(f_2))$ and hence $f = 3f_1 + f_2$ is the $11$-NZF described in Theorem \[TH: e=10\]. $\Box$
Proof of Theorem \[lm: 1,2,4-flow\] {#se:mod3-5}
===================================
As the preparation of the proof of Theorem \[lm: 1,2,4-flow\], we first need some necessary lemmas.
The first lemma is a stronger form of the famous Petersen’s theorem, and here we omit its proof (see Exercise 16.4.8 in [@BM2008]).
\[pp: matching1\] Let $G$ be a bridgeless cubic graph and $e_0\in E(G)$. Then $G$ has two perfect matchings $M_1$ and $M_2$ such that $e_0\in M_1$ and $e_0\notin M_2$.
We also need a splitting lemma due to Fleischner [@Fleischner1976].
Let $G$ be a graph and $v$ be a vertex. If $F\subset \delta_{G}(v)$, we denote by $G_{[v;F]}$ the graph obtained from $G$ by splitting the edges of $F$ away from $v$. That is, adding a new vertex $v^{*}$ and changing the common end of edges in $F$ from $v$ to $v^{*}$.
\[LE: splitting\] [([@Fleischner1976])]{} Let $G$ be a bridgeless graph and $v$ be a vertex. If $d_{G}(v)\geq 4$ and $e_{0}, e_{1},e_{2}\in \delta_{G}(v)$ are chosen in a way that $e_{0}$ and $e_{2}$ are in different blocks when $v$ is a cut-vertex, then either $G_{[v;\{e_{0}, e_{1}\}]}$ or $G_{[v;\{e_{0}, e_{2}\}]}$ is bridgeless. Furthermore, $G_{[v;\{e_{0}, e_{2}\}]}$ is bridgeless if $v$ is a cut-vertex.
Let $G$ be a signed graph. A path $P$ in $G$ is called a [*subdivided edge*]{} of $G$ if every internal vertex of $P$ is a $2$-vertex. The [*suppressed graph*]{} of $G$, denoted by $\overline{G}$, is the signed graph obtained from $G$ by replacing each maximal subdivided edge $P$ with a single edge $e$ and assigning $\sigma(e)=\sigma(P)$.
The following result is proved in [@Xu2005] which gives a sufficient condition when a modulo $3$-flow and an integer $3$-flow are equivalent for signed graphs.
\[[@Xu2005]\] \[TH: Xu-Zhang-1\] Let $G$ be a bridgeless signed graph. If $G$ admits a $\mathbb{Z}_3$-NZF, then it also admits a $3$-NZF.
Lemma \[TH: Xu-Zhang-1\] is strengthened in the following lemma, which will be served as the induction base in the proof of Theorem \[lm: 1,2,4-flow\].
\[lm: mod-3-flow\] Let $G$ be a bridgeless signed graph admitting a $\mathbb Z_{3}$-NZF. Then for any $e_0\in E(G)$ and for any $i\in \{1, 2\}$, $G$ admits a $3$-NZF such that $e_0$ has the flow value $i$.
Let $G$ be a counterexample with $\beta(G):=\sum_{v\in V(G)}|d_G(v)-2.5|$ minimum. Since $G$ admits a $\mathbb{Z}_3$-NZF, there is an orientation $\tau$ of $G$ such that for each $v\in V(G)$, $$\label{eq-1}
\partial \tau(v) :=\sum_{h\in H_G(v)}\tau(h) \equiv 0\pmod 3.$$
We claim $\Delta(G)\leq 3$. Suppose to the contrary that $G$ has a vertex $v$ with $d_G(v)\ge 4$. By Lemma \[LE: splitting\], we can split a pair of edges $\{e_1,e_2\}$ from $v$ such that the new signed graph $G'=G_{[v; \{e_1,e_2\}]}$ is still bridgeless. In $G'$, we consider $\tau$ as an orientation on $E(G')$ and denote the common end of $e_{1}$ and $e_{2}$ by $v^{*}$. If $\partial \tau(v^*)=0$, then $\beta(G')<\beta(G)$ and by Eq. (\[eq-1\]), $\partial \tau(u)\equiv 0 \pmod 3$ for each $u\in V(G')$, a contradiction to the minimality of $\beta(G)$. If $\partial \tau(v^*)\neq 0$, then we further add a positive edge $vv^{*}$ to $G'$ and denote the resulting signed graph by $G''$. Let $\tau''$ be the orientation of $G''$ obtained from $\tau$ by assigning $vv^*$ with a direction such that $\partial {\tau''}(v^*)\equiv 0 \pmod 3$. Then by Eq. (\[eq-1\]), $\partial {\tau''}(u)\equiv 0 \pmod 3$ for each $u\in V(G'')$. Since $\beta(G'')<\beta(G)$, we obtain a contradiction to the minimality of $\beta(G)$ again. Therefore $\Delta (G) \leq 3$.
Since $G$ is bridgeless, every vertex of $G$ is of degree $2$ or $3$. Note that the existence of the desired $3$-NZFs is preserved under the suppressing operation. Then the suppressed signed graph $\overline{G}$ of $G$ is also a counterexample, and $\beta(\overline{G})<\beta(G)$ when $G$ has some $2$-vertices. Therefore $G$ is cubic by the minimality of $\beta(G)$.
Since $G$ is cubic, by Eq. (\[eq-1\]), either $\partial \tau (v)=d_G(v)$ or $\partial \tau (v)=-d_G(v)$ for each $v\in V(G)$. By Lemma \[pp: matching1\], we can choose two perfect matchings $M_1$ and $M_2$ such that $e_0\notin M_1$ and $e_0\in M_2$. For $i=1, 2$, let $\tau_i$ be the orientation of $G$ obtained from $\tau$ by reversing the directions of all edges of $M_i$, and define a mapping $f_i: E(G) \to \{1,2\}$ by setting $f_i(e)=2$ if $e\in M_i$ and $f_i(e)=1$ if $e\notin M_i$. Then $f_1$ and $f_2$ are two desired $3$-NZFs of $G$ under $\tau_1$ and $\tau_2$, respectively, a contradiction.
Now we are ready to complete the proof of Theorem \[lm: 1,2,4-flow\].
[**Proof of Theorem \[lm: 1,2,4-flow\]**]{} We will prove by induction on $t=|B(G)|$, the number of cut-edges in $G$. If $t=0$, then $G$ is bridgeless and it is a direct corollary of Lemma \[lm: mod-3-flow\]. This establishes the base of the induction.
Assume $t>0$. Let $e=v_1 v_2$ be a cut-edge in $B(G)$ such that one component, say $B_1$, of $G-e$ is minimal. Let $B_2$ be the other component of $G-e$. Since $G$ admits a $\mathbb{Z}_3$-NZF, $\delta(G) \geq 2$. Thus $B_1$ is bridgeless and nontrivial. WLOG assume $v_i\in B_i$ ($i=1,2$). Let $B_i'$ be the graph obtained from $B_i$ by adding a negative loop $e_i$ at $v_i$. Then $B_i'$ admits a $\mathbb{Z}_3$-NZF since $G$ admits a $\mathbb{Z}_3$-NZF. By induction hypothesis, $B_2'$ admits a $5$-NZF $g_2$ with $g_2(e_2)=a\in \{1, 2\}$. By Lemma \[lm: mod-3-flow\], $B_1'$ admits a $3$-NZF $g_1$ such that $g_1(e_1)=a$. Hence we can extend $g_1$ and $g_2$ to a $5$-NZF $g$ of $G$ by setting $g(e)=2a$. Clearly $g$ is a desired $5$-NZF of $G$. $\Box$
Proof of Theorem \[thm: balanced-6-flow\] {#se:6-flow}
=========================================
In this section, we will complete the proof of Theorem \[thm: balanced-6-flow\], which is divided into two steps: first to reduce it from general flow-admissible signed graphs to cubic shrubberies (see Lemma \[lm: reduction\]); and then prove that every cubic shrubbery admits a balanced $\mathbb{Z}_2\times \mathbb{Z}_3$-NZF by showing a stronger result (see Lemma \[lm: water\]).
We first need some terminology and notations. Let $G$ be a graph. For an edge $e\in E(G)$, [*contracting*]{} $e$ is done by deleting $e$ and then (if $e$ is not a loop) identifying its ends. For $S\subseteq E(G)$, we use $G/S$ to denote the resulting graph obtained from $G$ by contracting all edges in $S$.
For a path $P$, let $End(P)$ and $Int(P)$ be the sets of the ends and internal vertices of $P$, respectively. For $U_1, U_2\subseteq V(G)$, a $(U_1, U_2)$-path is a path $P$ satisfying $|End(P)\cap U_i|=1$ and $Int(P)\cap U_i=\emptyset$ for $i=1,2$; if $G_1$ and $G_2$ are subgraphs of $G$, we write $(G_1,G_2)$-path instead of $(V (G_1),V(G_2))$-path. Let $C=v_1\cdots v_r v_1$ be a circuit. A [*segment*]{} of $C$ is the path $v_i v_{i+1}\cdots v_{j-1}v_j \pmod r$ contained in $C$ and is denoted by $v_iCv_j$ or $v_jC^-v_i$. An $\ell$-circuit is a circuit with length $\ell$.
For a plane graph $G$ embedded in the plane $\Pi$, a [*face*]{} of $G$ is a connected topological region (an open set) of $\Pi\setminus G$. If the boundary of a face is a circuit of $G$, it is called a [*facial circuit*]{} of $G$. Denote $[1,k] = \{1,2,\dots,k\}$.
Shrubberies
-----------
Let $G$ be a signed graph and $H$ be a connected signed subgraph of $G$. An edge $e\in E(G)\setminus E(H)$ is called a [*chord*]{} of $H$ if both ends of $e$ are in $V(H)$. We denote the set of chords of $H$ by ${\cal C}_G(H)$ or simply ${\cal C}(H)$, and partition ${\cal C}(H)$ into $$\mathcal{U}(H)=\mathcal{U}_G(H)=\{e\in \mathcal{C}(H) : H+e \mbox{ is unbalanced}\} \mbox{ and } \mathcal{B}(H)=\mathcal{B}_G(H)=\mathcal{C}(H)\setminus \mathcal{U}(H).$$ In particular, if $H$ is a circuit $C$ that either is unbalanced or satisfies $|\mathcal{U}(C)|+|V_2(G)\cap V(C)|\geq 2$, then it is *removable*.
A signed graph $G$ is called a *shrubbery* if it satisfies the following requirements:
- $\Delta (G)\leq 3$;
- every signed cubic subgraph of $G$ is flow-admissible;
- $|\delta_G(V(H))|+\sum_{x\in V(H)}(3-d_G(x))+2|\mathcal{U}(H)|\geq 4$ for any balanced and connected signed subgraph $H$ with $|V(H)|\geq 2$;
- $G$ has no balanced $4$-circuits.
By the above definition, the following result is straightforward.
\[PROP: inheritance\] Every signed subgraph of a shrubbery is still a shrubbery.
Let $G'$ be an arbitrary signed subgraph of $G$. Obviously, $G'$ satisfies (S1), (S2) and (S4). We will show that $G'$ satisfies (S3).
Let $H$ be a balanced and connected signed subgraph of $G'$ with $|V(H)|\ge 2$. Let $A_1=\delta_G(V(H))\setminus \delta_{G'}(V(H))$ and $A_2={\cal C}_G(H)\setminus {\cal C}_{G'}(H)$. Then $$\sum_{x\in V(H)}(3-d_{G'}(x))-\sum_{x\in V(H)}(3-d_{G}(x))=|A_1|+2|A_2|.$$ Since ${\cal U}_{G'}(H) \subseteq {\cal U}_G(H)$, we have $$|{\cal U}_G(H)|-|{\cal U}_{G'}(H)|\leq |A_2|.$$ Since $G$ is a shrubbery, $$|\delta_{G'}(V(H))|+\sum_{x\in V(H)}(3-d_{G'}(x))+2|\mathcal{U}_{G'}(H)|\ge |\delta_G(V(H))|+\sum_{x\in V(H)}(3-d_G(x))+2|\mathcal{U}_G(H)|\ge 4.$$
Therefore $G'$ satisfies (S3) and thus is a shrubbery.
Proposition \[PROP: inheritance\] will be applied frequently in the proof of Lemma \[lm: water\] and thus it will not be referenced explicitly.
The following two theorems and Lemma \[lm: flow extention\] will be applied to reduce Theorem \[thm: balanced-6-flow\].
[([@Seymour81])]{}\[6-flow\] Every ordinary bridgeless graph admits a $6$-NZF.
[([@Tutte54])]{}\[group-flow\] Let $A$ be an abelian group of order $k$. Then an ordinary graph admits a $k$-NZF if and only if it admits an $A$-NZF.
Let $G$ be an ordinary oriented graph, $T\subseteq E(G)$ and $A$ be an abelian group. For any function $\gamma: T\to A$, let ${\cal F}_{\gamma}(G)$ denote the number of $A$-NZF $\phi$ of G with $\phi(e)=\gamma(e)$ for every $e\in T$. For every $X\subseteq V(G)$, let $\alpha_X: E(G)\to \{-1,0,1\}$ be given by the rule $$\alpha_X(e)=\left\{
\begin{array}{rl}
1 & \mbox{if $e\in \delta_G(X)$ is directed toward $X$}\\
-1 & \mbox{if $e\in \delta_G(X)$ is directed away $X$}\\
0 & \mbox{otherwise}.
\end{array}
\right.$$ For any two functions $\gamma_1, \gamma_2$ from $T$ to $A$, we call $\gamma_1, \gamma_2$ [*similar*]{} if for every $X\subseteq V(G)$, the following holds $$\sum_{e\in T}\alpha_X(e)\gamma_1(e)=0 \mbox{ if and only if } \sum_{e\in T}\alpha_X(e)\gamma_2(e)=0.$$
\[Seymour\] (Seymour - Personal communication). Let $G$ be an ordinary oriented graph, $T\subseteq E(G)$ and $A$ be an abelian group. If the two functions $\gamma_1, \gamma_2: T\to A$ are similar, then ${\cal F}_{\gamma_1}(G)={\cal F}_{\gamma_2}(G)$.
We proceed by induction on the number of edges in $E(G)\setminus T$. If this set is empty, then ${\cal F}_{\gamma_i}(G)\leq 1$ and ${\cal F}_{\gamma_i}(G)=1$ if and only if $\gamma_i$ is an $A$-NZF of $G$ for $i=1,2$. Thus, the result follows by the assumption. Otherwise, choose an edge $e\in E(G)\setminus T$. If $e$ is a cut-edge, then ${\cal F}_{\gamma_i}(G)=0$ for $i=1,2$. If $e$ is a loop, then we have inductively that $${\cal F}_{\gamma_1}(G)=(|A|-1){\cal F}_{\gamma_1}(G-e)=(|A|-1){\cal F}_{\gamma_2}(G-e)={\cal F}_{\gamma_2}(G).$$ Otherwise, applying induction to $G-e$ and $G/e$ we have $${\cal F}_{\gamma_1}(G)={\cal F}_{\gamma_1}(G/e)-{\cal F}_{\gamma_1}(G-e)={\cal F}_{\gamma_2}(G/e)-{\cal F}_{\gamma_2}(G-e)={\cal F}_{\gamma_2}(G).$$
The following lemma directly follows from Lemma \[Seymour\].
\[lm: flow extention\] Let $G$ be an ordinary oriented graph and $A$ be an abelian group. Assume that $G$ has an $A$-NZF. If $G$ has a vertex $v$ with $d_G(v)\leq 3$ and $\gamma: \delta_G(v)\to A \setminus \{0\}$ satisfies $\partial \gamma (v)=0$, then there exists an $A$-NZF $\phi$ such that $\phi|_{\delta_G(v)}=\gamma$.
Let $f$ be an $A$-NZF of $G$. Since $d_G(v)\leq 3$, $f|_{\delta_G(v)}$ is similar to $\gamma$. Thus by Lemma \[Seymour\], we have ${\cal F}_{\gamma}(G)={\cal F}_{f|_{\delta_G(v)}}(G)\neq 0$. Therefore there exists an $A$-NZF $\phi$ such that $\phi|_{\delta_G(v)}=\gamma$.
Now we can reduce Theorem \[thm: balanced-6-flow\].
\[lm: reduction\] The following two statements are equivalent.
- Every flow-admissible signed graph admits a balanced $\mathbb{Z}_2\times \mathbb{Z}_3$-NZF.
- Every cubic shrubbery admits a balanced $\mathbb{Z}_2\times \mathbb{Z}_3$-NZF.
\(i) $\Rightarrow$ (ii). By (S2), every cubic shrubbery is flow-admissible, and thus (ii) follows from (i).
\(ii) $\Rightarrow$ (i). Let $G$ be a counterexample to (i) with $\beta (G)=\sum_{v\in V(G)} |d_G(v)-2.5|$ minimum. Since $G$ is flow-admissible, it admits a $k$-NZF $(\tau,f)$ for some positive integer $k$ and thus $V_1(G)=\emptyset$. Furthermore, by the minimality of $\beta(G)$, $G$ is connected and $V_2(G)=\emptyset$ otherwise the suppressed signed graph $\overline{G}$ of $G$ is also flow-admissible and has smaller $\beta(\overline{G})$ than $\beta(G)$. We are going to show that $G$ is a cubic shrubbery and thus admits a balanced $\mathbb{Z}_2\times \mathbb{Z}_3$-NZF by (ii), which is a contradiction to the fact that $G$ is a counterexample. By the definition of shrubberies, we only need to prove (I)-(III) in the following.
[**I.**]{} $G$ is cubic.
Suppose to the contrary that $G$ has a vertex $v$ with $d_G(v)\neq 3$. Then $d_G(v) \geq 4$. Let $\{e_1,e_2\} \subset \delta_G(v)$ and let $G'=G_{[v; \{e_1,e_2\}]}$. Denote the new common end of $e_{1}$ and $e_{2}$ in $G'$ by $v^{*}$. If $\partial f(v^*)=0$, let $G''=G'$. If $\partial f(v^*)\neq 0$, we further add a positive edge $vv^{*}$ with the direction from $v$ to $v^*$ and assign $vv^*$ with the weight $\partial f(v^*)$. Let $G''$ be the resulting signed graph. In both cases, $G''$ is flow-admissible and $\beta(G'')<\beta(G)$. By the minimality of $\beta(G)$, $G''$ admits a balanced $\mathbb{Z}_2\times \mathbb{Z}_3$-NZF, and thus so does $G$, a contradiction. This proves I.
[**II.**]{} $|\delta_G(V(H))|+2|\mathcal{U}(H)|\geq 4$ for any balanced and connected signed subgraph $H$ with $|V(H)|\geq 2$.
Suppose to the contrary that $H$ is such a subgraph with $|\delta_G(V(H))|+2|\mathcal{U}(H)|\leq 3$. Let $X=V(H)$. Then $H'=G[X]-\mathcal{U}(H)$ is a balanced and connected signed subgraph of $G$. WLOG assume that all edges of $H'
$ are positive. Let $G_1=G/E(H')$. Then $G_1$ is also flow-admissible.
Since $|\delta_G(X)|+2|\mathcal{U}(H)|\leq 3$, it follows from the choice of $G$ and Proposition \[flow admissible\] that either $|\mathcal{U}(H)|=0$ and $ |\delta_G(X)|\in \{2, 3\}$ or $|\mathcal{U}(H)|=1$ and $|\delta_G(X)|=1$. Let $x$ be the contracted vertex in $G_1$ corresponding to $E(H')$. Then $d_{G_1}(x)=|\delta_G(X)|+2|\mathcal{U}(H)|\in \{2, 3\}$ and $\beta(G_1)<\beta(G)$ since $|X|=|V(H)|\ge 2$. By the minimality of $\beta(G)$, $G_1$ admits a balanced $\mathbb Z_{2} \times \mathbb Z_{3}$-NZF $(\tau_1, f_1)$, where $\tau_1$ is the restriction of $\tau$ on $G_1$.
Let $H_X$ be the set of the half edges of each edge in $\delta_G(X)\cup \mathcal{U}(H)$ whose end is in $X$. Then $|H_X|=|\delta_G(X)|+2|\mathcal{U}(H)|=2$ or $3$. We add a new vertex $y$ to $H'+H_X$ such that $y$ is the common end of all $h\in H_X$, and denote the new graph by $G_2$. Since $G$ is flow-admissible, $G_2$ is a bridgeless ordinary graph and thus admits a balanced $\mathbb{Z}_2\times \mathbb{Z}_3$-NZF by Theorems \[6-flow\] and \[group-flow\]. Let $\tau_2$ be the restriction of $\tau$ on $G_2$ and define $\gamma (h) = f_1(e_h)$ for each $h\in H_X$. Note that $\tau_2(h)=\tau_1(h)$ for each $h\in H_X$. Since $(\tau_1,f_1)$ is a balanced $\mathbb{Z}_2\times \mathbb{Z}_3$-NZF of $G_1$, we have $\partial \gamma (y) =-\partial f_1(x)= 0$. By Lemma \[lm: flow extention\], there is a balanced $\mathbb{Z}_2\times \mathbb{Z}_3$-NZF $(\tau_2,f_2)$ of $G_2$ such that $f_2|_{\delta_{G_2}(y)}=\gamma=f_1|_{\delta_{G_1}(x)}$. Thus $(\tau_1,f_1)$ can be extended to a balanced $\mathbb{Z}_2\times \mathbb{Z}_3$-NZF of $G$, a contradiction.
This proves (II).
[**III.**]{} $G$ has no balanced $4$-circuits.
Suppose to the contrary that $G$ has a balanced $4$-circuit $C$. Then we may assume that all edges of $C$ are positive. Let $G'=G/E(C)$. Then $\beta(G')<\beta(G)$. By the minimality of $\beta(G)$, $G'$ admits a balanced $\mathbb Z_{2} \times \mathbb Z_{3}$-NZF, say $(f_1', f_2')$. Since $C$ is a circuit with all positive edges and $|E(C)| = 4$ and since $|\mathbb Z_{2} \times \mathbb Z_{3}|=6$, it is easy to extend $(f_1', f_2')$ to a balanced $\mathbb{Z}_2\times \mathbb{Z}_3$-NZF of $G$, a contradiction. This proves (III) and thus completes the proof of the lemma.
Nowhere-zero watering
---------------------
In this subsection, we will prove that every shrubbery admits a nowhere-zero watering (Lemma \[lm: water\]). We need some preparations.
\[th: 1-negative\] [([@MW1966])]{} Let $G$ be a $2$-connected graph with $\Delta(G)\leq 3$ and let $y_1, y_2, y_3\in V(G)$. Then either there exists a circuit of $G$ containing $y_1, y_2, y_3$, or there is a partition of $V(G)$ into $\{X_1, X_2, Y_1, Y_2, Y_3\}$ with the following properties:
- $y_i\in Y_i$ for $i=1,2,3$;
- $\delta_G(X_1, X_2)=\delta_G(Y_i, Y_j)=\emptyset$ for $1\leq i < j \leq 3$;
- $|\delta_G(X_i, Y_j)|=1$ for $i=1,2$ and $j=1,2,3$.
Let $H$ be a contraction of $G$ and let $x\in V(G)$. We use $\hat{x}$ to denote the vertex in $H$ which $x$ is contracted into.
\[thm: linkage\] [([@Lu; @Luo; @Zhang])]{} Let $G$ be a 2-connected signed graph with $|E_N(G)| = \epsilon(G)=k\ge 2$, where $E_N(G)=\{x_1y_1, \dots, x_ky_k\}$. Then the following two statements are equivalent.
- $G$ contains no two edge-disjoint unbalanced circuits.
- The graph $G$ can be contracted to a cubic graph $G'$ such that either $G'-\{ \hat{x}_1 \hat{y}_1, \dots, \hat{x}_k\hat{y}_k \}$ is a $2k$-circuit $C_1$ on the vertices $\hat{x}_1, \dots, \hat{x}_k, \hat{y}_1, \dots, \hat{y}_k$ or can be obtained from a $2$-connected cubic plane graph by selecting a facial circuit $C_2$ and inserting the vertices $\hat{x}_1, \dots, \hat{x}_k, \hat{y}_1, \dots, \hat{y}_k$ on the edges of $C_2$ in such a way that for every pair $\{ i, j \} \subseteq [1,k]$, the vertices $\hat{x_i}, \hat{x_j}, \hat{y_i}, \hat{y_j}$ are around the circuit $C_1$ or $C_2$ in this cyclic order.
\[lm: flow extention2\] [([@Jaeger1992])]{} Let $G$ be an ordinary oriented graph and $A$ be an abelian group. Then $G$ is connected if and only if for every function $\beta: V(G) \to A$ satisfying $\sum_{v\in V(G)} \beta(v) = 0$, there exists $\phi: E(G) \to A$ such that $\partial \phi = \beta$.
Let $G$ be a signed graph with an orientation. A *nowhere-zero watering* (briefly, NZW) of $G$ is a mapping $f: E(G)\to \mathbb Z_{2} \times \mathbb Z_{3}-\{(0, 0)\}$ such that $$\partial f(v) = (0,0) ~\mbox{if} ~ d_G(v) = 3 ~\mbox{and}~ \partial f(v) = (0,\pm 1) ~\mbox{if}~ d_G(v) = 1,2.$$ Similar to flows, the existence of an NZW is also an invariant under switching operation.
\[lm: remove circuit\] Let $G$ be a shrubbery and let $C$ be a removable circuit of $G$. Then for every NZW $f'=(f_1',f_2')$ of $G-V(C)$, there exists an NZW $f=(f_1,f_2)$ of $G$ so that $f(e) = f'(e)$ for every $e\in E(G')$ and $\operatorname{{\rm supp}}(f_1)=\operatorname{{\rm supp}}(f_1')\cup E(C)$.
We first extend $f'$ to $f: E(G)\to \mathbb Z_{2} \times \mathbb Z_{3}$ as follows where $\alpha_e$ is a variable in $\mathbb{Z}_3$ for every $e \in \mathcal{U}(C)$. $$f(e) = \left\{
\begin{array}{ll}
(0,\pm 1) &~\mbox{if $e \in \delta(V(C))$}\\
(1,0) &~\mbox{if $e\in E(C)$}\\
(0,1) &~\mbox{if $e\in \mathcal{B}(C)$}\\
(0,\alpha_e) &~\mbox{if $e\in \mathcal{U}(C)$.}
\end{array}
\right.$$ Since every $v \in V(G)\setminus V(C)$ adjacent to a vertex in $V(C)$ has degree less than three in $G'$, we may choose values $f(e)$ for each edge $e \in \delta(V(C)$ so that $f$ satisfies the boundary condition for a watering at every vertex in $V(G)\setminus V(C)$. Obviously by the construction $\partial f_1(v) = 0$ for every $v \in V(C)$. So we need only adjust $\partial f_2(v)$ for $v \in V(C)$ to obtain a watering. We distinguish the following two cases.
Case 1: $C$ is unbalanced.
In this case $\mathcal{B}(C) = \emptyset$. Choose arbitrary $\pm 1$ assignments to the variables $\alpha_e$. Since $C$ is unbalanced, for every vertex $u\in V(C)$, there is a function $\eta^u: E(C)\to \mathbb{Z}_3$ so that $\partial \eta_u(u)=1$ and $\partial \eta_u(v)=0$ for any $v\in V(C)\setminus \{u\}$. Now we may adjust $f_2$ by adding a suitable combination of the $\eta^u$ functions so that $f$ is an NZW of $G$, as desired.
Case 2: $C$ is balanced.
WLOG we may assume that every edge of $C$ is positive and every unbalanced chord is oriented so that each half edge is directed away from its end. In this case, each negative chord $e$ contributes $-2f_2(e) = \alpha_e$ to the sum $\sum_{v\in V(C)}\partial f_2(v)$. For every $v \in V(C)\cap V_2(G)$, let $\beta_v$ be a variable in $\mathbb{Z}_3$. Since $|\mathcal{U}(C)|+|V_2(G)\cap V(C)|\geq 2$, we can choose $\pm 1$ assignments to all of the variables $\alpha_e$ and $\beta_v$ so that the following equation is satisfied: $$\sum_{v\in V(C)}\partial f_2(v) = \sum_{v\in V(C)\cap V_2(G)}\beta_v.$$ By Lemma \[lm: flow extention2\], we may choose a function $\phi: E(C) \rightarrow \mathbb{Z}_3$ so that $$\partial \phi(v) = \left\{
\begin{array}{rl}
\beta_v - \partial f_2(v) &~\mbox{if $v\in V(C)\cap V_2(G)$}\\
-\partial f_2(v) &~\mbox{if $v\in V(C)\setminus V_2(G)$}.
\end{array}
\right.$$ Now modify $f$ by adding $\phi$ to $f_2$ and then $f$ is an NZW of $G$, as desired.
A *theta* is a graph consisting of two distinct vertices and three internally disjoint paths between them. A theta is *unbalanced* if it contains an unbalanced circuit. By the definition, the following observation is straightforward.
\[ob: no theta\] Let $G$ be a signed graph containing no unbalanced thetas and $\Delta(G) \leq 3$. Then for any unbalanced circuit $C$ and any $x\in V(G)\setminus V(C)$, $G$ contains no two internal disjoint $(x,C)$-paths.
\[lm: water\] Every shrubbery has an NZW. Furthermore, if $G$ is a shrubbery with an unbalanced theta or a negative loop and $\varepsilon\in \{-1,1\}$, then $G$ has an NZW $f = (f_1, f_2)$ such that $\sigma (\operatorname{{\rm supp}}(f_1))=\varepsilon$.
Let $G$ be a minimum counterexample with respect to $E(G)$. Then $G$ is connected.
\[cl: 2-connect\] $G$ is $2$-connected, and thus contains no loops.
[*Proof of Claim \[cl: 2-connect\].*]{} Suppose to the contrary that $G$ has a cut vertex. Since $\Delta(G)\leq 3$, $G$ contains a cut edge $e=v_1 v_2$. Let $G_i$ be the component of $G-e$ containing $v_i$. By the minimality of $G$, each $G_i$ admits an NZW $f^i=(f_1^i, f_2^i)$, and $\partial f_2^i(v_i)\neq 0$ since $d_{G_i}(v_i)\leq 2$. Thus we can obtain an NZW $f=(f_1, f_2)$ of $G$ by setting $f(e)=(0, 1)$ and $f|_{E(G_i)}=f^i$ or $-f^i$ according to the orientation of $e$ and the values of $\partial f_2^1(v_1)$ and $\partial f_2^2(v_2)$. Further, if $G$ contains an unbalanced theta or a negative loop, so does one component of $G-e$, say $G_1$. By the minimality of $G$, we choose $f^1$ such that $\sigma (\operatorname{{\rm supp}}(f_1^1))=\epsilon \cdot \sigma (\operatorname{{\rm supp}}(f_1^2))$. Hence $\sigma (\operatorname{{\rm supp}}(f_1))=\epsilon \cdot \sigma (\operatorname{{\rm supp}}(f_1^2))\cdot \sigma (\operatorname{{\rm supp}}(f_1^2))=\epsilon$, a contradiction. $\Box$
\[cl: remove\] $G$ has no removable circuit $C$ with one of the following properties:
[(A)]{} $G-V(C)$ contains an unbalanced theta.
[(B)]{} $G-V(C)$ is balanced and $\sigma(C)=\epsilon$.
[*Proof of Claim \[cl: remove\].*]{} Suppose the claim is not true. By the minimality of $G$, there exists an NZW $f'=(f_1', f_2')$ of $G-V(C)$ such that $\sigma (\operatorname{{\rm supp}}(f_1'))=\epsilon \cdot \sigma(C)$ in Case (A) and $\sigma (\operatorname{{\rm supp}}(f_1'))=1$ in Case (B). By Lemma \[lm: remove circuit\], $f'$ can be extended to an NZW $f=(f_1, f_2)$ of $G$ such that $\operatorname{{\rm supp}}(f_1)=\operatorname{{\rm supp}}(f_1')\cup E(C)$. Obviously, $\sigma (\operatorname{{\rm supp}}(f_1))=\sigma(\operatorname{{\rm supp}}(f_1'))\cdot \sigma(C)=\epsilon$, a contradiction. $\Box$
\[cl: 2-edge-cut\] Let $X\subset V(G)$ such that $|X| \geq 2$, $G[X]$ is balanced and $|\delta_G(X)|=2$. If $G-X$ either contains an unbalanced theta, or is balanced and contains a circuit, then $X\subseteq V_2(G)$ and $G[X]$ is a path.
[*Proof of Claim \[cl: 2-edge-cut\].*]{} Suppose the claim fails. Let $X\subset V(G)$ be a minimal set with the above properties. Recall that $G$ is $2$-connected by Claim \[cl: 2-connect\]. Since $|\delta_G(X)|=2$, $G[X]$ is connected. If $G[X]$ is a path, then $X\subseteq V_2(G)$. Thus $G[X]$ is not a path. Since $G[X]$ is connected, we have $X\cap V_3(G) \not = \emptyset$. Hence $X$ is nontrivial and $G[X]$ is $2$-connected by the minimality of $X$. By (S3), $X$ contains two vertices of $V_2(G)$. Let $C$ be a circuit in $G[X]$ containing at least two $2$-vertices. Then $C$ is removable and thus by Claim \[cl: remove\]-(A), $G-X$ contains no unbalanced theta. By the hypothesis, $G-X$ is balanced and contains a circuit.
Denote $\delta_G(X) = \{e_1, e_2\}$. Since both $G[X]$ and $G-X$ are balanced, by possibly replacing $\sigma_G$ by an equivalent signature, we may assume that $\sigma_G(e_1) \in \{-1,1\}$ and that $\sigma_G(e) = 1$ for every other edge $e \in E(G)$. Obviously, if $\sigma_G(e_1)=1$ then $G$ is an ordinary graph and so we get a contradiction to Claim \[cl: remove\]-(B) since $C$ is removable and balanced. Hence $\sigma_G(e_1)=-1$ and $e_1$ is the only negative edge in $G$.
Let $C'$ be an unbalanced circuit, which contains $e_1$. Then $C'$ is removable, $G-V(C')$ is balanced, and $\sigma(C') = -1$. By Claim \[cl: remove\]-(B), we have $\epsilon=1$. Since $C$ is removable and $\sigma_G(C)=1=\epsilon$, $G-V(C)$ is unbalanced by Claim \[cl: remove\]-(B) again. We may choose $C'$ such that $V(C') \cap V(C) = \emptyset$. Note that $e_1$ is the unique negative edge of $G$. $C'$ contains the edge cut $\{e_1, e_2\}$. Let $x \in V(C')\cap X$ and $C''$ be a circuit in $G-X$. Then there are two internal disjoint $(x,C'')$-paths $P_1$ and $P_2$ in $G- V(C)$ such that $e_i\in P_i$ for $i = 1,2$. Then $P_1\cup P_2\cup C''$ is an unbalanced theta in $G- V(C)$. This is a contradiction to Claim \[cl: remove\]-(A). $\Box$
\[cl: 3-edge-cut\] Let $X\subset V(G)$ such that $|X| \geq 2$, $G[X]$ is balanced and $|\delta_G(X)|\leq 3$. For any two distinct ends $x_1, x_2$ in $X$ of $\delta_G(X)$, there is an $(x_1,x_2)$-path in $G[X]$ containing at least one vertex in $V_2(G)$.
[*Proof of Claim \[cl: 3-edge-cut\].*]{} Let $x_1x_1', x_2x_2'\in \delta_G(X)$, and $B_i$ be the maximal $2$-connected subgraph of $G[X]$ containing $x_i$ for $i=1, 2$. Then every edge in $\delta_{G[X]}(V(B_i))$ is a bridge of $G[X]$, so $|\delta_{G}(V(B_i))|\leq |\delta_G(X)|$ since $G$ is $2$-connected.
If $V(B_1)\cap V(B_2)\neq \emptyset$, then $|V(B_1)\cap V(B_2)|\ge 2$ since $\Delta(G)\leq 3$, and thus $B_1=B_2$ by their maximality. By (S3), there is a vertex $y_1\in V(B_1)\cap V_2(G)$. Since $B_1$ is $2$-connected, it has a $(y_1,x_1)$-path $P_1$ and a $(y_1,x_2)$-path $P_2$ that are internally disjoint. Thus $P_1\cup P_2$ is a desired path.
If $V(B_1)\cap V(B_2)= \emptyset$, then for some $i\in \{1, 2\}$, say $i=1$, $|\delta_G(V(B_1))|=2$ since $|\delta_{G}(V(B_j))|\leq |\delta_G(X)|\leq 3$ for $j=1,2$. Let $y_2\in V(B_1)$ be the end of the unique edge in $\delta_G(V(B_1))\setminus \{x_1x_1'\}$ and $P_3$ be a $(y_2,x_2)$-path in $G[X]$. If $x_1\in V_2(G)$, then every $(x_1,x_2)$-path is a desired path. If $x_1\in V_3(G)$, then $|V(B_1)|\ge 2$ and thus $B_1$ has a vertex $y_3\in V_2(G)\setminus \{y_2\}$ by (S3). Since $B_1$ is $2$-connected, it has an $(y_3,x_1)$-path $P_4$ and a $(y_3,y_2)$-path $P_5$ which are internally disjoint. Thus $P_3\cup P_4\cup P_5$ is a desired path. $\Box$
\[cl:3-vertices\] $G$ contains no two disjoint unbalanced circuits $C_1$ and $C_2$ such that $V_3 \subseteq V(C_1)\cup V(C_2)$.
[*Proof of Claim \[cl:3-vertices\].*]{} Suppose the claim fails. Let $C_1$ and $C_2$ be two disjoint unbalanced circuits such that $V_3 \subseteq V(C_1)\cup V(C_2)$. Then every vertex of $G'= G-E(C_1\cup C_2)$ is of degree at most $2$. By Claim \[cl: remove\]-(A), $G - V(C_i)$ contains no unbalanced theta for each $i= 1,2$. Thus every nontrivial component of $G'$ is a path with one end in $V(C_1)$ and the other end in $V(C_2)$. Since $G$ is $2$-connected and $\Delta(G) \leq 3$, there are at least two $3$-vertices in each $C_i$.
When $\epsilon=-1$, choose $x_1, x_2$ from $V_3(G)\cap V(C_1)$ such that the segment $P=x_1C_1x_2$ contains all vertices of $V_3(G)\cap V(C_1)$. Let $P_i$ be the path in $G'$ with one end $x_i$ and $y_i$ be the other end of $P_i$ for $i = 1,2$. Since $C_2$ is unbalanced, there is a segment, say $y_1C_2y_2$, of $C_2$ such that the circuit $C=P\cup P_1\cup P_2\cup y_1C_2y_2$ is unbalanced, and thus $C$ is removable. This contradicts Claim \[cl: remove\]-(B) since $G-V(C)$ is a forest (which is balanced).
When $\epsilon=1$, by the minimality of $G$ and since $G''=G-V(C_1\cup C_2)$ is a forest, $G''$ admits an NZW $f'= (f_1', f_2')$ with $\operatorname{{\rm supp}}(f_1')=\emptyset$. By applying Lemma \[lm: remove circuit\] twice, we extend $f' = (f_1', f_2')$ to an NZW $f = (f_1, f_2)$ of $G$ such that $\operatorname{{\rm supp}}(f_1)=E(C_1)\cup E(C_2)$. So $\sigma(\operatorname{{\rm supp}}(f_1)) =\sigma (C_1) \cdot \sigma (C_2)=1$, a contradiction.
\[cl: 2-disjoint\] $G$ contains no two disjoint unbalanced circuits.
[*Proof of Claim \[cl: 2-disjoint\].*]{} Suppose to the contrary that $C_1$ and $C_2$ are two disjoint unbalanced circuits of $G$. By Claim \[cl:3-vertices\], $V_3(G)\setminus V(C_1\cup C_2)\neq \emptyset$.
Let $x\in V_3(G)\setminus V(C_1\cup C_2)$. By Claim \[cl: remove\]-(A) and Observation \[ob: no theta\], there exists a $2$-edge-cut of $G$ separating $x$ from $V(C_1\cup C_2)$. Let $\{e_1,e_2\}$ be such a $2$-edge-cut. Let $$\mathcal{F}=\{e_1\}\cup \{e\in E(G) : \{e,e_1\} \mbox{ is a $2$-edge-cut of $G$}\}$$ and $\mathcal{B}$ be the set of all nontrivial components of $G-\mathcal{F}$. Note that every member of $\cal B$ is $2$-connected. Since $d_G(x) = 3$, there is a $B_0\in \mathcal{B}$ containing $x$. Obviously $B_0$ doesn’t contain $C_1$ or $C_2$, so $|\mathcal{B}| \geq 2$.
Let $B \in \mathcal{B}$. Then $|\delta_G(B)| = 2$. If $B$ is balanced, then by (S3), $B$ contains at least two $2$-vertices and thus contains a circuit containing at least two $2$-vertices which is removable. If $B$ is unbalanced, then $B$ contains an unbalanced circuit which is also is removable. Thus each $B \in \mathcal{B}$ contains a removable circuits. Since $|\mathcal{B}| \geq 2$, by Claim \[cl: remove\]-(A), $B$ is an unbalanced circuit if it is unbalanced. Therefore every $B\in \mathcal{B}$ is either balanced or is an unbalanced circuit. In particular, $C_1$ and $C_2$ are two distinct members of $ \mathcal{B}$ and $|\mathcal{B}| \geq 3$.
Since $G$ is $2$-connected, there is a circuit that contains all edges in $\mathcal{F}$ and goes through every $B \in \mathcal{B}$. We choose such a circuit $C$ with the following properties:
\(1) $\sigma(C) = \epsilon$ (the existence of $C$ is guaranteed since $C_1$ is unbalanced);
\(2) subject to (1), $|V_2(G)\cap V(C -V(C_1))|$ is as large as possible;
\(3) subject to (1) and (2), $|E_N(G)\cap E(C -V(C_1))|$ is as small as possible.
Since each $B$ is either balanced or is an unbalanced circuit, $G- V(C)$ is balanced. Since $\sigma(C) = \epsilon$, by Claim \[cl: remove\]-(B), $C$ is not removable and thus $C$ is balanced.
Let $B \in \mathcal{B}\setminus \{C_1\}$. If $B$ is balanced or is unbalanced but not a circuit of length $2$, then it contains a $2$-vertex. Thus by (2) $C$ contains at least one $2$-vertex in $B$. If $B$ is an unbalanced circuit of length $2$, then by (3), $C$ contains the positive edge in $B$. In this case, since $C$ is balanced, the other edge in $B$ (which is negative) belongs to $\mathcal{U}(C)$. Therefore every $B \in \mathcal{B}\setminus \{C_1\}$ contributes at least $1$ to $|\mathcal{U}(C)|+|V_2(G)\cap V(C)|$. Since $|\mathcal{B}\setminus \{C_1\}| \geq 2$, we have $|\mathcal{U}(C)|+|V_2(G)\cap V(C)|\geq 2$. Hence $C$ is a removable circuit, a contradiction. $\Box$
\[cl: theta\] $G$ contains an unbalanced theta and $\epsilon=1$.
[*Proof of Claim \[cl: theta\].*]{} We first show that $G$ contains an unbalanced theta. Suppose that $G$ contains no unbalanced theta. If $G$ is unbalanced, $G$ contains an unbalanced circuit. If $G$ is balanced, $|V_2(G)|\geq 4$ by (S3) and thus it has a circuit containing at least two $2$-vertices since $G$ is $2$-connected. Hence $G$ has a removable circuit $C$ in either case. By the minimality of $G$, $G -V(C)$ has an NZW and by Lemma \[lm: remove circuit\], we may extend this to a desired NZW of $G$, a contradiction. Therefore $G$ contains an unbalanced theta.
The existence of unbalanced thetas implies that $\epsilon \in \{-1,1\}$. Let $C$ be an unbalanced circuit. By Claim \[cl: 2-disjoint\], $G$ contains no two disjoint unbalanced circuits, and thus $G-V(C)$ is balanced. By Claim \[cl: remove\]-(B), $\epsilon \neq \sigma(C)=-1$, so $\epsilon=1$. $\Box$
\[cl: k=1\] $|E_N(G)|\ge 2$.
[*Proof of Claim \[cl: k=1\].*]{} By Claim \[cl: theta\], $G$ is unbalanced. Suppose to the contrary that $E_N(G)=\{e_0\}$. Let $P$ be the maximal subdivided edge of $G$ containing $e_0$. Let $y_0, y_1$ be the two ends of $P$. Then $Int(P)\subseteq V_2(G)$ and $y_0, y_1\in V_3(G)$. Let $G'=G-Int(P)$ if $Int(P) \not = \emptyset$; Otherwise, let $G' = G - e_0$.
We claim that $G'$ is $2$-connected. Otherwise, let $B$ be the maximal $2$-connected subgraph of $G'$ containing $y_1$. Then $B\neq G'$ and $B$ is nontrivial since $d_G(y_1) = 3$. By the maximality of $B$, $\delta_{G'}(V(B))\neq \emptyset$ in which each edge is a bridge of $G'$. Thus $y_0\in V(G-V(B))$. Since $G$ is $2$-connected by Claim \[cl: 2-connect\], $\delta_G(V(B))$ is a $2$-edge-cut of $G$. Note that $B$ is balanced and $G-V(B)$ is balanced and contains circuits since $y_0\in V_3(G)$. By Claim \[cl: 2-edge-cut\], $V(B)\subseteq V_2(G)$, which contradicts the fact $y_1\in V_3(G)$.
[*(i) $G'$ contains no circuit $C$ with $V(C) \cap \{y_0,y_1\} \not = \emptyset$ and $|V(C)\cap V_2(G)| \geq 2$.*]{}
[*Proof of (i).*]{} Otherwise, $C$ is a removable circuit such that $G-V(C)$ is balanced and $\sigma(C) = 1 =\epsilon$, a contradiction to Claim \[cl: remove\]-(B).
Since $G'$ is a balanced shrubbery, $|V_2(G')|\geq 4$ by (S3) and thus at least two of them, say $y_2$ and $y_3$, also belong to $V_2(G)$. Note $\{y_2,y_3\}\cap \{y_0,y_1\} =\emptyset$. By $(i)$, there is no circuit in $G'$ containing $\{y_1,y_2,y_3\}$. Thus by Theorem \[th: 1-negative\], there is a partition of $V(G')$ into $\mathcal{I}=\{X_1, X_2, Y_1, Y_2, Y_3\}$ such that $y_i\in Y_i$ ($i=1,2,3$), $\delta_{G'}(X_1, X_2)=\delta_{G'}(Y_i, Y_j)=\emptyset$ ($1\leq i < j \leq 3$), and $\delta_{G'}(X_i, Y_j)=e_{ij}$ ($i=1,2$; $j=1,2,3$). For each $Z\in \mathcal{I}$, since $G'$ is $2$-connected and $|\delta_{G'}(Z)|\leq 3$, $G'[Z]$ is connected.
Since $G'$ is $2$-connected and $|\delta_{G'}(Y_j)|= 2$ for $j \in \{2,3\}$, we have the following statement.
[*(ii) For any $\{i,j\} = \{2,3\}$, there is a circuit $C_i$ in $G' - Y_j$ containing $y_1$ and all the edges in $\{e_{11}, e_{1i}, e_{2i}, e_{21}\}$. We choose $C_i$ such that $|V(C_i)\cap V_2(G)|$ is as large as possible. Then by (i), $|V(C_i)\cap V_2(G)| \leq 1$.*]{}
[*(iii) $y_0 \not \in Y_2\cup Y_3$, $Y_2=\{y_2\}$, and $Y_3=\{y_3\}$.*]{}
[*Proof of (iii).*]{} Let $j \in \{2,3\}$. We first show $|Y_j| = 1$ if $y_0\notin Y_j$. WLOG suppose to the contrary $y_0 \not \in Y_3$ and $|Y_3| \geq 2$. Since $y_0 \not \in Y_3$, $|\delta_G(Y_3)|= 2$. By (ii) $C_2$ is a circuit in $G'-Y_3$. Since $G'[Z]$ is connected for each $Z \in \mathcal{I}$, $G'-Y_3$ is connected. Thus there is a $(y_0, C_2)$-path $P'$ in $G'-Y_3$, so $P'\cup P\cup C_2$ is an unbalanced theta in $G - Y_3$. By Claim \[cl: 2-edge-cut\], $Y_3\subseteq V_2(G)$. Thus $G[Y_3]$ is a path and $Y_3\subset V(C_3)$. By the choice of $C_3$, $V(C_3)$ contains at most one $2$-vertex. This implies $|Y_3| = 1$.
Now we show $y_0 \not \in Y_2\cup Y_3$. Otherwise WLOG, assume $y_0 \not \in Y_3$ and $y_0\in Y_2$. Then $Y_3 = \{y_3\}$ and $y_3 \in V_2(G)$. By (S4), $C_3$ is not a balanced $4$-circuit, and thus there is a set $Z\in \{Y_1, X_1, X_2\}$ such that $|V(C_3)\cap Z|\ge 2$. Since $|\delta_G(Z)|=3$, by Claim \[cl: 3-edge-cut\], $(V(C_3)\cap V_2(G))\cap Z\neq \emptyset$. Thus $|V(C_3)\cap V_2(G)|\ge |(V(C_3)\cap V_2(G))\cap Z|+|\{y_3\}|\ge 2$, a contradiction to $(ii)$. This shows $y_0 \not \in Y_2\cup Y_3$ and thus $|Y_2| = |Y_3| = 1$.
[*(iv) $|X_i|=1$ if $y_0\notin X_i$ for any $i\in \{1,2\}$ and thus $ y_0 \in X_1\cup X_2$.* ]{}
[*Proof of (iv).*]{} Suppose that for some $i\in \{1,2\}$ $y_0\notin X_i$ and $|X_i| \ge 2$. WLOG assume $i =1$. Let $x_{1j}$ be the end of $e_{1j}$ in $X_1$ for $j=1,2,3$. Since $\Delta(G)\leq 3$, $x_{11}\neq x_{1j}$ for some $j \in \{2,3\}$. Note that $x_{11}, x_{1j}\in V(C_j)$. Since $|\delta_G(X_1)|= 3$ and $G[X_1]$ is balanced, by Claim \[cl: 3-edge-cut\], $(V(C_j)\cap V_2(G))\cap X_1\neq \emptyset$ by the choice of $C_j$. Since $y_j\in V(C_j)\cap V_2(G)$ by $(iii)$, $|V(C_3)\cap V_2(G)|\ge |(V(C_3)\cap V_2(G))\cap X_1|+|\{y_j\}|\ge 2$, a contradiction to $(ii)$.
If $y_0 \not \in X_1\cup X_2$, then $|X_1| = |X_2| = 1$. By (iii), $G[Y_2\cup Y_3\cup X_1\cup X_2]$ is a balanced $4$-circuit, a contradiction to (S4). Therefore $y_0 \in X_1\cup X_2$.
By (iv), WLOG assume $y_0 \in X_1$. Then by (iv) and (iii), $|X_2| = |Y_2| = |Y_3| = 1$. Denote $X_2=\{x_2\}$.
[*(v) $Y_1=\{y_1\}$.*]{}
[*Proof of (v).*]{} Suppose to the contrary that $Y_1\neq \{y_1\}$. Then $|Y_1|\ge 2$ and $G[Y_1]$ is balanced. Let $C_4$ be a circuit containing all the edges in $\{e_{11}, e_{12}, e_{22}, e_{21}\}$ and $|V(C_4)\cap V_2(G)|$ is as large as possible. Since $G[Y_1]$ is balanced and $|\delta_G(Y_1)|= 3$, by Claim \[cl: 3-edge-cut\], $V(C_4)\cap Y_1\cap V_2(G) \not = \emptyset$. Since $y_2 \in V(C_4)$, $|V(C_4)\cap V_2(G)| \geq 2$. Since $\delta_G(Y_1) \cap C(V) = \{e_{11}, e_{21}\}$ and $|\delta_G(Y_1)| = 3$, $G- V(C_4)$ is balanced. Thus $C_4$ is a removable circuit, a contradiction to Claim \[cl: remove\]-(B). This completes the proof of $(v)$.
Now we can complete the proof of the claim. Let $x_{11}$, $x_{12}$ and $x_{13}$ be the ends of $e_{11}$, $e_{12}$ and $e_{13}$ in $X_1$, respectively. By (S4), $G[\{x_{12},x_{13},x_2,y_2,y_3\}]$ is not a $4$-circuit, so $x_{12}\neq x_{13}$.
If $G'[X_1]$ contains two internally disjoint $(y_0,x_{12})$-path and $(y_0,x_{13})$-path, then $G'$ has a circuit $C_5$ which contains all vertices in $\{y_0, x_{12}, y_2, x_2, y_3, x_{13}\}$ and $\{y_2,y_3\}\subset V(C_5)\cap V_2(G)$, a contradiction to (i). Hence $G[X_1]$ has a cut-edge separating $y_0$ from $\{x_{12},x_{13}\}$.
Let $B_1$ be the maximal $2$-connected subgraphs in $G[X_1]$ containing $y_0$. Then every edge in $\delta_{G[X_1]}(B_1)$ is a cut-edge of $G[X_1]$ by the maximality of $B_1$. Since each $\delta_{G[X_1]}(B_1)$ is a cut-edge of $G[X_1]$ and since $G'$ is $2$-connected and $|\delta_{G'}(X_1)|=3$, $|\delta_{G[X_1]}(B_1)|=1$ or $2$. Since $G[X_1]$ has a cut-edge separating $y_0$ from $\{x_{12},x_{13}\}$, $x_{12}$ and $x_{13}$ are in the same component of $G[X_1]-B_1$. Denote this component by $B_2$.
Let $P'$ be an $(x_{12}, x_{13})$-path in $G[B_2]$. Then $C_6 = P'\cup x_{12}y_2x_2y_3x_{13}$ is a balanced circuit containing at least two $2$-vertices in $G$ ($y_2$ and $y_3$) and thus $C_6$ is a removable circuit of $G$. If $B_1$ has a circuit $C'$ containing $y_0$, then there is $(y_1,C')$-path $P''$ in $G'-V(C_6)$. Recall that $P$ is the maximal subdivided edge in $G$ containing the only negative edge $e_0$. Thus $P\cup P''\cup C'$ is an unbalanced theta in $G'- V(C_6)$, a contradiction to Claim \[cl: remove\]-(A). This implies $B_1$ is trivial and $V(B_1) = \{y_0\}$.
Let $z$ be the neighbor of $y_0$ in $B_2$. Then $ \delta_{G[X_1]}(B_2) = \{y_0z, e_{12}, e_{13}\}$. Since $x_{12}\neq x_{13}$, $z\neq x_{1j}$ for some $j\in \{2,3\}$. Since $|\delta_G(B_2)|=3$, by Claim \[cl: 3-edge-cut\], $G[B_2]$ has a $(z,x_{1j})$-path containing at least one vertex in $V_2(G)$. Note $G[B_2] = G'[B_2]$. Thus $G'$ has a circuit containing $y_0$ and at least two vertices in $V_2(G)$, a contradiction to (i). This completes the proof of Claim \[cl: k=1\]. $\Box$
By Claim \[cl: k=1\], $\epsilon(G)=|E_N(G)|\ge 2$. Denote $\epsilon(G)=k$. By Claim \[cl: 2-connect\] and Theorem \[thm: linkage\], we can choose a minimum subset $S\subseteq E(G)\setminus E_N(G)$ such that $H=G/S$ satisfies the following properties:
- $\Delta(H)\leq 3$;
- $H-N(H)-\cup_{e\in N(H)} Int(P_e)$ is a $2$-connected planar graph with a facial circuit $C$, where $P_e$ is the maximal subdivided edge in $H$ containing $e$;
- $x_1, \dots ,x_k, x_{k+1}, \dots , x_{2k}$ are pairwise distinct and lie in that cyclic order on $C$, where $E_N(H)=E_N(G)=\{e_1, \dots, e_k\}$ and $x_i, x_{k+i}$ are the two ends of $P_{e_i}$ for each $i\in [1,k]$.
For each $v\in V(H)$, let $G_v$ denote the corresponding component of $G-E(H)$. Clearly, $G_v$ is $2$-connected by the minimality of $S$. Moreover, $S=\cup_{v\in V(H)}E(G_v)$ and $E(G)=E(H)\cup S.$
\[cl: k=2\] $k=2$ and $|Int(P_{e_1})|+|Int(P_{e_2})|=1$.
[*Proof of Claim \[cl: k=2\].*]{} Since $k \geq 2$, it is easy to see by Claim \[cl: 2-edge-cut\] and by the minimality of $S$ that if $d_H(x) = 2$ then $G_x= \{x\}$. We first construct a circuit $C_H$ in the following cases. If there are distinct $i,j\in [1,k]$ such that $|Int(P_{e_i})|=|Int(P_{e_j})|=0$, let $C_H=C$; If $|Int(P_{e_i})|+|Int(P_{e_{i+1}})|\ge 2$ for some $i\in [1,k]$, let $C_H=C - E(x_iCx_{i+1}) -E(x_{i+k}Cx_{i+k+1})+P_{e_i} + P_{e_{i+1}}$. Note that $G_v$ is $2$-connected for any $v\in V(H)$, $\Delta(H)\leq 3$ and $\Delta(G)\leq 3$. Then $C_H$ can be extended to a removable circuit $C_G$ of $G$ and $G-V(C_G)$ is also balanced, a contradiction to Claim \[cl: remove\]-(B). So the claim holds. $\Box$
WLOG assume that $Int(P_{e_1})=\emptyset$ and $Int(P_{e_2})=\{y\}$ by Claim \[cl: k=2\]. Then $P_{e_1}=x_1x_3$ and $P_{e_2}=x_2yx_4$. Denote $A_i=x_i C x_{i+1}$ $\pmod 4$ for $i\in [1,4]$, ${C}_1=P_{e_1}\cup A_1\cup P_{e_2}\cup A_3$, and ${C}_2=P_{e_1}\cup A_4\cup P_{e_2}\cup A_2$. Note that both $C_1$ and $C_2$ contain the $2$-vertex $y$.
\[cl: H=G\] $H=G$ and $V_2(G)=\{y\}$.
[*Proof of Claim \[cl: H=G\].*]{} As noted in the proof of Claim \[cl: k=2\], $G_y = \{y\}$. Let $x \in V(C)$. WLOG assume $x \in V(C_1)$. Suppose that $G_x$ is nontrivial. Then $G_x$ is balanced and $|\delta_G(G_x)|\leq 3$. Since $G_x$ is $2$-connected, by Claim \[cl: 3-edge-cut\], ${C}_1$ can be extended to a circuit $C$ of $G$ such that $C$ contains the $2$-vertex $y$ and one $2$-vertex in $G_x$. Thus $C$ is balanced and removable and $G-V(C)$ is balanced, a contradiction to Claim \[cl: remove\]-(B). Hence, $G_x$ is trivial.
Assume that there exists a vertex $u\in (V(G)\setminus (V(C)\cup \{y\})) \cap V_2(G)$. Since $G$ is $2$-connected, there are two internal disjoint $(u,C)$-paths $Q_1$ and $Q_2$ with $v_1$ and $v_2$ the end vertices in $C$ respectively. Since $\Delta(G) \leq 3$, $v_1\not = v_2$. Let $C_3=Q_1 \cup Q_2 \cup v_1 C v_2$ and $C_4 \in \{{C}_1, {C}_2\}$ such that $V(C_4)\cap \{v_1, v_2\}
\neq \emptyset$. Then $C'=C_3 \Delta C_4$ is a circuit containing two $2$-vertices and the two negative edges. Thus $C$ is balanced and removable and $G-V(C')$ is balanced, which contradicts Claim \[cl: remove\]-(B). Thus $V_2(G)=\{y\}$.
Let $x$ be a $3$-vertex in $ V(H)\setminus V(C)$. If $G_x$ is nontrivial, then $G_x$ is balanced and $|\delta_G(G_x)| = 3$. By (S3), $G_x$ contains a $2$-vertex, a contradiction to the fact that $y$ is the only $2$-vertex in $G$. Thus $G_x$ is trivial and therefore $H=G$. $\Box$
\[cl: segment\] $Int(A_i)\neq \emptyset$ for each $i\in [1,4]$.
[*Proof of Claim \[cl: segment\].*]{} Suppose to the contrary that there is some $i\in [1,4]$, say $i=1$, such that $Int(A_1)= \emptyset$. Then $A_1$ is a chord in $\mathcal{U}(C_2)$. Since $C_2$ contains the $2$-vertex $y$, ${C}_2$ is a removable circuit of $G$, a contradiction to Claim \[cl: remove\]-(B) since $G-V(C_2)$ is balanced. $\Box$
**The final step.**
By Claim \[cl: segment\], let $y_1\in Int(A_1)$ be the neighbor of $x_1$. Let $Q$ be the component of $G-E(C)$ containing $y_1$. Since $d_G(y_1)=3$ by Claim \[cl: H=G\], $Q$ is nontrivial. Obviously, $V(Q)\cap \{x_1,x_2,x_3,x_4\}=\emptyset$ since $\Delta(G)=3$.
If there is a vertex $y_2$ in $V(Q)\cap (Int(A_2)\cup Int(A_3))$, let $P$ be a $(y_1,y_2)$-path in $Q$. Since $\Delta(G)\leq 3$, $C_3=P\cup y_1Cy_2$ is a circuit containing $x_2$. Then $C'={C}_2 \bigtriangleup C_3$ is a circuit of $G$ containing $y$ and the chord $x_1y_1 \in \mathcal{U}(C')$. Thus $C'$ is a removable circuit of $G$, a contradiction to Claim \[cl: remove\]-(B) since $G-V(C')$ is balanced.
If $V(Q)\cap (Int(A_2)\cup Int(A_3))=\emptyset$, then $V(Q)\cap V(C)\subseteq Int(A_4)\cup Int(A_1)$. Note that $|V(Q)\cap V(C)|\ge 2$ since $G$ is $2$-connected. Let $y_2, y_3\in V(Q)\cap V(C)$ be two ends of a segment $P'$ of $A_4\cup A_1$ such that the length of $P'$ is as large as possible. By Claim \[cl: H=G\], $G'=G-x_1x_3-y$ is a $2$-connected planar graph with a facial circuit $C$, and so $T'=\delta_{G'}(V(P'))\cap E(C)$ is a $2$-edge-cut of $G'$. Let $T=T'$ if $y_2, y_3\in Int(A_1)$, and otherwise $T=T'\cup \{x_1x_3\}$. Then $T$ is an edge-cut of $G$ with $|T| \leq 3$ and the component of $G-T$ containing $y_2$ is balanced and doesn’t contain $y$. By (S3), this component contains a $2$-vertex (distinct from $y$), which contradicts $V_2(G)=\{y\}$ by Claim \[cl: H=G\]. This completes the proof of Lemma \[lm: water\].
[s2]{}
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[^1]: This research project has been partially supported by an NSA grant H98230-14-1-0154, an NSF grant DMS-1264800
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---
abstract: 'We formulate $\mathcal{N}=2$ global supersymmetric Lagrangians of self-interacting vector multiplets in terms of variant multiplets, whose non-propagating fields are replaced with gauge three-forms. Setting the three-forms on-shell results in a dynamical generation of the parameters entering the scalar potential. As an application, we study how gauge three-forms may determine the partial breaking of $\mathcal{N}=2$ supersymmetry and how they affect the low energy effective description.'
author:
- 'Niccolò Cribiori$^a$ and Stefano Lanza$^{b}$'
title: |
[DFPD-2018/TH/02]{}
\
\
On the dynamical origin of parameters in $\mathcal{N}=2$ Supersymmetry
---
*$^a$ Institute for Theoretical Physics, TU Wien,\
Wiedner Hauptstrasse 8-10/136, A-1040 Vienna, Austria\
$^b$ Dipartimento di Fisica e Astronomia “Galileo Galilei”, Università degli Studi di Padova\
& I.N.F.N. Sezione di Padova, Via F. Marzolo 8, 35131 Padova, Italy*
Introduction and motivation
===========================
In four spacetime dimensions gauge three-forms do not carry any propagating degrees of freedom, nevertheless they can induce non-trivial physical effects. Their importance was recognized in cosmology, where they have been employed to provide a dynamical way to generate the cosmological constant [@Hawking:1984hk; @Brown:1987dd; @Brown:1988kg; @Duncan:1989ug; @Bousso:2000xa]. More recently, gauge three-forms have been used as tools to investigate gauge axionic shift symmetries, both for introducing new inflationary models [@Kaloper:2008fb; @Kaloper:2011jz; @Marchesano:2014mla; @Dudas:2014pva; @Valenzuela:2016yny] and for addressing the strong $CP$-problem [@Dvali:2005an; @Dvali:2004tma; @Dvali:2005zk; @Dvali:2013cpa; @Dvali:2016uhn; @Dvali:2016eay]. Indeed gauge three-forms have been embedded in theories enjoying global or local supersymmetry [@Duff:1980qv; @Stelle:1978ye; @Aurilia:1980xj; @Gates:1980ay; @Gates:1980az; @Buchbinder:1988tj; @Binetruy:1996xw; @Ovrut:1997ur; @Kuzenko:2005wh; @Nishino:2009zz; @Duff:2010vy; @Groh:2012tf; @Aoki:2016rfz; @Nitta:2018yzb]. In this context, they might provide new insights for nonlinear realizations of supersymmetry [@Farakos:2016hly; @Buchbinder:2017vnb; @Buchbinder:2017qls; @Kuzenko:2017vil] and for their natural coupling to membranes which, firstly explored in cosmology [@Brown:1987dd; @Brown:1988kg], was also extended in supersymmetric theories [@Ovrut:1997ur; @Bandos:2010yy; @Bandos:2011fw; @Bandos:2012gz; @Bandos:2018gjp].
More generically, gauge three-forms can be seen as counterparts of constant parameters appearing in four-dimensional effective theories, for which they provide a dynamical origin. For example, in [@Bousso:2000xa; @Bielleman:2015ina; @Farakos:2017jme; @Farakos:2017ocw; @Carta:2016ynn; @Herraez:2018vae] the parameters appearing in the $F$-term potential of Type II string theory compactified over Calabi-Yau three-folds were interpreted as expectation values of the field-strengths of some gauge three-forms. It is in fact of prominent importance that all the parameters of the effective field theories stemming from string theory, but the string length, can be understood as expectation values of some fields and, in this respect, the presence of gauge three-forms may come to help.
In this spirit, in [@Farakos:2017jme; @Farakos:2017ocw] four-dimensional theories with $\mathcal{N}=1$ global and local supersymmetry have been analysed. A procedure has been given to construct Lagrangians encoding gauge three-forms, which are on-shell equivalent to a generic class of chiral models. In particular, the chiral superfields in the latter are substituted in the former by variant versions, which contain gauge three-forms as highest components. Once such three-forms are integrated out, parameters are introduced which contribute to the superpotential of the given chiral model.
In this work we elaborate on these results in two directions. First of all, we extend the procedure of [@Farakos:2017jme; @Farakos:2017ocw] in order to incorporate also $\mathcal{N}=1$ vector superfields. This will allow us to construct models in which the Fayet–Iliopoulos parameter is generated dynamically. Secondly, we investigate $\mathcal{N}=2$ rigid supersymmetric Lagrangians for $\mathcal{N}=2$ vector multiplets and we reformulate them in terms of new variant multiplets containing gauge three-forms as non-propagating degrees of freedom. Since an $\mathcal{N}=2$ vector superfield contains three real auxiliary fields, as a consequence of the procedure we propose, for any such a superfield three real parameters are going to be generated dynamically. On the contrary to the $\mathcal{N}=1$ case, in which only specific parameters of a given superpotential can be interpreted as vacuum expectation values of the field strength of gauge three-forms, the approach in the $\mathcal{N}=2$ case is covering a more general situation, in which the entire potential has a dynamical origin. In other words, an $\mathcal{N}=2$ off-shell Lagrangian for variant vector multiplets will contain no parameters at all, since they are all going to be introduced when integrating out the non-propagating gauge three-forms.
As an application, the partial breaking of global supersymmetry is reviewed [@Antoniadis:1995vb; @Ferrara:1995xi; @Bagger:1996wp; @Fujiwara:2004kc; @Fujiwara:2005hj; @Ambrosetti:2009za; @Kuzenko:2015rfx; @Antoniadis:2017jsk; @Antoniadis:2018blk; @Farakos:2018aml] and reconstructed in the formulation with variant multiplets. The mutual orientation of the gauge three-forms will dictate whether the original, off-shell $\mathcal{N}=2$ supersymmetry may be partially broken to $\mathcal{N}=1$, once the gauge three-forms are set on-shell. We then construct an effective field theory with partially broken supersymmetry, which leads to an action of the Born–Infeld type [@Bagger:1996wp; @Rocek:1997hi], in which the supersymmetry breaking parameter appearing in front of the Lagrangian is generated dynamically. The information contained in this effective description is entirely encoded into boundary terms, which are necessary in the presence of gauge three-forms.
Throughout this work we use the superspace conventions of [@Wess:1992cp] and, even in the case of extended supersymmetry, most of the calculation are performed at the $\mathcal{N}=1$ superspace level for convenience.
Three-forms in $\mathcal{N}=1$ global supersymmetry {#sec:N1case}
===================================================
In four dimensions, gauge three-forms can be accommodated inside the auxiliary components of $\mathcal{N}=1$ superfields. Variant formulations of chiral and vector superfields have been constructed in [@Gates:1983nr], where the usual complex or real scalar auxiliary fields are exchanged with (the Hodge-dual of) field strengths of these three-forms. In this section, following the method introduced in [@Farakos:2017jme], Lagrangians are constructed for variant superfields, which are on-shell equivalent to the standard ones. One of the advantages of dealing with these alternative Lagrangians resides in the fact that the parameters appearing inside them, as for example in the superpotential or in the Fayet–Iliopoulos term, are going to be generated dynamically as vacuum expectation values. In addition, the variant $\mathcal{N}=1$ superfields introduced in this section will be essential ingredients to construct variant $\mathcal{N}=2$ superfields in the next sections.
Double three-form chiral multiplets and dynamical generation of the linear superpotential {#sec:N1caseC}
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Three-form multiplets are chiral multiplets whose non-propagating degrees of freedom are encoded into gauge three-forms, rather than complex scalar fields. Both single and double three-form multiplets can be constructed [@Gates:1980ay; @Gates:1980az; @Gates:1983nr] in which, respectively, one or two real non-propagating degrees of freedom are replaced by three-forms. In [@Farakos:2017jme] it was shown how to dynamically pass from ordinary chiral multiplets to three-form multiplets at the Lagrangian level. The construction is reviewed here for the case of the double three-form multiplet and by looking at a simple example.
Consider a chiral multiplet $X$. It can be expanded in chiral coordinates in superspace as $$X = \varphi + \sqrt 2 \, \theta \psi +\theta^2 f,$$ where $\varphi$ is a complex scalar, $\psi$ a Weyl fermion and $f$ a complex scalar auxiliary field.[^1] The most general Lagrangian, up to two derivatives, which can be constructed solely in terms of this ingredient is $$\label{N1L}
\mathcal{L}= \int d^4\theta \,K(X, \bar X) + \left(\int d^2 \theta\, W(X) + c.c\right)\,,$$ where $K(X,\bar X)$ is the Kähler potential and $W(X)$ is the superpotential, which is a holomorphic function of $X$. Without loss of generality, the superpotential can be rewritten as $$W(X) = c\, X + \hat W(X)\,,$$ where $c$ is a complex constant and the function $\hat W(X)$ is holomorphic. The bosonic components of are $$\label{N1bosA}
\mathcal{L} \big|_{\text{bos}} = -K_{\varphi \bar \varphi} \partial_m \varphi \partial^m \bar \varphi + K_{\varphi \bar \varphi} f \bar f+ \left[(c + \hat W_\varphi(\varphi)) f + \text{c.c.}\right]\,$$ and, setting the auxiliary field $f$ on-shell $$f = -\frac{\bar c+\bar{\hat{W}}_{\bar \varphi}}{K_{\varphi\bar\varphi}}\,,$$ the Lagrangian becomes $$\label{N1bosON}
\mathcal{L} \big|_{\text{bos, on-shell}} = -K_{\varphi \bar \varphi} \partial_m \varphi \partial^m \bar \varphi - \mathcal{V}(\varphi,\bar\varphi)\,,$$ where the scalar potential is $$\mathcal{V}(\varphi,\bar\varphi) = \frac{1}{K_{\varphi\bar\varphi}}\left|c + \hat W_\varphi(\varphi)\right|^2.$$
As shown in [@Farakos:2017jme], the Lagrangian can be thought of as originating from a parent Lagrangian for the double three-form multiplet, in which the parameter $c$ is generated dynamically. To construct such a Lagrangian we can start from $$\label{N1LB}
\mathcal{L}= \int d^4\theta K(X, \bar X) + \left(\int d^2 \theta\, \left(\Phi X + \frac14 \bar D^2 (\Sigma \bar \Phi) \right)+\int d^2 \theta\, \hat W(X) + {\rm c.c.}\right)\,,$$ where, with respect to the , the linear part of the superpotential has been promoted as $$\int d^2\theta\, c\,X \rightarrow \int d^2 \theta\, \left( \Phi X + \frac14 \bar D^2 (\Sigma \bar \Phi) \right)\,.$$ Here $\Phi$ is a chiral superfield with no kinetic terms, which will ultimately play the role of Lagrange multiplier, while $\Sigma$ is a complex linear multiplet, namely a complex scalar multiplet which is constrained by $$\label{ConS}
\bar D^2 \Sigma = 0.$$ Its superspace expansion is $$\begin{split}
\Sigma = &\sigma + \sqrt 2\theta \psi + \sqrt{2} \bar\theta \bar\rho - \theta \sigma_m \bar\theta \mathcal{B}^{m} + \theta^2 \bar s+ \sqrt 2 \theta^2\bar\theta \bar\zeta \\
&-\frac{i}{\sqrt{2}} \bar\theta^2 \theta \sigma^m \partial_m \bar\rho + \theta^2\bar\theta^2 \left(\frac{i}{2} \partial_m \mathcal{B}^{m} -\frac14 \Box \sigma \right),
\end{split}
\label{CompS}$$ where $\sigma$ and $s$ are complex scalar fields, $\psi$, $ \rho$ and $\zeta$ Weyl fermions, while the complex vector $\mathcal{B}^m$ can be interpreted as being the Hodge-dual of a complex three-form $\mathcal{B}_3 = \frac{1}{3!}\mathcal{B}_{mnl} dx^m \wedge dx^n \wedge dx^l$ as $$\mathcal{B}^m=\frac{1}{3!}\epsilon^{mnlp}\mathcal{B}_{nlp}\,.$$
As a consistency check, it is possible to integrate out $\Sigma$ from and recover the original Lagrangian . Since the superfield $\Sigma$ is constrained, it is not possible to take directly its variation. However, the constraint can be solved as $$\Sigma = \bar D_{\dot \alpha} \bar \Psi^{\dot \alpha}\,,$$ with $\bar \Psi^{\dot \alpha}$ an unconstrained spinorial superfield. The variation with respect to $\bar \Psi^{\dot \alpha}$ produces $$\bar D_{\dot \alpha} \bar \Phi = 0$$ and, since $\Phi$ chiral, the only possibility is that $$\label{N1solX}
\Phi=c\,,$$ with $c$ an arbitrary complex constant. Plugging into we thus obtain the Lagrangian .
On the other hand, it is possible to integrate out from both the Lagrange multiplier $\Phi$ and the chiral multiplet $X$. The variation with respect to $X$ gives the superspace equations of motion $$\label{N1solXb}
\begin{split}
\Phi &= \frac14 \bar D^2 K_X - \hat W_X\,,
\end{split}$$ while the variation with respect to the Lagrange multiplier replaces the old chiral superfield with a new one, which is expressed in terms of the complex linear multiplet $\Sigma$ $$\label{N1solPhi}
\begin{split}
X &= -\frac14 \bar D^2 \bar \Sigma \equiv S \,.
\end{split}$$ The superfield $S$ is called *double three-form multiplet* and it has been constructed dynamically from the Lagrangian . It is chiral and it can be expanded in superspace as $$S = \varphi^S + \sqrt 2 \, \theta \psi^S + \theta^2 f^S.$$ Its components, in terms of those of $\Sigma$, are (see Table \[tab:comp1\]) $$\begin{aligned}
\label{compSintermsofSigma1}
\varphi^S &= s,\\
\label{compSintermsofSigma2}
\psi^S_\alpha &= \zeta_\alpha + \frac{1}{2} i \sigma^m_{\alpha \dot \beta}\partial_m \bar \psi^{\dot \beta}\,, \\
\label{compSintermsofSigma3}
f^S &=-i{}^*\! \bar F_4 = -i \partial_m \bar{\mathcal{B}}^m,\end{aligned}$$ with $${}^*\!F_4 = \frac{1}{4!}\varepsilon^{klmn} F_{klmn},\qquad F_{klmn} = 4 \partial_{[k} \mathcal{B}_{lmn]}\,.$$ In addition, with respect to the standard chiral superfield $X$, the multiplet $S$ is invariant under the shift $$\label{Sgauge}
\Sigma \rightarrow \Sigma + L_1 + i L_2,$$ where $L_1$ and $L_2$ are real linear superfields. As a consequence, the complex three-form $\mathcal{B}_{klm}$ undergoes a gauge transformation of the type $$\mathcal{B}_{klm} \rightarrow \mathcal{B}_{klm} + 3\partial_{[k} \left( \Lambda_1 + i \Lambda_2\right)_{lm]}\,.$$ where $\Lambda_{1\,mn}$ and $\Lambda_{2\,mn}$ are components of real gauge two-forms. In this sense, the complex linear superfield $\Sigma$ contains a gauge three-form among its components. Moreover, with an appropriate gauge choice, it is possible to set $\psi_\alpha = \rho_\alpha=0$ and the fermionic component of $S$ becomes $\psi^S_\alpha = \zeta_\alpha$. Therefore, the double three-form multiplet $S$ shares the same degrees of freedom as a chiral multiplet, even though its auxiliary component is not a complex scalar field, but it is the Hodge-dual of the field strength of the gauge three-form $\mathcal{B}_{klm}$.
Plugging and into , the desired Lagrangian in terms of $S$ is obtained $$\label{N1L3}
\mathcal{L}= \int d^4\theta\, K(S, \bar S) + \left(\int d^2 \theta\, \hat W(S) + c.c\right)+ \mathcal{L}_{\text{bd}}\,,$$ where $$\label{N1Sbd}
\mathcal{L}_{\rm bd}=\frac14 \left(\int d^2\theta \bar D^2 - \int d^2\bar\theta D^2\right) \left[\left(\frac14 D^2 K_{\bar S} - \bar{\hat{W}}_{\bar S}\right) \Sigma\right]
+\text{c.c.} \,$$ are boundary terms which are necessary to ensure the correct variation of the action with respect to the gauge three-form [@Brown:1987dd; @Brown:1988kg; @Groh:2012tf]. The bosonic components of are $$\label{N1bosB}
\mathcal{L} \big|_{\text{bos}} = -K_{S \bar S} \partial_m s \partial^m \bar s - \frac{1}{4!} K_{S \bar S} F^{klmn} \bar F_{klmn} + \left[-i \hat{W}_S(s) {}^*\!\bar F_4 + c.c.\right] + \mathcal{L}_{\text{bd}},$$ where $$\label{N1bosBbd}
\mathcal{L}_{\text{bd}} = \frac{1}{3!} \partial_k \left[i \mathcal{B}_{lmn} \left(-iK_{S \bar S} \bar F^{klmn} - \varepsilon^{klmn}\bar{\hat{W}}_S(s) \right)\right] + c.c.\,.$$ As it can be shown from , indeed the boundary terms cancel those originating from the variation of the action defined by with respect to the gauge three-form. The potential of the parent Lagrangian can be obtained by setting the gauge three-form on-shell: $$\label{N1bosFsola}
\partial_k \left[iK_{S \bar S} F^{klmn} - \varepsilon^{klmn}\hat{W}_S(s)\right] = 0$$ whence $$\label{N1bosFsol}
iF_{klmn} = \frac{c+\hat{W}_S(s)}{K_{S \bar S}} \varepsilon_{klmn}\,,$$ with $c$ an arbitrary complex constant. Plugging the solution in we obtain exactly the model . The advantage of starting from the Lagrangian , rather than , is that no supersymmetry breaking parameter appears. In fact, in the complex constant $c$ is dynamically generated by solving the equation of motion for the gauge three-form. In other words, the constant $c$ has been promoted to the vacuum expectation value of the particular combination which appears in or, equivalently, the choice of such a parameter has been traded for the specification of the boundary condition for the gauge three-form.
--------------- ----------- ----------------- ---------------------
Superfield Spin-$0$ Spin-$\frac12$ Non-propagating
\[0.5ex\] $X$ $\varphi$ $\psi_\alpha$ $f$
$S$ $s$ $\psi^S_\alpha$ $\mathcal{B}_{mnp}$
--------------- ----------- ----------------- ---------------------
: The off-shell degrees of freedom of the ordinary chiral superfield $X$ and of the double three-form multiplet $S$. The complex auxiliary field $f$ of $X$ is replaced by a complex gauge three-form in the variant version.[]{data-label="tab:comp1"}
Three-form vector multiplet and dynamical realization of the Fayet–Iliopoulos term {#sec:N1caseV}
----------------------------------------------------------------------------------
In this subsection, we extend the procedure of [@Farakos:2017jme] to the case of a Fayet–Iliopoulos parameter, which is going to be dynamically generated as vacuum expectation value of a real gauge three-form. The discussion is again performed at the Lagrangian level and a parent Lagrangian for a variant vector multiplet will be introduced. Such a multiplet, which has been previously constructed for example in [@Gates:1983nr; @Antoniadis:2017jsk], accomodates a real gauge three-form as non-propagating field, in analogy with .
Given a vector multiplet $V$, the minimal Lagrangian which can be built is $$\label{VLagS}
\mathcal{L} =\left(\frac14 \int d^2\theta\, W^\alpha W_\alpha + \text{c.c.}\right)+ \xi \int d^4\theta\, V$$ where $\xi \in \mathbb{R}$ is the so called Fayet–Iliopoulos parameter. Focusing only on the bosonic sector for simplicity, the components reads $$\label{VLagOffSa}
\mathcal{L} |_{\text{bos}}= - \frac 14 F^{mn}F_{mn}+\frac12 {\rm D}^2 +\frac{\xi}{2}\, {\rm D}\,,$$ where we have neglected the total derivative which involves the gauge field and defined $F_{mn} \equiv 2 \partial_{[m} v_{n]}$. Setting the auxiliary field ${\rm D}$ on-shell $${\rm D} = -\frac{\xi}{2}\,,$$ we get $$\label{VLagOnSa}
\mathcal{L}|_{\text{bos, on-shell}} = - \frac 14 F^{mn}F_{mn} -\frac{\xi^2}{8}$$ with a constant, semi-positive definite potential $\mathcal{V} = \frac{\xi^2}{8}$.
In the spirit of the previous discussion, instead of considering , we can start from the parent Lagrangian $$\begin{split}
\label{VLagMaster}
\mathcal{L} =&\frac14 \left(\int d^2\theta\, W^\alpha W_\alpha + \text{c.c.}\right) -\frac18 \int d^2\theta\,\bar D^2 (\Lambda V) -\frac18 \int d^2\bar\theta\, D^2 (\Lambda V)
\\
&+\frac18 \left[ \int d^2\theta \bar D^2 (\Lambda \Sigma)+\int d^2\bar\theta D^2 (\Lambda \bar \Sigma)\right]\,,
\end{split}$$ which is obtained by promoting the Fayet–Iliopoulos parameter $\xi$ to a real Lagrangian multiplier $\Lambda$ and conveniently adding new terms which contain the complex linear multiplet $\Sigma$ encoding the three-form.
First, we check that from this Lagrangian we get the usual Lagrangian . This can be achieved by eliminating the dependence in on the gauge three-form. By integrating out the unconstrained spinorial superfields $\Psi^\alpha$ and $\bar\Psi^{\dot \alpha}$, such that $\Sigma = \bar D_{\dot \alpha} \bar \Psi^{\dot \alpha}$, we get $$D_\alpha \Lambda = 0, \quad \bar D_{\dot\alpha} \Lambda = 0\,,$$ whence $\Lambda$ is just a real constant $\xi$ $$\Lambda = \xi\,.$$ Once inserted into , the ordinary vector multiplet Lagrangian is recovered.
Let us now follow a second path in order to express the parent Lagrangian solely in terms of a new real three-form multiplet. The variation of the Lagrangian with respect to the Lagrangian multiplier $\Lambda$ gives $$\label{VsolV}
V = \frac{\Sigma + \bar\Sigma}{2}\equiv U,$$ which is a real multiplet containing a gauge three-form as auxiliary degree of freedom [@Gates:1983nr; @Antoniadis:2017jsk]. This can be understood as follows. We recall that $\Sigma$ contains in its expansion a complex vector $\mathcal{B}^m$ $$\frac14 \bar{\sigma}^{m\,\dot\alpha \alpha}
\left[D_\alpha,\bar{D}_{\dot{\alpha}}\right] \Sigma| = \mathcal{B}^m \equiv B^m + i C^m\,,$$ where $B^m \equiv {\rm Re}\, \mathcal{B}^m$ and $C^m \equiv {\rm Im}\, \mathcal{B}^m$. As discussed before, we can interpret $C^m$ as the Hodge dual of a three-form $C_{mnp}$ $$C^m = \frac{1}{3!}\varepsilon^{mnpq} C_{npq}\,,$$ whose four-form field strength $G_4$ components are defined as $$G_{mnpq} = 4 \partial_{[m} C_{npq]}\,.$$ Using we get $$\label{Vvara}
\begin{split}
\frac14 \bar{\sigma}^{m\,\dot\alpha \alpha}\left[D_\alpha,\bar{D}_{\dot{\alpha}}\right] U| &= B^m\\
\frac{1}{16} D^2 \bar D^2 U| &= -\frac12 {}^*\! G_4 - \frac{i}{2} \partial_m B^m
\end{split}$$ where we have introduced ${}^*\! G_4$, which is the Hodge-dual of the field strength $G_{mnpq}$ (see Table \[tab:comp2\]). Comparing with the usual projection of an ordinary real multiplet $V$ $$\label{Vvarb}
\begin{split}
\frac14 \bar{\sigma}^{m\,\dot\alpha \alpha}\left[D_\alpha,\bar{D}_{\dot{\alpha}}\right] V| &= v^m\\
\frac{1}{16} D^2 \bar D^2 V| &= \frac{\rm D}{2}-\frac{i}{2} \partial_mv^m
\end{split}$$ we recognize that, in the variant formulation , the auxiliary field ${\rm D}$ of the ordinary vector multiplet is replaced with the Hodge-dual of the field strength of the three-form $C_{mnp}$, namely $${\rm D} \rightarrow -\, {}^*\! G_4\,.$$
Moreover, the Lagrangian is invariant under the shift $$\label{Ugauge}
\Sigma \rightarrow \Sigma + \Phi + i L$$ with $\Phi$ and $L$ being, respectively, an arbitrary chiral and real linear multiplet. This in turns induces a gauge transformation for the vector three-form multiplet of the standard form $$U \to U + \Phi + \bar \Phi.$$ and reflects on the gauge transformation for the three-form $C_{mnp}$ as $$C_{mnp} \rightarrow C_{mnp} + 3\partial_{[m} \Lambda_{np]}$$ where $\Lambda_{mn}$ is an arbitrary real gauge two-form. This enforces the interpretation of $C_{mnp}$ as components of a gauge three-form.
Therefore the multiplet is the counterpart of the chiral double three-form multiplet and, in the following, we will dub it *real* (or *vector*) *three-form multiplet*.
--------------- ---------------- ---------- -----------------
Superfield Spin-$\frac12$ Spin-$1$ Non-propagating
\[0.5ex\] $V$ $\lambda$ $v^m$ ${\rm D}$
$U$ $\lambda_U$ $B^m$ $C_{mnp}$
--------------- ---------------- ---------- -----------------
: The fundamental off-shell component fields of the ordinary real superfield $V$ and of the real three-form multiplet $U$. The real auxiliary field ${\rm D}$ of the multiplet $V$ is here replaced by the real gauge three-form $C_{mnp}$.[]{data-label="tab:comp2"}
The variation of the Lagrangian with respect to the vector multiplet $V$ produces the superspace equations of motion $$\label{VsolLam}
\Lambda = \frac12\left(D^\alpha W_\alpha+\bar D_{\dot\alpha} \bar W^{\dot \alpha}\right) = D^\alpha W_\alpha\,,$$ whose lowest component is $$\Lambda| = 2\, {}^*\! G_4\,.$$ Substituting and in we get $$\label{VLag3f}
\mathcal{L} = \left(\frac14 \int d^2\theta\, W^\alpha W_\alpha + \text{c.c.} \right)+ \mathcal{L}_{\rm bd}$$ where the boundary terms are given by $$\begin{split}
\label{VLagBd}
\mathcal{L}_{\rm bd} =&-\frac18 \int d^2\theta\,\bar D^2 (\Lambda V) -\frac18 \int d^2\bar\theta\, D^2 (\Lambda V) \\
&+\frac18 \left[ \int d^2\theta\, \bar D^2 (\Lambda \Sigma)+\int d^2\bar\theta\, D^2 (\Lambda \bar \Sigma)\right]\\
=&\, \frac{1}{64} [D^2,\bar D^2] \left(\Lambda (\bar \Sigma - \Sigma)\right)|\,.
\end{split}$$ In components, this Lagrangian is $$\label{VLagOffSb}
\mathcal{L} =- \frac 14 F^{mn}F_{mn} -\frac{1}{2\cdot 4!} G^{mnpq} G_{mnpq} + \mathcal{L}_{\rm bd}\,,$$ with $$\label{VLagbdc}
\mathcal{L}_{\rm bd} = \frac{1}{3!}\partial_m \left(G^{mnpq} C_{npq}\right)\,.$$ Setting the gauge three-form on-shell we immediately get $$G_{mnpq} = \frac{\xi}{2} \varepsilon_{mnpq}$$ with $\xi$ a real constant. Plugging this solution back into , we obtain the on-shell Lagrangian . We conclude then that the role of the gauge three-form in the parent theory is to dynamically generate the Fayet–Iliopoulos parameter $\xi$ as expectation value of ${}^*\!G_4$. As for the previous case, this dynamically generated parameter is related to the scale of supersymmetry breaking.
To sum up, we have shown how it is possible to reformulate generic Lagrangians involving chiral and vector superfields by means of only one ingredient, that is a complex linear superfield $\Sigma$. In the off-shell formulation of these Lagrangians, the non-propagating degrees of freedom are encoded into gauge three-forms and, as a consequence, the parameters contributing to the breaking of supersymmetry are generated dynamically as vacuum expectation values. In the following we extend the discussion to the case of $\mathcal{N}=2$ supersymmetry.
$\mathcal{N}=2$ supersymmetry {#sec:N2Formalism}
=============================
Before examining how gauge three-forms can be embedded into rigid $\mathcal{N}=2$ supersymmetric theories, we briefly review known facts about the construction of $\mathcal{N}=2$ Lagrangians within the superspace approach. We are interested in particular in how to rephrase the expressions in the language of $\mathcal{N}=1$ superspace.
$\mathcal{N}=2$ chiral and vector multiplets {#sec:N2FormalismMult}
--------------------------------------------
The basic bricks which we shall need in the next section to build $\mathcal{N}=2$ Lagrangians are the chiral and vector (or reduced chiral) multiplets. They will be defined in an $\mathcal{N}=2$ superspace equipped, along with the space-time coordinates, with two sets of fermionic coordinates $\theta^\alpha$ and $\tilde \theta^\alpha$ associated to the two supersymmetry generators $Q^\alpha$ and $\tilde Q^\alpha$. The algebra satisfied by the $\mathcal{N}=2$ superspace derivatives without central charges is $$\begin{aligned}
&\{ D_\alpha, \bar D_{\dot \alpha} \} = \{ \tilde D_\alpha, \bar{\tilde D}_{\dot \alpha}\} = -2i \sigma^m_{\alpha \dot \alpha}\partial_m,\\
&\{D_\alpha, D_\beta\} = \{\tilde D_\alpha, \tilde D_\beta\} = \{D_\alpha \tilde D_\beta\} = \{D_\alpha, \bar{\tilde D}_{\dot \beta}\}=0,
\end{aligned}$$ where $\tilde D_\alpha$ generates the second supersymmetry.
The $\mathcal{N}=2$ chiral multiplet $\mathcal{A}$ can be represented by a superfield, which is chiral along the two fermionic directions $$\label{N2Achirconst}
\bar D_{\dot \alpha} \mathcal{A} = 0\,,\quad \bar {\tilde D}_{\dot \alpha} \mathcal{A} = 0\,.$$ It has $16+16$ off-shell degrees of freedom, which are encoded in three $\mathcal{N}=1$ chiral superfields $X$, $\Phi$ and $W_\alpha$. It can be expanded in the $\tilde \theta$ coordinates as $$\label{N=2VecMult}
\begin{aligned}
\mathcal{A}(y,\theta, \tilde \theta)
&= X(y,\theta) + \sqrt 2 \tilde\theta^\alpha W_\alpha(y,\theta) +\tilde \theta^2 \left(\frac{i}{2}\Phi(y,\theta)+\frac14 \bar D^2\bar X\right),
\end{aligned}$$ where $y$ collects the chiral spacetime coordinates. The supersymmetry transformations of the $\mathcal{N}=1$ components of $\mathcal{A}$ along the $\tilde \theta$ coordinates are given by $$\begin{aligned}
\tilde\delta X &= \sqrt 2 \eta^\alpha W_\alpha\,,\\[2mm]
\tilde \delta W &= i\sqrt 2 \sigma^m\bar\eta \partial_m X + \sqrt 2 \eta \left(\frac i2 \Phi+\frac14 \bar D^2 \bar X\right)\,,\\[2mm]
\tilde \delta \Phi &= 2\sqrt 2 i \left(\frac14 \bar D^2(\bar\eta\bar W)-i\bar\eta\bar\sigma^m\partial_m W\right).\end{aligned}$$ We stress that here the chiral superfield $W_\alpha$ does not satisfy any Bianchi identities and it cannot be explicitly written in terms of an $\mathcal{N}=1$ real potential. In other words, $W_\alpha$ does not represent the usual field strength of an $\mathcal{N}=1$ vector multiplet. The auxiliary components of $\mathcal{A}$ are defined as the projections $$\begin{aligned}
-\frac14 D^2\mathcal{A} | &= f\,,
\\
-\frac14 \tilde D^2\mathcal{A} | &= \frac{i}{2} \varphi -\bar f\,,
\\
-\frac14 D \tilde D\mathcal{A} |&= \frac{{\rm D}}{\sqrt{2}}\,.
\end{aligned}$$
It is possible to rephrase this construction in a manifestly $\rm SU(2)_R$-covariant manner. We first collect the superspace coordinates $\theta$ and $\tilde \theta$ into a $\rm SU(2)_R$ doublet $$\theta_i =\left(
\begin{array}{c}
\theta_\alpha\\
\tilde \theta_\alpha
\end{array}
\right)$$ and we define the superspace derivatives $D^{ij} = D^{i\,\alpha} D^j_{\alpha}$. The superspace expansion of $\mathcal{A}$ is then $$\mathcal{A} = X + \sqrt 2 \theta_i \lambda^i + \theta_{i} \theta_j \mathbb{Y}^{ij}+\dots,$$ where $\lambda^i$ is the $\rm SU(2)_R$ doublet containing the fermions, while $\mathbb{Y}^{ij} = -\frac14 D^{ij}\mathcal{A}|$ is a matrix containing the auxiliary fields $$\mathbb{Y} = \begin{pmatrix}f & \frac{{\rm D}}{\sqrt{2}} \\ \frac{{\rm D}}{\sqrt{2}} & \frac{i}{2} \varphi -\bar f \end{pmatrix},$$ which defines an $\rm SU(2)_R$ triplet $\vec{Y}$ as $$-2i \mathbb{Y} \equiv (\vec{\sigma} \cdot \vec{Y}) \sigma^2\,.$$ More explicitly, $\vec{Y}$ reads $$\label{TripletGen}
\vec{Y} = \begin{pmatrix}
2\, \text{Im} f + \frac{\varphi}{2}\\ - 2\,\text{Re} f+ \frac{i \varphi}{2} \\ \sqrt{2} D
\end{pmatrix} \,,$$ whose entries are generically complex.
The $\mathcal{N}=2$ vector (or reduced chiral) superfield $\mathcal{A}_D$ can be obtained from the chiral multiplet by imposing the constraint given in [@Grimm:1977xp; @deRoo:1980mm], which results in the reduction of its off-shell degrees of freedom to $8+8$. This is equivalent to set $\Phi=0$ directly in the superspace expansion [@Ambrosetti:2009za], giving $$\label{ChiralC}
\begin{split}
\mathcal{A}_D(y,\theta, \tilde \theta) &= X_D(y,\theta) + \sqrt 2 \tilde\theta^\alpha W_{D\alpha}(y,\theta) + \frac14\tilde \theta^2 \bar D^2\bar X_D
\end{split}$$ and by requiring also that the condition is preserved by the second supersymmetry, $\tilde \delta\, \Phi=0$, for consistency. From this requirement one gets the Bianchi identities $$D^\alpha W_{D\,\alpha} = \bar D_{\dot \alpha} \bar W^{\dot \alpha}_D\,,$$ which imply that $W_D$ can be expressed as the field strength of a gauge potential real superfield $V_D$ $$W_{D\,\alpha} = -\frac14 \bar D^2 D_\alpha V_D\,.$$ We also stress that, for a reduced multiplet, the auxiliary field triplet $\vec{Y}$ of is real. Adopting a manifestly $\rm SU(2)_R$ invariant notation, in which we define the doublet of fermions $${\bm \Psi} = \left(
\begin{array}{c}
\lambda\\
\psi
\end{array}
\right)$$ and the supersymmetry parameters $${\bm \eta} = \left(
\begin{array}{c}
\eta_1\\
\eta_2
\end{array}
\right),$$ the supersymmetry transformations of the vector multiplet can be written as
$$\begin{aligned}
\delta \varphi &= \sqrt{2} {\bm\eta} {\bm\Psi}\,, \label{Susy2VarPhi}
\\
\delta v^m &= i ({\bm\eta} \sigma^m \bar {\bm\Psi} + \bar {\bm \eta} \bar \sigma^m {\bm \Psi})\,, \label{Susy2VarB}
\\
\delta {\bm \Psi} &= i \sqrt{2} \sigma^m \bar {\bm \eta} \partial_m \varphi + \sigma^{mn} {\bm \eta}
F_{mn}+ \frac{i}{\sqrt{2}} (\vec{\sigma} \cdot \vec{Y}) {\bm \eta}\,, \label{Susy2VarPsi}
\\
\delta \vec{Y} &= \sqrt{2}\, \bar {\bm \eta} \vec{\sigma}^m\, \partial_m {\bm \Psi} + \text{h.c.}\,. \label{Susy2VarY}\end{aligned}$$
Structure of the $\mathcal{N}=2$ Lagrangian {#sec:N2Lag}
-------------------------------------------
In this section we review the structure and the properties of $\mathcal{N}=2$ supersymmetric Lagrangians for an arbitrary number of vector multiplets.
For simplicity, let us start by considering the case of a single vector multiplet . We define a holomorphic, but otherwise general, prepotential $F(\mathcal{A}_D)$, in terms of which a manifestly $\mathcal{N}=2$ Lagrangian can be built as the integral over the chiral $\mathcal{N}=2$ superspace $$\label{LN=2b}
\mathcal{L} =\frac i2 \int d^2\theta d^2\tilde \theta F(\mathcal{A}_D)+{\rm c.c.}\,.$$
By using the expansion and integrating over the fermionic coordinates $\tilde \theta$, the Lagrangian can be recast in the more familiar $\mathcal{N}=1$ language as $$\label{LN=1}
\mathcal{L} = \left(\frac14 \int d^2\theta \tau(X) W^\alpha W_\alpha + {\rm c.c.}\right) +\int d^4\theta K(X,\bar X)\,,$$ where $$\tau(X) =- i F_{XX}, \qquad K(X,\bar X) = \frac{i}{2}\left(X \bar F_{\bar X} - \bar X F_X \right)\,$$ and $\partial_X \partial_{\bar X}K = \text{Im}\, F_{XX}$ is the metric of the special Kähler scalar manifold. This Lagrangian has been written in a so called electric frame, in which only electric vector fields are present. Alternatively, with an $\rm SL(2,\mathbb{R})$ electro-magnetic duality transformation, it is possible to bring it into a magnetic frame, in which the electric vectors are exchanged with their magnetic dual. In fact, let us consider, rather than , the Lagrangian $$\label{LN=2}
\mathcal{L} = \frac{i}{2} \int d^2 \theta d^2 \tilde \theta \left[F( \mathcal{A}) - \mathcal{A}_D \mathcal{A} \right] + {\rm c.c.}\,,$$ where $\mathcal{A}$ is a chiral multiplet, while $\mathcal{A}_D$ is the magnetic dual of a vector multiplet. By integrating out $\mathcal{A}$, the Lagrangian is expressed entirely in the magnetic frame while, by integrating out $\mathcal{A}_D$, the constraint on $\mathcal{A}$ is imposed which reduces it to an $\mathcal{N}=2$ vector multiplet. In this sense, $\mathcal{A}_D$ can be thought of as a Lagrange multiplier. This can be most readily seen by rewriting in the language of $\mathcal{N}=1$ superspace $$\label{LN=1b}
\begin{aligned}
\mathcal{L} &= -\frac i4 \int d^2\theta \left( F_{XX} W^\alpha W_\alpha - 2 W_D^\alpha W_\alpha \right)\\
& - \frac i2 \int d^4 \theta F_X \bar{X} + \frac 14 \int d^2 \theta \Phi \left( X_D - F_X\right) + {\rm c.c.}\,.
\end{aligned}$$ The equations of motion of the $\mathcal{N}=1$ superfields $X_D$ and $V_D$ contained in $\mathcal{A}_D$ give, respectively, the Bianchi identities of $W_\alpha$ and the constraint which reduces $\mathcal{A}$ to a $\mathcal{N}=2$ vector multiplet: $$\begin{aligned}
\label{deltaVD} \delta V_D: &\qquad D^\alpha W_\alpha = \bar D_{\dot \alpha} \bar W^{\dot \alpha} \,,\\
\label{deltaXD} \delta X_D: &\qquad \Phi=0\,.\end{aligned}$$ We note that we cannot vary with respect to $W_D$, being it constrained by the Bianchi identities, but it is indeed necessary to vary with respect to the real, unconstrained potential $V_D$. Hence, we immediately recognize that the integration of the constrained superfield $\mathcal{A}_D$ has the role to set the constraints on $\mathcal{A}$, so that is identified with .
As proposed in [@Antoniadis:1995vb], however, a third possibility is to work in a frame which contains both electric and magnetic vectors at the same time. In fact the Lagrangian can be supplemented with both electric and magnetic Fayet–Iliopoulos parameters. Introducing the complex $\vec{E}$ and the real $\vec{M}$ parameters, we can add new couplings linear in $\vec{Y}$ and $\vec{Y}_D$ to obtain $$\label{N2Lag}
\mathcal{L} = \frac{i}{2} \int d^2 \theta d^2 \tilde \theta \left[F( \mathcal{A}) - \mathcal{A}_D \mathcal{A} \right] + \frac12 \left(\vec{E}\cdot \vec{Y}+ \vec{M}\cdot \vec{Y}_D\right)+\text{c.c.}\,.$$ These correspond respectively to electric and magnetic abelian gaugings of the theory and, but for those appearing possibly in the prepotential, they are the only parameters which are compatible with $\mathcal{N}=2$ supersymmetry. In this sense, therefore, the case of extended supersymmetry is more constrained with respect to the $\mathcal{N}=1$ situation, in which a large class of parameters can enter the superpotential. In addition, in order to preserve the R-symmetry, we assume that $\vec{E}$ and $\vec{M}$ transform as triplets under $\rm SU(2)_R$. We note that the auxiliary fields $\vec{Y}_D$ transform as total derivatives under the supersymmetry transformation, in contrast to $\vec{Y}$, which transform as total derivative only once the superfield $\mathcal{A}_D$ is integrated out. The integration of $X_D$ gives indeed $$\delta X_D: \qquad \Phi = 4 (M^2 + i M^1)\,,$$ which means that $\Phi$ is a constant superfield. For consistency, the condition $\tilde \delta\, \Phi=0$ has to be imposed again.
The Lagrangian can be easily generalized to the case of an arbitrary number of self-interacting chiral multiplets $\mathcal{A}^\Lambda$, with $\Lambda,\Sigma,\ldots=1,\ldots, N$, accompanied with an equal number of vector multiplets $\mathcal{A}_{D\,\Lambda}$ setting the constraints on $\mathcal{A}^\Lambda$: $$\label{N2NLag}
\mathcal{L} = \frac{i}{2} \int d^2 \theta d^2 \tilde \theta \left[F( \mathcal{A}) - \mathcal{A}_{D\,\Lambda} \mathcal{A}^\Lambda \right] + \frac12 \left(\vec{E}_\Lambda\cdot \vec{Y}^\Lambda+ \vec{M}^\Lambda\cdot \vec{Y}_{D \Lambda}\right)+\text{c.c.}\,.$$ After the Lagrange multiplier $\mathcal{A}_D$ is integrated out, the equations of motion of the auxiliary fields give $$\vec{Y}^\Lambda = - 2\, \mathcal{N}^{\Lambda \Sigma}\left(\text{Re}\, \vec{E}_\Sigma + \text{Re}\, F_{\Sigma \Gamma}\vec{M}^\Gamma\right)+2i\vec{M}^\Lambda\,,$$ where we have defined the metric of the special Kähler scalar manifold $\mathcal{N}_{\Lambda\Sigma} = \text{Im}\, F_{\Lambda\Sigma}$, together with its inverse $\mathcal{N}^{\Lambda\Sigma}=({\mathcal{N}_{\Lambda\Sigma}})^{-1}$. Substituting this expression for the auxiliary fields back into the Lagrangian, the following scalar potential is produced $$\label{VpotN=2}
\mathcal{V} = \mathcal{N}^{\Lambda \Sigma}\left(\text{Re}\, \vec{E}_\Lambda + F_{\Lambda\Gamma}\vec{M}^\Gamma\right)\cdot\left(\text{Re}\, \vec{E}_\Sigma + \bar F_{\Sigma\Delta}\vec{M}^\Delta\right)+2 \text{Im}\, \vec{E}_\Lambda \cdot \vec{M}^\Lambda\,.$$ Notice that $9N$ real parameters are appearing in the scalar potential. However $3N$ of them, namely those encoded in $\text{Im}\, \vec{E}_\Lambda$, are contributing solely as an additive constant to the Lagrangian and thus they can be disregarded, as long as we are focusing only on rigid supersymmetric theories.
In the following sections, by using variant $\mathcal{N}=1$ multiplets containing gauge three-forms, we will be able to rephrase the Lagrangians and in terms of peculiar $\mathcal{N}=2$ vector multiplets which dynamically generate the gauging parameters $\vec{E}$ and $\vec{M}$.
Three-forms in $\mathcal{N}=2$ global supersymmetry {#sec:N23form}
===================================================
We extend now the procedure introduced in [@Farakos:2017jme] and reviewed in Section \[sec:N1case\] to $\mathcal{N}=2$ Lagrangians of the kind of . As in Section \[sec:N1case\], the Lagrangian will be traded for an alternative one, which contains gauge three-forms and where the gauging parameters appear only when these gauge three-forms are set on-shell. We here examine the case where the $\mathcal{N}=2$ Lagrangian is built out of a single abelian vector multiplet. The generalization to an arbitrary number $N$ of vector superfields is reported in the Appendix .
The case of a single vector multiplet {#sec:N23form1}
-------------------------------------
Let us consider the case of a single vector multiplet $\mathcal{A}$, whose $\mathcal{N}=2$ Lagrangian is , endowed with Fayet–Iliopoulos parameters $\vec{E}$ and $\vec{M}$. In order to recast this Lagrangian in an $\mathcal{N}=1$ form, which is convenient for an analysis similar to that carried in Section \[sec:N1case\], we use the $\rm SU(2)$ R-symmetry of the theory and rotate the parameters such that $$\label{N2LagGauging}
\Re \vec{E} =\left(0,-e,\frac{\xi}{2\sqrt{2}}\right)\,, \qquad \vec{M} = (0,-m,0)\,,$$ with $e$, $m$ and $\xi$ real constants. Indeed, the $\rm SU(2)_R$ covariance of the Lagrangian ensures that there is no loss of generality in the choice . We have also discarded the imaginary part of $\vec{E}$, since we have already shown that it contributes only as an additive constant to the theory.
The Lagrangian , after integrating out the constrained superfield $\mathcal{A}_D$, can be written in $\mathcal{N}=1$ language as $$\label{N2LagB}
\begin{aligned}
\mathcal{L} =& \int d^4\theta\, K(X,\bar X)+ \left(\frac14 \int d^2\theta\, \tau (X) W^{\alpha} W_\alpha +\text{c.c.}\right)+
\\
& + \left( \int d^2\theta\,W(X)+\text{c.c.}\right) + \xi \int d^4\theta\, V\,,
\end{aligned}$$ with the superpotential $W(X)$ given by $$\label{N2W}
W(X)= e X + m F_X(X)\,.$$ The bosonic components of are $$\label{N2LScom}
\begin{split}
\mathcal{L}\big|_{\text{bos}} =& -\text{Im}\, F_{XX}\, \partial_m \varphi\, \partial^m \bar \varphi -\frac 14 \text{Im}\, F_{XX} F^{mn}F_{mn} - \frac18 \text{Re}\, F_{XX}\, \varepsilon_{klmn} F^{kl} F^{mn} + \\
&+ \text{Im}\, F_{XX} f \bar f +\frac12\text{Im}\, F_{XX} {\rm D}^2 + (e + m F_{XX}) f + (e + m \bar F_{XX}) \bar f + \frac{\xi}{2} {\rm D}\,
\end{split}$$ and, integrating out the auxiliary fields $f$ and ${\rm D}$, we arrive at $$\label{N2LScomOS}
\begin{split}
\mathcal{L}\big|_{\text{bos}} =& -\text{Im}\, F_{XX}\, \partial_m \varphi\, \partial^m \bar \varphi -\frac 14 \text{Im}\, F_{XX} F^{mn}F_{mn} - \frac18 \text{Re}\, F_{XX}\, \varepsilon_{klmn} F^{kl} F^{mn} - \mathcal{V} (\varphi,\bar\varphi) \,,
\end{split}$$ with the scalar potential $$\label{N2LV}
\begin{split}
\mathcal{V} (\varphi,\bar\varphi) = \frac{1}{ \text{Im}\, F_{XX}} |e+m F_{XX}|^2 + \frac{\xi^2}{8\, \text{Im}\,F_{XX}}\,.
\end{split}$$
We would like to generate dynamically the gauging parameters $e$, $m$ and $\xi$ entering the scalar potential and the superpotential. To this purpose, we shall perform a two steps procedure. First, we will trade the vector multiplet $\mathcal{A}_D$ in for its variant version $$\label{ChiralS}
\begin{split}
\mathcal{S}_D(y,\theta, \tilde \theta) &= S_D (y,\theta) + \sqrt 2 \tilde\theta^\alpha W_{D\,\alpha}(y,\theta) + \frac14 \tilde \theta^2 \bar D^2\bar S_D\,,
\end{split}$$ where $W_{D\,\alpha} =-\frac14 \bar D^2 D_\alpha U_D$. Here, the ordinary chiral multiplet $X$ and vector multiplet $V$ of are replaced with and respectively. On the one hand, this will allow for promoting the magnetic parameters $\vec{M}$ to be dynamical; on the other, it will allow for establishing an off-shell correspondence between the $\mathcal{N}=2$ multiplets and . Then, after integrating out the variant $\mathcal{S}_D$ multiplet, we will proceed to the second step. It consists in an additional trading, which exchanges the remaining $\mathcal{A}$ multiplet with another variant chiral multiplet of the kind of . The final Lagrangian, setting the residual three-forms on-shell, will coincide with the on-shell Lagrangian .
### First step: generating the magnetic parameters {#first-step-generating-the-magnetic-parameters .unnumbered}
We start recalling that, in $\mathcal{N}=1$ language, reads $$\label{Alt_LN=1a}
\begin{aligned}
\mathcal{L} &= -\frac i4 \int d^2\theta \left( F_{XX} W^\alpha W_\alpha - 2 W_D^\alpha W_\alpha \right)\\
&\quad - \frac i2 \int d^4 \theta F_X \bar{X} + \frac 14 \int d^2 \theta \Phi \left( X_D - F_X\right) + \frac12 \left(\vec{E}\cdot \vec{Y}+ \vec{M}\cdot \vec{Y}_D\right)+ {\rm c.c.}\,.
\end{aligned}$$ In order to reinterpret the magnetic gauging parameters as originating from vacuum expectation values of gauge three-forms, we first promote the coupling $\vec{M}\cdot \vec{Y}_D$ to a full dynamical entity. Hence we rewrite as $$\label{Alt_LN=1D}
\begin{aligned}
\mathcal{L} &= \bigg\{-\frac i4 \int d^2\theta \left( F_{XX} W^\alpha W_\alpha - 2 W_D^\alpha W_\alpha \right)
\\
& \quad\quad- \frac i2 \int d^4 \theta F_X \bar{X} + \frac 14 \int d^2 \theta \Phi \left( X_D - F_X\right) + \frac12 \vec{E}\cdot \vec{Y}+ {\rm c.c.}\bigg\}
\\
&\quad+\left\{\int d^2 \theta\, \left(\Lambda_1^D X_D + \frac14 \bar D^2 (\Sigma_{1D} \bar \Lambda_1^D) \right)+ {\rm c.c.}\right\}
\\
&\quad+\left\{\frac18 \int d^2\theta\,\bar D^2 [\Lambda_2^D (\Sigma_{2D} - V_D)] + \text{c.c.} \right\},
\end{aligned}$$ As explained in Section \[sec:N1caseC\] (see ), the third line provides the gauge three-form inside the $\mathcal{N}=1$ chiral superfield in $\mathcal{A}_D$, while the fourth, as in (see ), provides it for the vector superfield. In particular, $\Lambda_1^D$ and $\Lambda_2^D$ are respectively a chiral and a real superfield which play the role of Lagrange multipliers, while $\Sigma_1^D$ and $\Sigma_2^D$ are complex linear multiplets containing the gauge three-forms. As a consistency check, from the equations of motion of $\Sigma_{1D}$ and $\Sigma_{2D}$ we get $$\label{Alt_LambdaOS}
\Lambda_1^D = -M^2-i M^1\,,\qquad \Lambda_2^D= 2\sqrt{2} M^3\,,$$ with $\vec{M}$ arbitrary real integration constants, recovering .
On the contrary, in order to obtain an $\mathcal{N}=2$ Lagrangian which contains gauge three-forms in place of the auxiliary fields $\vec{Y}_D$, we have to integrate out the chiral superfield $X_D$ and vector superfield $V_D$, as well as the Lagrange multipliers $\Lambda_1^D$ and $\Lambda_2^D$. The variations with respect to the Lagrange multipliers $\Lambda_1^D$ and $\Lambda^D_2$ give the relations $$\begin{aligned}
\delta \Lambda_1^D: \quad X_D &= -\frac14 \bar D^2 \bar\Sigma_{1D} \equiv S_D\,, \label{Alt_XDsol}
\\
\delta \Lambda_2^D: \quad V_D &= \frac{\Sigma_{2D} + \bar \Sigma_{2D}}{2} \equiv U_D\,, \label{Alt_VDsol}\end{aligned}$$ which trade, respectively, the ordinary $\mathcal{N}=1$ chiral multiplet $X_D$ and vector multiplet $W_D^\alpha$ for a double and a vector three-form multiplet. Indeed, the variations with respect to the ordinary $\mathcal{N}=1$ superfields $X_D$ and $V_D$ give $$\begin{aligned}
\label{Alt_L1Dsol}
\delta X_D: \quad \Lambda^D_1 &= -\frac 14 \Phi\,,
\\
\label{Alt_L2Dsol}
\delta V_D: \quad \Lambda^D_2 &= - {\rm Im} (D^\alpha W_\alpha)\,.\end{aligned}$$ Plugging (\[Alt\_XDsol\]-\[Alt\_L2Dsol\]) in , we get $$\label{Alt_LN=1Db}
\begin{aligned}
\mathcal{L} &= \bigg\{-\frac i4 \int d^2\theta \left( F_{XX} W^\alpha W_\alpha - 2 W_D^\alpha W_\alpha \right)
\\
& \quad\quad- \frac i2 \int d^4 \theta F_X \bar{X} + \frac 14 \int d^2 \theta \Phi \left( S_D - F_X\right) + \frac12 \vec{E}\cdot \vec{Y}+ {\rm c.c.}\bigg\} + \mathcal{L}_{\rm bd}^{(D)}
\end{aligned}$$ with $$\label{Alt_LN=1Dbd}
\begin{split}
\mathcal{L}_{\rm bd}^{(D)}= &-\frac14 \left(\int d^2\theta \bar D^2 - \int d^2\bar\theta D^2\right) \left[\left(-\frac 14 \Phi \right) \bar\Sigma_{1D}\right]\\
&+\frac{1}{16} \left(\int d^2\theta \bar D^2 - \int d^2\bar\theta D^2\right) \left[ - \text{Im} (D^\alpha W_\alpha) \Sigma_{2D} \right] +\text{c.c.} \, .
\end{split}$$
Before moving on and integrating out the superfields $S_D$ and $U_D$, let us notice that the Lagrangian can be recast in a manifest $\mathcal{N}=2$ form as $$\label{Alt_LN=2}
\mathcal{L} = \frac{i}{2} \int d^2 \theta d^2 \tilde \theta \left[F( \mathcal{A}) - \mathcal{S}_D \mathcal{A} \right] + \frac12 \vec{E}\cdot \vec{Y} + {\rm c.c.} + \mathcal{L}_{\rm bd}^{(D)}\,.$$ As desired, we have promoted the usual reduced chiral Lagrange multiplier $\mathcal{A}_D$ appearing in to $\mathcal{S}_D$, which is a multiplet of the variant type as in . It contains a chiral double three-form multiplet $S_D$ and a real vector three-form mulitplet $U_D$, whose non propagating degrees of freedom are given by $$\label{Alt_Y3f}
\begin{split}
-\frac14 D^2\mathcal{S}_D | &= -i {}^*\!\bar F_{4}^D = - \frac{i}{3!} \varepsilon^{mnpq} \partial_{m} \bar{\mathcal{B}}_{ npq}^D \,,
\\
-\frac14 \tilde D^2\mathcal{S}_D | &= -i {}^*\!F_{4}^D = - \frac{i}{3!} \varepsilon^{mnpq} \partial_{m} \mathcal{B}_{ npq}^D \,,
\\
-\frac14 D \tilde D\mathcal{S}_D |&=-\frac{1}{\sqrt{2}} {}^*\!G_{4}^D = -\frac{1}{3! \sqrt{2}} \varepsilon^{mnpq} \partial_{m} C_{ npq}^D\,,
\end{split}$$ with $\mathcal{B}^D_{mnp}$ and $C_{mnp}^D$ the components of a complex and a real gauge three-form respectively.
Resuming the previous discussion, in order to retrieve a Lagrangian formulated solely in terms of the multiplet $\mathcal{A}$, the Lagrange multiplier $\mathcal{S}_D$ has to be integrated out. Owing to the presence of the gauge three-forms, in comparison with (\[deltaVD\],\[deltaXD\]), the variations with respect to $S_D$ and $U_D$ differ only in their auxiliary components. In fact, integrating out $C_{mnp}^D$ and ${\mathcal{B}}_{ npq}^D$, we get $$\label{Alt_DPhi_OS}
{\rm Im}\, {\rm D} = \sqrt{2} M^3\, \qquad \Phi = 4 (M^2 +i M^1)$$ consistently with , and . In other words, the choice of a variant Lagrange multiplier results in a dynamical generation of the magnetic Fayet–Iliopoulos parameters $\vec{M}$ which appear in . In order to make contact with we choose $\vec{M} = (0,-m,0)$ and, as a consequence, the Lagrangian becomes $$\label{Alt_LN=1c}
\begin{aligned}
\mathcal{L} &=
-\frac i4 \int d^2\theta F_{XX} W^\alpha W_\alpha - \frac i2 \int d^4 \theta F_X \bar{X} + m \int d^2 \theta F_X + \frac12 \vec{E}\cdot \vec{Y}+ {\rm c.c.}.
\end{aligned}$$ This Lagrangian, in comparison with and recalling the component structure of the chiral multiplet , suggests that integrating out $\mathcal{S}_D$ results in constraining $\mathcal{A}$ as $$\label{Alt_A}
\mathcal{A}(y,\theta, \tilde \theta) = X(y,\theta) + \sqrt 2 \tilde\theta^\alpha W_\alpha(y,\theta) +\tilde \theta^2 \left(-2i m+\frac14 \bar D^2\bar X\right).$$ Due to the presence of gauge-three forms, therefore, a parameter related to a magnetic Fayet–Iliopoulos gauging has been inserted dynamically into the expression of the $\mathcal{N}=2$ vector superfield. This parameter will play an important role when studying the mechanism of supersymmetry breaking, as shown in the next section.
### Second Step: generating the electric parameters {#second-step-generating-the-electric-parameters .unnumbered}
We have just presented a recipe in order to dynamically produce the magnetic gauging parameters, but for the moment nothing has be done on the electric gauging parameters $\vec{E}$. If we insist on considering the multiplet $\mathcal{A}$ an ordinary one as in , then the only choice is to add the electric gauging by hand from the start, as in . However, if we relax this request there is still another option: we may assume that $\mathcal{A}$ could be a variant multiplet of the kind of , which endows gauge three-forms in its non-propagating components, and allow for the dynamical generation of the electric gauging parameters as well. This represents the second step of the procedure we are proposing. We promote then to $$\label{Alt_N2Master}
\begin{aligned}
\mathcal{L} =& \int d^4\theta\, K(X,\bar X)+\left(\frac14 \int d^2\theta\, \tau(X) W^{\alpha} W_\alpha + \text{c.c.}\right)+\\
&+\left\{\int d^2\theta \Lambda_1 X
+\frac14 \int d^2\theta\, \bar D^2\left(\Sigma_1 \bar \Lambda_1\right)+ m \int d^2 \theta F_X+\text{c.c.}\right\}+\\
&+\left\{\frac18 \int d^2\theta\,\bar D^2 [\Lambda_2 (\Sigma_2 - V)] + \text{c.c.} \right\},
\end{aligned}$$ The second line provides the exchange between the chiral multiplet $X$ and its three-form counterpart: $\Lambda_1$ is a chiral Lagrange multiplier and $\Sigma_1$ a complex linear multiplet. The third line trades the vector multiplet $W^\alpha$ for a vector three-form multiplet: here $\Lambda_2$ is a real Lagrange multiplier and $\Sigma_2$ a complex linear multiplet. The Lagrangian truly reproduces . This can be most readily seen from the integration of the complex linear superfields $\Sigma_1$ and $\Sigma_2$, which sets $$\label{Alt_Lambda_sol}
\Lambda_1 = e\,, \qquad \Lambda_2 = \xi\,,$$ with $e$ and $\xi$ arbitrary real constants. Plugging in , we reobtain , ensuring the equivalence of the two descriptions. In order to re-express instead in terms of the three-form multiplets, we have to integrate out both the Lagrange multipliers $\Lambda_1$, $\Lambda_2$ and the ordinary $\mathcal{N}=1$ superfields $X$ and $V$. The variations with respect to the Lagrange multipliers $\Lambda_1$ and $\Lambda_2$ give $$\begin{aligned}
\delta \Lambda_1: \quad X &= -\frac14 \bar D^2 \bar\Sigma_1 \equiv S\,, \label{Alt_Xsol}\\
\delta \Lambda_2: \quad V &= \frac{\Sigma_2 + \bar \Sigma_2}{2} \equiv U\,, \label{Alt_Vsol}\end{aligned}$$ which trade the $\mathcal{N}=1$ multiplets for their three-form counterparts, while the variations with respect to $X$ and $V$ produce $$\begin{aligned}
\label{Alt_L1sol}
\delta X: \quad \Lambda_1 &= \frac 14 \bar{D}^2 K_{X} - m F_{XX}+ \frac{i}4 F_{XXX} W^\alpha W_\alpha\,,
\\
\label{Alt_L2sol}
\delta V: \quad \Lambda_2 &= \text{Im}\, F_{XX} D^\alpha W_\alpha + \frac12 \left[(D^\alpha \tau) W_\alpha - (\bar D_{\dot \alpha} \bar \tau) \bar W^{\dot \alpha} \right]\,.\end{aligned}$$ Inserting (\[Alt\_Xsol\]–\[Alt\_L2sol\]) in we get $$\label{Alt_N2LD}
\mathcal{L} = \int d^4\theta K(S,\bar S) + \left( \frac14 \int d^2\theta \tau(S) W^{\alpha} W_{\alpha}+ m \int d^2 \theta F_S+ \text{c.c.}\right) +\mathcal{L}_{\rm bd},$$ with the boundary terms $$\label{Alt_N2LDbd}
\begin{split}
\mathcal{L}_{\rm bd}= &-\frac14 \left(\int d^2\theta \bar D^2 - \int d^2\bar\theta D^2\right) \left[\left(\frac 14 \bar{D}^2 K_{S} - m F_{SS} + \frac{i}4 F_{SSS} W^\alpha W_\alpha \right) \bar\Sigma_1\right]\\
&+\frac{1}{16} \left(\int d^2\theta \bar D^2 - \int d^2\bar\theta D^2\right) \left[ \left(\text{Im}\, F_{SS} D^\alpha W_\alpha +\frac12 W^\alpha D_\alpha \tau + \frac12 \bar W_{\dot \alpha} \bar D^{\dot \alpha} \bar \tau \right) \Sigma_2 \right]
\\
& +\text{c.c.} \, .
\end{split}$$ The bosonic components of are $$\label{Alt_N2LScom}
\begin{split}
\mathcal{L}\big|_{\text{bos}} =& -\text{Im}\, F_{SS}\, \partial_m s\, \partial^m \bar s -\frac 14 \text{Im}\, F_{SS} F^{mn}F_{mn} - \frac18 \text{Re}\, F_{SS}\, \varepsilon_{klmn} F^{kl} F^{mn} \\
&+ \text{Im}\, F_{SS} {}^*\! {F}_4 {}^*\!\bar{{F}}_4 +\frac12 \text{Im}\, F_{SS} ({}^*\!G_4)^2 +\left(-i m F_{SS} {}^*\!\bar{{F}}_4 + {\rm c.c.}\right)+ \mathcal{L}_{\rm bd}
\end{split}$$ with $$\label{Alt_N2Lbdcom}
\begin{split}
\mathcal{L}_{\rm bd} = &\left\{\frac{1}{3!} \partial_k\left[ B_{lmn}\left(\text{Im}\, F_{SS}\bar F^{klmn}-i m\, \varepsilon^{klmn} \bar F_{SS}\right)\right]+\text{c.c.}\right\}\\
&+\frac{1}{3!}\partial_k \left(\text{Im}\, F_{SS}\, C_{lmn} G^{klmn}\right) \, .
\end{split}$$ By setting the gauge three-forms on shell as $$\begin{aligned}
\label{Alt_Fos}
F_{klmn} &=-\frac{i}{{\rm Im}\, F_{SS}}(e+m F_{SS})\varepsilon_{klmn}\,,
\\
\label{Alt_Gos}
G_{klmn} &= -\frac{\xi}{2\, \text{Im}\, F_{SS}}\, \varepsilon_{klmn}\,,\end{aligned}$$ the same potential as is recovered.
We notice finally that the Lagrangian can also be recast in $\mathcal{N}=2$ superspace as $$\label{N2LagSS}
\mathcal{L} = \frac{i}{2} \int d^2 \theta d^2 \tilde \theta F( \mathcal{S}) +\text{c.c.} + \mathcal{L}_{\text{bd}}\,.$$ provided that we introduce the $\mathcal{N}=2$ multiplet defined as $$\label{Alt_S}
\mathcal{S}(y,\theta, \tilde \theta) = S(y,\theta) + \sqrt 2 \tilde\theta^\alpha W_\alpha(y,\theta) +\tilde \theta^2 \left(-2 i m+\frac14 \bar D^2\bar S\right).$$ where $S$ is that introduced and $W_{\alpha} = -\frac14 \bar D^2 D_\alpha U$, with $U$ as in . This multiplet provides a direct off-shell correspondence with the chiral multiplets or .
### Summary of the results {#summary-of-the-results .unnumbered}
In this section we have presented a two steps procedure which allows for a dynamical generations of the parameters entering the superpotential and the scalar potential of generic $\mathcal{N}=2$ globally supersymmetric models. The first step leads to a dynamical generation of the magnetic gauging parameters. Once they are generated by integrating out the corresponding gauge three-forms, we can proceed to a second step. This amounts to promote also the electric gauging parameters to expectation values of four-form field strengths. The final result is the off-shell Lagrangian . In other words, the inclusion of parameters in the original Lagrangian has been traded for a boundary condition problem. Indeed the two steps can be performed at the same time, by conveniently combining and , but we preferred to keep them separate for the plainness of the discussion. In the next section we are going to apply the formalism developed so far to analyse the mechanism of spontaneous partial breaking of supersymmetry.
An alternative procedure
------------------------
In the previous subsection we proposed a two steps procedure in order to dynamically generate parameters in generic $\mathcal{N}=2$ supersymmetric theories. In particular, the magnetic Fayet–Iliopoulos parameter $m$ is generated first, while its electric counterparts are introduced in a second step. Even though the steps can be performed at the same time, in the sense that they do not clash, they generate different parts of the superpotential and of the scalar potential. One can however wonder if it would be possible to generated dynamically these quantities solely in one passage, following a logic closely related to that of [@Farakos:2017jme]. It turns out indeed that the procedure of [@Farakos:2017jme] can be applied directly to the $\mathcal{N}=2$ case and the entire superpotential can be generated dynamically in a single step, but only for a particular class of models. In the present subsection we depict this alternative procedure for a system with a generic number of vector superfields, the minimal case of one single superfield being somehow trivial, as it is going to be clear in a while. The reader interested in the analysis of the partial breaking of supersymmetry can skip this subsection at first.
We start from the Lagrangian $$\label{N2NNMaster}
\begin{aligned}
\mathcal{L} =& \int d^4\theta\, K(X,\bar X)+\left(\frac14 \int d^2\theta \tau_{\Lambda \Sigma}(X) W^{\Lambda\alpha} W^\Sigma_\alpha + c.c\right)\\
&+\left\{\int d^2\theta \Lambda_{1\Sigma} X^\Sigma-\frac14 \int d^2\theta\, \bar D^2\left[ \Sigma_{1\Lambda} \,\mathcal{N}^{\Lambda\Sigma}(\Lambda_{1\Sigma}-\bar \Lambda_{1\Sigma})\right]+\text{c.c.}\right\}\\
&+\left\{\frac18 \int d^2\theta\,\bar D^2 [\Lambda_{2\Gamma} (\Sigma^\Gamma_2 - V^\Gamma)] + \text{c.c.} \right\}\,,
\end{aligned}$$ where the second line comes from trading the superpotential term in for an expression introducing gauge three-forms dynamically: $\Lambda_{1\Sigma}$ are chiral superfields which play the role of Lagrange multipliers and $\Sigma_{1\Lambda}$ are complex linear multiplets. The third line, instead, is the dynamical promotion of the Fayet–Iliopoulos term: $\Sigma_{2}^\Gamma$ are complex linear multiplets while $\Lambda_{2\Gamma}$ are real Lagrange multipliers superfields.
The equations of motion for $\Sigma_{1\Lambda}$, $$D_\alpha[\mathcal{N}^{\Lambda \Sigma}(\Lambda_{1\Sigma}- \bar \Lambda_{1\Sigma})]=0,$$ impose indeed that $\Lambda_{1\Sigma} = e_\sigma + m^\Gamma F_{\Sigma \Gamma}$. On the other hand, integrating out the complex linear superfield $\Sigma_2$, we immediately get that $\Lambda_{2\Sigma}$ are just real constants $\xi_{\Sigma}$. Plugging these back into the Lagrangian , we must obtain again . It is however immediate to realize that, for this to happen, we have to require the prepotential $F(\mathcal{A})$ to be a degree–two homogeneous function of its argument, in order that $F_{\Lambda\Sigma}X^\Sigma = F_\Lambda$ and the superpotential is recovered. This restriction is avoided in the two steps procedure we presented before.[^2]
Despite this fact, we can still produce a Lagrangian which contains gauge three-forms in place of the auxiliary fields $\vec{Y}^\Lambda$. To this purpose, we have to integrate out the chiral superfields $X^\Lambda$ and the vector superfields $V^\Lambda$, as well as the Lagrange multipliers $\Lambda_{1\Sigma}$ and $\Lambda_{2\Sigma}$. The variation with respect to the Lagrange multipliers $\Lambda_{1\Sigma}$ re-expresses the chiral superfields $X^\Lambda$ as double three-form multiplets as $$\label{N2Xsoll}
\begin{aligned}
X^\Lambda &= \frac14 \bar D^2 \left[(N^{\Lambda \Gamma} (\Sigma_{1\Gamma}-\bar \Sigma_{1\Gamma})\right]\equiv S^\Lambda\,.
\end{aligned}$$ Notice that this relationship is non-linear in $S^\Lambda$ and it might not be possible in general to solve it and obtain an explicit expression for these superfields, but we can nevertheless understand their main properties. We can calculate first of all the lowest components $$S^\Lambda | = \mathcal{N}^{\Lambda \Gamma} \left(-\frac14 \bar D^2 \bar \Sigma_{1\,\Gamma} \big|\right)+\text{fermions} \equiv s^\Lambda +\text{fermions}\,.$$ We notice then that is left invariant by the gauge transformations $$\Sigma_{1\, \Gamma} \to \Sigma_{1\, \Gamma} + \tilde L_\Gamma+\bar F_{\Gamma \Delta} L^\Delta,$$ where $\tilde L_\Lambda$ and $ L^\Lambda$ are real linear superfields. In order to be compatible with this gauge transformation, the gauge three-forms in $S^\Lambda$ have to appear in $$-\frac14 D^2 S^\Lambda | = -i \mathcal{N}^{\Lambda \Sigma}\, {}^*\!\mathcal{F}_{4\Sigma} + \text{fermions}\,,$$ within the specific combination of four-forms $$\mathcal{F}_{\Lambda\, klmn} \equiv \tilde F_{\Lambda \,klmn}+\bar F_{\Lambda \Sigma} F^\Sigma_{klmn}\,.$$ In other words, in order to preserve the associated gauge invariance, which is necessary for the matching of the degrees of freedom, the complex gauge-three forms in $S^\Lambda$ are divided into two real parts, ${}^*\!F^\Lambda_4$ and ${}^*\!\tilde F_{4\,\Lambda}$ which appear in the Lagrangian combined inside ${}^*\!\mathcal{F}_{4\,\Lambda}$. This is more involved with respect to the analogous case in the previous subsection, but it is essential in order to reconstruct the correct on-shell form of the Lagrangian.
The variation with respect to the real Lagrange multipliers $\Lambda_{2\Sigma}$, instead exchanges the usual vector multiplets $V^\Lambda$ with their variant versions , as in the previous discussion. The variations with respect to the chiral superfields $X^\Lambda$ and the vector superfields $V^\Lambda$ give respectively $$\label{N2L1soll}
\begin{aligned}
\Lambda_{1\, \Sigma}&= \frac 14 \bar{D}^2 K_{\Sigma} - \frac14 \tau_{\Sigma \Gamma\Delta} W^{\Gamma\, \alpha} W^\Delta_{\alpha} \\
&- \frac18 \bar D^2 \left[(\Sigma_{1\,\Pi}-\bar \Sigma_{1\,\Pi}) (\Lambda_{1\,\Gamma}-\bar \Lambda_{1\,\Gamma}) \mathcal{N}^{\Pi\Lambda} \mathcal{N}^{\Delta\Gamma} \tau_{\Sigma \Lambda\Delta}\right]\,,\\
\Lambda_{2\,\Sigma}&= \text{Re}\, \tau_{\Sigma\Gamma} D^{\alpha} W^{\Gamma}_\alpha + \frac12 \left( D^\alpha \tau_{\Sigma \Gamma} W^{\Gamma}_ \alpha + \bar D_{\dot \alpha} \bar \tau_{\Sigma \Gamma} \bar W^{\Gamma \dot \alpha} \right)\,.
\end{aligned}$$ Substituting , and into we obtain a new Lagrangian fully re-expressed in terms of the three-form multiplets $$\label{N2LDD}
\mathcal{L} = \int d^4\theta\, K(X,\bar X)+\left(\frac14 \int d^2\theta \tau_{\Lambda\Sigma}(X) W^{\Lambda\,\alpha} W^\Sigma_\alpha + c.c\right)+\mathcal{L}_{\rm bd},$$ with $$\label{N2LDbdd}
\begin{split}
\mathcal{L}_{\rm bd}= &-\frac14 \left(\int d^2\theta \bar D^2 - \int d^2\bar\theta D^2\right) \left[\Lambda_{1\,\Sigma} \mathcal{N}^{\Sigma\Gamma} \bar\Sigma_{1\,\Gamma}\right]\\
&+\frac{1}{16} \left(\int d^2\theta \bar D^2 - \int d^2\bar\theta D^2\right) \left( \Lambda_{2\,\Gamma} \Sigma^\Gamma_2\right) +\text{c.c.} \,.
\end{split}$$ Its bosonic components are $$\label{N2NLScomm}
\begin{split}
\mathcal{L}\big|_{\text{bos}} =& -\mathcal{N}_{\Lambda\Sigma}\, \partial_m s^\Lambda\, \partial^m \bar s^\Sigma -\frac 14 \mathcal{N}_{\Lambda\Sigma} F^{\Lambda\, mn}F^\Sigma_{mn} - \frac18 \text{Re}\, F_{\Lambda\Sigma}\, \varepsilon_{klmn} F^{\Lambda\, kl} F^{\Sigma mn} \\
&+ \mathcal{N}^{\Lambda\Sigma} {}^*\!\mathcal{F}_{4\,\Lambda} {}^*\!\bar{\mathcal{F}}_{4\,\Sigma} +\frac12 \mathcal{N}_{\Lambda\Sigma} {}^*\!G_4^\Lambda {}^*\!G_4^\Sigma+ \mathcal{L}_{\rm bd}
\end{split}$$ with $$\label{N2Lbdcom}
\begin{split}
\mathcal{L}_{\rm bd} = &\left\{-\frac{1}{3!} \partial_k \left[\varepsilon^{klmn} \mathcal{N}^{\Lambda\Sigma} \left(\tilde A_{\Lambda \,lmn} + \bar F_{\Lambda\Sigma\Gamma} A^\Gamma_{lmn}\right) {}^*\!\bar{ \mathcal{F}}_{4\Sigma}\right]+\text{c.c.}\right\}
\\
&-\frac{1}{3!}\partial_k \left(\varepsilon^{klmn} \mathcal{N}_{\Lambda\Sigma}\, B^\Lambda_{lmn} {}^*\! G_4^\Sigma\right) \, .
\end{split}$$
We notice that in no gauging parameter appears. Indeed, the entire superpotential is dynamically generated once the gauge three-forms are set on-shell. In fact, from , the equations of motion for the real gauge three-forms $A^\Lambda_3$ and $\tilde A_{3\,\Lambda}$ give $$\begin{split}
\text{Re}\, \left(\mathcal{N}^{\Lambda\Sigma} {}^*\!\bar{\mathcal{F}}_{4\,\Sigma}\right) =- m^\Lambda\,,\qquad \text{Re}\, \left(\mathcal{N}^{\Sigma\Gamma} \bar F_{\Lambda\Sigma} {}^*\!\bar{ \mathcal{F}}_{4\,\Gamma}\right) = e_\Lambda\,,
\end{split}$$ with $e_\Lambda$ and $m^\Lambda$ arbitrary real constants. These equations can be recast as $$\label{N2Nsolcalff}
{}^*\! \mathcal{F}_{4\,\Lambda} = - i(e_\Lambda+ \bar F_{\Lambda\Sigma} m^\Sigma)\,.$$ The equation of motion for the three-forms $B_3^\Lambda$ gives $$\label{N2Nsolhh}
{}^*\! G^\Lambda_4 = \frac12 \mathcal{N}^{\Lambda\Sigma} \xi_\Sigma$$ with $\xi_\Sigma$ arbitrary real constants. Plugging and in we obtain directly the same potential as , for the particular choice .
Dynamical Partial Supersymmetry Breaking {#sec:DPSB}
========================================
In this section we examine how the partial breaking of supersymmetry is realized in terms of the gauge three-forms. We discuss to original model of [@Antoniadis:1995vb] using our formalism and then we construct an effective theory capturing its low energy description, along the lines of [@Bagger:1996wp; @Rocek:1997hi; @Antoniadis:2017jsk]. Through the entire discussion, all the parameters are going to be generated dynamically.
Dynamical Antoniadis–Partouche–Taylor model
-------------------------------------------
We consider the Lagrangian which, as we have shown, reproduces the scalar potential . This is the same scalar potential of [@Antoniadis:1995vb], therefore we can argue that is a different off-shell completion of the same on-shell model. In [@Antoniadis:1995vb] it is discussed how the scalar potential admits vacua in which the $\mathcal{N}=2$ supersymmetry is partially spontaneously broken and a massless goldstino is present in the spectrum, together with a massive scalar and a massive fermion. We refer therefore the reader to [@Antoniadis:1995vb; @Antoniadis:2017jsk] for additional details concerning the nature of the vacua of , while in the present subsection we concentrate on the analysis of the supersymmetry transformations of the fermions, in our formalism, in order to identify the goldstino. With respect to [@Antoniadis:1995vb], we are going to give conditions on the gauge three-forms which are valid off-shell and which match the result of [@Antoniadis:1995vb], when going on-shell.
To tell what amount of supersymmetry is preserved, it is necessary to examine how the supersymmetry variations of the fermions behave. We then focus on $$\label{SusyVar2PsiPB}
\delta {\bm \Psi} =\frac{i}{\sqrt{2}} (\vec{\sigma} \cdot \vec{Y}) {\bm \eta} + \ldots\,.$$ and we remind that a goldstino transforms with a shift under the broken supersymmetry. It is clear that, for generic values of $\vec{Y}$, the whole $\mathcal{N}=2$ supersymmetry is non-linearly realized and the vacuum is not supersymmetric. In order for the vacuum to preserve $\mathcal{N}=1$ supersymmetry, it is therefore necessary that a linear combination of the fermions $\psi$ and $\lambda$ transforms homogeneously, namely without a shift in any of the two supersymmetry parameters. This can happen if and only if the matrix $$\label{SusyVar2Matr}
\vec{\sigma} \cdot \vec{Y} = \begin{pmatrix} Y^3 & Y^1-i Y^2 \\Y^1 + i Y^2 & -Y^3\end{pmatrix}$$ has at least one zero eigenvalue. A necessary condition is that its determinant $$\label{detsy}
{\rm det}\, ( \vec{\sigma} \cdot \vec{Y} ) = - \vec{Y} \cdot \vec{Y}$$ vanishes. As pointed out in [@Antoniadis:1995vb], when the magnetic Fayet–Iliopoulos parameter is turned off in , then $\vec{Y}$ is real and the quantity is always positive, but for the trivial case in which $\vec{Y}=0$ and supersymmetry is totally preserved. When the magnetic gauging are inserted, however, $\vec{Y}$ acquires a non-vanishing imaginary part and the matrix $ \vec{\sigma} \cdot \vec{Y} $ can be degenerate. In this case, the vacuum can preserve $\mathcal{N}=1$ supersymmetry. This is how the partial breaking $\mathcal{N}=2 \rightarrow \mathcal{N}=1$ was originally conceived in [@Antoniadis:1995vb], but here we adopt a slightly different perspective, in which the mechanism is sourced by a certain choice of boundary conditions, compatible with the equations of motions.
Considering , the auxiliary fields $\vec{Y}$ obtained form the variant multiplets depend on the gauge three-forms as $$\label{Y3forms}
\vec{Y} =-2 \begin{pmatrix}
{\rm Re}\, {}^*\! F_4 + m\\ -{\rm Im}\,{}^*\! F_4 +i m\\ \frac{1}{\sqrt{2}} {}^*\! G_4 \end{pmatrix}\,.$$ Therefore, it is possible to realise that the matrix $$\label{SusyVar2Matr4f}
\vec{\sigma} \cdot \vec{Y} = -2 \begin{pmatrix} \frac{1}{\sqrt{2}} {}^*\! G_4 & {}^*\! F_4 + 2m\\{}^*\! \bar{F}_4 & -\frac{1}{\sqrt{2}} {}^*\! G_4 \end{pmatrix}$$ is degenerate when $$\frac12 ({}^*\! G_4)^2 = -|{}^*\! F_4|^2 - 2m\, {}^*\! \bar{F}_4 \,.$$ This equation can be solved by $$\label{DegenF4}
{\rm Im}\,{}^*\! {F}_4 =0 \qquad {\rm and} \qquad \frac12 ({}^*\! G_4)^2 = -2m\,{\rm Re}\,{}^*\! {F}_4 - |{}^*\! F_4|^2\,,$$ which is are off-shell conditions on the gauge three-forms for the partial breaking to occur. When going on-shell by using and , $\vec{Y}$ becomes $$\label{Y3formsOS}
\vec{Y} = \frac{2}{\text{Im}\, F_{SS}}\begin{pmatrix}
0\\ e+ m \bar F_{SS} \\ -\frac{\xi}{2\sqrt{2}} \end{pmatrix}$$ and, in order for to be satisfied, we need $$\label{PB_Vacuum}
\frac{e}{m} = - \text{Re}\, F_{SS}\,,\quad \quad -\frac{\xi}{2\sqrt{2}\, m} = \text{Im}\, F_{SS}$$ so that gives $$\vec{Y} = 2m\begin{pmatrix}
0\\ -i \\ 1
\end{pmatrix},$$ which matches the result in [@Antoniadis:1995vb]. For this choice of $\vec{Y}$, the matrix is degenerate. The on-shell supersymmetry transformations are indeed $$\begin{aligned}
\delta \psi = i m (-\eta_1+\eta_2)+\ldots\,,\quad
\delta \lambda = i m (\eta_1-\eta_2)+\ldots
\end{aligned}$$ and we recognize that the combination $\psi+\lambda$ transforms linearly $$\delta (\psi+\lambda) = 0 +\ldots,$$ while the combination $\psi-\lambda$ transforms non-linearly $$\delta (\psi-\lambda) = -2im (\eta_1-\eta_2) +\ldots,$$ signaling the presence of a goldstino and the spontaneous breaking of one of the two supersymmetries. In our formalism, therefore, the partial breaking of supersymmetry is a consequence of the boundary conditions and . A different choice of the boundary conditions would generically lead to a different amount of broken supersymmetry.
The low energy effective description
------------------------------------
In this subsection we use the formalism we have developed in order to construct an effective description for the previous model with partially broken supersymmetry, along the lines of [@Bagger:1996wp; @Rocek:1997hi; @Antoniadis:2017jsk; @Dudas:2017sbi]. We will give evidence that the boundary terms that we have been including so far in all the Lagrangians are not solely an artifact of the formalism, but they contain important physical information.
The starting point is the Lagrangian , or equivalently . For convenience, however, this time we choose the boundary conditions so as the solutions to the equations of motion and now read $$\begin{aligned}
\label{Alt_Fos2}
F_{klmn} &=-\frac{i}{{\rm Im}\, F_{SS}}\left(e+i \frac{\xi}{2\sqrt 2}+m F_{SS}\right)\varepsilon_{klmn}\,,
\\
\label{Alt_Gos2}
G_{klmn} &= 0\,.\end{aligned}$$ The Lagrangian associated to these new boundary conditions and the one studied in the previous sections are related by a $\rm SU(2)_R$ transformation and therefore their physical properties are not changed.[^3] The scalar potential is given by $$\mathcal{V} = \frac{1}{\text{Im}\, F_{SS}}\left| e + i\frac{\xi}{2\sqrt 2}+m F_{SS}\right|^2$$ and differs from only for an irrelevant additive constant. In particular the conditions and for the existence of a vacuum with partial breaking of supersymmetry are again satisfied, but the triplet of auxiliary fields $\vec Y$ is now rotated to $$\vec{Y} = \frac{2}{\text{Im}\, F_{SS}}\begin{pmatrix}
\frac{\xi}{2\sqrt{2}}\\ e+ m \bar F_{SS} \\ 0 \end{pmatrix}\,,$$ which, setting the three-forms on-shell, becomes $$\vec{Y} = 2m\begin{pmatrix}
-1\\ -i \\ 0
\end{pmatrix}.$$ The spectrum in this vacuum is the same as before: it contains a massless goldstino, a massive scalar and a massive fermion, whose masses are proportional to $F_{SSS}$.
It is possible then to construct an effective theory for this setup by restricting the analysis to an energy regime well below the scale given by the masses of the massive fields, or equivalently by taking the formal limit $F_{SSS}\to \infty$. To this purpose, we first expand the Lagrangian around the vacuum. In particular we assume that the choices
\[BG\_Vac\] $$\begin{aligned}
F_{SS}^{(0)} &= - \frac{e}{m} - i\frac{\xi}{2 \sqrt{2} m}\,, \label{BG_VacA}
\\
F_{klmn}^{(0)} &=-\frac{i}{{\rm Im}\, F_{SS}^{(0)}}\left(e+i\frac{\xi}{2\sqrt 2}+m F_{SS}\right)\varepsilon_{klmn}\,,\label{BG_VacB}
\\
G_{klmn}^{(0)} &= 0\label{BG_VacC}\end{aligned}$$
hold for a particular background value $S_0$ of the superfield $S$, set $S= S_0 + \tilde S$ and expand around $S_0$. Expanding then , along with the boundary terms , using , we arrive at the following effective Lagrangian $$\label{Lfluct}
\begin{aligned}
\mathcal{L} &= \left\{F_{SS}^{(0)} \int d^2 \theta\, \left( \frac{i}{8} \tilde{S} \bar D^2 \bar{\tilde{S}} - \frac{i}4 W^{\alpha} W_{\alpha}+ m \tilde S \right)+\left(e+ i \frac{\xi}{2\sqrt{2}}\right)\int d^2 \theta\, \tilde{S}+ \text{c.c.} \right\}\\
&\quad\,+\mathcal{L}_{\rm bd} +\ldots\,,
\end{aligned}$$ with $$\begin{split}
\mathcal{L}_{\rm bd}= &\frac14 \left(\int d^2\theta \bar D^2 - \int d^2\bar\theta D^2\right) \left[ {\rm Im}\, F_{SS}^{(0)} \left(\frac 14 {D}^2 {\tilde S} \right) {\tilde\Sigma}_1\right]+\text{c.c.} \, ,
\end{split}$$ where the dots stand for higher order terms in the fluctuations. We notice that the coupling linear in $\tilde S$ with the complex electric parameter $e + i\frac{\xi}{2\sqrt 2}$ is generated from the boundary terms , after we set $$\Lambda_1 = \Lambda_1^{(0)} + \tilde \Lambda_1 = \left(e + i\frac{\xi}{2\sqrt 2}\right)+ \tilde \Lambda_1\,.$$ The presence of this coupling is crucial for the existence of the effective theory and it is indeed a pure boundary term contribution.
We take now the limit of infinite mass. By setting to zero the divergent part of the equations of motion of the fluctuations, the following constraint is produced $$\label{constr1}
\frac i8 S \bar D^2 \bar S - \frac i4 W^\alpha W_\alpha + m S =0$$ where from now on we omit the tilde on the fields. This constraint can be recast into the form $$S = \frac{W^\alpha W_\alpha}{-4im + \frac12 \bar D^2 \bar S},$$ which can be solved iteratively [@Bagger:1996wp; @Rocek:1997hi]. The solution is $$S =\frac{i}{4m} \left\{W^2 - \bar D^2\left[\frac{W^2 \bar W^2}{(4m)^2+ A + \sqrt{(4m)^4+2 (4m)^2A+ B^2}}\right]\right\},$$ where we have defined $$\begin{aligned}
A =\frac{D^2 W^2 + \bar D^2 \bar W^2}{2}\,, \qquad B =\frac{D^2 W^2 - \bar D^2 \bar W^2}{2}\,.\end{aligned}$$ It is known [@Rocek:1997hi; @Ferrara:2014oka; @Antoniadis:2017jsk; @Dudas:2017sbi] that, in the language of $\mathcal{N}=2$ superspace, the constraint is a consequence of a nilpotent constraint imposed on top of the original vector superfield, namely $\mathcal{S}^2=0$. This constraint is removing the entire $\mathcal{N}=1$ chiral superfield $S$ from $\mathcal{S}$ and expresses it as a function of the remaining $\mathcal{N}=1$ vector superfield $W_\alpha$.
By implementing the constraint into , the low energy effective action reduces $$\begin{aligned}
\label{LeffBG}
\mathcal{L} &= \left( \frac{i e}{4m} - \frac{\xi}{8 \sqrt{2} m}\right)\int d^2 \theta\,\left\{W^2 - \bar D^2\left[\frac{W^2 \bar W^2}{(4m)^2+ A + \sqrt{(4m)^4+2 (4m)^2A+ B^2}}\right]\right\}\\
&\quad\, + \text{c.c.}\,.
\end{aligned}$$ We stress that, from our perspective, this action is entirely contained in the boundary terms , which captures therefore the effective description of the model in the infrared regime. As a consequence, when considering four-dimensional Lagrangian with gauge three-forms, it is important to include the appropriate boundary terms, since they can contain non-trivial physical information. The on-shell bosonic components of can be recast into the form of a Born–Infeld action $$\label{LDBI}
\mathcal{L} = -\frac{m\xi }{\sqrt 2}\left(1-\sqrt{-\det \left(\eta_{mn}+F_{mn}\right)}\right)+\frac{me}{4}\epsilon^{mnpq}F_{mn}F_{pq},$$ where we have rescaled $F_{mn} \to \sqrt 2 m F_{mn}$ in order to have canonically normalized kinetic terms. We stress that all the parameters appearing in this Lagrangian have been generated dynamically, as vacuum expectation values of gauge three-forms. In particular, the product $m \xi$ may be related to the tension of the D3 brane described by , for which we have provided a dynamical origin. We notice finally that, by choosing a boundary condition in which $m=0$, the partial breaking of supersymmetry does not occur and this effective description does not hold.
Conclusion
==========
In this work we have studied $\mathcal{N}=2$ global supersymmetric models in which the parameters entering the superpotential and the scalar potential have a dynamical origin. A systematic procedure has been given in order to trade standard $\mathcal{N}=2$ multiplets for variant versions, in which gauge-three forms appear as non-propagating degrees of freedom. When going on-shell, parameters are generated in the theory as integration constants for the gauge three-forms. In other words, the choice of parameters in the original Lagrangian is traded for the problem of specifying certain boundary conditions for the gauge three-forms, compatibly with their equations of motion.
Our results may be relevant, first of all, for understanding the origin of parameters in effective theories which come from string theory. It is known in fact that string theory does not have any free parameter, but the string length and therefore eventual additional parameters appearing in models originating from string theory have to be interpreted as expectation values of certain fields. In this context, four dimensional theories preserving $\mathcal{N}=2$ supersymmetry may appear for example when compactifying string theory on Calabi–Yau three folds.
The results presented in this work can be of interest also for the study of supersymmetric theories from a pure four-dimensional point of view, in particular for understanding the relation between off-shell and on-shell formulations of extended supersymmetry, which has not been completely clarified at present. We have presented in fact novel off-shell multiplets and Lagrangians which reproduce correctly known on-shell setups.
As possible further directions, it would be important to extend the analysis to local supersymmetry and explore the coupling of the gauge three-forms to membranes already in four dimensions as in [@Bandos:2018gjp]. We leave these developments for future work.
Acknowledgements {#acknowledgements .unnumbered}
----------------
N.C. is supported by an FWF grant with the number P 30265. S.L. is grateful for hospitality to Istituto de Física Teórica UAM-CSIC, Madrid, where this work was completed. We thank Fotis Farakos, Dmitri Sorokin, Gabriele Tartaglino-Mazzucchelli and especially Gianguido Dall’Agata and Luca Martucci for discussions and comments on the manuscript.
Conventions {#app:Conv}
===========
The components of a three-form $A_3$ are defined as $$A_3 = \frac{1}{3!} A_{kmn} d x^k\, \wedge d x^m\, \wedge d x^n\,,$$ whose field strength is defined as $$F_4 \equiv d A_3\,, \qquad F_4 = \frac{1}{4!} F_{klmn} d x^k\, \wedge d x^l\, \wedge d x^m\,\wedge d x^n$$ with components $$F_{klmn} = 4\, \partial_{[k} A_{lmn]}\,.$$ The Hodge-dual of a four-form field strength $F_4$ is $${}^*\!F_4 = \frac{1}{4!}\varepsilon^{klmn} F_{klmn} = \frac{1}{3!} \varepsilon^{klmn} \partial_{[k} A_{lmn ]}\,$$ and in our conventions $$\varepsilon_{klmn} \varepsilon^{pqrs} = - 4! \delta^{p}_{[k} \delta^{q}_{l} \delta^{r}_{m}\delta^{s}_{n]}\,.$$
Component structure of ${\cal N}=1$ superfields {#app:Super}
===============================================
Here we collect the component structures of the $\mathcal{N}=1$ multiplets considered throughout this work.
The chiral multiplet $X$ is defined by $$\bar{D}_{\dot\alpha} X = 0
\label{eq:ChiralA}$$ and its component expansion is $$X = \varphi + \sqrt{2} \theta \psi + \theta^2 f + i \theta \sigma^m \bar\theta \partial_m \varphi -\frac{i}{\sqrt{2}} \theta^2 \partial_m \psi \sigma^m \bar\theta + \frac 14 \theta^2\bar\theta^2 \Box \varphi,
\label{ChiralB}$$ where $\varphi$ and $f$ are complex scalar fields, while $\psi$ is a Weyl spinor. The independent components of $\Phi$ can be defined by the projections $$\label{ChiralComp}
\begin{aligned}
\Phi | &= \varphi \,, \\
D_\alpha \Phi &|= \sqrt 2 \psi_\alpha \,,\\
-\frac14 D^2 \Phi| &= f \,,
\end{aligned}$$ where the vertical line means that the quantity is evaluated at $\theta=\bar\theta=0$.
The real scalar multiplet $V$ is defined by $V = \bar V$ and it has the following component structure $$\begin{split}
V =& \, u + i \theta \chi - i \bar\theta \bar\chi + {i} \theta^2 \bar\varphi - {i} \bar\theta^2 {\varphi} - \theta \sigma^m \bar\theta v_m \\
& +i\theta^2\bar{\theta} \left( \bar{\lambda}
+\frac{i}{2}\bar{\sigma}^m\partial_m \chi\right)-i\bar{\theta}^2 \theta \left( \lambda+\frac{i}{2}{\sigma}^m\partial_m \bar{\chi}\right)+\frac12\theta^2\bar\theta^2 \left({\rm D}-\frac12\Box u \right) \, ,
\label{VectorB}
\end{split}$$ where $u$ and $D$ are real scalar fields, $\varphi$ is a complex scalar field, $v_m$ is a real vector field and $\chi$ and $\lambda$ are Weyl spinors. The independent components of $V$ can be defined by projections $$\label{Vprojections}
\begin{aligned}
V | &= u \,,\\
D_\alpha V| &= i \chi_\alpha\,,\\
\frac 14 \bar{\sigma}^{\dot\alpha \alpha}_m [D_\alpha, \bar{D}_{\dot\alpha}] V| &= v_m\,,\\
\frac{i}{4} D^2 V| &= \bar\varphi\,,\\
-\frac14 \bar D^2 D_\alpha V| &= -i\lambda_\alpha\,,\\
\frac{1}{16} D^2 \bar{D}^2 V| &= \frac12\left({\rm D} - i \partial^m v_m \right) \, .
\end{aligned}$$
The real linear multiplet $L$ is a real multiplet which, in addition, satisfies the condition $$D^2 L = 0\,, \qquad \bar{D}^2 L = 0 \,.
\label{RealLMdef}$$ Its component expansion is $$\begin{split}
L = &l + i \theta \eta -i \bar\theta \bar\eta - \frac12 \theta \sigma_m \bar\theta \varepsilon^{mnpq} \partial_{n}\Lambda_{pq} \\
&+ \frac12 \theta^2\bar\theta \bar\sigma^m \partial_m \eta - \frac12 \bar\theta^2 \theta \sigma^m \partial_m \bar\eta -\frac14 \theta^2\bar\theta^2 \Box l \, ,
\end{split}
\label{RealLM}$$ where $l$ is a real scalar, $\Lambda_{mn}$ is a rank 2 antisimmetric tensor and $\eta$ is a Weyl spinor.\
The independent components of $L$ can be defined by projections $$\begin{aligned}
L | &= l \,, \\
D_\alpha L| &= i \eta_\alpha\,,\\
\frac12\bar{\sigma}^{m\,\dot\alpha\,, \alpha}\left[D_\alpha,\bar{D}_{\dot{\alpha}}\right] L| &= \varepsilon^{mnpq} \partial_{n}\Lambda_{pq} \,.
\end{aligned}$$
The complex linear multiplet $\Sigma$ satisfies the condition $$\bar{D}^2 \Sigma = 0 \,.
\label{ComplexLMdef}$$ Its component expansion is $$\begin{split}
\Sigma = &\sigma + \sqrt 2\theta \psi + \sqrt{2} \bar\theta \bar\rho - \theta \sigma_m \bar\theta \mathcal{B}^{m} + \theta^2 \bar s+ \sqrt 2\theta^2\bar\theta \bar\zeta \\
&-\frac{i}{\sqrt{2}} \bar\theta^2 \theta \sigma^m \partial_m \bar\rho + \theta^2\bar\theta^2 \left(\frac{i}{2} \partial_m \mathcal{B}^{m} -\frac14 \Box \sigma \right).
\end{split},
\label{eq:ComplexLM}$$ where $\sigma$ and $\bar s$ are complex scalars, $\rho$, $\psi$ and $\xi$ are Weyl spinors and $\mathcal{B}^{m}$ is a complex vector which is Hodge dual to the three-form $$\mathcal{B}^{m} = \frac{1}{3!} \varepsilon^{mnpq} \mathcal{B}_{npq}.
\label{ComplexLMvector}$$ The components of $\Sigma$ can be defined by the projections $$\label{SigmaCom}
\begin{aligned}
\Sigma | &= \sigma \,,\\
D_\alpha \Sigma| &= \sqrt 2 \psi_\alpha\,,\\
\bar D_{\dot \alpha} \Sigma| &= \sqrt 2 \bar \rho_{\dot\alpha}\,,\\
\frac14 \bar{\sigma}^{m\,\dot\alpha \alpha}
\left[D_\alpha,\bar{D}_{\dot{\alpha}}\right] \Sigma| &= \mathcal{B}^{m}\,,\\
-\frac14 D^2 \Sigma| &= \bar s\,,\\
\bar D_{\dot \alpha} D^2 \Sigma| & = -4\sqrt 2 \bar \zeta_{\dot \alpha} +2\sqrt 2 i \, \partial_m \psi^\alpha \sigma^m_{\alpha \dot \alpha}\,,\\
\frac{1}{16} \bar{D}^2 D^2 \Sigma| &= i \partial_m \mathcal{B}^m \,.
\end{aligned}$$
Case with $N$ vector multiplets {#app:Nvec}
===============================
The procedure outlined in Section \[sec:N23form1\] can be generalized to the case involving an arbitrary number $N$ of vector multiplets. We recall that the $\mathcal{N}=2$ Lagrangian equipped with both electric and magnetic gauging parameters is $$\tag{\ref{N2NLag}}
\mathcal{L} = \frac{i}{2} \int d^2 \theta d^2 \tilde \theta \left[F( \mathcal{A}) - \mathcal{A}_{D\,\Lambda} \mathcal{A}^\Lambda \right] + \frac12 \left(\vec{E}_\Lambda\cdot \vec{Y}^\Lambda+ \vec{M}^\Lambda\cdot \vec{Y}_{D \Lambda}\right)+\text{c.c.}\,,$$ which, in $\mathcal{N}=1$ language, translates into $$\label{AltN_LN=1a}
\begin{aligned}
\mathcal{L} &= -\frac i4 \int d^2\theta \left( F_{\Lambda \Sigma} W^{\Lambda\,\alpha} W_\alpha^\Sigma - 2 W_{D\,\Lambda}^\alpha W_\alpha^\Lambda \right)\\
&\quad - \frac i2 \int d^4 \theta F_\Lambda \bar{X}^\Lambda + \frac 14 \int d^2 \theta \Phi^\Lambda \left( X_{D\,\Lambda} - F_\Lambda\right) +\frac12 \left(\vec{E}_\Lambda\cdot \vec{Y}^\Lambda+ \vec{M}^\Lambda\cdot \vec{Y}_{D\, \Lambda}\right)+ {\rm c.c.}\,.
\end{aligned}$$ We use the $\rm SU(2)$ R-symmetry of the theory to rotate the parameters such that $$\label{N2NLagGauging}
\Re \vec{E}_\Lambda =\left(0,-e_\Lambda,\frac{\xi_\Lambda}{2\sqrt{2}}\right)\,, \qquad \vec{M}^\Lambda = (0,-m^\Lambda,0)\,,$$ with $e_\Lambda$, $m^\Lambda$ and $\xi_\Lambda$ real constants. In this way the Lagrangian can be expressed in the $\mathcal{N}=1$ Language as $$\begin{aligned}
\label{N2NLagB}
\mathcal{L} = &\int d^4\theta\, K(X,\bar X)+ \left(\frac14 \int d^2\theta\, \tau_{\Lambda \Sigma} (X) W^{\Lambda\alpha} W^\Sigma_\alpha+\text{c.c.}\right)\\
&+\left(\int d^2\theta\,W(X)+\text{c.c.}\right) + \xi_\Lambda \int d^4\theta\,V^\Lambda \,,
\end{aligned}$$ with $$\tau_{\Lambda \Sigma}(X) =- i F_{\Lambda \Sigma}, \qquad K = \frac{i}{2}\left(X^\Lambda \bar F_{ \Lambda} - \bar X^{\Lambda} F_\Lambda \right),$$ and the superpotential $W(X)$ is given by $$\label{N2NW}
W(X)= e_\Lambda X^\Lambda + m^\Lambda F_\Lambda(X)\,.$$ The bosonic components are $$\label{N2NLScom}
\begin{split}
\mathcal{L}\big|_{\text{bos}} =& -\mathcal{N}_{\Lambda \Sigma}\, \partial_m \varphi^\Lambda\, \partial^m \bar \varphi^\Sigma -\frac 14 \mathcal{N}_{\Lambda \Sigma} F^{\Lambda\, mn}F^\Sigma_{mn} - \frac18 \text{Re}\, F_{\Lambda \Sigma}\, \varepsilon_{klmn} F^{\Lambda\,kl} F^{\Sigma mn} \\
&+ \mathcal{N}_{\Lambda \Sigma} f^\Lambda \bar f^\Sigma +\frac12 \mathcal{N}_{\Lambda \Sigma} D^\Lambda D^\Sigma\\
& + (e_\Lambda + F_{\Lambda \Sigma} m^\Sigma) f^\Lambda + (e_\Lambda +\bar F_{\Lambda \Sigma} m^\Sigma) \bar f^\Lambda + \frac{1}{2} \xi_\Lambda D^\Sigma\,.
\end{split}$$ Integrating out the auxiliary fields $f^\Lambda$ and $D^\Lambda$ we arrive at $$\label{N2NLScomOS}
\begin{aligned}
\mathcal{L}\big|_{\text{bos}} =& -\mathcal{N}_{\Lambda \Sigma}\, \partial_m \varphi^\Lambda\, \partial^m \bar \varphi^\Sigma -\frac 14 \mathcal{N}_{\Lambda\Sigma} F^{\Lambda\,mn}F^\Sigma_{mn} \\
& - \frac18 \text{Re}\, F_{\Lambda \Sigma}\, \varepsilon_{klmn} F^{\Lambda\, kl} F^{\Sigma\, mn} - \mathcal{V}(\varphi, \bar \varphi)\,,
\end{aligned}$$ with the scalar potential $$\label{N2NV}
\begin{split}
\mathcal{V}(\varphi, \bar \varphi) =\; &\mathcal{N}_{\Lambda \Sigma} m^\Lambda m^\Sigma + \mathcal{N}^{\Lambda \Sigma} (e_\Lambda + m^\Gamma \text{Re}\,F_{\Gamma\Lambda})(e_\Sigma + m^\Delta \text{Re}\, F_{\Sigma \Delta})+\frac18 \mathcal{N}^{\Lambda \Sigma} \xi_\Lambda \xi_\Sigma\,,
\end{split}$$ which coincides with for the particular choice .
The first step of the procedure consists in promoting the magnetic gauging parameters $\vec{M}^\Lambda$ to be dynamical. This may be achieved by trading the Lagrange multiplier vector multiplets $\mathcal{A}_{D\,\Lambda}$ for the variant three-form multiplets . At $\mathcal{N}=1$ level, we promote then the term $\vec{M}^\Lambda\cdot \vec{Y}_{D\, \Lambda}$ in to a full dynamical entity as $$\label{AltN_LN=1D}
\begin{aligned}
\mathcal{L} &= \bigg\{-\frac i4 \int d^2\theta \left( F_{\Lambda\Sigma} W^{\Lambda\,\alpha} W_\alpha^\Sigma - 2 W_{D\,\Lambda}^\alpha W_\alpha^\Lambda \right)-
\\
& \quad\quad- \frac i2 \int d^4 \theta F_\Lambda \bar{X}^\Lambda + \frac 14 \int d^2 \theta \Phi^\Lambda \left( X_{D\,\Lambda} - F_\Lambda\right) + \frac12 \vec{E}_\Lambda \cdot \vec{Y}^\Lambda+ {\rm c.c.}\bigg\}+
\\
&\quad+\left\{\int d^2 \theta\, \left(\Lambda_1^{D\,\Pi} X_{D\,\Pi} + \frac14 \bar D^2 (\Sigma_{1 {D\,\Pi}} \bar \Lambda_1^{D\,\Pi}) \right)+ {\rm c.c.}\right\}
\\
&\quad+\left\{\frac18 \int d^2\theta\,\bar D^2 [\Lambda_2^{D\,\Pi} (\Sigma_{2{D\,\Pi}} - V_{D\,\Pi})] + \text{c.c.} \right\}.
\end{aligned}$$ The third line provides the trading between the $\mathcal{N}=1$ chiral multiplets $X_{D\,\Pi}$ and a double three-form multiplet, while the fourth line the one between the $\mathcal{N}=1$ vector multiplets $W_{D\,\Pi}^\alpha$ and their three-form counterparts. Integrating the complex linear superfields $\Sigma_{1{D\,\Pi}}$ and $\Sigma_{2{D\,\Pi}}$ gives $$\label{AltN_LambdaOS}
\Lambda_1^{D\,\Pi} = -M^{2\,\Pi}-i M^{1\,\Pi}\,,\qquad \Lambda^{D\,\Pi}_2= 2\sqrt{2} M^{3\,\Pi}\,,$$ with $\vec{M}^{\Pi}$ arbitrary real constants, establishing the equivalence between and . On the other hand, the variations with respect to the Lagrange multipliers $\Lambda_1^{D\,\Pi}$ and $\Lambda^{D\,\Pi}_2$ give the relations $$\begin{aligned}
\delta \Lambda_1^{D\,\Pi}: \quad X_{D\,\Pi} &= -\frac14 \bar D^2 \bar\Sigma_{1D\,\Pi} \equiv S_{D\,\Pi}\,, \label{AltN_XDsol}
\\
\delta \Lambda_2^{D\,\Pi}: \quad V_{D\,\Pi} &= \frac{\Sigma_{2D\,\Pi} + \bar \Sigma_{2D\,\Pi}}{2} \equiv U_{D\,\Pi}\,, \label{AltN_VDsol}\end{aligned}$$ and those with respect to the ordinary $\mathcal{N}=1$ superfields $X_{D\,\Pi}$ and $V_{D\,\Pi}$ result in $$\begin{aligned}
\label{AltN_L1Dsol}
\delta X_{D\,\Pi}: \quad \Lambda^{D\,\Pi}_1 &= -\frac 14 \Phi^\Pi\,,
\\
\label{AltN_L2Dsol}
\delta V_{D\,\Pi}: \quad \Lambda^{D\,\Pi}_2 &= - {\rm Im} (D^\alpha W^\Pi_\alpha)\,.\end{aligned}$$ Plugging (\[AltN\_XDsol\]-\[AltN\_L2Dsol\]) in , we get $$\label{AltN_LN=1Db}
\begin{aligned}
\mathcal{L} &=\bigg\{-\frac i4 \int d^2\theta \left( F_{\Lambda\Sigma} W^{\Lambda\,\alpha} W_\alpha^\Sigma - 2 W_{D\,\Lambda}^\alpha W_\alpha^\Lambda \right)-
\\
& \quad\quad- \frac i2 \int d^4 \theta F_\Lambda \bar{X}^\Lambda + \frac 14 \int d^2 \theta \Phi^\Lambda \left( S_{D\,\Lambda} - F_\Lambda\right) + \frac12 \vec{E}_\Lambda \cdot \vec{Y}^\Lambda+ {\rm c.c.}\bigg\}+ \mathcal{L}_{\rm bd}^{(D)}
\end{aligned}$$ with $$\label{AltN_LN=1Dbd}
\begin{split}
\mathcal{L}_{\rm bd}^{(D)}= &-\frac14 \left(\int d^2\theta \bar D^2 - \int d^2\bar\theta D^2\right) \left[\left(-\frac 14 \Phi^\Lambda \right) \bar\Sigma_{1D\,\Lambda}\right]\\
&+\frac{1}{16} \left(\int d^2\theta \bar D^2 - \int d^2\bar\theta D^2\right) \left[ - \text{Im} (D^\alpha W^\Lambda_\alpha) \Sigma_{2D\,\Lambda} \right] +\text{c.c.} \, .
\end{split}$$
This Lagrangian can also be rewritten in $\mathcal{N}=2$ superspace as $$\label{AltN_LN=2}
\mathcal{L} = \frac{i}{2} \int d^2 \theta d^2 \tilde \theta \left[F( \mathcal{A}) - \mathcal{S}_{D\,\Lambda} \mathcal{A}^\Lambda \right] + \frac12 \vec{E}_{\Lambda}\cdot \vec{Y}^\Lambda + {\rm c.c.} + \mathcal{L}_{\rm bd}^{(D)}\,.$$ where the $\mathcal{N}=2$ Lagrange multiplier $\mathcal{S}_{D\,\Lambda} $ is $$\label{ChiralSDN}
\begin{split}
\mathcal{S}_{D\,\Lambda}(y,\theta, \tilde \theta) &= S_{D\,\Lambda} (y,\theta) + \sqrt 2 \tilde\theta^\alpha W_{{D\,\Lambda}\alpha}(y,\theta) + \frac14\tilde \theta^2 \bar D^2\bar S_{D\,\Lambda}\,.
\end{split}$$ where $W_{{D\,\Lambda}\alpha} = -\frac14 \bar D^2 D_\alpha U_{D\,\Lambda}$, with $S_{D\,\Lambda}$ and $U_{D\,\Lambda}$ defined respectively in and .
In order to re-express the Lagrangian only in terms of the chiral multiplets $\mathcal{A}^\Lambda$, the variant Lagrange multipliers $\mathcal{S}_{D\,\Lambda}$ have to be integrating out. This amounts in integrating out the complex linear superfields $\Sigma_{1 {D\,\Lambda}}$ and $\Sigma_{2 {D\,\Lambda}}$ which, as in and , results in setting $$\label{AltN_DPhi_OS}
{\rm Im}\, {\rm D}^\Pi = \sqrt{2} M^{3\,\Pi}\, \qquad \Phi^\Pi = 4 (M^{2\,\Pi} +i M^{1\,\Pi})$$ If we choose $\vec{M}^\Pi = (0,-m^\Pi,0)$, the Lagrangian reads $$\label{AltN_LN=1c}
\begin{aligned}
\mathcal{L} &=
-\frac i4 \int d^2\theta F_{\Lambda\Sigma} W^{\Lambda\,\alpha} W^\Sigma_\alpha - \frac i2 \int d^4 \theta F_\Lambda \bar{X}^\Lambda + m^\Lambda \int d^2 \theta F_\Lambda + \frac12 \vec{E}_\Lambda\cdot \vec{Y}^\Lambda+ {\rm c.c.}.
\end{aligned}$$ In comparison with and recalling the component structure of the chiral multiplet , this suggests that integrating out $\mathcal{S}_{D\,\Lambda}$ constraints $\mathcal{A}^\Lambda$ to be the reduced chiral multiplet $$\label{AltN_A}
\mathcal{A}^\Lambda(y,\theta, \tilde \theta) = X^\Lambda(y,\theta) + \sqrt 2 \tilde\theta^\alpha W^\Lambda_\alpha(y,\theta) +\tilde \theta^2 \left(-2im^\Lambda+\frac14 \bar D^2\bar X^\Lambda\right).$$
The second step is to generate dynamically also the electric gauging parameters $\vec{E}$, by promoting the reduced chiral multiplets $\mathcal{A}^\Lambda$ to three-form multiplets as well. We convert then to $$\label{AltN_N2Master}
\begin{aligned}
\mathcal{L} =& \int d^4\theta\, K(X,\bar X)+\left(\frac14 \int d^2\theta\, \tau_{\Lambda\Sigma}(X) W^{\Lambda\,\alpha} W^\Sigma_\alpha + \text{c.c.}\right)+\\
&+\left\{\int d^2\theta \Lambda_{1\Pi} X^\Pi
+\frac14 \int d^2\theta\, \bar D^2\left(\Sigma_{1\Pi} \bar \Lambda_1^\Pi\right)+ m^\Pi \int d^2 \theta F_\Pi+\text{c.c.}\right\}+\\
&+\left\{\frac18 \int d^2\theta\,\bar D^2 [\Lambda_{2\Pi} (\Sigma^\Pi_2 - V^\Pi)] + \text{c.c.} \right\}\,.
\end{aligned}$$ As a check of consistency between the Lagrangians and , we may integrate out the complex linear superfields $\Sigma_1$ and $\Sigma_2$, obtaining $$\label{AltN_Lambda_sol}
\Lambda_{1\Lambda} = e_\Lambda\,, \qquad \Lambda_{2\Lambda} = \xi_\Lambda\,,$$ with $e_\Lambda$ and $\xi_\Lambda$ arbitrary real constants. Inserting in , we in fact re-obtain . Let us now recast the Lagrangian only in terms of the three-form multiplets. The variations with respect of the Lagrange multipliers $\Lambda_{1\Pi}$ and $\Lambda_{2\Pi}$ give $$\begin{aligned}
\delta \Lambda_{1\Pi}: \quad X^\Pi &= -\frac14 \bar D^2 \bar\Sigma_1^\Pi \equiv S^\Pi\,, \label{AltN_Xsol}
\\
\delta \Lambda_{2\Pi}: \quad V^\Pi &= \frac{\Sigma_2^\Pi + \bar \Sigma_2^\Pi}{2} \equiv U^\Pi\,, \label{AltN_Vsol}\end{aligned}$$ which trade the $\mathcal{N}=1$ multiplets for their three-form counterparts. The variations with respect to $X^\Pi$ and $V^\Pi$ give explicit expressions for the Lagrange multipliers $\Lambda_{1\Pi}$ and $\Lambda_{2\Pi}$: $$\begin{aligned}
\label{AltN_L1sol}
\delta X^\Pi: \quad \Lambda_{1\Pi} &= \frac 14 \bar{D}^2 K_{\Pi} - \frac{1}{4} \tau_{\Sigma\Lambda\Pi} W^{\Sigma\,\alpha} W^\Lambda_\alpha- m^\Sigma F_{\Sigma\Pi}\,,
\\
\label{AltN_L2sol}
\delta V^\Pi: \quad \Lambda_{2\Pi} &= \mathcal{N}_{\Pi\Sigma} D^\alpha W^\Sigma_\alpha + \frac{1}{2} \left[ (D^\alpha \tau_{\Pi\Sigma}) W_\alpha^\Sigma - (\bar D_{\dot \alpha} \tau_{\Pi\Sigma}) \bar W^{\dot \alpha\, \Sigma} \right]\,.\end{aligned}$$ Inserting (\[AltN\_Xsol\]–\[AltN\_L2sol\]) in we get $$\label{AltN_N2LD}
\mathcal{L} = \int d^4\theta K(S,\bar S) + \left( \frac14 \int d^2\theta \tau_{\Lambda\Sigma}(S) W^{\Lambda\,\alpha} W^\Sigma_\alpha + \text{c.c.}\right) +\mathcal{L}_{\rm bd},$$ with $$\label{AltN_N2LDbd}
\begin{split}
\mathcal{L}_{\rm bd}= &-\frac14 \left(\int d^2\theta \bar D^2 - \int d^2\bar\theta D^2\right) \left[\Lambda_{1 \Pi} \bar\Sigma_1^\Pi\right]\\
&+\frac{1}{16} \left(\int d^2\theta \bar D^2 - \int d^2\bar\theta D^2\right) \left( \Lambda_{2\Pi} \Sigma^\Pi_2 \right) +\text{c.c.} \, .
\end{split}$$ with $\Lambda_{1\Pi}$ and $\Lambda_{2\Pi}$ specified by and . The bosonic components of the Lagrangian are $$\label{AltN_N2LScom}
\begin{split}
\mathcal{L}\big|_{\text{bos}} =&-\mathcal{N}_{\Lambda \Sigma}\, \partial_m \varphi^\Lambda\, \partial^m \bar \varphi^\Sigma -\frac 14 \mathcal{N}_{\Lambda \Sigma} F^{\Lambda\, mn}F^\Sigma_{mn} - \frac18 \text{Re}\, F_{\Lambda \Sigma}\, \varepsilon_{klmn} F^{\Lambda\,kl} F^{\Sigma mn} \\
&+ \mathcal{N}_{\Lambda \Sigma} {}^*\! {F}_4^\Lambda {}^*\!\bar{{F}}_4^\Sigma +\frac12 \mathcal{N}_{\Lambda \Sigma} {}^*\!G_4^\Lambda {}^*\!G_4^\Sigma+\left(-i m^\Lambda F_{\Lambda\Sigma} {}^*\!\bar{{F}}_4^\Sigma + {\rm c.c.}\right)+ \mathcal{L}_{\rm bd}
\end{split}$$ with $$\label{AltN_N2Lbdcom}
\begin{split}
\mathcal{L}_{\rm bd} = &\left\{\frac{1}{3!} \partial_k\left[ B_{lmn}^\Lambda\left( \mathcal{N}_{\Lambda\Sigma}\bar F^{\Sigma\,klmn}- i m^\Sigma\, \varepsilon^{klmn} \bar F_{\Lambda\Sigma}\right)\right]+\text{c.c.}\right\}\\
&+\frac{1}{3!}\partial_k \left(\mathcal{N}_{\Lambda\Sigma}\, C_{lmn}^\Lambda G^{\Sigma\,klmn}\right) \, .
\end{split}$$ Setting the gauge three-forms on shell as $$\begin{aligned}
\label{AltN_Fos}
F_{klmn}^\Lambda &=-i \mathcal{N}^{\Lambda\Sigma} (e_\Sigma+m^\Pi F_{\Pi\Sigma})\varepsilon_{klmn}\,,
\\
\label{AltN_Gos}
G_{klmn}^\Lambda &= -\frac12 \mathcal{N}^{\Lambda\Sigma}\, \xi_\Sigma\, \varepsilon_{klmn}\,,\end{aligned}$$ the same potential as is recovered.
The Lagrangian can also be fully re-expressed in $\mathcal{N}=2$ language as $$\label{N2LagNSS}
\mathcal{L} = \frac{i}{2} \int d^2 \theta d^2 \tilde \theta F( \mathcal{S}) +\text{c.c.} + \mathcal{L}_{\text{bd}}\,.$$ provided that we introduce the $\mathcal{N}=2$ chiral multiplets defined as $$\label{AltN_S}
\mathcal{S}^\Lambda(y,\theta, \tilde \theta) = S^\Lambda(y,\theta) + \sqrt 2 \tilde\theta^\alpha W^\Lambda_\alpha(y,\theta) +\tilde \theta^2 \left(-2im^\Lambda+\frac14 \bar D^2\bar S^\Lambda\right).$$ with $S^\Lambda$ as in and $W_{\alpha}^\Lambda = -\frac14 \bar D^2 D_\alpha U^\Lambda$, where $U^\Lambda$ is defined in .
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[^1]: The reader is referred to appendix \[app:Super\] for more details on the component structure of the superfields introduced here and in the following.
[^2]: We notice that, in the case of a single vector superfield, the homogeneity fixes $F(\mathcal{A})=\frac i2 \mathcal{A}^2$. Its second derivative is then constant and part of the following discussion becomes trivial.
[^3]: In terms of the standard formalism for treating $\mathcal{N}=2$ Lagrangians, with respect to the previous $\rm SU(2)$ gauge choice, the Fayet–Iliopoulos parameter $\xi$ has become now the imaginary part of the electric gauging parameter $e$.
|
---
abstract: 'The periodic motion of a classical point particle in a one-dimensional double-well potential acquires a surprising degree of complexity if friction is added. Finite uncertainty in the initial state can make it impossible to predict in which of the two wells the particle will finally settle. For two models of friction, we exhibit the structure of the basins of attraction in phase space which causes the final-state sensitivity. Adding friction to an integrable system with more than one stable equilibrium emerges as a possible “route to chaos” whenever initial conditions can be specified with finite accuracy only.'
author:
- Joshua Oldham and Stefan Weigert
date: 23 June 2015
title: Friction Causing Unpredictability
---
Department of Mathematics, University of York\
York YO10 5DD, United Kingdom
Introduction
============
The difficulty to reliably predict the behaviour of a classical systems is usually related to the existence of fractal structures in the mathematical model describing the system. Conservative non-integrable systems such as three interacting planetary bodies [@poincare99] and chaotic dissipative systems such as Lorenz’s model of the atmosphere [@lorenz63] provide two well-known cases in point. Consider adding a third body to the integrable system of two planetary bodies which interact through gravitation. The KAM theorem [@arnol'd89] describes how the original foliation of the system’s phase-space into tori is being replaced gradually by a highly intricate mixed phase space. Finite balls of initial conditions will no longer contain trajectories on tori only but also others which separate at an exponential rate. The non-linearity present in Lorenz’s model gives rise to a strange attractor [@ruelle+71]. Its properties dominate the long-term evolution of the system since trajectories with neighboring initial conditions are likely to visit rather different regions of phase space at comparable later times. This phenomenon has been called *final state sensitivity* [@grebogi+83].
An actual macroscopic physical system, however, cannot exhibit fractal structures in a strict sense: the system would need to match its mathematical description on *arbitrary* length scales [@mandlbrot77] but classical models break down on the molecular or atomic level. Experimentally observed structures may be highly intricate over many – but not all – orders of magnitude. Nevertheless, finite intricacies are sufficient to make reliable long-term predictions impossible when combined with initial states which are known only approximately.
Adding friction to conservative non-integrable model systems washes out fractal structures on the finest scales but remnants may continue to exist. This has been shown for a spherical pendulum with three stable equilibrium positions, in the presence of gravity [@motter+13]. When friction is added, basins of attraction with highly intricate boundaries emerge. If the initial state is not known exactly the fine structure of the basin boundaries – in spite of not being fractal – already prevents any reliable prediction of the equilibrium position near which the pendulum will come to rest.
The purpose of this paper is to identify a different mechanism which also results in an unpredictable final state if initial conditions are known only with finite accuracy. Starting from an *integrable* system ** with multiple stable equilibria, we will show that the addition of friction may *create* basins of attraction with intricate boundaries, leading to a situation which resembles the one of the pendulum just described. The main difference is that this mechanism does not rely on pre-existing fractal structures: instead, incorporating friction *causes* intricate structures to emerge in an originally integrable system.
Sec. \[sec:Final-state-sensitivity\] of this paper provides an initial, qualitative explanation of why adding friction to an integrable system with two or more stable equilibria may cause final-state sensitivity. Then, in Sec. \[sec:Model-systems\], we investigate the structure of basins of attraction generated by two different types of friction when acting on a particle moving in a piece-wise constant double-well potential. Finally, we summarize and discuss our results in Sec. \[sec:Summary\].
Final-state sensitivity in a double-well potential\[sec:Final-state-sensitivity\]
=================================================================================
To illustrate how friction creates final state sensitivity, we consider a classical particle moving along a straight line in the presence of a symmetric double-well potential. The system is described by the Hamiltonian function $$H(p,q)=\frac{p^{2}}{2m}+W(q)\,,\quad p,q\in\mathbb{R\,},\label{eq: Hamiltonian}$$ where $q$ and $p$ denote position and the momentum of the particle, respectively. The minima of the potential $W(q)$ are located at $q=q_{\pm}$ , separated by a barrier of height $W_{0}\equiv W(0)$ which defines the *critical* energy, $E_{c}\equiv W_{0}$. The phase-space diagram of the system is shown in Fig. \[fig: double well flow (general)\], displaying
![Phase-space structure of a double-well potential $W(q)$ with minima $L$ and $R$, located within wells ${\cal L}$ and ${\cal R}$ (shaded areas), respectively; a particle with neighbouring initial conditions $\mathbf{z},\mathbf{z}^{\prime}$ (full and dashed lines, respectively) may end up in different wells when subjected to periodic, dissipative “kicks” which reduce its momentum and thus its energy. \[fig: double well flow (general)\] ](doublewellflowgeneral.png)
the familiar types of trajectories. The minima $L$ and $R$ of the potential are *stable fixed points* each surrounded by *periodic orbits* with energies $E$ not exceeding the critical value, $0<E<E_{c}\equiv W_{0}$. For $E=E_{c}$, the particle may rest at the *unstable fixed point* at $q=0$, or travel on one of the two *separatrices* connected to it. The trajectories with energy above the critical value, $E>E_{c}$ are *periodic*, encircling both minima on a single round trip. With a single degree of freedom and the energy $H(p,q)$ as a conserved quantity, the system is *integrable*, leading to the global foliation of its phase space into one-dimensional tori.
Adding friction will modify all trajectories except when then particle initially rests at one of the three fixed points. If located on a separatrix or on any periodic trajectory with energy less than $E_{c}$, dissipation will cause the particle to “spiral” into either the left or the right minimum of the potential $W(q)$, depending on its original position relative to the origin, $q=0$. The particle cannot escape from a well once it has been trapped, and the fixed points $L$ and $R$ turn into *attractors*.
The destiny of a particle with initial energy $E>E_{c},$ however, it is not immediately obvious since it may end up in either well. Friction will inevitably “draw” the particle towards the location of the separatrices of the unperturbed system. At some time, the energy of the particle will drop below the critical value $E_{c}$. The position of the particle relative to the origin at the time of the drop will determine whether it becomes trapped in the left or in the right well.
For simplicity, let us assume that friction acts at discrete times only, repeatedly reducing the momentum of the particle by a constant factor. Suppose that for initial conditions $\mathbf{z}=(p,q)^{T}$, the particle will – after a possibly long time – settle in the *right* well as illustrated in Fig. \[fig: double well flow (general)\]. Intuitively, a slightly smaller initial momentum (see $\mathbf{z}^{\prime}$ in the figure) could cause the particle to negotiate the barrier one less time and to settle in the *left* well instead. The slight change in the initial condition has thus altered the long-term behaviour of the system. Therefore, the finally state of a particle may become unpredictable from a *practical* point of view, i.e. whenever its initial conditions are known to lie within a small but finite volume of phase space only.
More formally, the non-Hamiltonian equations of motion map an initial state $\mathbf{z}(t_{0})$ to a new value $\mathbf{z}(t)$ at time $t$, $$\mathbf{z}(t_{0})\mapsto\mathbf{z}(t),\quad\mathbf{z}\equiv\begin{pmatrix}p\\
q
\end{pmatrix}\,,\label{eq: general dynamics}$$ leading to a *decrease* of the energy defined in (\[eq: Hamiltonian\]): $E_{0}\to E<E_{0}$. To ascertain whether a particle with initial state $\mathbf{z}(t_{0})$ ends up near $L$ or near $R$, one needs to determine the earliest time $t$ such that its energy $E$ falls below the critical value, $$E<E_{c}\,.\label{eq: energy condition}$$ Repeating this calculation for *all* initial conditions will divide the phase space into two disjoint sets known as *basins of attraction* which encode whether the particle ends up in well ${\cal L}$ or ${\cal R}$. Let us investigate the structure of their boundaries for two models of friction, using a particularly simple double-well potential.
Piece-wise constant double-well potential with friction\[sec:Model-systems\]
============================================================================
The double-well potential considered here is based on a “particle in a box” defined by two infinitely high potential walls at $q=\pm\ell$ which restrict motion to a line segment of length $2\ell$. The particle bounces off the walls elastically resulting in an instantaneous reversal of its momentum: $p\to-p$; its position $q=\pm\ell$ remains unchanged when hitting a wall. A piece-wise constant potential,
$$W(q)=\begin{cases}
W_{0}\,, & |q|<\varepsilon\,,\\
0\,, & \varepsilon<|q|<\ell\,,
\end{cases}\label{eq:flat double well}$$
models the smooth double-well. For simplicity, we take an arbitrarily thin potential barrier, corresponding to $\varepsilon\to0$. The only impact of this “infinitesimal” barrier is to confine the particle in a well once its energy drops below the critical value $W_{0}$, thus creating the wells ${\cal L}$ and ${\cal R}$. Two continuous sets of potential minima exist because the bottom of the potential is flat.
A widespread method to investigate *non-integrable* systems is to start with an integrable system and add a perturbation, be it a time-independent potential term as in the KAM theorem or a deterministic time-dependent force [@chirikov79]. To support our claim that friction generically causes final state sensitivity, we will model it in two different ways which are inspired by these approaches. In the first case, the elastic collisions of the particle with the boundary walls are made inelastic (cf. Sec. \[sub:Inelastic-collisions\]) while an impulsive friction force is applied periodically in the second case (cf. Sec. \[sub:Periodic-damping\]). The first model depends on a single parameter only, the *coefficient of restitution.* The second model depends on two parameters, the *frequency* and the *strength* of the dissipative “kick.”
Inelastic collisions\[sub:Inelastic-collisions\]
------------------------------------------------
The motion of the particle in the piece-wise constant double well (\[eq:flat double well\]) consists of free motion between the walls interspersed with momentum-reversing elastic collision at the walls. The dynamics changes fundamentally upon replacing the elastic collisions at the walls by inelastic ones, characterized by a *coefficient of restitution*, $r\in(0,1)$: $$\mathbf{z}\mapsto\mathbf{z}^{\prime}=\mathbf{R\cdot}\mathbf{z}\text{\,,\quad\ensuremath{\mathbf{R}\equiv\left(\begin{array}{cc}
-r & 0\\
0 & 1
\end{array}\right)}\qquad for}\quad q=\pm\ell\,.\label{eq: restitution}$$ This minor change turns the conservative system into a dissipative one and – as we will see – is sufficient to create an embryonic form of final-state sensitivity.
The long-term dynamics of the particle does not depend on its initial *position*: all particles with fixed momentum $p_{0}$ but arbitrary position $q_{0}\in(-\ell,\ell)$ will experience the same amount of friction, only to end up in same well. Thus, let us assume that the particle starts out with positive initial momentum $p_{0}>p_{c}$, beings located at $q_{0}=\ell_{-}$, i.e. just to left of the right wall. Then, the initial state $\mathbf{z}_{0}=(p_{0},\ell_{-})$ at time $t_{0}$ evolves according to $$\mathbf{z}(t_{n})=\mathbf{R}^{n}\cdot\mathbf{z}_{0}=\begin{pmatrix}(-r)^{n}p_{0}\\
\ell_{-}
\end{pmatrix}\,,\qquad n\in\mathbb{N}_{0}\,,\label{eq: restitution evolution}$$ with the times $t_{n}$ being defined by particle returning to its initial position $\ell_{-}$. Monitoring the value of its momentum at the walls is sufficient to determine the well which will trap the particle. The particle will be trapped in well ${\cal R}$, for example, if its last collision at the right wall makes its energy drop below the critical value $E=E_{c}$ due to $p\mapsto(-r)p$.
For positive initial momentum $p_{0}$ the particle will hit the right wall first. The well to finally trap the particle is determined by the number of collisions $n(E_{0})$ before it drops below $E_{c}$. Denoting the energy of the particle after $n$ collisions by $E_{n}$, we need to find the number $n(E_{0})$ such that the energy of the particle falls below $E_{c}$ for the first time, $$n(E_{0})=\min_{n\in\mathbf{N}}\,\left\{ E_{n}<E_{c}\right\} \,,\quad\mbox{with }E_{0}>E_{c}\,.\label{eq: energy constraint (inelastic)}$$ Using Eq. (\[eq: restitution evolution\]) the number $n(E_{0})$ is easily found to be $$n(E_{0})=\left\lceil \frac{1}{2}\frac{\ln\left(E_{c}/E_{0}\right)}{\ln r}\right\rceil =\left\lceil \frac{\ln\left(p_{c}/p_{0}\right)}{\ln r}\right\rceil \,,\label{eq: LR condition}$$ where the initial moment defines the initial energy, $E_{0}=p_{0}^{2}/2m$, and $\left\lceil x\right\rceil $ is the ceiling function extracting the smallest integer greater or equal to the number $x$. If $n$ is odd (even), a particle with positive momentum $p_{0}$ will end up in the well on the right (left). The basins of attraction for the wells ${\cal L}$ and ${\cal R}$ are given by alternating horizontal bands in phase space shown in Fig. \[fig: basins of attraction (inelastic wall model)\]. The widths of the bands decrease with decreasing friction (and they increase with energy $E$ which the figure does not show due to the limited momentum range).
If the initial conditions $(p_{0},q_{0})$ of a particle are known *exactly*, then the deterministic dynamics leads to a unique and well-defined final state which can be predicted with certainty. However, limited precision of the initial conditions may results in a genuine indeterminacy of the final state. Assume that the initial state of the particle is only known to lie inside a rectangle with sides $\Delta q>0$ and $\Delta p>0$, centered about the point $\mathbf{z}_{0}$. Trajectories with initial momenta $p_{0}$ and $p_{0}^{\prime}\equiv rp_{0}$ are bound to end up in different wells. Thus, if the inaccuracy in momentum exceeds this value, $$\Delta p>p_{0}-p_{0}^{\prime}\,,$$ the uncertainty rectangle will cut across *at least two* adjacent basins of attraction. In other words, given the initial momentum $p_{0}$ and any finite uncertainty $\Delta p$ about it, the prediction of the final state becomes impossible for a coefficient of restitution in the interval $$1-\frac{\Delta p}{p_{0}}<r<1\,,\label{eq: interval for r}$$ since the rectangle with sides $\Delta q$ and $\Delta p$ will contain trajectories destined for the wells ${\cal L}$ and ${\cal R}$. We conclude that sufficiently weak inelasticity prevents the reliable prediction of the final state. In this well-defined sense, adding friction to an integrable system provides a mechanism which prevents accurate long-term predictions.
Periodic damping\[sub:Periodic-damping\]
----------------------------------------
Now we turn to a model where friction is caused by a periodic, dissipative force which acts during a short time interval only. It will be convenient to consider the limit of an *instantaneous* action which multiplies the momentum of the particle by a constant factor $\gamma\in(0,1)$ at times $T_{k}=kT$, with $k\in\mathbb{N}$, and a free parameter $T$. This approach is analogous to periodically kicking a system with a deterministic force which, for a particle moving freely on a ring known as a “rotor,” produces *deterministically chaotic* motion [@arnol'd89]. Since our model depends on *two* independent parameters, $\gamma$ and $T$, we expect more complicated basins of attraction compared to the model with inelastic reflections.
To construct the basins of attraction of the wells ${\cal L}$ and ${\cal R}$, we need to determine when, for arbitrary initial conditions $(p_{0},q_{0})^{T}$, the energy of the particle falls below the critical value for the first time We then record whether, at that moment of time, it is located to the left or to the right of the origin, i.e. within ${\cal L}$ or ${\cal R}$. For simplicity, the particle is assumed to begin its journey at time $t=0^{+}$, i.e. just after $t=0$, with *positive* momentum $p_{0}>p_{c}$ and arbitrary initial position $q_{0}\in(-\ell,\ell$).
The particle moves freely during intervals of length $T$, with perfectly elastic collisions occurring at the boundary walls which only change the sign of its momentum. An expression for its time evolution in closed form can be found if we “unfold” the trajectory by imagining identical copies of the double-well to be arranged along the position axis. Instead of being reflected at the right wall located at $q=\ell$, the particle enters the next double well, which occupies the range $(\ell,3\ell$), and continues to move to the right, etc. In this setting, the momentum does not change its sign when the particle moves from one double well to the adjacent one. The sign of its momentum in the original double well is negative (or positive) if the particle has reached the $s^{th}$ copy of the double well, with $s\in\mathbb{N}$ being odd (or even).
To determine the dynamics of the system over one period of length $T$, we combine the free motion with the dissipative kicks:
1. during the motion of the particle from $t=0^{+}$ to just before the first kick at time $t=T$, its phase-space coordinates are given by $$\mathbf{z}(t)=\begin{pmatrix}p_{0}\\
q_{0}+p_{0}t/m
\end{pmatrix}\equiv\mathbf{F}(t)\text{\ensuremath{\cdot}}\mathbf{z}_{0}\,,\qquad\mathbf{F}(t)=\begin{pmatrix}1 & 0\\
t/m & 1
\end{pmatrix}\,,\qquad t\in(0^{+},T^{-})\,,\label{eq: free motion}$$ where $q\in(0,\infty)$ due to the unfolding;
2. the dissipative kick at time $T$ reduces the momentum of the particle by the factor $\gamma\in(0,1)$, $$\mathbf{z}(T^{+})=\begin{pmatrix}\gamma p(T^{-})\\
q(T^{-})
\end{pmatrix}\mathbf{\equiv D}\cdot\mathbf{z}(T^{-})\,,\qquad\mathbf{D}=\left(\begin{array}{cc}
\gamma\ & 0\\
0 & 1
\end{array}\right)\,,\qquad t\in(T^{-},T^{+}).\label{eq: first kick}$$
To obtain the *actual* position and momentum of the particle inside the box at time $t$, we map (or “fold back”) the expression $\mathbf{F}(t)\cdot\mathbf{z}$ to the interval $q\in(-\ell,\ell)$, by writing
$$\mathbf{z}(t)=\begin{pmatrix}(-)^{s(t)}p_{0}\\
\left[\left(q_{0}+p_{0}t/m\right)\!\!\!\!\!\!\mod2\ell\right]-\ell
\end{pmatrix}\,,\qquad t\in(0^{+},T^{+})\,,\label{eq: actual coordinates after one period}$$
where the value of the integer $s(t)$ is determined by writing $q+pt/m=\overline{q}(t)+2\ell s$, with $\overline{q}(t)\in(-\ell,+\ell)$. The momentum $p$ changes sign whenever the “unfolded” coordinate passes through the values $\ell,3\ell,5\ell,\ldots$
The time evolution of the initial state $\mathbf{z}_{0}$ from time $t=0^{+}$ to $t=T_{k}^{+}\equiv(kT)^{+}$, i.e. just after the kick with label $k$, follows from concatenating Eqs. (\[eq: free motion\]) and (\[eq: first kick\]) $k$ times, $$\mathbf{z}(T_{k}^{+})=\left(\mathbf{D}\cdot\mathbf{F}(T^{-})\right)^{k}\cdot\mathbf{z}_{0}\equiv\begin{pmatrix}\gamma & 0\\
\gamma T/m & 1
\end{pmatrix}^{k}\cdot\mathbf{z}_{0}=\begin{pmatrix}\gamma^{k} & 0\\
\sigma_{k}(\gamma)T/m & 1
\end{pmatrix}\cdot\mathbf{z}_{0}\,,\label{eq: periodic damping time evolution}$$ where $$\sigma_{k}(\gamma)=\frac{1-\gamma^{k}}{1-\gamma}\,,\qquad k\in\mathbb{N}\,.\label{eq: geometric series for gamma}$$ In analogy to Eq. (\[eq: actual coordinates after one period\]), the “true” coordinates of the particle inside the box are obtained as $$\mathbf{z}(T_{k}^{+})=\begin{pmatrix}(-)^{s(t)}\gamma^{k}p_{0}\\
\left[\left(q_{0}+\sigma_{k}(\gamma)p_{0}T/m\right)\mod2\ell\right]-\ell
\end{pmatrix}\,,\label{eq: actual coordinates after k periods}$$ assuming that, after $k$ kicks, the energy $E_{k}=p_{k}^{2}/2m$ of the particle has not yet dropped below the critical value $E_{c}$.
We are now in the position to determine which initial conditions $\mathbf{z}_{0}$ will send the particle to the left and the right well, respectively. Using Eq. (\[eq: actual coordinates after k periods\]), we first determine the smallest value of $k$ which reduces the energy of the particle below the critical value, $E_{k}<E_{c}$, or $$k_{c}=\left\lceil \frac{1}{2}\frac{\ln(E_{c}/E_{0})}{\ln\gamma}\right\rceil =\left\lceil \frac{\ln(p_{c}/p_{0})}{\ln\gamma}\right\rceil ,\qquad k_{c}\in\mathbb{N}\,,\label{eq: critical k (periodic damping)}$$ assuming, of course, that $p_{0}>p_{c}$. This relation structurally resembles the result (\[eq: LR condition\]), with the number $k_{c}$ of dissipative kicks playing the role of the number of inelastic collisions $n_{c}.$ The sign of the position coordinate after $k_{c}$ kicks, $q(T_{k_{c}}^{+})$, follows from Eq. (\[eq: actual coordinates after k periods\]) and determines whether the particle is trapped in ${\cal L}$ or ${\cal R}$. The explicit dependence of $\mathbf{z}(T_{k}^{+})$ on the initial *position* implies that changes in $q_{0}$ may also produce different final states, in contrast to the model studied in Sec. \[sub:Inelastic-collisions\].
Fig. \[fig: magnified basins (kicked)\], which has been generated numerically on the basis of Eq. (\[eq: actual coordinates after k periods\]), illustrates these conclusions. The first vertical bar visualizes the basins of attraction associated with the wells $\mathcal{L}$ and $\mathcal{R}$, respectively. The expected dependence on both initial momentum and position becomes clearly visible in the magnifications which also reveal that the boundaries of the apparently irregular basins of attractions are not fractal.
The boundaries of the basins can be found directly from Eq. (\[eq: actual coordinates after k periods\]): all initial conditions $(p_{0},q_{0})$ mapped to a fixed value of position at time $kT$ are located on lines of the form $$p_{0}(q_{0})=-\frac{m}{\sigma_{k}(\gamma)T}q_{0}+\mbox{const}\simeq-\frac{1}{k}\frac{m}{T}q_{0}+\mbox{const}\,,\qquad\gamma\apprle1\,,$$ using $\sigma_{k}(\gamma)\simeq(1/k)+{\cal O}(1-\gamma)$, which holds for weak damping, i.e. for $\gamma$ approaching the value one from below. Consequently, the boundaries of the basins of attraction are *straight lines* in phase space just as for the model with inelastic reflections off the wall. The lines are no longer horizontal but their slope approaches the value zero if a large number of kicks is required for the particle to settle in a well.
Assume once again that the initial conditions of the particle can be prepared with *finite precision* only, i.e. they lie inside a phase-space rectangle with area $\Delta q\Delta p>0$ and center $\mathbf{z}_{0}$. For any finite imprecision one can always find a damping strength $\gamma$ such that at least one basin boundary crosses the rectangle; this is sufficient to prevent the prediction of the well to finally trap the particle. For large initial momenta $p_{0}$, the reasoning behind the derivation of the inequality (\[eq: interval for r\]) also applies here since the strips constituting the basins of attraction will, typically, have almost horizontal boundaries. Thus, for any initial conditions $(p_{0},q_{0})$ and finite uncertainties, damping strengths within the interval $$1-\frac{\Delta p}{p_{0}}<\gamma<1\label{eq: interval for gamma}$$ correspond to a situation with an unpredictable final state. Occasionally, the uncertainty rectangle with sides $\Delta q$ and $\Delta p$ may cover an area where a slight change in initial *position* causes the particle to reach different wells, which only increases the final state sensitivity.
Summary and conclusions\[sec:Summary\]
======================================
We have shown that adding friction to an integrable one-dimensional double-potential well causes its dynamics to exhibit a rudimentary form of final-state sensitivity. For simplicity, the well has been modeled as a “box” divided into two regions by a thin wall. A particle has been subjected to two types of dissipative forces which, by reducing its initial energy, cause the particle to necessarily settle in one the wells after a finite, possibly long time. The main result of our study is that adding friction to an integrable system produces basins of attraction with finely structured boundaries.
If the particle collides *inelastically* with the confining walls the resulting basins foliate the phase space of the system into horizontal layers of variable width which get narrower for decreasing friction. Any ball of initial conditions which extends beyond more than one band prevents us from predicting with certainty the well in which the particle will finally settle. *Periodic dissipative kicks* create basins of attraction with slightly more intricate boundaries, due to their additional position dependence. Since the particle must settle in a well after *finite* time the observed structures cannot be fractal. In practice, however, it is crucial whether the initial conditions can be specified with sufficient accuracy to avoid a spread across basins which send the particle to different final states.
These model systems demonstrate that adding friction to an integrable system with multiple stable equilibria can have a fundamental impact on long-term predictability. The motion is not “deterministically random” which would require fractal phase-space structures. However, if the accuracy of the initial conditions falls below a specific threshold, the final state of the system cannot be predicted reliably. Experimentally, the precision required for a reliable long-term prediction may well be out of reach.
We expect our conclusions to be structurally stable in the sense that they should not depend on the model of friction used. Any dissipative mechanism will, firstly, contract all initial conditions into a small phase-space region which is energetically just above the barrier of the double well; secondly, the energy of the particle will drop below $E_{c}$ in a way which depends sensitively on the initial conditions. Continuous Stokes friction, for example, is thus likely to generate similar basins of attraction.
To systematically study the creation of basins of attraction with intricate boundaries in more general, smooth potential wells, we suggest to exploit the existence of action-angle variables $(I,\varphi)$ in integrable systems. The energy $E$ represents a convenient starting point to study the impact of friction forces, when expressed as a function of the action $I$, $$I(E)=\oint p(q,E)dq\,.\label{eq: general action}$$ Suitable perturbations are easily added to the new form of the Hamiltonian, $H=H(I)$, once position $q$ and momentum $p$ have been mapped to $(I,\varphi$) by means of a canonical transformation.
Finally, we highlight a natural application of the effective unpredictability of a final state due to friction, given sufficiently imprecise initial conditions. It arises upon introducing a larger number of identical potential wells arranged on a ring, (37 or 38, say), mimicking a one-dimensional roulette wheel. Including periodic dissipative kicks provides a surprisingly simple explanation why a finite spread in initial momenta and positions is sufficient to generate random outcomes, the working hypothesis underlying any gambling. This approach should be contrasted with models of an actual roulette wheel where unpredictable trajectories arise through a multitude of effects: motion in a bent annulus-shaped region embedded into three dimensions, the presence of gravity, rolling resistance and a (presumably) non-integrable time-dependent potential.
Acknowledgments {#acknowledgments .unnumbered}
---------------
JO is grateful for financial support through a “Summer-2014 Publication Studentship”, provided by the Department of Mathematics at the University of York, UK.
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V. I. Arnol’d: *Mathematical Methods of Classical Mechanics*. Springer Science & Business Media New York (1989)
D. Ruelle, F. Takens: *On the Nature of Turbulence*. Comm. Math. Phys. **20**, 167192 (1971)
C. Grebogi, S. W. McDonald, E. Ott, J. A. Yorke: *Final State Sensitivity: An Obstruction to Predictability*. Phys. Lett. A **99**, 415-418 (1983)
B. Mandlbrot: *Fractals: Form, Chance and Dimension*. Freeman, San Francisco, CA (1977)
A. Motter, M. Gruiz, G. Karolyi, T. Tel: *Doubly Transient Chaos: The Generic Form of Chaos in Autonomous Dissipative Systems*. Phys. Rev. Lett. **111**, 194101 (2013)
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---
abstract: 'The hypercentral Constituent Quark Model (hCQM) for the baryon structure is reviewed and its applications are systematically discussed. The model is based on a simple form of the quark potential, which contains a Coulomb-like interaction and a confinement, both expressed in terms of a collective space coordinate, the hyperradius. The model has only three free parameters, determined in order to describe the baryon spectrum. Once the parameters have been fixed, the model, in its non relativistic version, is used to predict various quantities of physical interest, namely the elastic nucleon form factors, the photocouplings and the helicity amplitudes for the electromagnetic excitation of the baryon resonances. In particular, the $Q^2$ dependence of the helicity amplitude is quite well reproduced, thanks to the Coulomb-like interaction. The model is reformulated in a relativistic version by means of the Point Form hamilton dynamics. While the inclusion of relativity does not alter the results for the helicity amplitudes, a good description of the nucleon elastic form factors is obtained.'
author:
- |
M.M. Giannini\
Dipartimento di Fisica dell’Università di Genova\
and\
I.N.F.N., Sezione di Genova\
E. Santopinto\
I.N.F.N., Sezione di Genova
title: The hypercentral Constituent Quark Model and its application to baryon properties
---
Introduction
============
The quark model has been introduced fifty years ago [@gm; @zw] as a realization of the $SU(3)$ symmetry and it has been used with success for the description of many important properties of hadrons, as the existence of multiplets, their quantum numbers and the magnetic moments [@rhd; @kok]. The idea of quarks as effective particles (Constituent Quarks) emerged very early [@morp65] and was further developed with the introduction of the colour quantum numbers.
Here we shall concentrate ourselves on Constituent Quark Models (CQM) for baryons.
After the pioneering work of Isgur and Karl (IK) [@ik] a series of CQM followed: the relativized Capstick-Isgur model (CI) [@ci], the algebraic approach (BIL) [@bil], the hypercentral CQM (hCQM) [@pl; @nc; @es], the chiral Goldstone Boson Exchange model ($\chi$CQM) [@olof; @ple1; @ple2; @ple3], the Bonn instanton model (BN) [@bn1; @bn2; @bn3; @bn4] and the interacting quark-diquark model [@diq].
All models reproduce the baryon spectrum, which is the first quantity to be approached when building a model for the baryon structure, but have been widely used to describe baryon properties. In some cases the calculations referred to as a CQM one are performed using a simple h.o. wave function for the internal quark motion either in a nonrelativistic (HO) or relativistic framework (rHO).
The photocouplings for the excitation of the baryon resonances have been calculated in various models, among others we quote HO [@cko], IK [@ki], CI [@cap], BIL [@bil], hCQM [@aie] (for a comparison among these and other previous approaches see e.g. [@aie; @cr2]). The calculations reproduce the overall trend, but the strength is systematically lower than the data. The fact that quite different models lead to similar results can be ascribed to their common $SU(6)$ structure.
As for the nucleon elastic form factors there are the calculations performed by BIL [@bil; @bil2] with the algebraic method and by the Rome group [@card_95; @card_00; @demelo_N] within a light front approach based on the CI model. The hCQM has been firstly applied in the nonrelativistic version with Lorentz boosts [@mds; @rap] and then it has been reformulated relativistically [@ff_07; @ff_10]. A quite good description of the elastic form factors is achieved also using the GBE [@wagen; @boffi] and the BN [@mert] models, both being fully relativistic. The same happens for the interacting quark-diquark model [@diq], specially in its relativistic version [@diq2].
A sensible test of both the energy and the short range properties of the quark structure is provided by the $Q^2$ behaviour of the helicity amplitudes for the electromagnetic excitation to the baryon resonances.
In the HO framework, there are various calculations of the transverse helicity amplitudes, among them we quote refs. [@cko; @ki; @cl; @warns; @sw], while a systematic rHO approach has been used by [@ck]. A light cone calculation, using the CI [@ci] model, has been performed [@cap] and then successfully applied to the $\Delta$ [@card_ND] and Roper excitations [@card_Roper]. For more recent light cone approaches, see ref. [@azn-bur] and references therein. The algebraic method has been also used for the calculation of the transverse helicity amplitudes [@bil]. The hCQM, in its nonrelativistic version, has produced nice predictions for the transverse excitation of the negative parity resonances [@aie2] and, recently, for both the transverse and longitudinal helicity amplitudes of all resonances having a sensible excitation strength [@sg]. The calculation of the helicity amplitudes in a relativistic hCQM is in progress and some preliminary results for the $\Delta$ resonance are now available [@fb22]. Helicity amplitudes have been calculated also by the Bonn group, both for the nonstrange [@mert; @ronn] and strange resonances [@caut].
The models have been applied also to the decays of baryons. The strong decays have been quite soon calculated with the IK model [@ki] and in its relativized versions [@cr; @cr3]. There are also calculations in other models, namely BIL [@bil3], GBE [@melde]. As for the hCQM, there are some preliminary calculations [@bad]. There are also calculations of the semileptonic decays of baryons in the BN model [@mig].
Finally we quote calculations of the axial nucleon form factors in the GBE [@boffi; @gloz] and BN [@mert].
A review of Constituent Quark Models
====================================
Nonrelativistic approach
------------------------
The possibility of a nonrelativistic description of the internal quark dynamics was considered very early [@morp65] after the introduction of the quark model. In this framework, one can introduce the three-quark wave function $\Psi_{3q} $, factorized according to the various degrees of freedom: $$\Psi_{3q}~=~\theta_{colour} ~\chi_{spin}~\Phi_{flavour}~\psi_{space}.
\label{3q}$$ In agreement with the Pauli principle, the wave function $\Psi_{3q} $ must be totally antisymmetric for the exchange of any quark pair. Baryons must be colour singlets and the corresponding wave function $\theta_{colour}$ is by itself antisymmetric, therefore the remaining factors must be completely symmetric. Actually a symmetric quark model has been formulated before the introduction of the colour quantum numbers and the symmetric three-quark states have been classified [@ko; @hd].
Early Lattice QCD calculations [@wil] showed that the quark interaction can be split into a long range part, which is spin and flavour independent and contains confinement, and a short range spin-dependent one [@deru]. This means that one can assume the dominant part to be $SU(6)$ invariant and the wave function of Eq. (\[3q\]) becomes $$\Psi_{3q}~=~\theta_{colour} ~\Phi_{SU(6)}~\psi_{space}.
\label{3q_2}$$ In order to satisfy Pauli principle, the product $$~\Phi_{SU(6)}~\psi_{space}$$ must be symmetric and then both factors $\Phi_{SU(6)}$ and $\psi_{space}$ must have the same permutation symmetry, that is symmetric (S), antisymmetric (A) or one of the two mixed symmetry types (MS, MA), which are distinguished by the symmetry or the antisymmetry with respect to a quark pair.
It should be reminded that each quark belongs to the fundamental $SU(6)$ representation with dimension 6 and that with three quarks one can obtain the following $SU(6)$-representations: $$SU(6): 6 \otimes 6 \otimes 6= 20 \oplus 70 \oplus 70 \oplus 56,$$ the corresponding symmetry type is, respectively, A, M, M, S.
The spin and flavour content of each $SU(6)$-representation is well defined, since the three $SU(6)$ representations can be decomposed according to the following scheme
$$20~=~^4 \underline{1}~+~^2 \underline{8},$$
$$56~=~^2 \underline{8}~+ ~^4 \underline{10} ,$$
$$70~=~^2 \underline{1}~+~^2 \underline{8}~+~^4 \underline{8}~+~^2
\underline{10} .
\\$$
The suffixes in the r.h.s. denote the multiplicity $2S+1$ of the $3q$ spin states and the underlined numbers are the dimensions of the $SU(3)$ representations. This means for instance that the $56$ representation contains a spin-$1/2$ $SU(3)$ octect and a spin-$3/2$ $SU(3)$ decuplet.
The various baryon resonances can be grouped into $SU(6)$-multiplets, the energy differences within each multiplet being at most of the order of $15\%$ as in the case of $N-\Delta$ mass difference and of the splittings within the $SU(3)$ multiplets. In Fig. \[baryon\] we report the experimental non strange baryon spectrum, including only the three- and four- star states [@pdg10]. The notation for the $SU(6)$-multiplets is $(d, L^P)$, where $d $ is the dimension of the $SU(6)$-representation, $ L$ is the total orbital angular momentum of the three-quark state describing the baryon and $P$ the corresponding parity. An alternative but equivalent notation is $ L^P_t$, where t is the symmetry type of the $SU(6)$ representation.
The fact that the $4-$ and $3-$star non strange resonances can be arranged in $SU(6)$ multiplets indicates that the quark dynamics has a dominant $SU(6)$ invariant part accounting for the average multiplet energies, while the splittings within the multiplets are obtained by means of a $SU(6)$ violating interaction, which can be spin and/or isospin dependent and can be treated as a perturbation.
![ (Color online) The experimental spectrum of the non strange three- and four-star resonances [@pdg10]. The states are reported in columns with the same parity P and grouped into $SU(6)$-multiplets.[]{data-label="baryon"}](spectrum){width="4.5in"}
The various constituent quark models are quite different, but they have a simple general structure in common, since in any case, analogously to what stated above, the quark interaction $V_{3q}$ can be split into a spin-flavour independent part $V_{inv}$, which is $SU(6)$-invariant and contains the confinement interaction, and a $SU(6)$-dependent part $V_{sf}$, which contains spin and eventually flavour dependent interactions $$\label{v3q}
V_{3q}~=~V_{inv}~+~V_{sf}.$$
$CQM$ Kin. Energy $V_{inv}$ $V_{sf}$ ref.
---------------- -------------- ------------------------ -------------- ---------
Isgur-Karl nonrel. h.o. + shift OGE [@ik]
Capstick-Isgur rel. string +coul-like OGE [@ci]
$U(7) B.I.L.$ $M^2$ vibr + L Gürsey-Rad [@bil]
Hypercentral nonrel./rel. $O(6)$: lin + hyp.coul OGE [@pl]
Glozman-Riska rel. h.o. / linear GBE [@olof]
Bonn rel. linear + $3$ body instanton [@bn1]
quark-diquark nonrel./rel. linear + Coulomb spin-isospin [@diq]
: Illustration of the features of various CQMs
\[cqm\]
In Table \[cqm\] we report a list of the Constituent Quark Models and their main features. The order is chronological.
The Isgur-Karl model
--------------------
The Isgur-Karl [@ik] model has some general features that are interesting also for other models, so it is worthwhile to devote to it some attention (more details can be found in [@mg]).
The kinetic energy T is assumed to be nonrelativistic $$T~=~\sum_i (m_i + \frac{\vec{p}_i^{~2}}{2m_i})~=~M_{tot} + \frac{\vec{P}^2}{2 M_{tot}} +T_{intr},$$ where $M_{tot}$ is the total mass of the three quarks, $\vec{P}$ is their total momentum and the intrinsic kinetic energy is expressed in terms of the momenta $\vec{p}_\lambda$ and $\vec{p}_\rho$, which are conjugated to the Jacobi coordinates $\vec{\rho}$ and $\vec{\lambda}$, $$\vec{\rho}~=~ \frac{1}{\sqrt{2}}(\vec{r}_1 - \vec{r}_2) ~,\\
~~~~\vec{\lambda}~=~\frac{1}{\sqrt{6}}(\vec{r}_1 + \vec{r}_2 - 2\vec{r}_3) ~.
\label{coord}$$ In the case of non strange baryons, all quark have the same mass m and then T assumes the form $$\label{kin}
T~=~3m + \frac{\vec{P}^2}{6m} + \frac{\vec{p}_\rho^{~2}}{2m} + \frac{\vec{p}_\lambda^{~2}}{2m};$$ in the case of quarks with different mass, the kinetic energy contains appropriate reduced masses [@ik].
The confining interaction is assumed to be a harmonic oscillator (h.o.) $$V_{ho}~=~\sum_{i<j} \frac{1}{2}~K~ (\vec{r}_i - \vec{r}_j)^2,$$ which, in terms of the Jacobi coordinates, becomes $$V_{ho}~=~\frac{3}{~2}~K~ (\rho^{2} + \lambda^{2}),$$ with $\rho^2=\vec{\rho}^{~2}$ and $\lambda^2=\vec{\lambda}^2$. The three-quark interaction is then given by two three-dimensional h.o. and the energy levels can be written as $E= (3 +N) \hbar \omega$, with $N=2n+l_\rho +l_\lambda$, where n is a non negative integer number and $l_\rho$ and $l_\lambda$ are the orbital angular momenta associated to the Jacobi coordinates. The h.o. parameter $\omega$ is given by $\sqrt{\frac{3K}{m}}$.
N $(d,L^P)$ $^2\underline{8}$ $^48$ $^210$ $^410$
--- --------------- ------------------- ----------------- ---------------- ----------------
0 $(56,0^+)$ $P_{11}(939)$ $P_{33}(1232)$
1 $(70,1^-)$ $S_{11}(1535)$ $S_{11}(1650) $ $S_{31}(1620)$
$D_{13}(1520)$ $D_{13}(1700) $ $D_{33}(1700)$
$D_{15}(1675) $
2 $(56,0^{+*})$ $P_{11}(1440)$ $P_{33}(1600)$
2 $(70,0^+)$ $P_{11}(1710)$ $P_{13}$ $P_{31}$
2 $(56,2^+)$ $P_{13}(1720)$ $P_{31}(1910)$
$F_{15}(1680)$ $P_{33}(1920)$
$F_{35}(1905)$
$F_{37}(1950)$
2 $(70,2^+)$ $P_{13}$ $P_{11}$ $P_{33}$
$F_{15}$ $P_{13}$ $F_{35}$
$F_{15}$
$F_{17}$
2 $(20,1^+)$ $P_{11}$
$P_{13}$
: The non strange three-quark states belonging to the h.o. shells up to N=2. The notation for the baryon resonances is $X_{2I 2J}$, where X=S,P,D,F,…denotes the pion wave in the decay channel, I and J are the isospin and the spin of the state, respectively. The numbers within parentheses are the masses of the 4- and 3- star resonances displayed in Fig. (\[baryon\]). The asterisk in the first N=2 configuration reminds that the spin-isospin structure is the same as in the ground state but it has a radial excitation. []{data-label="ho"}
The general structure of the h.o. space wavefunction is $$\psi_{NLt}(\vec{\rho},\vec{\lambda})=C_N P_N(\rho, \lambda) e^{-1/2 \alpha^2 (\rho^2+\lambda^2)} ~Y_{l_\rho}(\Omega_\rho) ~Y_{l_\lambda}(\Omega_\lambda),$$ where $\alpha^2 =\sqrt{3K m} / \hbar$, $C_N$ is a normalization factor, $P_N$ a polynomial of degree N and the spherical harmonics have to be combined to a definite total orbital angular momentum L; t (=A, M, S) is the symmetry type, the same as the $SU(6)$ states.
In Table \[ho\] we report the $SU(6)$ states that can be assigned to the first three shells. All the states reported in Fig. (\[baryon\]) fit very well into the scheme. The number of predicted states is however much larger than the observed 4- and 3- star states. The problem of such missing resonances is common to all CQMs and it has been suggested long time ago that some resonances may be observable in electroproduction experiments and not in strong interaction processes [@ki]. This statement is supported by the new states reported in the last edition of the PDG review [@pdg12].
In Table \[ho\] there is no room for the 3-star $D_{35}(1930)$. In fact, the total spin 5/2 can be obtained combining the L=1 total orbital angular momentum with the total spin 3/2 of the three quark, however the negative parity states belong to the 70-dimensional representation of $SU(6)$, which cannot contain a $\Delta$ state with total spin 3/2. In order to describe this state and some new 2-star negative parity resonances [@pdg12] one should introduce the N=3 shell and the number of missing resonances will be highly increased. Similarly, the shells with N greater than 2 are necessary for the resonances with high spin values,
An important observation regarding the h.o. spectrum is the level ordering, which, for any wo body potential is $0^+, 1^-, 0^+$, while experimentally the $1^-$ states are in average almost degenerate with the first $0^+$ excitation. Moreover, the spacing between two shells is the same over the whole spectrum and the levels are highly degenerate, since the energy depends on the h.o. quantum number N only.
In order to avoid the equal spacings and the degeneracy of the levels, in the Isgur-Karl model a shift potential U is added, which simply redefines the energies of the $SU(6)$ states, without any attempt to diagonalize it. In this way the energies of the $SU(6)$ configurations can be written as $$\begin{aligned}
\label{ik_en}
\begin{array} {ccl}
E(0^+_S) & = & E_0 ,\\
E(0^{+*}_S) & = & E_0+ 2\Omega -\Delta, \\
E(0^-_M) & = & E_0+ 2\Omega -\Delta/2, \\
E(2^+_S) & = & E_0+ 2\Omega -2 \Delta /5, \\
E(2^+_M) & = & E_0+ 2\Omega -\Delta/5 \\
E(1^+_A) & = & E_0+ 2\Omega,
\end{array}\end{aligned}$$ where $$\Omega = \hbar \omega -a_0/2+a_2/3, ~~~~~~~~~~\Delta=- 5/4 a_0+5/3 a_2 -1/3 a_4,$$ where the coefficients $a_m$ (m=0,2,4) are determined by the moments of the U potential $$a_m=3 (\frac{\alpha}{\sqrt{\pi}})^3 \int d^3\rho (\alpha \rho)^m U(\sqrt{2} \rho) e^{-\alpha^2 \rho^2}.$$ No explicit form is assumed for the potential U, but the three coefficients $a_m$ are used as free parameters to be fitted to the experimental spectrum. In this way also the position of the Roper N(1440) resonance is correctly described.
Having assumed the space wave function as given by a h.o. three-quark potential, one can build the various $SU(6)$ configurations to be identified, according to Table \[ho\], with the observed resonances. The states contained in each multiplet can be denoted as $$|B~ ^{2S+1}X_J \rangle_t,$$ where $B=N, \Delta$ for isospin $1/2,3/2$, respectively, S in the suffix is the total 3q spin, J the spin of the baryon state, X=S,P,D,…according to the total 3q orbital angular momentum and t is the symmetry type. For instance the nucleon is denoted by $|N~ ^{2}S_{1/2}\rangle_S$, the Roper resonance is $|N~ ^{2}S^*_{1/2}\rangle_S$, where the asterisk means that the state is the first radial excitation of the nucleon, the $\Delta$ is $|\Delta~ ^{4}S_{3/2}\rangle _S$ and so on.
Since the quark interaction considered up to now is $SU(6)$ invariant, the energies given by Eqs. (\[ik\_en\]) are common to all the states in any $SU(6)$ multiplet, at variance with the experimental spectrum (see Fig. \[baryon\]). In order to describe the splittings within each multiplet, one has to introduce a $SU(6)$ violating interaction $V_{sf}$, which, in the case of the Isgur-Karl model [@ik] is given by the hyperfine interaction, in the form proposed in ref. [@deru] $$H_{hyp}
= \sum_{i<j} \frac{2 \alpha_S}{3m_i m_j} [\frac{8 \pi}{3} \vec{S}_i \cdot \vec{S}_j ~ \delta(\vec{r}_{ij}) +\frac{1}{r_{ij}^3} (\frac{3(\vec{S}_i \cdot \vec{r}_{ij})(\vec{S}_j \cdot \vec{r}_{ij})}{r_{ij}^2} -\vec{S}_i \cdot \vec{S}_j] ,
\label{hyp}$$ where $\vec{r}_{ij}=\vec{r}_{i}-\vec{r}_{j}$. Eq. (\[hyp\]) is the spin dependent part of the One Gluon Exchange (OGE) interaction between two quarks, the spin independent part being a Coulomb-like term $1/r_{ij}$, which can be considered implicitly taken into account in the shift potential U. The structure of Eq. (\[hyp\]) is the same as the Breit-Fermi term in the higher order Coulomb potential for electrons in atoms. The OGE interaction is in principle valid for short interquark distances, however, it is used just for the determination of the form of the spin-dependent quark interaction and the strong coupling constant $\alpha_S$ is considered as a free parameter to be fitted to the $N - \Delta$ mass difference.
The hyperfine interaction is diagonalized in the h.o. basis, using as unperturbed energies the ones given by Eqs. (\[ik\_en\]). Its matrix elements, in the case of u and d quarks, are given in terms of the quantity (see also Appendix 2 of Ref. [@mg]) $$\delta =\frac{4 \alpha_S \alpha^3}{3 \sqrt{2 \pi} m^2} ,$$ which is substantially the $N - \Delta$ mass difference and can be fixed to about 300 MeV. As for the remaining free parameters, the h.o. constant is fitted to the proton r.m.s. radius, obtaining $\alpha^2 = 1.23$ fm$^2$, the other parameters are determined by comparison of the theoretical spectrum with the experimental one. The resulting description of the spectrum is quite good, both for non strange and strange resonances [@ik].
An important consequence of the introduction of the hyperfine interaction is that the baryon states are superpositions of $SU(6)$ configurations. For instance, the nucleon is expanded as $$|N \rangle = a_S |N ^2S_{1/2} \rangle _S + a'_S |N ^2S^*_{1/2}\rangle_S + a_M |N ^2S_{1/2}\rangle_M + a_D |N ^4D_{1/2}\rangle_M,
\label{N}$$ with $ a_S=0.931, a'_S=-0.274, a_M=-0.233, a_D=-0.067$ [@mg]; the asterisk in the second term of Eq. (\[N\]) means that the spin-isospin part is the same as the first term, but the space part corresponds to a radially excited wave function. The Roper resonance has a similar expansion, with the dominant component given by $|N ^2S'_{1/2} \rangle_S$: $ a_s=0.281, a'_s=-0.960, a_M=--0.003, a_D=-0.001$. The $\Delta$ resonance is given by $$|\Delta \rangle = b_S |\Delta ^4S_{3/2} \rangle_S + b'_S |\Delta ^4S^*_{3/2}\rangle_S + b_D |\Delta ^4S_{3/2}\rangle_S + b'_D |\Delta ^2D_{3/2} \rangle _M ,$$ with $ b_S=0.963, b'_S=0.231, b_D=-0.119, b'_D=0.075$. It is well known that with pure $SU(6)$ configurations the E2 electromagnetic $N - \Delta$ transition vanishes [@bm]. However, because of the hyperfine interaction, the $\Delta$ state acquires a non-zero D-wave component and then a small quadrupole strength arises [@dg; @ikk]. The theoretical estimate of the ratio $$R=- \frac{G_{E2}}{G_{M_1}}
\label{E2}$$ is about $-0.02$ [@ikk; @dg], which compares favourably with the experimental value [@pdg12].
The behaviour of the nucleon electromagnetic form factors in the Isgur-Karl model is dominated by the Gauss factor $e^{-\frac{q^2}{6 \alpha^2}}$, therefore it is too strongly damped for medium-high values of the square momentum of the virtual photon $q^2$. On the contrary, the neutron charge form factor is nicely described and this is due to the hyperfine interaction [@iks]. In fact, if the nucleon state is the symmetric $SU(6)$ configuration $|N ^2S_{1/2} \rangle _S$, the charge form factor is proportional to the total charge of the three-quark system; the hyperfine interaction introduces in the nucleon state a mixed symmetry component $|N ^2S_{1/2} \rangle _M$, giving rise to a non zero charge form factor for the neutron.
The Capstick-Isgur model
------------------------
This model [@ci] is the extension to the baryon sector of the relativized model for mesons formulated in ref. [@gi]. The three-quark hamiltonian is written as $$H= T + V_{3q},$$ where T is the relativistic kinetic energy $$T=\sum_{i=1}^3 \sqrt{p_i^2 + m_i^2},
\label{relkin}$$ the three quark potential $V_{3q}$ is separated into two terms, according to Eq. (\[v3q\]). In the nonrelativistic limit $$V_{3q} \rightarrow V_{si} + V_{sd},$$ where the spin dependent interaction is $$V_{si} = V_{string} + V_{coul}.$$ The first term is the three-body adiabatic potential generated by the quantum ground state in a $Y-$shaped string configuration and provides confinement; $V_{string}$ is given by [@ci] $$V_{string} = C_{qqq} + b \sum_{i=1}^3 |\vec{r}_i - \vec{r}_{junction}|,$$ where $C_{qqq}$ is an overall constant energy shift and b is the string tension. For practical purposes, $V_{string}$ is split into two-and three-body effective terms $$V_{string} = C_{qqq} + f b \sum_{i<j}r_{ij} + V_{3b},$$ where $$V_{3b}= b (\sum_{i=1}^2 |\vec{r}_i - \vec{r}_{junction}| -f \sum_{i<j}r_{ij}).$$ The parameter f is chosen to be 0.5493 [@dosch] in order to minimize the expectation value of $V_{3b}$ in the h.o. ground state of the baryon. In this way $V_{3b}$ is a small correction and can be treated perturbatively and $V_{string}$ becomes very close to $1/2 b \sum_{i<j}r_{ij}$.
The potential $V_{coul}$, in the nonrelativistic limit, is given by $$V_{coul}= \sum_{i<j} -\frac{2 \alpha_S(r_{ij})}{3 r_{ij}};$$ in ref. [@ci] the momentum (or space) dependence of the strong coupling constant $\alpha_S(r_{ij})$ is properly taken into account.
The spin dependent potential $V_{sd}$, again in the nonrelativistic limit, is $$V_{sd}= V_{hyp}+ V_{so},$$ where $V_{hyp}$ is the hyperfine interaction of Eq. (\[hyp\]) and $V_{so}$ is a spin-orbit interaction containing in particular a Thomas precession term. Please note that the sum of $V_{coul}$ and $V_{hyp}$, together with the corresponding Thomas precession spin-orbit, derive from the nonrelativistic limit of the OGE interaction.
In order to avoid the nonrelativistic approximation one has, according to the discussion reported in ref. [@ci], to introduce appropriate $\sqrt{\frac{E}{m}}$ factors and to smear the interactions over a two quark distribution $$\rho_{ij}(\vec{r}_i-\vec{r}_j)= \frac{\sigma_{ij}^3}{\pi^{3/2}} e^{-\sigma_{ij}^2 (\vec{r}_i-\vec{r}_j)^2},$$ in particular for the contact term in the hyperfine interaction. In this way the factors $m_i$ are substituted with the corresponding energies and the momentum dependence of the interaction is taken into account.
The three-body equation, with the relativistic kinetic energy, is solved by means of a variational approach in a large h.o. basis. The result is a good description of the baryon spectrum, including strange, charm and bottom resonances [@ci].
As already mentioned in the introduction, the model by Capstick-Isgur has been successfully applied to the calculation of the electromagnetic amplitudes for the transitions to the $\Delta$ [@card_ND] and Roper resonances [@card_Roper].
The U(7) model
--------------
The typical feature of this model [@bil] is to describe the state of a three quark system by means of a group theoretic approach.
In order to describe the space degrees of freedom, the model uses the method of bosonic quantization, similarly to what has been done in the Interacting Boson model in nuclear physics [@AI] and in molecular physics [@iac] as well. The idea is to consider a string-like model with a Y-shaped configuration, in which the vectors $\vec{r}_i (i=1,2,3)$ denote the end points of the string configuration. To this end, one introduces two vector boson operators defined in terms of the Jacobi coordinates of Eq. (\[coord\]) $\vec{\rho}$, $\vec{\lambda}$, together with their conjugate momenta $\vec{p_\rho}$, $\vec{p_\lambda}$ $$b_{\rho, m} = \frac{1}{\sqrt{2}} (\rho_m + i p_{\rho, m}), ~~~~~~~~~~b^\dagger_{\rho, m} = \frac{1}{\sqrt{2}} (\rho_m - i p_{\rho, m}),$$ $$b_{\lambda, m} = \frac{1}{\sqrt{2}} (\lambda_m + i p_{\lambda, m}), ~~~~~~~~~~b^\dagger_{\lambda, m} = \frac{1}{\sqrt{2}} (\lambda_m - i p_{\lambda, m})$$ (with $m=-1,0,+1$) and an auxiliary scalar boson $s$, $s^\dagger$. The bilinear forms $G_{\alpha \alpha'} =c^\dagger_\alpha c_{\alpha'}$, where $c^\dagger_\alpha$ ($\alpha=1, \ldots 7$) is one of the seven creation operators, generate the Lie algebra of U(7). The choice of U(7) is in agreement with the usual prescription that any problem with $\nu$ space degrees of freedom should be written in terms of the Lie algebra $U(\nu+1)$ [@iac93] and all the states are assigned to the totally symmetric representation $[N]$ of $U(\nu+1)$, N being the maximum number of shells. The physical states can be constructed by applying a suitable product of boson operators to the vacuum state $$\frac{1}{\mathcal{N}} (b^\dagger_\rho)^{n_\rho} (b^\dagger_\lambda)^{n_\lambda} (s^\dagger)^{N-n_\rho-n_\lambda},$$ $\mathcal{N}$ is a normalization factor.
The squared mass operator $M^2$ is expressed as the most general combination of $G_{\alpha \alpha'}$, with the condition of being at most quadratic, preserving angular momentum and parity and transforming as a scalar under the permutation group. The general form of $M^2$ contains several models of baryons structure, including single particle (i.e. h.o.) and collective string models. The calculations are performed choosing the latter model,with the consequence that $M^2$ contains also a term of the type $b^\dagger b^\dagger s s + h.c.$, which causes a spread of the wave function over many h.o. shells. In order to make more transparent the interpretation of the results, the mass operator is rewritten in terms of vibrational and rotational contributions to the baryon spectrum, using a procedure already introduced for the Interacting Boson Model [@kl] $$M^2 = M_0^2 + M_{vib}^2 + M_{rot}^2 + M_{vib-rot}^2;$$ in this way the baryon excitation spectrum is determined by vibrations and rotations of the string-like configuration. There seems to be no evidence of excitations due to the term $M_{vib-rot}^2$, therefore it is omitted and the remaining two terms, according to the discussion reported in [@bil], are simplified obtaining the mass formula $$M^2 = M_0^2 + N[ \kappa_1 n_u + \kappa_2 (n_v + n_w)] + \alpha L,
\label{M_{inv}}$$ where $\kappa_1, \kappa_2, \alpha$ are free parameters, L is the total orbital angular momentum and $n_u, n_v, n_w$ are the eigenvalues of number operators of the type $c^\dagger c$, labeling the vibrational energy levels.
Eq. (\[M\_[inv]{}\]) describes the $SU(6)$ invariant part of the interaction. The splittings within the multiples are introduced with reference to the internal part of the state (see Eq. (\[3q\])). The corresponding algebraic structure is $$G_i = SU_c(3)\otimes SU_s(2) \otimes SU_f(3),$$ describing the colour, spin and flavour degrees of freedom, respectively. Baryons are colour singlets and then only the spin-flavour degrees of freedom contribute to the energy splittings. The mass squared operator $M^2_{sf}$ is written in a Gürsey-Radicati form [@rad] $$\begin{aligned}
\label{rad}
\begin{array} {ccl}
M^2_{sf}& = &a [C_2(SU(6)_{sf})-45 ] + b [C_2(SU(3)_f)-9 ] + \\
& & + b' [C_2(SU_I(2))-\frac{3}{4} ] + b'' [C_1(U_Y(1))-1 ] + \\
& & + b''' [C_2(U_Y(1))-1] + c [C_2(SU_S(2))-\frac{3}{4} ].
\end{array}\end{aligned}$$ The quantities denoted as $C_n(X), n=1,2$ are the Casimir operators of the Group X; the constants in Eq. (\[rad\]) are chosen in order that each term vanishes in the nucleon ground state. For non strange baryons the hypercharge Y is equal to 1 and the b and b’ terms can be grouped into a single term, therefore one can use the simplified form $$\label{simpl}
M^2_{sf} = a [C_2(SU(6)_{sf})-45 ] + b [C_2(SU(3)_f)-9 ] +
c [C_2(SU_S(2))-\frac{3}{4} ].$$
The seven parameters in Eqs. (\[M\_[inv]{}\]) and (\[simpl\]) are obtained fitting the non strange baryon spectrum and the results are very good. In the model the number of shell is not limited, therefore one can describe well also the four star resonances with higher values of the spin, such as $G19, H19, G19, I 1 11$ and $H3 11$. The extension of the model to the strange baryons is presented in ref. [@bil_s].
The model allows to calculate also the elastic nucleon form factors [@bil2] and the electromagnetic transition amplitudes for photo-and electroproduction [@bil; @bil2], provided that a form for the charge distribution along the string is assumed. In this way both the elastic and inelastic form factors are adequately described.
The Goldstone Boson Exchange Model
-----------------------------------
The model is based on the consideration that QCD exhibits an approximate chiral symmetry which is spontaneously broken [@olof]. As a consequence of such spontaneous symmetry breaking, quarks acquire an effective mass and Goldstone bosons emerge, which are indentified with the pseudoscalar meson octet. Therefore it is assumed that baryons are considered as a system of three constituent quarks with an effective quark-quark interaction $V_{3q}$, which is split into parts according to Eq. (\[v3q\]). While different forms are assumed for $V_{inv}$ in the various versions of the model, the spin-flavour part is always chosen as an exchange of pseudoscalar (Goldstone) bosons between two quarks. The simplest form of this chiral interaction can be written as [@olof] $$\label{chir}
H_{\chi} \sim \sum_{i<j} V(\vec{r}_{ij}) \vec{\lambda}^F_i \cdot \vec{\lambda}^F_j \vec{\sigma}_i \cdot \vec{\sigma}_j ,$$ where $\vec{\lambda}^F_i $ are the flavour Gell-Man matrices and $\vec{\sigma}_i $ the quark spin operators. The interaction has a Yukawa form containing a spin-spin and a tensor part. The spin-spin part is given by $$\label{yuk}
V_P(\vec{r}_{ij}) = \frac{g^2}{4 \pi} \frac{1}{3} \frac{1}{4 m_i m_j} \vec{\sigma}_i \cdot \vec{\sigma}_j \vec{\lambda}^F_i \cdot \vec{\lambda}^F_j [\mu^2 \frac{e^{-\mu r_{ij}}}{r_{ij}} - 4 \pi \delta(\vec{r}_{ij})],$$ where $m_i$ are the quark masses and P labels the exchanged boson of mass $\mu$; the $\delta$ is actually smeared out by the finite size of quarks and mesons. The flavour structure of the quark-quark interaction is then $$V_{octet}(r_{ij}) = \sum_{a=1}^3 V_\pi (r_{ij}) \vec{\lambda}^a_i \cdot \vec{\lambda}^a_j + \sum_{b=4}^7 V_K (r_{ij}) \vec{\lambda}^b_i \cdot \vec{\lambda}^b_j + V_\eta (r_{ij}) \vec{\lambda}^8_i \cdot \vec{\lambda}^8_j .
\label{oct}$$
In ref. [@olof] $V_{inv}$ is assumed to be a h.o. potential and the boson exchange interaction is considered in the chiral limit, in which case the masses of all the three quarks are equal and $ V_\pi =V_K=V_\eta$. Treating the interaction as a perturbation, the masses of the baryons can be expressed in terms of a limited number of radial integrals, which are used as parameters in order to fit the experimental values. Already in this simplified approach, a reasonable description of the spectrum is obtained, in particular the boson exchange interaction leads to the correct ordering between the excited $0^+$ and the first negative parity levels. As a further improvement also deviations from the chiral limit and the contribution of the tensor part of the boson exchange interaction are considered.
A substantial improvement of the model has been started in ref. [@ple1], in the sense that the confinement interaction is assumed to be linear and $V_{3q}$ is inserted into a Faddeev equation for the three quark system to be solved numerically. The $V_{sf}$ is given by Eq. (\[oct\]), to which the exchange of the singlet meson ($\eta '$) is added. In the first application, only nucleon and $\Delta$ states are considered [@ple1; @ple2]. The interquark potential is then: $$V(r_{ij}) = V_{octet}(r_{ij}) +V_{singlet}(r_{ij}) +C r_{ij} ;
\label{gbe}$$ in $V_{octet}$ only the $\pi$ and $\eta$ potentials contribute and the singlet potential is $$V_{singlet}(r_{ij}) = \frac{2}{3} \vec{\sigma}_i \cdot \vec{\sigma}_j V_{\eta '} (r_{ij}).$$ The quark-eta coupling constant is assumed equal to the quark-pion one, deduced from the pion-nucleon interaction. Keeping the meson masses equal to their physical values, there remain only four free parameters, namely the two parameters which determine the smearing of the $\delta$ function in Eq. (\[yuk\]), the $\eta '$ coupling constant and the strength of the linear confinement C. The result is a good description of the 14 lowest N and $\Delta$ states, respecting the ordering displayed by the experimental spectra.
A unified description of both non strange and strange baryons is finally achieved in [@ple3], where the interaction of Eq. (\[gbe\]) is used together with a relativistic kinetic energy as in Eq. (\[relkin\]). The three-quark wave equation is solved by means of a variational approach. Again a quite satisfactory description of the low-lying light and strange baryons is achieved, respecting in particular the already mentioned relative ordering of the positive and negative parity states.
The Bonn model
--------------
The authors start from the consideration that the nonrelativistic approach seems to be completely inadequate for the description of the internal motion of quarks with small constituent masses and therefore they introduce a relativistic formulation [@bn1].
The relativistic formulation is performed within quantum field theory and is based on the six-point Green’s function $G_x$ describing three interacting quarks. The infinite series of Feynman diagrams, necessary in order to describe a bound state, is rearranged in the same way used by Bethe-Salpeter for the two-particle case. The result is that $G_x$ obeys to an integral equation containing two irreducible kernels $K^{(2)}$ and $K^{(3)}$, describing the two- and three-particle interactions, respectively. Introducing the momentum space representation $G_P$ of $G_x$, the integral equation can be written in concise form as $$G_P= G_{0P} - i G_{0P} K_P G_P,$$ where $G_{0P}$ is the three quark propagator and $K_P$ the total kernel, or equivalently as $$(G_{0P}^{-1} + i K_P) G_P = I,
\label{G6}$$ showing that $G_P$ is the resolvent of the pseudohamiltonian $H_P$ $$H_P = (G_{0P}^{-1} + i K_P).
\label{pH}$$
The idea is to extract from $G_P$ the baryon contributions, meant as real bound states of three quarks with positive energy $\sqrt{\vec{P}^2+M^2}$. To this end, $G_P$ is expanded in a Laurent series, which, near a pole, gives $$G_P(p_\rho,p_\lambda;p'_\rho,p'_\lambda) = -i \frac{\chi_{\overline{P}}(p_\rho,p_\lambda) \overline{\chi}_{\overline{P}}(p_\rho,p_\lambda)}{P^2-M^2+i\epsilon} .
\label{laur}$$ The quantity $\chi_{\overline{P}}(p_\rho,p_\lambda)$ is the Fourier transform of the Bethe-Salpeter (BS) amplitude $\chi_{\overline{P}}(x_1,x_2,x_3)$ for the bound state $|\overline{P} \rangle $, defined as transition amplitudes between the state $|\overline{P} \rangle $ and the vacuum $ |0 \rangle $ $$\chi_{\overline{P}}(x_1,x_2,x_3) = \langle 0|T(\Psi_{a_1}(x_1) \Psi_{a_2}(x_2) \Psi_{a_3}(x_3) ) |\overline{P} \rangle ,$$ where $\Psi_{a_i}(x_i)$ is the quark field and $a_i$ denotes the Dirac, flavour and colour indices. Thanks to translational invariance, one gets a BS amplitude $\chi_{\overline{P}}(\rho,\lambda)$ which depends on relative coordinates only and, in momentum space, on the conjugate momenta $p_\rho,p_\lambda$; here $\rho$ and $\lambda$ are tetravectors, whose spatial part coincide practically with the Jacobi coordinates defined in Eq. (\[coord\]).
The factorization property of the pole residue allows then to extract the BS amplitude $\chi_{\overline{P}}(p_\rho,p_\lambda)$, which satisfies the equation $$\chi_{\overline{P}} = -i G_{0\overline{P}}~ K_ {\overline{P}}~ \chi_{\overline{P}} ,
\label{BS}$$ where $K_{\overline{P}}$ has, as mentioned before, two- and three-body contributions $K^{(2)}$ and $K^{(3)}$, respectively.
In principle the BS equation (\[BS\]) allows a covariant description of baryons as bound states of three quarks in the framework of QCD. However it cannot be used practically because the single quark propagators and the kernels $K^{(2)}$, $K^{(3)}$ are only formally defined in perturbation theory as an infinite sum of Feynman diagrams and are not deducible from QCD. Moreover, the dependence on the relative energy (or relative time) leads to a complicate analytical structure. Therefore an adequate parametrization is necessary, to be introduced after having having obtained a six-dimensional reduction of the full eight dimensional BS equation, the so called Salpeter equation, trying to preserve the covariance of the theory and keep it as close as possible to the quite successful nonrelativistic quark model.
This can be achieved following the lines of what has already performed in the covariant quark model for meson case using an instantaneous $q\overline{q}$ BS equation [@bn-sal]. To this end, the quark propagators are assumed to be given by their free forms with effective constituent masses. Moreover the kernels $K^{(2)}$ and $K^{(3)}$ are approximated by effective interactions which are instantaneous in the baryon rest frame, that is $$K^{(3)}_P(p_\rho,p_\lambda; p'_\rho,p'_\lambda) ~ = ~V^{(3)}_P(\vec{p_\rho},\vec{p_\lambda}; \vec{p'_\rho},\vec{p'_\lambda}),
\label{k3}$$ $$K^{(2)}_{\frac{2}{3}P+p_\lambda}p_\rho, p'_\rho) ~ = ~V^{(2)}_P(\vec{p_\rho},;\vec{p'_\rho}),
\label{k2}$$ where $P=(M,0)$. This instantaneous approximation can be formulated in a covariant way following the method proposed in ref. [@wm]. The reduction to the six dimensional Salpeter equation can be performed more easily if only the three-body kernel is present; the introduction of the two body kernel is possible provided that it is substituted by a suitable effective interaction $V_{eff}$. In any case the reduction to the Salpeter amplitude $\Phi_M$ is achieved by means of an integration over the energy variables, a procedure which, thanks to Eqs. (\[k3\]) and (\[k2\]) affects the BS amplitude only $$\Phi_M(\vec{p}_\rho,\vec{p}_\lambda)=\int \frac{dp_\rho^0}{2 \pi}\frac{dp_\lambda^0}{2 \pi} \chi_M(p_\rho^0,p_\lambda^0,\vec{p}_\rho,\vec{p}_\lambda),$$ where $\chi_M$ is the BS amplitude in the rest frame.
Finally the Salpeter equation is written in a Hamiltonian formulation $$\mathcal{H}_M \Phi_M^\Lambda= M \Phi_M^\Lambda,$$ where $\Lambda$ is a projector operator over positive energy states.
In order to perform explicit calculations of the baryon spectrum one has to assume some specific form of the hamilton operator $\mathcal{H}_M$. In agreement with the previous discussion, $\mathcal{H}_M$ contains two-and three-body potentials [@bn2].
The confining three-body potential is chosen within a string-like picture, where the quarks are connected by gluonic strings (flux tubes) and the potential increases linearly with a collective radius $r_{3q}$ $$V^{(3)}_{conf}(\vec{r}_1,\vec{r}_2,\vec{r}_3)\sim r_{3q}(\vec{r}_1,\vec{r}_2,\vec{r}_3).$$ There are three different ways to define $r_{3q}$ [@bn2]. The first one is the $Y$-type [@carl] $$r_{3q}=r_Y=min \sum_{i=1}^3 |\vec{r}_i-\vec{r}_0|,$$ $\vec{r}_0$ is the position where the flux tubes can merge and is chosen in order to minimize $r_{3q}$. A second possibility is given by the so called $\Delta$-type $$r_{3q}=r_\Delta=\sum_{i<j}^3 |\vec{r}_i-\vec{r}_j|;$$ rescaling $r_\Delta$ by a factor f one gets a good approximation of $r_Y$ ([@bn2] and references quoted therein), provided that $1/2 < f < 1/\sqrt{3}$. The third choice is provided by the hypercentral one [@pl] $$r_{3q}=r_H=\sqrt{\vec{\rho}^2+\vec{\lambda}^2}.$$ Having tested that the structure of spectra depends only slightly on the choice of $r_{3q}$, the authors of ref. [@bn2] use the three-body $\Delta$-shape string potential rising linearly with $r_\Delta$, which provides the $SU(6)$ invariant part of the three-quark interaction.
The two-body potential is taken from the instanton interaction introduced by 't Hooft [@tH] in the $SU(2)$ case (extended to $SU(3)$ in ref. [@svz]), in which the $\delta$ term is smeared with an effective range $\lambda$. Such an interaction acts only on flavour antisymmetric states and therefore it does non act on $\Delta$ states, thereby leading to a $N-\Delta$ mass splitting.
The model hamiltonian depends on seven free parameters, which are used to describe both the non-strange [@bn2] and strange [@bn3] baryon sector. The results are quite satisfactory.
The model has been successfully applied to the description of the elastic form factors [@mert; @ronn], the helicity amplitudes for both the nonstrange [@ronn] and strange resonances [@caut], the semileptonic decays of baryons [@mig] and the axial form factors [@mert].
The interacting quark-diquark model
-----------------------------------
The Interacting quark quark model [@diq] and its relativistic version [@rel_diq] give a good reproduction of the spectrum, moreover they have much less missing resonances than a normal three quark model. In particular, we report here the rest frame mass operator of the Relativistic quark-diquark model : $$\begin{aligned}
M=E_0 +\sqrt{q^2+m^2_1}+\sqrt{q^2+m^2_1}+ M_{\rm dir}(r)+ M_{\rm cont} +M_{\rm ex}(r),\end{aligned}$$ where $E_0$ is a constant, $M_{\rm dir} (r)$ and $M_{\rm ex} (r)$, respectively, are the direct and the exchange diquark-quark interaction, $m_1$ and $m_2$ stand for the diquark and quark masses, where $m_1$ is either $m_S$ or $m_{AV}$ according if the mass operator acts on a scalar or an axial vector diquark, and $M_{\rm cont}(r)$ is a contact interaction. The direct term is a Coulomb-like interaction with a cutoff plus a linear confinement term $$\begin{aligned}
M_{\rm dir}= -\frac{\tau}{r}\left(1-e^{-\mu r}\right) + \beta r.\end{aligned}$$ A simple mechanism that generates a Coulomb-like interaction is the one-gluon exchange. One needs also an exchange interaction. This is indeed the crucial ingredient of a quark-diquark description of baryons and has the form $$\begin{aligned}
M_{\rm ex}(r)= (-1)^{l+1}2Ae^{-\sigma r}\left[A_S(\vec{s}_1\cdot\vec{s}_2)+A_I(\vec{t}_1\cdot\vec{t}_2)+
A_{SI}(\vec{s}_1\cdot\vec{s}_2)
(\vec{t}_1\cdot\vec{t}_2)\right],\end{aligned}$$ where $\vec{s}$ and $\vec{t}$ are the spin and the isospin operators. Moreover, we consider a contact interaction $$\begin{aligned}
M_{\rm cont}(r) = \left(\frac{m_1m_2}{E_1E_2}\right)^{1/2+\epsilon}\frac{\eta^3D}{\pi^{3/2}}e^{-\eta^2 r^2}
\delta_{L,0}\delta_{s_1,1}
\left(\frac{m_1m_2}{E_1E_2}\right)^{1/2+\epsilon},\end{aligned}$$ where $E_i=\sqrt{q^2+m^2_i}(i=1,2), \epsilon, \eta$ and $D$ are parameters of the model. The relativistic Interacting quark-diquark model is a relativistic version of the Interacting model of Ref. [@diq]. The Interacting quark-diquark model hamiltonian is $$\begin{aligned}
H&=&E_0+\frac{q^2}{2\mu}-\frac{\tau}{r}+\beta r+ [B+C \delta_{L,0}]\delta_{s_1,1}\nonumber\\
&+& (-1)^{l+1}2Ae^{-\alpha r}\left[(\vec{s}_1\cdot\vec{s}_2)+(\vec{t}_1\cdot\vec{t}_2)+(\vec{s}_1\cdot\vec{s}_2)
(\vec{t}_1\cdot\vec{t}_2)\right],\end{aligned}$$ where $\vec{s}_1$ and $\vec{s}_2 $ are the spin of the quark and of the diquark respectively, while $\vec{t}_1$ and $\vec{t}_2 $ the the same for the isospin. The contact interaction $ C \delta_{L,0}$ acts only on the spatial ground state, while the $\delta_{s_1,1}$ on the axial diquark.
![ (Color online) The experimental spectrum of the non strange three- and four-star resonances [@pdg10] in comparison with the results of the interacting quark-diquark model [@diq]. []{data-label="q-diq"}](q-diq){width="4.5in"}
The hypercentral Constituent Quark Model
========================================
The hyperspherical coordinates
------------------------------
The starting point of the hypercentral Constituent Quark Model (hCQM) is the introduction of the hyperspherical coordinates [@morp52; @sim; @baf], which are given by the angles ${\Omega}_{\rho}=({\theta}_{\rho},{\phi}_{\rho})$ and ${\Omega}_{\lambda}=({\theta}_{\lambda},{\phi}_{\lambda})$ together with the hyperradius, $x$, and the hyperangle, $\xi$, defined in terms of the absolute values $\rho$ and $\lambda$ of the Jacobi coordinates of Eq. (\[coord\]) $$x = \sqrt{\vec{\rho}^2 + \vec{\lambda}^2}, ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ \xi = \arctan {\frac{\rho}{\lambda}}.$$ The hyperradius x is a collective variable, which gives a measure of the dimension of the three-quark system, while the hyperangle $\xi$ reflects its deformation.
Using these variables, the nonrelativistic kinetic energy operator of Eq. (\[kin\]), after having separated the c.m. motion, can be written as $$- \frac{\hbar^2}{2m} (\Delta_\rho + \Delta_\lambda) = - \frac{\hbar^2}{2m} ( \frac{\partial ^2}{\partial x^2}+\frac{5}{x} \frac{\partial}{\partial x} + \frac{L^2(\Omega)}{x^2}).$$ The grand angular operator $L^2(\Omega)=L^2(\Omega_\rho, \Omega_\lambda,\xi)$ is the six-dimensional generalization of the squared angular momentum operator and is a representation of the quadratic Casimir operator of the rotation group in six dimensions O(6). Its eigenfunctions are the so called hyperspherical harmonics (h.h.) [@baf] $Y_{[\gamma]l_{\rho}l_{\lambda}}({\Omega}_{\rho},{\Omega}_{\lambda},\xi)$ $$L^2(\Omega_\rho, \Omega_\lambda,\xi)~Y_{[{\gamma}]l_{\rho}l_{\lambda}}({\Omega}_{\rho},{\Omega}_{\lambda},\xi)~=~-\gamma(\gamma+4) Y_{[{\gamma}]l_{\rho}l_{\lambda}}({\Omega}_{\rho},{\Omega}_{\lambda},\xi);
\label{grand}$$ the grand angular quantum number $\gamma$ is given by $\gamma = 2 n + l_\rho + l_\lambda$, where n is a nonnegative integer and $ l_\rho$, $l_\lambda$ are the angular momenta corresponding to the Jacobi coordinates of Eq. (\[coord\]).
The h.h. describe the angular and hyperangular part of the three-quark wave function and are written as [@baf] $${Y}_{[{\gamma}]l_{\rho}l_{\lambda}}
({\Omega}_{\rho},{\Omega}_{\lambda},\xi)~=~
{Y}_{l_{\rho}m_{\rho}}({\Omega}_{\rho})~{Y}_{l_{\lambda}
m_{\lambda}}({\Omega}_{\lambda})~~
^{(2)}{P_{\gamma}^{l_{\rho}l_{\lambda}}}~(\xi).\label{yg}$$ where the hyperangular functions $^{(2)}{P_{\gamma}^{l_{\rho}l_{\lambda}}}(\xi)$ are given in terms of trigonometric functions and Jacobi polynomials [@baf].
The h.h. form a complete orthogonal basis in the space of the functions of $\Omega_\rho, \Omega_\lambda,\xi $ and then any three-quark wave function can be expanded as a series of h.h.
$$\Psi(\vec{\rho},\vec{\lambda})~~=~~\sum_{\gamma,l_{\rho},l_{\lambda}}~
c_{[\gamma]l_{\rho}l_{\lambda}}~
{\psi}_{\gamma}(x)~{Y}_{[{\gamma}]l_{\rho}l_{\lambda}}(\Omega),
\label{span}$$
the hyperradial wave function ${\psi}_{\gamma}(x)$, depending on the hyperradius x only, is completely symmetric for the exchange of the quark coordinates.
The hypercentral approximation
------------------------------
An expansion similar to Eq. (\[span\]) is valid for any quark interaction, the first term depending on the hyperradius $x$ only: $$\Sigma_{i<j} ~V(r_{ij})~=~V(x) + \cdots .$$
Retaining the first term only one gets the so called hypercentral approximation. Such approximation has been applied with success to the description of few-nucleon systems [@hca; @ffs], while in the baryon case it has been shown that the matrix elements of the currently used two-body qq potentials in the 3q space exhibit an almost perfectly hypercentral behaviour [@has].
In the hypercentral approximation, the three-quark potential depends on the hyperradius x only and therefore it has a three-body character, since the dependence on the single pair coordinates cannot be disentangled from the third one. The possibility of three-body forces is strictly related to the existence of a direct gluon-gluon interaction, which is one of the fundamental features of QCD. The diagram shown in Fig. \[string\] a is the lowest order one leading to a non vanishing three-body interaction among quarks in a baryon, but of course many others can be considered. A three-quark mechanism is considered also in flux tube models, which have been proposed as a QCD-based description of quark interactions [@ip], leading to the Y-shaped three-quark configuration of Fig. \[string\]b [@carl], besides the standard $\Delta$-like two-body one of Fig. \[string\]c. Furthermore, a Born-Oppenheimer treatment of the confinement potential in a QCD motivated bag model leads quite naturally to three-body forces [@has; @hel] which increases linearly with some ’collective’ radius.
![ a) The lowest order diagram leading to a non zero three-body interaction of quarks in a baryon. b) The -Y-shaped string configuration. c) The $\Delta$-shaped string configuration.[]{data-label="string"}](3q){width="4in"}
In the hCQM the three-quark interaction is assumed to be hypercentral $$V_{3q}(\vec{\rho},\vec{\lambda})~=~V(x) ,$$ as a consequence, the three-quark wave function is factorized $$\psi_{3q}(\vec{\rho},\vec{\lambda})~=
\psi_{\gamma \nu}(x)~~
{Y}_{[{\gamma}]l_{\rho}l_{\lambda}}({\Omega}_{\rho},{\Omega}_{\lambda},\xi);
\label{psi}$$ the hyperradial wave function $\psi_{\gamma \nu}(x)$ is labeled by the grand angular quantum number $\gamma$ defined above and by the number of nodes $\nu$. The angular-hyperangular part of the 3q-state is completely described by the h.h. and is the same for any hypercentral potential. The dynamics is contained in the hyperradial wave function $\psi_{\gamma \nu}(x)$, which, because of the factorization Eq. (\[psi\]), is obtained as a solution of the hyperradial equation $$[~\frac{{d}^2}{dx^2}+\frac{5}{x}~\frac{d}{dx}-\frac{\gamma(\gamma+4)}{x^2}]
~~\psi_{\gamma \nu}(x)
~~=~~-2m~[E-V_{3q}(x)]~~\psi_{\gamma \nu}(x).
\label{hyrad}$$
The Eq. (\[hyrad\]) can be solved analytically in two cases. The first one is the six-dimensional harmonic oscillator (h.o.) $$\sum_{i<j}~\frac{1}{2}~k~(\vec{r_i} - \vec{r_j})^2~=~\frac{3}{2}~k~x^2~=
~V_{h.o}(x),
\label{eq:ho}$$ which is exactly hypercentral. The eigenvalues are given, as already mentioned, by $E= (3 +N) \hbar \omega$, where N can be written as $N=2 \nu + \gamma$ and the hyperradial wave functions are reported in Appendix A.
The second analytical case is given by the hyperCoulomb (hC) potential [@nc; @hyc; @breg; @sig; @sig2] $$V_{hyc}(x)= -\frac{\tau}{x}.
\label{hyc}$$
The eigenvalues of the hyperCoulomb problem can be obtained by generalizing to six dimensions the calculations performed in three dimensions, obtaining $$E_{n,\gamma}~=~-\frac{{{\tau}^2}m}{2n^2},
\label{en}$$ where $n~=~N+\frac{5}{2}$ is the principal quantum number and $N=\gamma+\nu$, where $\nu~= 0,1,2,~\ldots$ is the radial quantum number that counts the number of nodes of the wave function.
The fundamental reason why the three-body problem with h.o. or hC interaction is exactly solvable is that they have a dynamic symmetry, $U(6)$ and $O(7)$ respectively.
The dynamic symmetry $O(7)$ of the hC problem can be used to obtain the eigenvalues using purely algebraic methods. The hyperCoulomb Hamiltonian can be rewritten as [@breg]
$$H~=~-\frac{{{\tau}^2}m}{2~[C_{2}(O(7))+\frac{25}{4}]},$$
where $C_{2}(O(7))$ is the quadratic Casimir operator of $O(7)$. It can be shown [@sig2] that the eigenvalues of $C_{2}(O(7))$ are given by $(\nu+\gamma)(\nu+\gamma+5)$, obtaining $$E~=~-\frac{{{\tau}^2}m}{2~[(\nu+\gamma)(\nu+\gamma+5)+25/4]},$$ which coincides with Eq. (\[en\]).
The eigenfunctions of Eq. (\[hyrad\]) with the hyperCoulomb potential can be obtained analytically and are [@sig2] $$\psi_{\gamma \nu}(x)~~=~~
\left[\frac{\nu!~(2g)^6}{(2 \gamma + 2 \nu+5)(\nu+ 2 \gamma+4)!^3}
\right]^
{\frac{1}{2}}~(2gx)^{\gamma}~e^{-gx}~~
L^{2\gamma+4}_{\nu}(2gx)
,\label{eigom}$$ where for the associated Laguerre polynomials the notation of Ref. [@mf] is used and $g~=~\frac{\tau m}{\gamma+\nu+\frac{5}{2}}$. The explicit expression of the hyperradial wave functions are obtained in ref. [@sig2] and reported in Appendix A.
A complete solution of the hyperCoulomb problem, using the SO(7,2) dynamical group, has been worked out in ref. [@bis], where the states, the elastic and inelastic form factors have been also developed.
![ (Color online) Qualitative structure of theoretical spectra for the h.o. (left) and for the hC potentials (right), up to the first three shells. The energy units are arbitrary and different for the two potentials.[]{data-label="ho-hc"}](spectrum_ho-hc){width="4.5in"}
The hC potential has important features [@pl; @sig; @sig2]. The energy eigenvalues depend on $\nu + \gamma$ and then the negative parity states are exactly degenerate with the positive parity excitations, as is shown in Fig. (\[ho-hc\]). The observed Roper resonance is somewhat lower with respect to the negative parity baryon resonance, at variance with the prediction of any $SU(6)-$invariant two-body potential, therefore the hC potential provides a better starting point for the description of the spectrum. The spectrum of Fig. (\[ho-hc\]) shows that within the first three shell the hC potential exhibits two more states with respect to the h.o. The first extra level has positive parity and contains a further N and $\Delta$ state, thus enhancing slightly the number of theoretical states, but the second one has negative parity and allows to insert the recently observed states, already mentioned in Sec. 2.2.
Another interesting property of the hC potential is that the form factors calculated with its wave functions have a power-law behaviour [@nc; @sig; @sig2], leading to an improvement with respect to the widely used harmonic oscillator, for which the form factors are too strong damped for increasing momentum transfer.
![ (Color online) Qualitative structure of theoretical spectra for the potential of Eq. (\[conf\]), up to the first three shells. The energy units are arbitrary.[]{data-label="conf"}](spectrum_conf){width="4.in"}
The hypercentral Constituent Quark Model
----------------------------------------
The hC potential has interesting features, however it is not confining. In the hCQM, the $SU(6)$ conserving part of the potential is then assumed to be the sum of the hC interaction and a linear confinement term [@pl; @es] $$V_{inv}~=~-\frac{\tau}{x} + \alpha x;
\label{h_pot}$$ the hyperradial equation (\[hyrad\]) must be solved numerically and the presence of the confinement removes the degeneracies typical of the hC potential, as shown in Fig. (\[conf\]), however the general structure is only slightly modified with respect to the hC potential.
![ (Color online) The quark-antiquark potential calculated in LQCD [@LQCD] for static quarks in the $SU(3)$ limit. The constant $\beta$ is the inverse QCD coupling and $r_0\sim 0.5$ fm.[]{data-label="bali"}](bali){width="4.5in"}
The structure of the potential of Eq. (\[h\_pot\]) is formally similar to the quark-antiquark Cornell potential [@corn] widely used for the description of mesons. It is noteworthy that Lattice QCD calculations [@LQCD] are able to reproduce the Cornell potential, as it is seen in Fig. (\[bali\]), where the LQCD results for static quark-antiquark pairs in the $SU(3)$ limit are reported. The model potential of Eq. (\[h\_pot\]) can then be considered as the hypercentral approximation of the Cornell potential.
Having chosen the form for the hypercentral potential, the solutions of the hypercentral equation (\[hyrad\]) produce a series of wave functions $\psi_{\gamma \nu}(x)$ and then one can build up the model $SU(6)$ states. Taking advantage of the fact that also the h.o. potential is hypercentral, one can start from the states of the Isgur-Karl model [@ik], express them in terms of the hyperspherical coordinates and substitute the h.o. hyperradial wave functions with those determined by the potential of Eq. (\[h\_pot\]). The complete $SU(6)$ configurations for non strange baryons are reported in Appendix B.
![ (Color online) The spectrum obtained using the hCQM hamiltonian of Eq. (\[H\_hCQM\]). The free parameters are fitted to the experimental values of the 4\* and 3\* resonances reported in the PDG [@pdg10].[]{data-label="hCQM"}](hCQM){width="4.5in"}
In order to complete the model hamiltonian one has to add a term violating the $SU(6)$ symmetry. In hCQM such term is chosen to be the standard hyperfine interaction [@deru; @ik] of Eq. (\[hyp\]). The hamiltonian for the three quark system in the hCQM is then $$H_{hCQM}~=~3m + \frac{\vec{p}_\rho^{~2}}{2m} + \frac{\vec{p}_\lambda^{~2}}{2m}-\frac{\tau}{x} + \alpha x + H_{hyp}.
\label{H_hCQM}$$ In this way the baryon states are superpositions of the $SU(6)$ configurations reported in Appendix B.
Using the notation introduced with the Isgur-Karl model, the nucleon state can be written as $$\begin{aligned}
\label{nucl}
|N \rangle = &a_S |N ^2S_{1/2} \rangle _S + a'_S |N ^2S^*_{1/2} \rangle _S + a''_S |N ^2S^{**}_{1/2} \rangle _S+
\nonumber \\
&+ a_M |N ^2S_{1/2} \rangle _M + a_D |N ^4D_{1/2} \rangle _M ,\end{aligned}$$ with $ a_S=0.976, a'_S=-0.196, a''_S=-0.043, a_M=-0.051, a_D=-0.070$ [@pl; @es]; the asterisks in the second and third term of Eq. (\[nucl\]) mean that the spin-isospin part are the same as in the first term, but the space part corresponds to hyperradially excited wave functions. One should not forget that in the hCQM there are two hyperradial excitations of the nucleon within the first three shells (see Fig. (\[ho-hc\]). The Roper resonance has a similar expansion, with the dominant component given by $|N ^2S'_{1/2} \rangle _S$: $ a_s=0.183, a'_s=0.967, a''_s=0.185, a_M=0.021, a_D=-0.176$. The $\Delta$ resonance is given by $$|\Delta \rangle = b_S |\Delta ^4S_{3/2} \rangle _S + b'_S |\Delta ^4S'_{3/2} \rangle _S + b_D |\Delta ^4S_{3/2} \rangle _S + b'_D |\Delta ^2D_{3/2} \rangle _M ,$$ with $ b_S=0.976, b'_S=0.166, b''_S=-0.042, b_D=0.026, b'_D=-0.132$.
The quark mass is taken to be 1/3 of the proton mass, as prescribed in order to reproduce the proton magnetic moment (see e.g. [@mg]). In this way in the model there are only three free parameters: $\tau$, $\alpha$ and the strength of the hyperfine interaction, the latter being mainly determined form the $N-\Delta$ mass difference. The parameters are found by fitting the energies of the 4$*$ and 3$*$ non strange baryons relative to the nucleon and are given by $$\label{par}
\alpha= 1.16fm^{-2},~~~~\tau=4.59~.
%\label{param}$$ The resulting spectrum is reported in Fig. (\[hCQM\]).
Having fixed the parameters of the three-quark hamiltonian from the spectrum, the baryon states are completely determined and can be used for the calculations of various properties. In the rest of the paper, the results obtained by the present nonrelativistic hCQM are given by parameter free calculations, that is they are [**predictions**]{}.
An analytical model
-------------------
The hyperradial equation with the hypercentral potential of Eq. (\[h\_pot\]) cannot be solved analytically unless the linear confinement term is treated as a perturbation [@sig; @sig2]. This situation is reasonably valid for the lower states since they are confined in the low x region where the hC term is dominant. With this assumption, the perturbative contributions to the energies of the states are determined by the integral $$\int_{0}^{\infty}~{dx}~x^{6}~|\psi_{\gamma \nu}(x)|^{2},$$ which is given by [@sig; @sig2] $$\frac{1}{2g}
\frac{\nu!}{(\nu+2\gamma+5){(\nu+2\gamma+4)!}^3}
\left[2~\Gamma(\nu+2\gamma+5)\right]^{2}
\sum_{\sigma}
\frac{\Gamma(7+2\gamma+\sigma)}{\sigma!(\nu-\sigma)!^2
(\sigma-\nu+2)!^{2}}, \label{int}$$ where $$\nu-2~\leq~\sigma~\leq~\nu.$$
![ (Color online) The spectrum obtained using the analytical model. The free parameters are fitted to the experimentl values of the 4\* and 3\* resonances reported in the PDG [@pdg10].[]{data-label="solv"}](solv){width="4.5in"}
The energy eigenvalues are then $$E_{n,\gamma}~=~~-\frac{\tau^{2}m}{2n^{2}}+\frac{\alpha}{2m\tau}
[3n^2-\gamma(\gamma+4)-\frac{15}{4}],
\label{ener}$$ where, according to the definition introduced in Sec. 3.2, n is given by $\nu+\gamma+5/2$. This formula, to which a constant $E_0$ should be added, is used to reproduce the energies averaged over the states with the same quantum numbers $\nu,\gamma$. In the fitting procedure, the Roper resonance is not taken into account because the presence of the confinement pushes it upwards with respect to the negative parity resonances. Assuming a quark mass about 1/3 of the proton mass, the result of the fit leads to [@sig; @sig2] $$E_0=2.152 \mbox{\ GeV}, ~~~~\tau=6.39,~~~~\alpha=0.148 \mbox{\ fm}^{-2},$$ the latter two values are not much different from the ones of the non perturbative analysis reported in Eq. (\[par\]).
In order to describe the splittings within the $SU(6)$ multiplets, one has to add a $SU(6)$ violating interaction to the potential of Eq. (\[h\_pot\]). For simplicity, such interaction is assumed [@sig; @sig2] to contain a spin-spin term $$V^{S}(x)~=~A~e^{-\beta x}~
\sum_{i<j}~\vec{\sigma_{i}}\cdot\vec{\sigma_{j}}~
=~A~e^{-\beta x}~[2~S^2-\frac{9}{4}]
,\\
\label{spin}$$ where S is the total spin of the 3-quark system, and a tensor interaction $$V^{T}(x)~=~B~\frac{1}{x^3}~\sum_{i<j}~\left[\frac{\left(\vec{\sigma_{i}} \cdot
(\vec{r_{i}}-\vec{r_{j}})\right)
~\left(\vec{\sigma_{j}}\cdot(\vec{r_{i}}-\vec{r_{j}})\right)}
{|\vec{r_{i}}-\vec{r_{j}}|^{2}}
-\frac{1}{3}(\vec{\sigma_{i}}\cdot\vec{\sigma_{j}})\right].
\label{tens}$$
The parameters of the spin-spin interaction Eq. (\[spin\]) are determined from the $N(939)~-~\Delta (1232)$ and the $N(1535)~-~N(1650)$ splittings, obtaining $A~=~140.7~$MeV and $\beta~=~1.53 ~$fm$^{-1}$. We observe that, at variance with the OGE hyperfine interaction, the spin-spin term has a non-zero range, a feature that seems to be necessary since the hC wave functions are not so concentrated near the origin.
The effect of the tensor interaction Eq. (\[tens\]) is very small in the case of the hC wave functions, so there is no way to determine the parameter B directly from the spectrum. However, assuming for B a value of about 1/10 A, the description of the spectrum is slightly improved. The final results for the spectrum are reported in Fig. (\[solv\]).
The description of the spectrum is particularly good for the negative parity resonances. The analytical model has been also used for the calculation, without free parameters, of the electromagnetic transition amplitudes to some negative parity baryon resonances [@sig; @sig2].
The success of the 1/x model in describing the spectrum indicates that, at least for the inner states, the confinement is provided by the hC potential [@sig2], as further supported by ref. [@adam].
The baryon spectrum
===================
Results from the hCQM
---------------------
As discussed in the previous section, the free parameters of the hCQM are determined by fitting the masses of the 4\* and 3\* resonances [@pdg10] reported in Fig. (\[baryon\]). Of course the model can predict the masses of all other resonances belonging to the first three energy shells and the number of theoretical states exceeds the observed one, leading to the problem of the missing resonances, a problem in common with other CQM, in particular the h.o. one. In this respect, it is interesting to compare the number of states predicted by the two potentials, the h.o. (see Table \[ho\]) and the hCQM, which, as shown in Table \[comp\] are 30 and 39 respectively. These two values are certainly larger than the number of observed 4\* and 3\* states, however the situation becomes different if one considers separately the positive and negative parity states and if also the new results from PDG2012 [@pdg12] are taken into account.
------------- ------ ------ ----------- ------------- ----------- -------------
h.o. hCQM PDG10 PDG10 PDG12 PDG12
4\* + 3\* 4\*+3\*+2\* 4\* + 3\* 4\*+3\*+2\*
$N^+$ 14 15 5 8 6 11
$N^-$ 5 10 5 7 6 9
$ \Delta^+$ 9 10 6 7 6 7
$ \Delta^-$ 2 4 2 3 2 4
Total 30 39 18 25 20 31
------------- ------ ------ ----------- ------------- ----------- -------------
: The number of states predicted by the h.o. and hCQM models, reporting separately the positive and negative parity N and $\Delta$ states. In the last four columns, the number of states listed in the 2010 [@pdg10] and 2012 [@pdg12] PDG editions are reported.[]{data-label="comp"}
The positive parity states allowed by the h.o. model are abundant, but the negative ones are just what is necessary for the description of the observed 4\* and 3\* states of PDG2010 [@pdg10]. In the hCQM there are two positive parity states more and the negative parity ones are doubled, in agreement with the fact that in the hCQM spectrum (see Fig. \[conf\]) there are two extra levels, one $0^+_S$ with a P11 and a P33 state, and a $1^-_M$ level, where a further series of negative parity states can be settled.
In the last edition of the PDG [@pdg12], some new states are reported. This achievement has been possible also thanks to the availability of very precise cross section and polarization data from photoproduction experiments at CLAS [@clas] and by the most recent coupled- channel analysis of the Bonn-Gatchina group [@anis] (for a discussion see [@vb_new]). In particular there is a resonance $D_{13}(1875)$ with a 3\* status. The negative parity states allowed by the h.o. model are all already occupied and only the hCQM, with its further negative parity level, can describe such a state. Moreover, if one considers also the new 2\* states [@pdg12], there are 13 negative parity resonances, 9 of the N and 4 of the $\Delta$ type, to be compared with the allowed values of the hCQM, that is 10 and 4, respectively. Finally, the total number of observed 4\*, 3\* and 2\* resonances is 31, greater than the number allowed by the h.o. and not so far from 39, the hCQM value.
The comparison of the theoretical spectrum with all the 4\*, 3\* and 2\* listed in PDG [@pdg12] is shown in Fig. (\[hCQM\_12\]) [@gs15]. In this Figure two theoretical levels are not shown, that is one $N1/2$ and one $N3/2$ state belonging to the $(20, 1^+)$ $SU(6)$ multiplet; they are mixed by the hyperfine interaction only with the states with $\gamma+\nu$ greater than 2 and do not contribute to the structure of the nucleon.
![ (Color online) The spectrum obtained with the hCQM described in Sec. 3 in comparison with all the 4\*, 3\* and 2\* resonances reported in the 2012 edition of the PDG [@pdg12].[]{data-label="hCQM_12"}](hCQM_12){width="4.5in"}
The overall description of the spectrum is quite good, considering that the model has only three free parameters. As for the Roper resonance, it is practically degenerate with the negative parity states, thanks to the behaviour of the hC potential, which is only slighlty modified by the confinement term. However, the theoretical Roper mass is still to high with respect to the experimental data. This problem is common to various CQM and it has some time ago suggested the idea that the Roper is not simply a “breathing” mode of the nucleon, but it is a hybrid state qqqG [@lb], that is three quark plus a gluon component. However this model for the Roper has been ruled out by the recent results on the $\gamma* p \rightarrow N(1440)$ data [@ab_rop], which showed that the longitudinal electroexcitation is significantly non zero while the hybrid model predicts that it should be vanishing.
The theoretical states in the higher part of the spectrum are somewhat compressed as an effect of the hC interaction. However, the masses of the resonances in this region are determined with large uncertainties and, because of their strong decays, the states are expected to have large widths and to be partially overlapped. When comparing the theoretical baryon spectra with the experimental data one should not forget that up to now CQM models predict states with zero width, since no coupling to the continuum has been consistently introduced. Some time ago the Isgur-Karl model has been implemented with quark-meson couplings [@blask], allowing to calculate both the strong decay widths and the effects of the continuum on the resonance energies. This approach considers quark and meson degrees of freedom on the same footing, it would be desirable on the contrary to have an approach in which quark-antiquark pair mechanisms are consistently taken into account. In this respect an important improvement has been achieved by a recent work [@sb1; @bs; @sb2], in which an unquenched constituent quark model for baryons has been formulated and the quark-antiquark pair contributions are taken into account consistently.
The hCQM with isospin
---------------------
In the hCQM hamiltonian, the $SU(6)$ violating term is provided by the hyperfine interaction of Eq. (\[hyp\]), similarly to what happens with the Isgur-Karl model [@ik] and its semirelativistic extension [@ci]. However the $SU(6)$ violation can be given also by a flavour dependent term, which is more or less explicitly included in other approaches. In fact, in the algebraic model (BIL [@bil]) the quark energy is written in terms of Casimir operators of symmetry groups which are relevant for the three-quark dynamics (see Sec. 2.4). In particular, for the internal degrees of freedom the Gürsey-Radicati mass formula [@rad] is used, leading to an isospin dependent term which turns out to be important for the description of the spectrum. In the $\chi$CQM [@olof], the quark-quark interaction is provided by one meson exchange and therefore the corresponding potential is spin-flavour dependent and is crucial for the description of baryons up to 1.7 GeV. As for the BN [@bn2]) model, the $SU(6)$ violation arises from the instanton interaction which does not act on $\Delta$ states. Moreover, it has been pointed out that an isospin dependence of the quark potential can be obtained by means of quark exchange [@dm].
Therefore there are many motivations for the introduction of a flavour dependent term in the three-quark interaction and for this reason also in the case of the hCQM an isospin dependent term has been included in the quark interaction [@vass; @iso].
The $SU(6)$ violation coming from the hyperfine interaction is still present, with one important modification, namely in the spin-spin interaction the $\delta$-like term is substituted with a smearing factor given by a gaussian function of the quark pair relative distance [@vass]: $$\label{srho}
H_{S}=~A_{S}~\sum_{i<j}~\frac{1}{(\sqrt{\pi}\sigma_S)^{3}}~
e^{-\frac{r_{ij}^2}{\sigma_S^2}}~({\vec{s}}_{i}\cdot {\vec{s}}_{j}).$$
The remaining $SU(6)$ violation comes from two terms. The first one is isospin dependent $$\label{taurho}
{H}_{I}=
~A_{I} \sum_{i<j}\frac{1}{(\sqrt{\pi}\sigma_I)^3}~
e^{-\frac{{\bf r}^2_{ij}}{\sigma_I^2}}({\vec{t}}_i \cdot {\vec{t}}_j),$$\[st\] where $\vec{t}_i$ is the isospin operator of the i-th quark and $r_{ij}$ is the relative quark pair coordinate. The second one is a spin-isospin interaction, given by $$\label{si}
{H}_{SI}=
A_{SI}~\sum_{i<j}\frac{1}{(\sqrt{\pi}\sigma_{SI})^3}~e^{-\frac{r^2_{ij}}
{\sigma^2_{SI}}}({\vec{s}}_i \cdot {\vec{s}}_j)({\vec{t}}_i \cdot {\vec{t}}_j),$$ where $\vec{s}_i$ and $\vec{t}_i$ are respectively the spin and isospin operators of the i-th quark and $r_{ij}$ is the relative quark pair coordinate. The complete interaction is then given by $$\label{tot}
H_{int}~=~V(x) +H_{S} +
H_{I} +H_{SI},$$ where V(x) is the hypercentral potential of Eq. (\[h\_pot\]). The resulting spectrum for the 3\*- and 4\*- resonances is shown in Fig. (\[hCQM\_iso\_12\]).
![ (Color online) The spectrum obtained with the hCQM with the spin and isospin dependent interactions of Eq. (\[tot\]) in comparison with all the 4\*, 3\* and 2\* non strange resonances reported in the 2012 edition of the PDG [@pdg12].[]{data-label="hCQM_iso_12"}](hCQM_iso_12){width="4.5in"}
The $N-\Delta$ mass difference is no more due only to the hyperfine interaction. In fact, in this model its contribution is only about $35\%$, the remaining splitting coming from the spin-isospin term $(50\%)$ and from the isospin one $(15\%)$.
It should also be noted that the negative parity resonances are again well described. In this model however there is the correct inversion between the Roper and the negative parity resonances and this is almost entirely due to the spin-isospin interaction, as stated in Ref. [@olof]. In general, the position of the Roper resonance is reproduced in all models containing an isospin dependent interaction [@bil; @olof; @bn2].
Also the higher states are fairly described and slightly less compressed than in the standard hCQM.
The tensor part of the hyperfine interaction, which is omitted for simplicity in Eq. (\[tot\]), is taken into account in the calculation, however its contribution to the spectrum is negligible.
An extension to strange baryons
-------------------------------
The hypercentral interaction of Eq. (\[h\_pot\]) describes the average energies of the $SU(6)$ multiplets, while the splittings within each multiplets are generated by the hyperfine interaction Eq. (\[hyp\]) or by the spin-isospin interaction of Eq. (\[tot\]). In the latter case the flavour dependence is due only to the isospin operators, provided that the interest is limited to the non strange baryons. In order to describe the spectrum of strange baryons as well, it is necessary to introduce a flavour dependence which involves both isospin and strangeness. This can be achieved in the hCQM in a similar manner to the algebraic model [@bil] quoted in Sec. 2.4, that is describing the $SU(6)$ violation by means of a Gürsey-Radicati (GR) mass formula [@gr].
The original GR mass formula [@rad] can be rewritten in terms of Casimir operators [@gr] $$\label{grorigcasimir}
M=M_0+C~C_2[SU_S(2)]+D~C_1[U_Y(1)]+E~\left [C_2[SU_I(2)]-
\frac{1}{4}(C_1[U_Y(1)])^2 \right],$$ where $C_2[SU_S(2)]$ and $C_2[SU_I(2)]$ are the $SU(2)$ (quadratic) Casimir operators for spin and isospin, respectively, $C_1[U_Y(1)]$ is the Casimir for the $U(1)$ subgroup generated by the hypercharge $Y$.
However, in the framework of the CQM, the underlying symmetry is provided by $SU(6)$ and Eq. (\[grorigcasimir\]) is not the most general formula that can be written on the basis of a broken $SU(6)$ symmetry. It can then be generalized as follows [@gr] $$\begin{aligned}
\label{grfull}
M=M_0&+A~C_2[SU_{SF}(6)]+B~C_2[SU_F(3)]+C~C_2[SU_S(2)]+
\nonumber \\
&+D~C_1[U_Y(1)]+E~\left(C_2[SU_I(2)]-\frac{1}{4}(C_1[U_Y(1)])^2\right),\end{aligned}$$ where $M_0$ is the $SU_{SF}(6)$ invariant mass.
The idea is then to consider the energy levels provided by the hypercentral potential of Eq. (\[h\_pot\]) as the values of the central masses of the $SU_{SF}(6)$ multiplets and to use the generalized mass formula in order to describe the spin-flavour splittings within the multiplets [@gr]. The hamiltonian is assumed to be $$H = H_0 + H_{GR},
\label{HGR}$$ where $H_0$ is the hCQM hamiltonian without the hyperfine interaction $$H_0 = 3m + \frac{\vec{p}_\rho^{~2}}{2m} + \frac{\vec{p}_\lambda^{~2}}{2m} -\frac{\tau}{x} + \alpha x
\label{h0}$$ and $H_{GR}$ is given by the spin-flavour dependent part of Eq. (\[grfull\]) $$\begin{aligned}
\label{H_GR}
H_{GR} = & A~C_2[SU_{SF}(6)]+B~C_2[SU_F(3)]+C~C_2[SU_S(2)]+
\nonumber \\
&+D~C_1[U_Y(1)]+E~\left(C_2[SU_I(2)]-\frac{1}{4}(C_1[U_Y(1)])^2\right).\end{aligned}$$
$SU_{SF}(6)$ $C_2$ $SU_F(3)$ $C_2$
-------------- ---------------- ----------- -------
$[56]$ $\frac{45}{4}$ $[8]$ 3
$[70]$ $\frac{33}{4}$ $[10]$ 6
$[20]$ $\frac{21}{4}$ $[1]$ 0
: The eigenvalues of the quadratic Casimir operators for the groups $SU_{SF}(6)$ (left) and $SU_F(3)$ (right).
\[casim\]
In order to apply the generalized GR mass formula to the baryon spectrum it is necessary to assume that the coefficients A,B, …in Eq. (\[H\_GR\]) be the same in the various $SU_{SF}(6)$ multiplets. This actually seems to be the case, as shown by the algebraic approach to the baryon spectrum [@bil], where a formula similar to Eq. (\[H\_GR\]) has been applied.
The matrix elements of $H_{GR}$ are completely determined by the values of the various Casimir operators [@gr]: for the $SU_{SF}(6)$ and $SU_F(3)$ groups the values of the Casimir operator $C_2$ are reported in Table \[casim\], while for the $SU(2)$ and $U_Y(1)$ groups one has $$\langle C_2[SU_I(2)] \rangle = I(I+1), ~~~\langle C_1[U_Y(1)] \rangle = Y, ~~~\langle C_2[SU_S(2)] \rangle = S(S+1).$$
The mass of each baryon state is then $$\label{masses}
\langle B \vert H \vert B \rangle =E_{\gamma \nu}+ \langle B \vert H_{GR}
\vert B \rangle ,$$ where $E_{\gamma \nu}$ are the eigenvalues of the hypercentral potential of Eq. (\[h0\]).
The parameters $\alpha$ and $\tau$ of the hypercentral potential have been fitted in Sec. 3.3 in presence of the hyperfine interaction. Here the $SU(6)$ violation is provided by a different mechanism and then these parameters must fitted to the spectrum together with those introduced in Eq. (\[H\_GR\]).
Such fit can be performed in two ways. The first one is an analytical procedure which consists in choosing a limited number of well known resonances and expressing their mass differences in terms of the Casimir operator values. In this way a part of the unknown coefficients is evaluated directly, while the remaining ones is fitted to the experimental spectrum. A possible choice of resonance pairs is given by the following ones $$\begin{aligned}
\nonumber
\langle N(1650) S11 - N(1535) S11 \rangle & = & 3C, \\
\nonumber
\langle \Delta(1232) P11 - N(938) P11 \rangle & = & 9B + 3C + 3E,\\
\nonumber
\langle N(1535) S11 - N(1440) P11 \rangle & = & E_{10} - E_{01} +12 A, \\
\nonumber
\langle \Sigma(1193) P11 - N(938) P11 \rangle & = & 3/2 E - D, \\
\langle \Lambda(1116) P01 - N(938) P11 \rangle & = & - D - 1/2 E .
\label{analyt}\end{aligned}$$
.
Parameter (I) (II)
----------- --------------- ---------------
$\alpha$ $1.4 fm^{-2}$ $2.1 fm^{-2}$
$\tau$ 4.8 3.9
$A$ -13.8 MeV -11.9 MeV
$B$ 7.1 MeV MeV 11.7 MeV
$C$ 38.3 MeV 30.8 MeV
$D$ -197.3 MeV -197.3 MeV
$E$ 38.5 MeV 38.5 MeV
: The fitted values of the parameters of the Hamiltonian (\[HGR\]) [@gr]. Column (I) reports the values given by the analytical procedure of Eq. (\[analyt\]), while Column (II) contains the results of the direct fit to the experimental spectrum
\[param\]
![ (Color online) The spectrum obtained with the hCQM and the generalized Gürsey-Radicati formula of Eq. (\[grfull\]) in comparison with all the 4\* and 3\* N, $\Delta$ and $\Lambda$ resonances reported in the 2000 edition of the PDG [@pdg00]. The parameters given by the direct fit are used (see Table \[param\], column (II)).[]{data-label="hCQM_GR"}](hCQM_GR_PDG00){width="4.5in"}
![ (Color online) The same as in Fig. \[hCQM\_GR\] for the $\Sigma$, $\Xi$ and $\Omega$ resonances.[]{data-label="hCQM_GR2"}](hCQM_GR_2_PDG00){width="3.5in"}
In the second procedure all the parameters are fitted in order to reproduce the baryon spectrum. The resulting values of the parameters are reported in Table \[param\] and the corresponding spectrum is shown in Fig. \[hCQM\_GR\]. The overall description of the baryon spectrum of the 4\* and 3\* resonances is quite good, specially considering the simplicity of the model.
Using the analytical procedure the overall agreement with the spectrum is slightly worsened, but the strange sector is better described.
In both procedure, there is the need of a non zero value of the parameter A in order to reproduce the spectrum. An attempt to fit the data fixing A=0 has been tried, however the resulting parameters $\alpha$ and $\tau$ are quite different form those reported in Table \[param\]. Furthermore, the correct ordering of the Roper resonance and the negative parity resonances is lost. The presence of the Casimir $C_2[SU_{SF}(6)]$ is essential in order to shift down the energy of the first excited $0^+$ state with respect to the $1^-$, an effect which is similar to the one produced by the U-potential in the Isgur-Karl model and by the presence of a flavour dependent interaction in the previously quoted CQMs.
The electromagnetic excitation of baryon resonances
===================================================
The transition amplitudes
-------------------------
The electromagnetic excitation of the baryon resonances is an important source of information concerning the nucleon structure. The absorption of real photons is a direct measure of the excitation strength while the inelastic electron scattering is a probe of the excited nucleon structure at short distances. There are presently many experimental data taken at various laboratories (Jlab, Mainz, Bonn, …) but a systematic study of the electromagnetic excitation of the nucleon at high $Q^2$ is expected to be performed by the upgraded 12 GEV beam at Jlab [@wp; @wp2].
There is an intense theoretical and phenomenological activity which aims at extracting the transition amplitudes from the experimental data on photo- and electro-production of mesons off nucleons using mainly the Partial Wave Analysis [@brag07]. Various groups have devoted much effort in this sense, using different techniques. Among them we quote the analyses made by the following groups: George Washington University (SAID) [@gwu], Mainz University (MAID) [@uim; @dmt; @maid07], Dubna-Mainz-Taipei (DMT) [@dmt2; @dmt] Bonn-Gatchina (BnGa) [@bnga], EBAC at Jefferson Lab [@ebac], Jülich [@jul], Giessen [@gies], Zagreb-Tuzla [@zt].
From the theoretical point of view, the photo- and electro-excitations of the nucleon to the various baryon resonances are described by the helicity amplitudes, defined as the matrix elements of the electromagnetic interaction, $A_{\mu} J^{\mu}$, between the nucleon, $N$, and the resonance, $B$, states: $$\label{hel}
\begin{array}{rcl}
\mathcal{A}_{1/2}&=& \sqrt{\frac{2 \pi \alpha}{k_0}} \langle B, J', J'_{z}=\frac{1}{2}\ | J_{+}| N, J~=~
\frac{1}{2}, J_{z}= -\frac{1}{2}\
\rangle,\\
& & \\
\mathcal{A}_{3/2}&=& \sqrt{\frac{2 \pi \alpha}{k_0}} \langle B, J', J'_{z}=\frac{3}{2}\ |J_{+} | N, J~=~
\frac{1}{2}, J_{z}= \frac{1}{2}\
\rangle,\\
& & \\
\mathcal{S}_{1/2}&=& \sqrt{\frac{2 \pi \alpha}{k_0}} \langle B, J', J'_{z}=\frac{1}{2}\ | J_{0}| N, J~=~
\frac{1}{2}, J_{z}= \frac{1}{2}\
\rangle,\\
\end{array}$$ where $k_0$ is the photon energy and, for the transverse excitation, the photon has been assumed, without loss of generality, as left-handed.
The hCQM has been completely specified in Sec. 3.3. We recall that the Hamiltonian is given by Eq. (\[H\_hCQM\]) $$H_{hCQM}~=~3m + \frac{\vec{p}_\rho^{~2}}{2m} + \frac{\vec{p}_\lambda^{~2}}{2m}-\frac{\tau}{x} + \alpha x + H_{hyp},$$ where $\alpha = 1.16$ fm$^{-2}$, $\tau=4.59$ and the strength of the hyperfine interaction is fixed by the $N-\Delta$ mass difference.
The states of the various resonances have been explicitly built up and therefore they can be used for the calculation of any quantity of physical interest. In order to proceed to the calculation of the helicity amplitudes, one has to specify the current in the electromagnetic interaction. In the framework of the hCQM, the current $J^{\mu}$ is simply given by the sum of the quark currents $ j_\mu(i)$ $$J^{\mu} = \sum_{i=1}^3 j_\mu(i)
\label{qcurr}$$ and will be used in its nonrelativistic form [@cko; @ki] $$\rho(\vec{k})= \sum_{i=1}^3 e_i e^{\vec{k} \cdot \vec{r}_i}, ~~~~\vec{j}(\vec{k}) = \frac{1}{2m} \sum_{i=1}^3 (\vec{p}_i' + \vec{p}_i + i \vec{\sigma}^q(i) \times \vec{k}) e^{\vec{k} \cdot \vec{r}_i},
\label{q_curr}$$ where $e_i$ is the charge of the i-th quark $$e_i=\frac{1}{2} [\frac{1}{3} + \tau^q_3 (i) ],$$ $\vec{r}_i$ is the quark coordinate and $\vec{\sigma}^q(i)$, $\vec{\tau}^q(i)$ are, respectively, the quark spin and isospin operators.
In order to compare the theoretical results with the experimental data, the calculation should be performed in the rest frame of the resonance (see e.g. [@azn-bur-12]). The nucleon and resonance wave functions are actually calculated in their respective rest frames and, before evaluating the matrix elements given in Eqs. (\[hel\]), one should boost the nucleon to the resonance c.m.s.. However, in order to minimize the discrepancy between the nonrelativistic and the relativistic boosts in comparing with the experimental data, we can consider the Breit frame, as in refs. ([@aie2; @bil; @sg]). In this frame $\vec{p}_N = -\vec{p}_R=-\vec{k}/2$, where $\vec{p}_N$, $\vec{p}_R$ and $\vec{k}$ are, respectively, the nucleon, resonance and photon trimomenta. The relation of the latter with the momentum transfer squared $Q^2$ is given by: $${\vec{k}}^2 = Q^2 + \frac{(W^2 - M^2)^2}{2(M^2 + W^2) + Q^2},
\label{breit}$$ where $M$ is the nucleon mass, $W$ is the mass of the resonance and $Q^2~=
~{\vec{k}}^2- k_0^2$, $k_0$ being the photon energy.
![ (Color online) The electromagnetic excitation of nucleon resonances. The decay can occur in more than one pion. []{data-label="helamp"}](helamp){width="3.5in"}
Furthermore, one has to consider that the helicity amplitudes extracted from the photoproduction contain also the sign of the $\pi N N^*$ vertex (see Fig. \[helamp\]). The theoretical helicity amplitudes are then defined up to a common phase factor $\zeta$ $$A_{1/2,3/2}~=~\zeta~ \mathcal{A}_{1/2,3/2} ~~~~~~~~~~~S_{1/2}~=~\zeta~ \mathcal{S}_{1/2} .$$ The factor $\zeta$ can be taken [@sg] in agreement with the choice of ref. [@ki], with the exception of the Roper resonance, in which case the sign is in agreement with the analysis performed in [@azn07].
In the following Subsections the results for the photo- and electro-excitation of the baryon resonances calculated with the hCQM will be presented. For consistency reasons, in the calculations the values of $W$ given by the model are used instead of the phenomenological ones.
The calculations regarding the photocouplings have already been published in [@aie], those for the transverse excitation of the negative parity resonances in [@aie2], while a systematic study of all the electromagnetic excitations has been reported in [@sg].
It should be stressed that the results are obtained with a parameter free calculation, that is they are [**predictions of the model**]{}.
\[photo\]
------------- -------------- --------------- ------------- ---------------
$A_{1/2}^p $ $A_{1/2}^p $ $A_{3/2}^p$ $A_{3/2}^p$
$Resonance$ hCQM PDG hCQM PDG
P11(1440) $88$ $-65 \pm 4$
D13(1520) $-66$ $-24 \pm 9$ $67$ $ 166 \pm 5$
S11(1535) $109$ $90 \pm 30$
S11(1650) $69$ $53 \pm 16$
D15(1675) $1$ $19 \pm 8$ $2$ $15 \pm 9$
F15(1680) $-35$ $ -15 \pm 6$ $24$ $ 133 \pm 12$
D13(1700) $8$ $ -18 \pm 13$ $-11$ $ -2 \pm 24$
P11(1710) $43$ $ 9 \pm 22$
P13(1720) $94$ $ 18 \pm 30 $ $-17$ $ -19 \pm 20$
------------- -------------- --------------- ------------- ---------------
: Photocouplings (in units $10^{-3}$ GeV$^{-1/2}$) predicted by the hCMQ [@sg] in comparison with PDG data [@pdg12] for proton excitation to N\*-like resonances. The proton transitions to the S11(1650), D15(1675) and D13(1700) resonances vanish in the $SU(6)$ limit.
\[delta\]
------------- -------------- -------------- ------------- --------------
$A_{1/2}^p $ $A_{1/2}^p $ $A_{3/2}^p$ $A_{3/2}^p$
$Resonance$ hCQM PDG hCQM PDG
P33(1232) $-97$ $-135 \pm 6$ $-169$ $-250 \pm 8$
S31(1620) $30$ $27 \pm 11$
D33(1700) $81$ $104 \pm 5$ $70$ $85 \pm 2$
F35(1905) $-17$ $26 \pm 11$ $-51$ $-45 \pm 20$
F37(1950) $-28$ $-76 \pm 12$ $-35$ $-97 \pm 10$
------------- -------------- -------------- ------------- --------------
: The same as in Table \[photo\] but for the $N-\Delta$ excitation.
The photocouplings in the hCQM
------------------------------
The resonances which have been considered are those which, according to the PDG classification [@pdg12], have an electromagnetic decay with a three- or four- star status. This happens for twelve resonances, namely the positive parity $I=\frac{1}{2}$ states $$P11(1440), F15(1680), P11(1710),$$ the negative parity $I=\frac{1}{2}$ states $$D13(1520), S11(1535), S11(1650), D15(1675)$$ and the $I=\frac{3}{2}$ ones $$P33(1232), S31(1620), D33(1700), F35(1905), F37(1950).$$ Besides these states, we have considered also the two resonances D13(1700) and P13(1720), which are excited in an energy range particularly interesting for the phenomenological analysis.
\[photo\_n\]
------------- -------------- -------------- --------------- ------------- ---------------- ---------------
$A_{1/2}^n $ $A_{1/2}^n $ $A_{1/2}^n $ $A_{3/2}^n$ $A_{3/2}^n$ $A_{3/2}^n$
$Resonance$ hCQM PDG BnGa hCQM PDG BnGa
P11(1440) $58$ $40 \pm 10$ $43 \pm 12 $
D13(1520) $-1$ $-59 \pm 9$ $-49 \pm 8$ $-61$ $ -139 \pm 11$ $-113 \pm 12$
S11(1535) $-82$ $-46 \pm 27$ $-93 \pm 11$
S11(1650) $-21$ $-15 \pm 21$ $-25 \pm 20$
D15(1675) $-37$ $-43 \pm 12$ $-60 \pm 7$ $-51$ $-58 \pm 13$ $ -88 \pm 10$
F15(1680) $38$ $ 29 \pm 10$ $34 \pm 6$ $15$ $ -33 \pm 9$ $-44 \pm 9$
D13(1700) $12$ $ 0 \pm 50$ $70$ $ -3 \pm 44$
P11(1710) $-22$ $ -2 \pm 14$ $-40 \pm 20 $
P13(1720) $-48$ $ 1 \pm 15 $ $-80 \pm 50 $ $4$ $ -29 \pm 61$ $-44 \pm 9$
------------- -------------- -------------- --------------- ------------- ---------------- ---------------
: Photocouplings (in units $10^{-3}$ GeV$^{-1/2}$) predicted by the hCMQ [@sg] in comparison with PDG data [@pdg12] and the recent Bonn-Gatchina analysis [@bn_n] for neutron excitation to N\*-like resonances.
The proton and neutron photocouplings predicted by the hCQM [@aie] are reported in Tables \[photo\], \[delta\] and \[photo\_n\] in comparison with the PDG data [@pdg12] and the analysis by the Bonn-Gatchina group [@bn_n]. The overall behaviour is fairly well reproduced, but in general there is a lack of strength. The proton transitions to the S11(1650), D15(1675) and D13(1700) resonances vanish exactly in absence of hyperfine mixing and are therefore entirely due to the $SU(6)$ violation.
As already noted in Sec. 2.2, the hyperfine interaction is responsible for a deformation of the $\Delta$ resonance and therefore the ratio of Eq. (\[E2\]) is different from zero. This ratio can also be expressed in terms of the helicity amplitudes $$\label{REM}
R_{EM}~=~- ~\frac{G_{E}}{G_{M}}~=~ ~\frac{\sqrt{3} ~ A_{1/2}~-~A_{3/2}}{\sqrt{3}~A_{1/2}~+~3 ~A_{3/2}};$$ with the theoretical values reported in Table \[photo\], $R_{EM}$ turns out to be smaller than the experimental one. The point is that the E2 transition strength predicted by the hCQM is too low and a possible explanation of this result will be discussed later.
The results obtained with other calculations are qualitatively not much different [@aie; @cr2] and this is because the various CQM models have the same $SU(6)$ structure in common.
It should be reminded that in previous nonrelativistic calculations with h.o. wave functions [@cko], it was necessary to assume a proton radius of the order of 0.5 fm in order to ensure a vanishing $A_{1/2}^p$ for the resonances D13(1520) and F15(1680), whose peaks are absent in the forward photoproduction [@cko]. The proton radius calculated with the hCQM is actually 0.48 fm and this explains why the predictions of the hCQM do not differ too much from the other calculations. A too low proton radius is of course a problem if one wants to calculate the elastic form factors of the nucleon, but for the description of the helicity amplitudes it is beneficial and, as we shall see later, it plays an important role in the discussion concerning the mechanisms which are missing in any CQM.
There are now many new analyses concerning the neutron helicity amplitudes ([@bn_n] and references quoted therein). In Table \[photo\_n\] we report also the results of the Bn-Ga analysis [@bn_n]. The hCQM predictions are in fair agreement with these data, perhaps better than with the PDG ones.
![(Color on line) The P33(1232) helicity amplitudes predicted by the hCQM (full curves) $A_{3/2}$ (a), $A_{1/2}$ (b) and $S_{1/2}$ (c), in comparison with the data of ref. [@azn09] and with the Maid2007 analysis [@maid07] of the data by refs. [@joo02] and [@lav04]. The PDG points [@pdg12] are also shown. The figure is taken from ref. [@sg] (Copyright (2012) by the American Physical Society)[]{data-label="p33"}](ap32_t_p33 "fig:"){width="2.3in"} ![(Color on line) The P33(1232) helicity amplitudes predicted by the hCQM (full curves) $A_{3/2}$ (a), $A_{1/2}$ (b) and $S_{1/2}$ (c), in comparison with the data of ref. [@azn09] and with the Maid2007 analysis [@maid07] of the data by refs. [@joo02] and [@lav04]. The PDG points [@pdg12] are also shown. The figure is taken from ref. [@sg] (Copyright (2012) by the American Physical Society)[]{data-label="p33"}](ap12_t_p33 "fig:"){width="2.3in"} ![(Color on line) The P33(1232) helicity amplitudes predicted by the hCQM (full curves) $A_{3/2}$ (a), $A_{1/2}$ (b) and $S_{1/2}$ (c), in comparison with the data of ref. [@azn09] and with the Maid2007 analysis [@maid07] of the data by refs. [@joo02] and [@lav04]. The PDG points [@pdg12] are also shown. The figure is taken from ref. [@sg] (Copyright (2012) by the American Physical Society)[]{data-label="p33"}](ap12_s_p33 "fig:"){width="2.3in"}
The helicity amplitudes in the hCQM
-----------------------------------
As already mentioned, a systematic review of the hCQM predictions for the transverse and longitudinal helicity amplitudes and their comparison with the experimental data is reported in ref. [@sg]. Here we limit ourselves to some of the most important excitations.
![(Color on line) The P11(1440) proton transverse (a) and longitudinal (b) helicity amplitudes predicted by the hCQM (full curves), in comparison with the data of refs. [@vm09], [@azn09] and the Maid2007 analysis [@maid07] of the data by refs. [@fro99],[@joo02], [@lav04] and [@ung06]. The PDG point [@pdg12] is also shown. The figure is taken from ref. [@sg] (Copyright (2012) by the American Physical Society).[]{data-label="p11"}](ap12_t_p11 "fig:"){width="2.3in"} ![(Color on line) The P11(1440) proton transverse (a) and longitudinal (b) helicity amplitudes predicted by the hCQM (full curves), in comparison with the data of refs. [@vm09], [@azn09] and the Maid2007 analysis [@maid07] of the data by refs. [@fro99],[@joo02], [@lav04] and [@ung06]. The PDG point [@pdg12] is also shown. The figure is taken from ref. [@sg] (Copyright (2012) by the American Physical Society).[]{data-label="p11"}](ap12_s_p11 "fig:"){width="2.3in"}
The transition amplitudes for the excitation of the P33(1232) resonance are given in Fig. \[p33\].
The transverse excitation to the $\Delta$ resonance has a lack of strength at low $Q^2$, a feature in common with all CQM calculations. The medium-high $Q^2$ behavior is decreasing too slowly with respect to data, similarly to what happens for the nucleon elastic form factors [@mds; @ff_07]. As we shall see later, the nonrelativistic calculations are improved by taking into account relativistic effects. Since the $\Delta$ resonance and the nucleon are in the ground state $SU(6)$-configuration, we expect that their internal structures have strong similarities and that a good description of the $N-\Delta$ transition from factors is possible only with a relativistic approach. Such feature is further supported by the fact that the transitions to the higher resonances are only slightly affected by relativistic effects [@mds].
The Roper excitation is reported in Fig. \[p11\]. Because of the $\frac{1}{x}$ term in the hypercentral potential of Eq. (\[H\_hCQM\]), the Roper resonance can be included in the first resonance region, at variance with h.o. models, which predict it to be a 2 $\hbar \omega$ state. There are problems in the low $Q^2$ region, but for the rest the agreement is interesting, specially if one remembers that the curves are predictions and the Roper has been often been considered a crucial state, non easily included into a constituent quark model description. In particular, the longitudinal excitation is quite different from zero [@ab_rop], in agreement with the hCQM and at variance with the hybrid qqq-gluon model [@lb]. In the present model, the Roper is a hyperradial excitation of the nucleon.
![(Color on line) The D13(1520) proton helicity amplitudes predicted by the hCQM (full curves) $A_{3/2}$ (a), $A_{1/2}$ (b) and $S_{1/2}$ (c), in comparison with the data of refs. [@vm09], [@azn09], with the compilation reported in refs. [@fh; @ger] and the Maid2007 analysis [@maid07] of the data by refs. [@joo02] and [@lav04]. The PDG points [@pdg12]are also shown. The figure is taken from ref. [@sg] (Copyright (2012) by the American Physical Society)[]{data-label="d13"}](ap32_t_d13 "fig:"){width="2.3in"} ![(Color on line) The D13(1520) proton helicity amplitudes predicted by the hCQM (full curves) $A_{3/2}$ (a), $A_{1/2}$ (b) and $S_{1/2}$ (c), in comparison with the data of refs. [@vm09], [@azn09], with the compilation reported in refs. [@fh; @ger] and the Maid2007 analysis [@maid07] of the data by refs. [@joo02] and [@lav04]. The PDG points [@pdg12]are also shown. The figure is taken from ref. [@sg] (Copyright (2012) by the American Physical Society)[]{data-label="d13"}](ap12_t_d13 "fig:"){width="2.3in"} ![(Color on line) The D13(1520) proton helicity amplitudes predicted by the hCQM (full curves) $A_{3/2}$ (a), $A_{1/2}$ (b) and $S_{1/2}$ (c), in comparison with the data of refs. [@vm09], [@azn09], with the compilation reported in refs. [@fh; @ger] and the Maid2007 analysis [@maid07] of the data by refs. [@joo02] and [@lav04]. The PDG points [@pdg12]are also shown. The figure is taken from ref. [@sg] (Copyright (2012) by the American Physical Society)[]{data-label="d13"}](ap12_s_d13 "fig:"){width="2.3in"}
![(Color on line) The S11(1525) proton transverse (a) and longitudinal (b) helicity amplitudes predicted by the hCQM (full curve), in comparison with the data of refs. [@azn05_1] (open diamonds), [@azn09] (full diamonds), [@arm99] (crosses), [@den07] (open squares), [@thom01] (full squares), the Maid2007 analysis [@maid07] (full triangles) of the data by refs. [@joo02] and the compilation of the Bonn-Mainz-DESY data of refs. [@kru; @bra; @beck; @breu] (stars), presented in [@thom01]. The PDG point [@pdg12] (pentagon) is also shown. The figure is taken from ref. [@sg] (Copyright (2012) by the American Physical Society)[]{data-label="s11"}](ap12_t_s11 "fig:"){width="2.3in"} ![(Color on line) The S11(1525) proton transverse (a) and longitudinal (b) helicity amplitudes predicted by the hCQM (full curve), in comparison with the data of refs. [@azn05_1] (open diamonds), [@azn09] (full diamonds), [@arm99] (crosses), [@den07] (open squares), [@thom01] (full squares), the Maid2007 analysis [@maid07] (full triangles) of the data by refs. [@joo02] and the compilation of the Bonn-Mainz-DESY data of refs. [@kru; @bra; @beck; @breu] (stars), presented in [@thom01]. The PDG point [@pdg12] (pentagon) is also shown. The figure is taken from ref. [@sg] (Copyright (2012) by the American Physical Society)[]{data-label="s11"}](ap12_s_s11 "fig:"){width="2.3in"}
We consider now the excitations to some negative resonances [@aie2; @sg], namely the D13(1520) and the S11(1525) ones, reported in Figs. \[d13\] and \[s11\], respectively.
![(Color on line). The hCQM predictions for the transverse D13(1520) helicity amplitudes (full curve) in comparison with the h.o. results corresponding to two values of the proton radius: 0.5 fm (ho1) and 0.86 (ho2). The data are the same as in Fig. \[d13\].[]{data-label="ho_hel"}](ap32_d13_ho "fig:"){width="2.3in"} ![(Color on line). The hCQM predictions for the transverse D13(1520) helicity amplitudes (full curve) in comparison with the h.o. results corresponding to two values of the proton radius: 0.5 fm (ho1) and 0.86 (ho2). The data are the same as in Fig. \[d13\].[]{data-label="ho_hel"}](ap12_d13_ho "fig:"){width="2.3in"}
The agreement in the case of the S11 is remarkable, specially if one considers that the hCQM curve for the transverse transition has been published three years in advance [@aie2] with respect to the recent TJNAF data [@azn09], [@azn05_1], [@den07], [@thom01].
It is interesting to discuss the influence of the hyperfine mixing on the excitation of the resonances. Usually there is only a small difference between the values calculated with or without hyperfine interaction. In some cases, however the excitation strength vanishes in the $SU(6)$ limit, as already mentioned in Table \[photo\], the non vanishing result is then entirely due to the hyperfine mixing of states. In the case of the S11(1650) resonance, the resulting transverse and longitudinal excitations have a relevant strength.
The three helicity amplitudes of the D13(1700) resonance are again non zero because of the hyperfine mixing but the excitation strength is very low. Also in the case of the transverse excitation of the D15(1675), the strength is given by the hyperfine mixing, while the longitudinal amplitude $S_{1/2}$ vanishes also in presence of a $SU(6)$ violation.
The results of the hCQM, as reported in Figs. \[p33\], \[p11\], \[d13\], \[s11\] and in ref. [@sg] show some general features. First of all, there is a nice agreement with data at medium-high values of $Q^2$. This is mainly due to the presence of the $1/x$ term in the hCQM hamiltonian. In fact, the helicity amplitudes calculated with the analytical model of refs. [@sig; @sig2] have a $Q^2$ dependence very similar to the complete hCQM and numerical values only slightly different. The wave functions of the analytical model are exactly those determined by the $1/x$ potential, which gives then the main contribution to the transition strength.
Another important fact which helps in obtaining a good behaviour of the hCQM helicity amplitudes is the smallness of the resulting proton radius. As already mentioned, the r.m.s. radius of the proton calculated with the hCQM wave functions corresponding to the parameters of Eq. (\[par\]) is $0.48~$fm, which is very near to the value necessary in order to fit the $D13$ photocoupling [@cko].
Both these features, the presence of the hyperCoulomb term in the quark potential and the smallness of the proton radius, concur in order to obtain the results shown in the figures. This can be seen also looking at Fig. \[ho\_hel\], where the results of Fig. (\[d13\]) for the D13(1520) excitation are given in comparison with the h.o. curves corresponding to two values of the proton radius, namely the experimental one (about 0.86 fm) and the one fitted to the $A^p_{3/2}$ amplitude (about 0.5 fm). The curves with the correct proton radius are completely out of the experimental data because of the gaussian factor $e^{-\frac{Q^2}{6 \alpha^2}}$, typical of the h.o.; $\alpha$ is the h.o. constant (see Sec. 2.2) and its value corresponding to the experimental proton radius is 0.229 GeV. In the case of the smaller radius 0.5 fm, with $\alpha$ = 0.41 GeV, the $A^p_{3/2}$ amplitude is of course well reproduced but the h.o. curve for the $A^p_{1/2}$ amplitude is again far from the data. For this reason, we expect that the hCQM should be a good starting point for the description of the non perturbative components of the parton distribution.
There are in general discrepancies at low $Q^2$, displaying a lack of strength which is typical of all CQMs. Nevertheless, in many cases the $A^p_{1/2}$ amplitudes are better reproduced than the $A^p_{3/2}$ ones.
These shortcomings of the hCQM results could be ascribed to the non-relativistic character of the model. In fact, the electromagnetic excitation leads to a recoil of both the nucleon and the resonance and the effect is expected to increase with the momentum transfer $Q^{2}$, while the wave functions are calculated in the rest frame of each three-quark system. As it will be discussed in Sec. 6.2, it is possible to apply Lorentz boosts in order to bring the nucleon and the resonance to a common Breit frame, but this relativistic corrections produce only a slight modification of the hCQM results [@mds2].
There is a consensus on the fact that the missing strength at low $Q^2$ is due to the lack of quark-antiquark effects [@aie2], probably important in the outer region of the nucleon. This statement receives a strong support by the explicit calculations of the meson cloud contributions to the helicity amplitudes performed in the framework of dynamical models (see ref. [@dmt; @sato] and references therein).
In particular the Dubna-Taipei-Mainz (DMT) model introduces the pion cloud contribution to the electromagnetic excitation according to the mechanism shown in the upper left part of Fig. (\[dmt\]). In the same figure, the longitudinal $S_{1/2}$ and transverse $A_{3/2} $ and $A_{1/2} $ helicity amplitudes for the $N-\Delta$ are reported [@ts03]. The theoretical predictions of the hCQM (full curves) are compared with the results of the MAID fit [@maid99] and with the pion cloud contribution calculated by means of the DMT model (dashed curve) [@dmt]. The hCQM results are much lower than the experimental data, but the pion cloud contribution gives relevant contributions just where the hCQM is lacking. This is particularly evident in the case of the longitudinal $S_{1/2}$ amplitude: the hCQM predicts an almost vanishing value while the pion alone seems to be able to account for the data. Of course one cannot simply add the hCQM and pion contributions, since they are calculated in two different and inconsistent frameworks, but it is nevertheless interesting that the pion cloud seems to contribute systematically where the hCQM is lacking, as it can be seen also for the helicity amplitudes of many other resonances [@ts03].
![(Color on line). The pion cloud mechanism considered in the DMT model [@dmt] (upper left). The remaining figures report the hCQM predictions for the $N-\Delta$ excitation (full curves) in comparison with the meson cloud contribution (dashed curves) and the results of the MAID fit [@maid99] (full points and the fitted curves passing through them) [@ts03]. []{data-label="dmt"}](dmt){width="5in"}
In this way the emerging picture in connection with the electromagnetic excitation of the nucleon resonances is that of a small confinement zone of about $0.5$ fm surrounded by a quark-antiquark (or meson) cloud. The calculations of the meson cloud performed with the DMT model shows that this picture seems to be reasonable, but the problem is how to include in a consistent way the quark-antiquark pair creation mechanisms in the framework of the CQM. This goal can be achieved by unquenching the quark model and, as already mentioned earlier, an important improvement has been achieved by a recent work [@sb1; @bs; @sb2]. We shall come back on this point in the discussion.
The elastic form factors of the nucleon
=======================================
Introductory remarks
--------------------
An important aspect of the quark model predictions concerning the elastic form factors is their $Q^2$ dependence, which is strictly related to the form of the quark wave functions and then of the quark potential.
We can start studying the nucleon charge form factor in absence of the hyperfine mixing. The nucleon state (see App. B) can be written as $$|N \rangle = \psi_{00}(x) \Omega_{[0]00}\frac{1}{\sqrt{2}} (\chi_{MS} \phi_{MS} + \chi_{MA} \psi_{MA}),$$ where $\Omega_{[0]00} = 1/(4 \pi)^2 4/\sqrt{\pi}$.
The nucleon charge form factor is given by the matrix element of the charge density operator of Eq. (\[q\_curr\]) $$G^N_E(Q^2)~ = ~\langle N| 3 \frac{1+3\tau_0 (3)}{6}e^{-i a k\lambda_z} |N \rangle ,$$ where $\tau_0(3)$ is the third component of the isospin operator of the third quark, $a=\sqrt{\frac{2}{3}}$ and $Q^2=k^2$ in the Breit system, according to Eq. (\[breit\]). Introducing the hyperspherical coordinates and performing the integrals over the angle and hyperangle variables, one gets $$G^N_E(Q^2)~ = ~\frac{1+\tau_0}{2} F(k),$$ where $\tau_0$ is the third component of the nucleon isospin and $$F(k)=\frac{8}{a^2 k^2} \int_0^\infty dx~ x^3 ~\psi_{00}(x)^2 J_2(a k x),
\label{hff}$$ $J_2(z)$ being a Bessel function of integer order 2. Eq. (\[hff\]) can be inverted, obtaining an expression of the wave function in terms of the form factor $$\psi_{00}(x)^2=\frac{a^4}{8 x^2} \int_0^\infty dk~ k^3 ~F(k)~ J_2(a k x).
\label{hwf}$$ In this way, starting from any given form factor it is possible to obtain the appropriate wave function $ \psi_{00}(x)^2$ and then also the potential for which $ \psi_{00}(x)$ is the ground state. Assuming the dipole form factor $1/(1+b^2 k^2)$, the resulting potential is given by $$V_{dip}(x)= \frac{a^2}{2 b^2}[1- \frac{1}{2}(\frac{K_0(y)}{K_1(y)})^2- 4 \frac{K_0(y)}{y K_1(y)}-\frac{6}{y^2}],
\label{vdip}$$ where $y=ax/b$ and $K_0(y),K_1(y)$ are the modified Bessel function of the second kind. It is interesting to note that for large values of y, the potential assumes the form $$V_{dip}(x) \rightarrow \frac{a^2}{4 b^2}[1- \frac{7}{y}- \frac{9}{y^2}],
\label{vdip_2}$$ that is there is no confinement.
The nucleon charge form factor turns out to be proportional to the charge and therefore it is zero for the neutron. This happens as long as the space part of the state is completely symmetric. Because of the hyperfine interaction, also the state state $|N ^2S_{1/2} \rangle _M$, having mixed space symmetry, contributes to the nucleon (see Eq. (\[nucl\])), thereby generating a non-zero neutron form factor [@iks].
In h.o. models the ground state wave function is given by a gaussian $e^{-\frac{\alpha^2 x^2}{2 }}$ which leads to the form factor $e^{-\frac{k^2}{6 \alpha^2}}$, where $\alpha$ is the h.o. constant (see Sec. 2.2). Because of the hyperfine mixing, the nucleon state is given by a superposition of $SU(6)$ configurations of Eq. (\[nucl\]), but the dominant behaviour of the form factor is in any case a gaussian, which is too strongly damped with respect to the experimental data.
In the case of the purely hypercoulomb potential, the ground state wave function is given by an exponential function $e^{-gx}$ and the corresponding form factor is $$G^p_E(Q^2)|_{hC} ~ = ~\frac{1}{(1+\frac{k^2}{6 g^2})^{7/2}}.$$
The use of the hCQM with the confinement potential modifies this power-law behaviour, however, even taking into account the hyperfine mixing, the resulting form factor has a more realistic $Q^2$ behaviour with respect to the h.o. model.
As for the other nucleon form factors, in the $SU(6)$ limit, there is perfect scaling, similarly to the dipole fit, in the sense that $$G^p_E(Q^2)|_{hCQM} ~ = ~\frac{1}{\mu_p} G^p_M(Q^2)|_{hCQM} ~ = ~\frac{1}{\mu_n} G^n_M(Q^2)|_{hCQM}
\label{scal}$$ and $$G^n_E(Q^2)|_{hCQM} ~ = ~0.$$
![(Color on line) The ratio $R=\mu_p \frac{G^p_E(Q^2)}{G^p_M(Q^2)}$ as a function of the mometum transfer $Q^2$. The data are taken from [@milb; @jones00; @gayou01; @posp; @gayou02; @pun].[]{data-label="rap07"}](R_07){width="6in"}
The hyperfine mixing modifies scarcely the above results, with the already mentioned difference of producing a non zero neutron charge form factor,
Comparing the predictions of the hCQM with the experimental data of the nucleon elastic form factor, one is faced with serious discrepancies, which may be due to two important issues, namely the non-relativistic character of the model and the smallness of the proton radius.
In the last decade, new and important data on the proton elastic form factors have been obtained by polarization experiments at Jefferson Lab. In fact, if one performs elastic scattering of longitudinally polarized electrons from unpolarized protons, the recoiling proton has both longitudinal ($P_l$) and transverse ($P_t$) polarization with respect to the momentum transfer in the scattering plane [@pol]. From the ratio $P_t/P_l$, one obtains directly the ratio of the proton electric and magnetic form factors $$R~ = ~\mu_p \frac{G^p_E(Q^2)}{G^p_M(Q^2)}~ = ~-\mu_p \frac{P_t}{P_l} \frac{E_e+E_e'}{2M_p}
tan \frac{\theta_e}{2},
\label{ratio}$$ where $\mu_p$ is the proton magnetic moment, $E_e$ is the energy of the incident electron, $E'_e$ is the energy of the scattered electron, $\theta_e$ is the electron scattering angle and $M_p$ is the proton mass.
At variance with the dipole fit, this ratio is deviating strongly from unity and it is continuously decreasing towards zero, as shown in Fig. \[rap07\]. The measurements have been quite recently extended up to $Q^2 = 8.5$ GeV$^2$ [@puck10], but they will be discussed later, in comparison with a reanalysis [@puck12] of the last three points of ref. [@gayou02].
The interesting issue is the existence of a dip in the form factor, as it is suggested by the almost linear decrease of the ratio R. Of course, this unexpected result has triggered many theoretical calculations based on quark models attempting to explain the strong deviation of R from unity and discussing the possibility of a zero in the electric form factor of the proton.
It is interesting to remind, at least for historical purposes, that the idea of a dip in the electric form factor of the proton $G_E^p(Q^2)$ has been considered already at the early stage of the quark model [@mo_er]. The problem was that of reconciling the Pauli principle with the completely symmetric character of the three-quark wave function in the $\Delta$ resonance: in absence of the colour degrees of freedom, the only way out was a completely antisymmetric space function, which can be obtained introducing a factor $(r_{12}^2-r_{13}^2)(r_{23}^2-r_{21}^2)(r_{31}^2-r_{32}^2)$ in the wave function. In general, in presence of a completely antisymmetric space wave function, it is easy to show [@mo_er] that the charge density in the origin $\rho(0)$ is zero and therefore $\int dq q^2 G_E^p(q) =0$, which means that the form factor $G_E^p$ has a dip with a position strongly dependent on the model form factor. This idea has not been further considered since a ground state wave function with nodes is barely acceptable and the introduction of colour made it possible to have completely symmetric space functions.
Actually, there is a calculation, prior to the recent R data, which has predicted a dip in $G_E^p(Q^2)$. It has been shown [@holz96] that the simple Skyrme soliton model, with vector meson corrections, leads to a form factor with a dip at $Q^2~ \sim 1$ GeV$^2$ and that, after boosting the initial and final proton state to the Breit system, the zero is shifted up to $10$ GeV$^2$, a value highly compatible with the data of Fig. \[rap07\]. In later versions of the calculation the zero was pushed to $\sim 16$ GeV$^2$ [@holz00].
Furthermore, in ref. [@fjm], a calculation of the electromagnetic proton form factors using a relativistic light-cone constituent quark model has been performed. The plot of R derived from their results exhibits a strong decrease with $Q^2$ that is due to a zero in the electric form factor at $6$ GeV$^2$ [@mill].
As a final remark, we mention the semiphenomenological fit of the nucleon form factors performed in ref. [@ijl]. The parametrized formula contains an intrinsic contribution, that now can be ascribed to the quark core, and a vector meson cloud. For the charge form factor of the proton, the data were available only up to $3$ GeV$^2$, but if one extrapolates the fitted formula to higher momentum transfer, one is faced with the remarkable fact of a dip at about $9$ GeV$^2$. However, in ref. [@ijl], the intrinsic (quark core) contribution was not considered for the isovector Pauli form factor $F^V_2$. Its inclusion in $F^V_2$ has been performed in a subsequent paper [@bi], but in this case the zero, if present, is shifted beyond $10$ GeV$^2$.
Coming back to the hCQM, the ratio R, according to Eq. (\[scal\]), is identically 1 in the $SU(6)$ limit and it remains practically 1 also in presence of the hyperfine interaction. The hCQM described up to know is nonrelativistic and it is worthwhile to analyse the effect of introducing relativity, which, as we shall see in the next subsection, is quite beneficial for the description of the decrease of the ratio R.
Introducing relativity
----------------------
In order to describe the electromagnetic form factors both in the elastic and inelastic (helicity amplitudes) cases, one has to calculate matrix elements of the type $$\langle \Psi_F | J_\mu | \Psi_I \rangle ,
\label{matr}$$ where $ \Psi_I, \Psi_F$ are the initial and final baryon states, respectively and $J_\mu$ is the quark current of Eq. (\[qcurr\]). The wave functions of the three quark systems are determined in the respective rest frames, but the calculation of the matrix elements of Eq. (\[matr\]) requires that the quark states be boosted to the Breit system. As long as the momentum transfer increases, one expects that the effects of the relativistic transformations are more important and therefore relativistic corrections must be introduced. In the case of the excitation to the baryon resonances, the state B has a mass which can be much greater than the nucleon one and in this case the recoil is expected to be less relevant. However, for the ground states, nucleon and $\Delta$ in the $SU(6)$ limit, the relativistic corrections are unavoidable.
Relativistic corrections to the matrix elements can be introduced starting from the thee-quark wave function in momentum space in the rest frame $\psi(\vec{p}_\rho,\vec{p}_\lambda)$ and applying the Lorentz boosts [@mds] $$\Psi_I~ = ~\prod_{i=1}^3 B_i~u(p_i) ~\psi_I(\vec{p}_\rho,\vec{p}_\lambda),$$ where $B_i (i=1,2,3)$ are the usual Dirac boost operators that transform the quark spinors $u(p_i)$ from the nucleon rest frame to the Breit one; the quark momenta $p_i$ are in the baryon rest frame and the variables $\vec{p}_\rho,\vec{p}_\lambda$ are conjugate to the standard coordinates $\vec{\rho},\vec{\lambda}$, always in the rest frame. The final state is written in a similar way.
In order to get the first order corrections to the nonrelativistic approach, the matrix elements are expanded keeping the first order in the quark momenta, but any order in the momentum transfer [@mds].
![The charge (a) and magnetic (b) form factors of the proton. The dashed curves are the predictions of the nonrelativistic hCQM, the full curves are obtained taking into account the corrections given in Eqs. (\[GE\_b\]) and (\[GM\_b\]). The data are from a compilation of ref. [@card_95]. The figure is taken from ref. [@rap] (Copyright (2000) by the American Physical Society).[]{data-label="ff_b"}](p_ch_b "fig:"){width="2.9in"} ![The charge (a) and magnetic (b) form factors of the proton. The dashed curves are the predictions of the nonrelativistic hCQM, the full curves are obtained taking into account the corrections given in Eqs. (\[GE\_b\]) and (\[GM\_b\]). The data are from a compilation of ref. [@card_95]. The figure is taken from ref. [@rap] (Copyright (2000) by the American Physical Society).[]{data-label="ff_b"}](p_mag_b "fig:"){width="2.9in"}
![The ratio R predicted by the hCQM with the relativistic corrections [@mds2; @rap]; the data are from ref. [@jones00].[]{data-label="rap_b"}](R_b){width="4in"}
This approach can be applied to the calculation of the elastic form factors of the nucleon, obtaining simple analytical forms [@mds] $$G_E(Q^2)~ = ~ f_E G_E^{nr}(Q^2 \frac{M^2}{E^2}),
\label{GE_b}$$ $$G_M(Q^2)~ = ~f_M G_E^{nr}(Q^2 \frac{M^2}{E^2}),
\label{GM_b}$$ where $G_E^{nr}, G_M^{nr}$ are the electric and magnetic form factors predicted by the nonrelativistic hCQM, M is the nucleon mass, E its energy in the Breit frame and $f_E, f_M$ are known kinematic factors [@mds]: $$f_E~ = ~\frac{E}{M} (t_S)^2 t_I, ~~~~f_M~ = ~\frac{E}{M} (t_S)^2 t_I \frac{g_\sigma}{2m},$$ where $$t_S = \frac{1}{Mm} (E \eta_S -\frac{M}{E} \frac{Q^2}{12}), ~~~~t_I=\frac{Mm}{E \eta_I + \frac{MQ^2}{6E}}, ~~~~ g_\sigma=\frac{2}{3} \eta_I+ \frac{4}{9}M,$$ with $$\eta_S=(m^2+ \frac{M^2 Q^2}{36E^2})^{1/2}, ~~~~\eta_I=(m^2+ \frac{M^2Q^2}{9E^2})^{1/2}.$$ One sees from Eqs. (\[GE\_b\]) and (\[GM\_b\]) that, besides the multiplicative factors, the effect of relativity is to rescale the argument of the nonrelativistic form factors. It is just thanks to this mechanisms that in ref. [@holz96], the zero of the soliton form factor has been shifted to higher values.
In Fig. \[ff\_b\] (a) and (b) the results for the electric and magnetic form factor of the proton are shown. The predictions of the hCQM are higher than the data, mainly because the proton radius is about 0.5 fm, however the relativistic corrections are beneficial, although there is still a discrepancy with respect to the data.
![The transverse helicity amplitudes for the D13(1520) resonance. The full curves are the predictions of the hCQM, reported in [@aie2; @sg]. The dot-dashed (dashed) curves are the results with the relativistic corrections in the Breit (equal velocity) frame (From ref. [@mds2]).[]{data-label="helamp_b"}](helamp_b){width="4in"}
The interesting point in connection with Eqs. (\[GE\_b\]) and (\[GM\_b\]) is that the ratio R of Eq. (\[ratio\]) acquires a $Q^2$ dependence because of the factor $g_\sigma$ [@rap]. In Fig. \[rap\_b\], the ratio R obtained applying the boosts to the nonrelativistic hCQM is given by the full curve, which deviates from the horizontal line, that is the prediction of the nonrelativistic hCQM.
One can conclude from Fig. \[ff\_b\] that the decrease of R can be ascribed to relativistic effects [@mg00; @rap]. This can be understood if one considers that the nonrelativistic results for the form factors are obtained in the nucleon rest frame and that the charge and magnetization densities, which are equal in the rest frame, behave in a different manner when a Lorentz boost to the Breit frame is applied.
A similar approach can be applied also to the helicity amplitudes predicted by the hCQM. In this case one obtains again simple analytical forms [@mds2], which are omitted here for simplicity. The corrections have been calculated for the transverse excitation of various resonances, both in the Breit and in the equal velocity frames, showing that in any case the deviation from the nonrelativistic predictions are very small. As an example, we show in Fig. \[helamp\_b\] the results for the transverse helicity amplitudes of the D13(1520) resonance [@mds2]. A similar situation is valid also for the S11(1535), S11(1650), S31(1620) and D33(1700) states [@mds2].
The relativistic hCQM and the elastic form factors
==================================================
The relativistic formulation
----------------------------
The inclusion of relativistic effects applying Lorentz boosts to the nonrelativistic wave functions is an approximation which is beneficial, as shown in the preceding Section, but certainly not sufficient. What is needed is a relativistically covariant theory, which can be achieved by means of the Bethe-Salpeter approach or considering one of the forms of relativistic dynamics introduced by Dirac [@dirac].
A relativistic covariant theory is characterized by its behaviour under the transformations of the Poincaré group. To this end it is sufficient to consider the generators of the infinitesimal transformations $P_\mu (\mu=0,1,2,3)$ for the space-time translations and $J_i$ and $K_i (i=1,2,3)$ for the space rotations and boosts, respectively. These generators obey to the following commutation relations ($P_0=H$)
$$[P_\mu, P_\nu] = 0, ~~~[J_i, P_j] = \epsilon_{ijk} P_k, ~~~[J_i, H] =0,
\label{poi_1}$$
$$[J_i, J_j] = \epsilon_{ijk} J_k, ~~~[K_i, K_j] = ~- \epsilon_{ijk} J_k,
\label{poi_2}$$
$$[K_i, P_j] = \delta_{ij} H, ~~~[K_i, H]=iP_i, ~~~ [K_i, J_k] = \epsilon_{ikj} K_j.
\label{poi_3}$$
The problem of building a relativistic theory is equivalent to finding a solution to the Eqs. (\[poi\_1\]), (\[poi\_2\]) and (\[poi\_3\]). Some of the ten quantities $P_\mu, J_i, K_i$ are complicated by their dependence on the hamiltonian, while the remaining ones are interaction free. As shown in ref. [@dirac], there are three different types of solutions, called instant, point and front form, differing in the type and numbers of interaction free quantities. In practical calculations all of them can be used, but here we shall concentrate ourselves on the Point Form (PF), for which the angular momentum operators $J_i$ and the boosts $K_i$ are interaction free [@klink], while the four-momentum operator $P_\mu$ contain the interaction.
The general three quark states are defined on the product space $H_1 \otimes H_2 \otimes H_3$ of the one-particle spin-1/2, positive energy representations $\it{H_i}=\it{L^2}
(\it{R}^3) \times \it{S}^{1/2}$ (i=1,2,3) of the Poincaré group [@pv; @ff_10] and can be written as $$|p_1, p_2, p_3, \lambda_1, \lambda_2, \lambda_3 \rangle ~ = ~| p_1,\lambda_1 \rangle | p_2, \lambda_2 \rangle |p_3, \lambda_3 \rangle ,$$ where $p_i$ is the four-momentum of the i-th quark and $\lambda_i$ the z-projection of its spin. The states $| p_i,\lambda_i \rangle $, i=1,2,3 are given by the Dirac spinors $u(p_i)$. We introduce the so called velocity states [@klink; @pv] as $$|v, k_1, k_2, k_3, \lambda_1, \lambda_2, \lambda_3 \rangle ~ = ~U_{B(v)} | k_1, k_2, k_3, \lambda_1, \lambda_2, \lambda_3 \rangle _0 ,
\label{vel}$$ where the suffix 0 means that the quark three-momenta $\vec{k}_i$ satisfy the rest frame condition $\sum_{i=1}^3 ~\vec{k}_i~= ~ 0$. With canonical boosts $U_{B(v)}$, the transformed quark tetramomenta are given by $p_i = U_{B(v)} k_i$ and satisfy the relation $$\sum_{i=1}^3 ~p_i^{\mu}~=
~\frac{P^{\mu}}{M} \sum_{i=1}^3 ~\epsilon (\vec{k_i}),$$ where $\epsilon (\vec{k_i})$ is the rest frame quark energy, $ P^{\mu}$ is the observed nucleon 4-momentum and M its mass. In this way the conditions for the PF are satisfied [@klink; @melde], so that the rest frame quark momenta $\vec{k}_i$ and the quark spins undergo the same Wigner rotation when the boost is applied. A particularly appealing consequence is that the combination of the angular momentum states can be performed using the Clebsch-Gordan coefficients as in the nonrelativistic theory [@klink; @klink2].
The hCQM can now be formulated in a consistent relativistic approach. The model hamiltonian of Eq. (\[H\_hCQM\]) is substituted with the mass operator M [@ff_07] $$M~=~ \sum_{i=1}^3 \sqrt{\vec{k}_i^2 + m^2}-\frac{\tau}{x} + \alpha x + H_{hyp},
\label{mass}$$ the rest frame quark momenta $\vec{k}_i$ satisfy the condition $\sum_{i=1}^3 ~\vec{k}_i~= ~ 0$. In the calculation, the internal quark momenta are substituted with the Jacobi ones $\vec{p}_\rho = \frac{1}{\sqrt{2}}(\vec{k}_1-\vec{k}_2) $, $\vec{p}_\lambda = \frac{1}{\sqrt{6}} (\vec{k}_1+\vec{k}_2 -2 \vec{k}_3) $, which are compatible with the rest frame condition and undergo the same Wigner rotations as the internal momenta $\vec{k}_i$.
The mass operator M can be considered as the result of a Bakamjian-Thomas (BT) construction [@BT], in which the interaction is introduced by adding to the free mass operator $M_0 = \sum_{i=1}^3 \sqrt{\vec{k}_i^2 + m^2}$ an interaction $M_I$ in such a way that $M_I$ commutes with the non interacting Lorentz generators $J_k, K_i$. The momentum operator $P_\mu$ can be written as $MV_\mu$, where the velocity operator $V_\mu$ commutes with the total mass operator M, which is independent of the non interacting four velocity [@kp].
The mass operator M can be diagonalized and the resulting baryon spectrum turns out to be not much different from the nonrelativistic one, with only a slight modification of the fitted parameters $\alpha$ and $\tau$. One obtains in this way the three-quark wave functions $\psi(\vec{k}_\rho, \vec{k}_\lambda)$. The bound states in the rest frame are $$\psi(\vec{k}_\rho, \vec{k}_\lambda) | k_1, k_2, k_3, \lambda_1, \lambda_2, \lambda_3 \rangle _0 ,$$ which are then boosted to any reference frame according to the Eq. (\[vel\]) $$U_{B(v)} \psi(\vec{k}_\rho, \vec{k}_\lambda) | k_1, k_2, k_3, \lambda_1, \lambda_2, \lambda_3 \rangle _0 ~ = ~ \psi(\vec{k}_\rho, \vec{k}_\lambda) |v, k_1, k_2, k_3, \lambda_1, \lambda_2, \lambda_3 \rangle ,
\label{bar}$$ the baryon states in any frame are then given by superpositions of velocity states.
The elastic form factors in the relativistic theory
---------------------------------------------------
In order to describe the electromagnetic properties of the baryons, it is necessary to introduce the quark current. For the i-th quark the current is given by $$\overline{u}(p_i) j_\mu(i) {u}(p_i') ~ = ~ \overline{u}(p_i) e \gamma _\mu(i) {u}(p_i'),$$ ($u(p_i)$ are the quark spinors), with which one can calculate the matrix elements of the total quark current $J_\mu$ in the space of the single quark free spinor states: $$\begin{aligned}
\begin{array}{rcl}
\langle p_1, p_2, p_3, \lambda_1, \lambda_2, \lambda_3| J_\mu |p_1', p_2', p_3', \lambda_1', \lambda_2', \lambda_3' \rangle & ~ = ~& \\
& & \\
\sum_i \overline{u}(p_i) j_\mu(i) {u}(p_i') \overline{u}(p_j) {u}(p_j') \delta(p_j-p_j') \overline{u}(p_k) {u}
(p_k') \delta(p_k-p_k'), & &
\end{array}\end{aligned}$$ the current conservation is ensured by applying the simple substitution $J_\mu'= J_\mu - q_\mu (q \cdot J)/q^2$, where $q_\mu$ is the photon virtual momentum.
The elastic form factors are extracted from the matrix elements of the current $J_\mu'$ between two nucleon states of the type of Eq. (\[bar\]), provided that the boosts are chosen in such a way to transform the initial and final nucleon states to the Breit system. In this way one can calculate the predicted values of the nucleon form factors in the hCQM in the relativistic version [@ff_07]. The theoretical results are compared with the experimental data in Fig. \[ff\_nqf\].
The ratio R turns out to deviate from unity more strongly than in the semirelativistic analysis shown in Sec. 6.2, in which it becomes flat at a value of about 0.6.
The relativistic calculations are not very far from the experimental data and this is a nice achievement if one thinks that the curves in Fig. \[ff\_nqf\] are predictions, however the residual discrepancy indicates that something is missing in the theoretical description.
![The nucleon elastic form factors predicted by the relativistic hCQM (full line) [@ff_07]. The $G_M^p$ data are taken from the reanalysis made in [@brash] of the data from [@gmp]. The $G_E^p$ data are obtained from the $G_M^p$ ones and the fit [@brash] of the Jlab data on the ratio R; the data for $G_E^n$ and $G_M^n$ are from refs. [@gen] and [@gmn], respectively. The figure is taken from ref. [@ff_07] (Copyright (2007) by the American Physical Society).[]{data-label="ff_nqf"}](gep_nqf "fig:"){width="2.5in"} ![The nucleon elastic form factors predicted by the relativistic hCQM (full line) [@ff_07]. The $G_M^p$ data are taken from the reanalysis made in [@brash] of the data from [@gmp]. The $G_E^p$ data are obtained from the $G_M^p$ ones and the fit [@brash] of the Jlab data on the ratio R; the data for $G_E^n$ and $G_M^n$ are from refs. [@gen] and [@gmn], respectively. The figure is taken from ref. [@ff_07] (Copyright (2007) by the American Physical Society).[]{data-label="ff_nqf"}](gmp_nqf "fig:"){width="2.5in"} ![The nucleon elastic form factors predicted by the relativistic hCQM (full line) [@ff_07]. The $G_M^p$ data are taken from the reanalysis made in [@brash] of the data from [@gmp]. The $G_E^p$ data are obtained from the $G_M^p$ ones and the fit [@brash] of the Jlab data on the ratio R; the data for $G_E^n$ and $G_M^n$ are from refs. [@gen] and [@gmn], respectively. The figure is taken from ref. [@ff_07] (Copyright (2007) by the American Physical Society).[]{data-label="ff_nqf"}](gen_nqf "fig:"){width="2.5in"} ![The nucleon elastic form factors predicted by the relativistic hCQM (full line) [@ff_07]. The $G_M^p$ data are taken from the reanalysis made in [@brash] of the data from [@gmp]. The $G_E^p$ data are obtained from the $G_M^p$ ones and the fit [@brash] of the Jlab data on the ratio R; the data for $G_E^n$ and $G_M^n$ are from refs. [@gen] and [@gmn], respectively. The figure is taken from ref. [@ff_07] (Copyright (2007) by the American Physical Society).[]{data-label="ff_nqf"}](gmn_nqf "fig:"){width="2.5in"}
We should remind that the CQM accounts only for the contribution of a quark core and that the proton radius corresponding to the hCQM wave functions is much lower that the experimental one. In fact, according to the analysis of the helicity amplitudes for the electroexcitation of the nucleon resonances, the comparison with the experimental data and the evaluation of the pion cloud terms has shown that there is the need of a meson or quark-antiquark cloud surrounding the three valence quarks. In this respect it is interesting to note that the fit performed in ref. [@ijl] is based on expressions of the form factors including both an intrinsic and meson contribution; moreover the intrinsic contribution corresponds to a radius ranging from 0.34 fm to 0.55, according to the type of form factor used, dipole or monopole, respectively. There are various attempts in the literature to add meson contributions to the quark core ones, but the problem of a consistent treatment of the quark-antiquark pair creation mechanism, that is of unquenching the CQM, is very complicated and only recently there has been an important improvement for the case of baryons [@sb1; @bs; @sb2].
There is another relevant feature which is missing. The valence quarks can be viewed as effective degrees of freedom which take into account implicitly complicated interactions involving also gluons and quark-antiquark pairs. As a consequence of such interactions, the quarks can acquire a mass and also a size. This statement is supported by the analysis of the deep inelastic electron-proton scattering performed in ref. [@ricco] within the Bloom-Gilman duality: the inelastic scattering is due an elastic scattering on constituent quarks with an approximate scaling rule given by the quark form factor. Therefore it seems reasonable to take into account the possibility of quark form factors also in the description of the elastic electron-nucleon scattering.
![The nucleon elastic form factors obtained by the relativistic hCQM (full line) with quark form factors [@ff_07]. The data are as in Fig. \[ff\_qff\]. The figure is taken from ref. [@ff_07] (Copyright (2007) by the American Physical Society).[]{data-label="ff_qff"}](gep_qff "fig:"){width="2.5in"} ![The nucleon elastic form factors obtained by the relativistic hCQM (full line) with quark form factors [@ff_07]. The data are as in Fig. \[ff\_qff\]. The figure is taken from ref. [@ff_07] (Copyright (2007) by the American Physical Society).[]{data-label="ff_qff"}](gmp_qff "fig:"){width="2.5in"} ![The nucleon elastic form factors obtained by the relativistic hCQM (full line) with quark form factors [@ff_07]. The data are as in Fig. \[ff\_qff\]. The figure is taken from ref. [@ff_07] (Copyright (2007) by the American Physical Society).[]{data-label="ff_qff"}](gen_qff "fig:"){width="2.5in"} ![The nucleon elastic form factors obtained by the relativistic hCQM (full line) with quark form factors [@ff_07]. The data are as in Fig. \[ff\_qff\]. The figure is taken from ref. [@ff_07] (Copyright (2007) by the American Physical Society).[]{data-label="ff_qff"}](gmn_qff "fig:"){width="2.5in"}
A detailed expression of these quark form factors is not known, then one can assume some phenomenological form and use it in order to get a better fit of the elastic nucleon form factors. Presently the role of these form factors is actually to parametrize the intrinsic structure of the constituent quarks, but also any other effect which is not included in the theory, as for instance the quark-antiquark pair or meson cloud contributions.
![The ratio $\mu_p G_E^P(Q^2)/G_M^P(Q^2)$ obtained by the relativistic hCQM (full line) with quark form factors [@ff_07]. The data are as in Fig. \[rap07\]. The figure is taken from ref. [@ff_07] (Copyright (2007) by the American Physical Society).[]{data-label="rap_qff"}](rap_qff){width="3in"}
Quark form factors as superpositions of monopole and dipole functions has been used within the relativistic hCQM [@ff_07], leading to a very nice agreement with data. The results of the hCQM taking into account the quark form factors are shown in Fig. \[ff\_qff\], where, for the $G_E^p, G_M^P$ and $G_M^N$ form factors, the ratio to the dipole fit is reported. The fit is performed using the data for the form factors $G_E^N, G_M^P$ and $G_M^N$ and for the ratio $R= \mu_p G_E^p/ G_M^P$, while the curve for $G_E^p$ is derived from the ratio R and $G_M^P$. The resulting ratio R is given in Fig. \[rap\_qff\].
The theoretical description of the experimental data is quite accurate, specially considering the fact that any discrepancy is certainly enhanced by the fact that the ratio to the dipole form factor is presented.
The new data and the problem of a dip in the electric form factor of the proton
-------------------------------------------------------------------------------
Quite recently, the measurement of the ratio R has ben extended up to $Q^2= 8.5$ GeV$^2$ [@puck10]. The curves of ref. [@ff_07] have been extrapolated, without any new parameter fit, to $Q^2= 12$ GeV$^2$ and compared with the new data [@ff_10]. As shown in the left side of Fig. \[rap\_10\], the agreement of the extrapolation with the new data is quite good, although some doubt is cast because of the large error in the last point. The latter is compatible with a dip, while the theoretical curve actually continues to decrease, but there seems to be no indication of a zero at finite values of $Q^2$.
The high $Q^2$ behaviour of the proton charge form factor can be studied in an alternative way by introducing the ratio $$F_p~=~ Q^2\frac {F^p_2(Q^2)}{F^p_1(Q^2)},$$ where $F^p_1(Q^2)$ and $F^p_2(Q^2)$ are, respectively, the Pauli and Dirac proton form factors. According to the analysis of Ref. [@brodsky-farrar; @brodsky-lepage], such ratio should reach an asymptotic constant value. It is interesting to note that in correspondence of a zero of the proton form factor $G^p_E(Q^2)$, $F_p$ must cross the value $4 M_p^2$ [@ff_10], where $M_p$ is the proton mass. Using the curves for the ratio R and for the proton magnetic form factor $G^p_M(Q^2)$ evaluated in [@ff_10], it is easy to determine $F^p_1(Q^2)$ and $F^p_2(Q^2)$. The results are shown in the right side of Fig. \[rap\_10\].
![Left: The ratio $\mu_p G_E^P(Q^2)/G_M^P(Q^2)$ obtained by the relativistic hCQM (full line) with quark form factors [@ff_10]. The data are as in Fig. \[rap07\] and from [@puck10]. Right: the ratio $F_p= Q^2 F^p_2(Q^2)/F^p_1(Q^2)$ obtained by the relativistic hCQM (full line) with quark form factors [@ff_10]; the data are as in the left side. The figure is taken from ref. [@ff_10] (Copyright (2010) by the American Physical Society) []{data-label="rap_10"}](rap_10 "fig:"){width="2.5in"} ![Left: The ratio $\mu_p G_E^P(Q^2)/G_M^P(Q^2)$ obtained by the relativistic hCQM (full line) with quark form factors [@ff_10]. The data are as in Fig. \[rap07\] and from [@puck10]. Right: the ratio $F_p= Q^2 F^p_2(Q^2)/F^p_1(Q^2)$ obtained by the relativistic hCQM (full line) with quark form factors [@ff_10]; the data are as in the left side. The figure is taken from ref. [@ff_10] (Copyright (2010) by the American Physical Society) []{data-label="rap_10"}](F_p "fig:"){width="2.5in"}
Of course, the two plots in Fig. \[rap\_10\] contain the same information and no further comment is needed.
The new data by ref. [@puck10] have some problem of compatibility with the previous ones. In fact, the trend of the ratio was, mainly thanks to the last points of ref. [@gayou02], a linear decrease to a zero given by the fit [@ppv] $$R ~=~ 1.0587 - 0.14265 Q^2.$$ As for the data in ref. [@puck10], the last point, with a very large error, is compatible with such trend, but the other two definitely no. This question has been studied in ref. [@puck12], where the last points of ref. [@gayou02] have been reanalyzed, with the result that the measured values of R for $3.5$ GeV$^2 \leq Q^2 \leq 5.6$ GeV$^2$ should be increased, improving the consistency with the higher $Q^2$ data. The presence of a dip in the charge form factor of the proton is still an open problem, which will hopefully solved by the planned experiments at the 12 GeV upgrade of the Jefferson Lab electron accelerator.
Coming back to the hCQM, the theoretical curve reported in Fig. \[rap\_10\] shows no evidence of dip, in agreement with the new data and the modified [@gayou02] points. It should be noted that in ref. [@puck10] the present data have been fitted by a curve for which the dip, if any, is pushed well beyond $10$ GeV$^2$.
The problem of the R ratio has been dealt with also by other groups involved in the building of CQMs.
The Rome group [@card_00] have calculated the R ratio using the model of ref. [@gi] in the front from dynamics, obtaining a decrease with $Q^2$ given by the Melosh rotations. Using the GBE model with point form dynamics, the Pavia-Graz group [@wagen; @boffi] reproduced the form factor up to $4$ GeV$^2$ and a decrease of the ratio. A similar behaviour [@mert] is obtained by the Bonn group within the Bethe-Salpeter approach with instanton interaction. For further details, the reader is referred to refs. [@gao; @kees].
Relativistic transition form factors
------------------------------------
As quoted in Sec. 6.2, the helicity amplitudes seem to be practically unaffected by the application of Lorentz boosts, with the possible ecception of the $N-\Delta$ excitation. These statements can now be checked by calculating the helicity amplitudes with the relativistic hCQM described in the previous sections. These work is presently in progress and for the moment there are only preliminary results on the $N-\Delta$ transition, which is expected to be more sensitive to relativistic corrections, as it happens with the elastic form factor. One should not forget that in the $SU(6)$ limit, the nucleon and the $\Delta$ states are degenerate and belong to the lowest shell.
.[]{data-label="fb22"}](N-Delta_12 "fig:"){width="2.5in"} .[]{data-label="fb22"}](N-Delta_32 "fig:"){width="2.5in"}
The procedure for the calculation of the $N-\Delta$ helicity amplitude is the same as in the case of the elastic form factor, apart from the fact that in this case the final baryon state is given by a $\Delta$. The results [@fb22] are given in Fig. \[fb22\].
As expected, the calculations in the fully consistent relativistic hCQM produce a sensible improvement of the results. There is still a lack of strength at low $Q^2$, which, as mentioned in Sec. 5.3, is due to the missing quark-antiquark pair creation mechanisms.
For the remaining resonances, the approximate inclusion of relativistic effects performed in Sec. 6.2 is certainly preliminary and a program of a systematic relativistic calculation of both the longitudinal and transverse helicity amplitudes is desirable. In fact there is an important experimental program at the 12 GeV upgrade of Jlab and the availability of a consistent quark model will be useful for the analysis of the high $Q^2$ behaviour of inelastic processes, which are expected to provide valuable information on the short distance properties of the nucleon [@wp; @wp2].
Summary and conclusions
=======================
The hCQM provides a good basis for the description of many baryon properties, such as the spectrum, the transition amplitudes for the photo- and electro-excitation of the resonances and the elastic nucleon form factors.
The model potential (see Eq. (\[H\_hCQM\])) has, as it happens in all CQMs, a dominant $SU(6)$-invariant part and a perturbative term producing the $SU(6)$ violation, a structure which is present already in the early Lattice QCD calculations [@wil; @deru]. The spin-flavour independent potential with its hC and linear confinement terms has strong similarities with the Cornell one [@corn] and the static $q \overline{q}$ interaction derived from LQCD in the $SU(3)$ limit [@LQCD], while the $SU(6)$ violating term is based on the One-Gluon-Exchange mechanism [@deru].
The model has only three free parameters, which are fixed in order to describe the experimental non strange baryon spectrum, obtaining a good agreement in particular for the negative parity resonances [@aie2]. Once the parameters have been determined, all the baryon states can be consistently constructed and used for parameter-free calculations.
Among these there are the predictions of the hCQM for the helicity amplitudes, which are fairly described. The medium-high $Q^2$ behaviour is generally well reproduced, while there is a lack of strength al low $Q^2$, a feature which is in common with all CQMs. The results for the 1/2 amplitudes are usually better than for the 3/2 ones; this is particularly true for the S11(1535) transverse excitation, whose behaviour has been predicted [@aie2] three years in advance with respect to data [@azn09], [@azn05_1], [@den07], [@thom01]. It should be reminded that the calculated proton radius turns out to be $0.48~$fm; such a value was obtained by the fit of the $D13$ photocoupling [@cko], while in the hCQM case is a consequence of the fit to the spectrum.
It should be observed that the hC term 1/x [@pl; @sig; @sig2] plays a crucial role in obtaining a good agreement with the experimental data. First of all, there is a perfect degeneracy between the first negative parity level $1^-_M$ and the first hyperradial excitation $0^+_S$, a feature which is only slightly modified by the confinement term and is at variance with any two-body potential. This fact provides a better starting point for the description of the Roper resonance, although its precise position can be given only if flavour dependent interactions are considered. Moreover, the particular structure of the 1/x spectrum allows to include within the first three shells all the observed 4- and 3- star negative parity resonances, while in the h.o. case one should involve the $3 \hbar \omega$ leveles, the more so if one considers also the 2-star states [@gs15]. Finally, the presence of the hC potential leads to more realistic quark wave functions, which allow a better description of the helicity amplitudes [@aie2; @sg]. This statement is supported by the fact that the analytical model [@sig; @sig2], where the confinement is treated as a perturbation, makes use only of the hC wave functions and the resulting helicity amplitudes are very similar to those predicted by the complete model of Eq. (\[H\_hCQM\]).
The nonrelativistic hCQM has been extended in order to include also isospin dependent interactions [@iso] and to describe strange baryon resonances [@gr].
The above results have been obtained in the nonrelativistic version of the model and are quite satisfactory, apart from the elastic nucleon form factor. In fact, as long as the transfer momentum $Q^2$ is increased, the theoretical predictions deviate from the experimental data, both because of the small theoretical proton radius and because of an incorrect account of the recoil in passing from the rest frame, where the calculations are performed, to the Breit frame which is more appropriate for the interpretation of the form factors as related to the charge and magnetic distributions. In principle, these considerations are valid also for the inelastic transitions to the resonances, but in this case, because of the large masses of the resonances, the recoil is less important, apart from the lightest resonance, that is the $\Delta(1232)$. On the other hand, one should not forget that the nucleon and the $\Delta(1232)$ are, in the $SU(6)$ limit, both in the ground state configuration. Actually, the relativistic effects turn out to be important for both the elastic nucleon form factors and for the electromagnetic excitation of the $\Delta(1232)$ resonance.
Relativity can be introduced in the hCQM calculations of the form factors, both elastic and inelastic, by simply applying Lorentz boost to the nonrelativistic three quark states [@mds; @rap; @mds2]. As expected, the helicity amplitudes for the higher resonances are not much affected by this relativistic corrections. On the contrary, the boosted elastic nucleon form factors are improved but are still not in agreement with the experimental data.
A consistent way of constructing a relativistic quark model is provided by the forms of relativistic dynamics introduced by Dirac [@dirac]. We have chosen the Point Form (PF), which allows to combine the angular momentum states in the same way as in the nonrelativistic case and considered a mass operator according to a Bakamjian-Thomas (BT) construction [@BT] and containing the quark interaction of the hCQM, that is a hC potential and a linear confinement. Refitting the parameters, which turn out to be very similar to the nonrelativistic ones, it is possible to predict the elastic form factors [@ff_07; @ff_10], which are now very near to the data. Also the predictions for the $\Delta$ excitation are improved if the relativistic PF is used. A very good description of the elastic form factors is achieved if quark form factors are introduced [@ff_07].
An important issue in connection with the elastic nucleon form factors is the observed behaviour of the ratio $R=\mu_p \frac{G^p_E(Q^2)}{G^p_M(Q^2)}$ with increasing values of the momentum transfer. In the nonrelativistic model, this ratio is practically one, while already the simple application of Lorentz boosts leads to a ratio R which deviates from one, agreement with the idea that the depletion is a relativistic effect [@mg00; @rap]. In fact the relativistic hCQM, with the inclusion of quark form factors, is able to describe the behaviour of R as well [@ff_07]. As for the possible presence of a dip in the electric form factor, the situation is not yet clear and hopefully it will be settled by the future Jlab experiments. Extending the relativistic hCQM calculations up to $12$ GeV$^2$, there seems to be no indication of a zero in the proton electric form factor.
The hCQM shares with all other models the drawback that the predicted resonance states have zero width, that is there is no coupling to the continuum. To this end one can introduce some mechanism for the production of mesons and in this way it is possible to calculate the baryon decays as for instance in refs. [@ki; @cr; @cr3; @bil3; @melde]. However the coupling to the continuum is important also for its consequences on the spectrum, in particular for the possible presence of meson cloud or quark-antiquark components in the states. The latter feature can be introduced by simply considering higher Fock components in the baryon states, either in the form of a meson-baryon or $4q \overline{q}$ configurations. A more consistent procedure is preferable and this can be achieved by generalizing the quark interaction in order to take into account the coupling to the continuum and then perform ab initio calculations of the various quantities of interest. An attempt in this direction is the work of ref. [@blask], where however quark and meson-baryon degrees of freedom are treated on the same footing. An important progress in this direction is provided by the recent unquenching of the CQM formulated in refs. [@sb1; @bs; @sb2], in which in particular the quark-antiquark pair contributions to the spin and the flavor asymmetry in the proton have been calculated consistently. As already mentioned, the quark-antiquark pair creation mechanism is expected to be the main contribution to the electromagnetic excitation strength at low $Q^2$. In any case, these works open the way to a consistent quark model calculation of the baryon spectrum, the strong decays, the meson electroproduction and the elastic form factors.
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Appendix A. The hyperradial wave functions {#appendix-a.-the-hyperradial-wave-functions .unnumbered}
==========================================
The hyperradial wave functions $\psi_{\gamma \nu} (x)$ are solutions of the hyperradial Schödinger equation (\[hyrad\]) of the text $$[~\frac{{d}^2}{dx^2}+\frac{5}{x}~\frac{d}{dx}-\frac{\gamma(\gamma+4)}{x^2}]
~~\psi_{\gamma \nu}(x)
~~=~~-2m~[E-V_{3q}(x)]~~\psi_{\gamma \nu}(x).$$ they are normalized according to the condition $$\int_0^\infty dx ~x^5 |\psi_{\gamma \nu} (x)|^2~=~1.$$ The reduced wave function $u_{\gamma \nu}(x)$ is defined as $$\psi_{\gamma \nu} (x)~=~\frac{u_{\gamma \nu}(x)}{x^{5/2}}$$ and obeys to the reduced equation $$\frac{{d}^2u_{\gamma \nu}(x)}{dx^2}-[\gamma (\gamma+4)-\frac{15}{4}]\frac{u_{\gamma \nu}(x)}{x^2}
+\frac{2m}{ \hbar^2}~[E-V_{3q}(x)]~~u_{\gamma \nu}(x)~~=~~0;
\label{red}$$ the behaviour for small values of x is given by $$u_{\gamma \nu}(x) \sim x^{5/2+\gamma}.$$
---------- ------- ------------ ------------------------------------------------- ----------- -----------------------------------------------------------------------
$\gamma$ $\nu$ $N_{h.o.}$ $\psi_{\gamma \nu} (x)$ (h.o.) $ N_{hC}$ $\psi_{\gamma \nu} (x)$ (hC)
0 0 0 $\alpha^3 $ 0 $\frac{(2g)^3}{\sqrt{5!}} e^{-gx}$
1 0 1 $\frac{\alpha^4}{\sqrt{3}} x $ 1 $\frac{(2g)^4}{\sqrt{7!}} x e^{-gx}$
0 1 2 $\frac{\alpha^3}{\sqrt{3}} (\alpha^2 x^2 - 3) $ 1 $\frac{(2g)^3}{\sqrt{7!}} \sqrt{6} (5-2 g x) e^{-gx}$
2 0 2 $\alpha^5 x^2$ 2 $\frac{(2g)^5}{3\sqrt{8!}} x^2 e^{-gx}$
1 1 3 - 2 $\frac{(2g)^4}{3 \sqrt{7!}} x (7-2 g x) e^{-gx}$
0 2 4 - 2 $\frac{(2g)^3}{3 \sqrt{2} \sqrt{6!}} (30-24 g x + 4 g^2 x^2) e^{-gx}$
---------- ------- ------------ ------------------------------------------------- ----------- -----------------------------------------------------------------------
: The hyperradial wave functions $\psi_{\gamma \nu} (x)$ for the harmonic (h.o.) and hyperCoulomb (hC) potentials, up to the second energy shell. In the h.o. case $\alpha=\sqrt{3 K m}/\hbar$, $N_{h.o.} = 2 \nu + \gamma$ and an overall common factor is omitted. For the hC case $N_{hC}= \nu + \gamma$ and the constant g is given by $\tau m/(\nu+\gamma+5/2)$. \[wf\]
In the case of the h.o. interaction $V_{3q}(x)=3/2 k x^2$, the asymptotic behaviour is determined by the gaussian factor $$u_{\gamma \nu}(x) \sim e^{-\alpha^2 x^2/2},$$ for the reduced wave function one can then assume the form, apart from a normalization factor, $$u_{\gamma \nu}(x) ~=~ x^{5/2+\gamma} e^{-\alpha^2 x^2/2} P(x).$$ In order that Eq. (\[red\]) be satisfied, the function P must obey to $$\frac{d^2 P}{dy^2} + (\frac{a}{y}-1) \frac{dP}{dy}+\nu \frac{P}{y} ~=~0,
\label{hyconf}$$ where the variable $y=\alpha^2 x^2$ has been introduced and $$a~=~3+\gamma, ~~~~~~~-\nu ~=~\frac{3+\gamma}{2} - \frac{\epsilon}{4 \alpha^2},$$ with $\epsilon=\frac{2m}{ \hbar^2} E$. Eq. (\[hyconf\]) is the hypergeometric confluent equation and P is then given by the hypergeometric confluent function $F(-\nu, 3+\gamma,y)$. In order to preserve the correct behaviour for small and large x, $\nu$ has to be a non negative integer and P(y) is a polynomial of order $\nu$. The energy values are then given by $$E~=~(3 + 2\nu + \gamma) \hbar \omega,$$ with $\omega=\sqrt{\frac{3K}{m}}$.
The h.o. wave functions for the first three shells are reported in Table \[wf\].
For the hC potential $-\frac{\tau}{x}$, one can proceed in an analogous way. The reduced equation (\[red\]) is written as $$\frac{{d}^2u_{\gamma \nu}(x)}{dx^2}-[\gamma (\gamma+4)-\frac{15}{4}]\frac{u_{\gamma \nu}(x)}{x^2}
+\frac{2m}{ \hbar^2}~[E-V_{3q}(x)]~~u_{\gamma \nu}(x)~=~0.
\label{redhC}$$ In this case the asymptotic behaviour is determined by the factor $e^{-gx}$, where $$g^2~=~-\sqrt{2 m E},$$ ($\hbar$ is taken equal to 1). One can then assume $$u_{\gamma \nu}(x) ~=~ x^{5/2+\gamma} e^{-g x} P(x).$$ Eq. (\[red\]) is satisfied if the function P obeys to $$\frac{d^2 P}{dy^2} + (\frac{a}{y}-1) \frac{dP}{dy}+\nu \frac{P}{y} ~=~0,
\label{conf}$$ where the variable y is now defined as y=2gx and the parameters a and $\nu$ are given by $$a~=~5+2 \gamma, ~~~~~~~-\nu ~=~\frac{5}{2} + \gamma - \frac{m \tau}{g }.$$ From the last formula one obtains finally [@sig2] $$g~=~\frac{m \tau}{\frac{5}{2}+ \gamma + \nu}$$ and $$E~=~- \frac{g^2}{2m}~=~-\frac{m \tau^2}{2[\frac{25}{4}+(\gamma+\nu)(\gamma+\nu+5)]}.$$
The quantities P(y) are associated Laguerre polynomials (see Eq. (\[eigom\]) in the text). Also for the hC wave functions for the first three shells are reported in Table \[wf\].
Appendix B. The baryon states {#appendix-b.-the-baryon-states .unnumbered}
=============================
The baryon states are superpositions of $SU(6)-$configurations, which can be factorized as follows (see Eq. (\[3q\]) of the text) : $$\label{psi_tot1}
\Psi_{3q} = \theta_{colour} \cdot \chi_{spin} \cdot \Phi_{isospin}
\cdot \psi_{3q}(\vec{\rho},\vec{\lambda}).$$
As already mentioned in the text, the various parts must be combined in order to have a completely antisymmetric three-quark wave function. To this end it is necessary to study the behaviour of the different factors with respect to the permutations of three objects (that is with respect to the group $S_3$). In general, any three particle wave function belongs to one of the following symmetry types: antisymmetry (A), symmetry (S), mixed symmetry with symmetric pair (MS) and mixed symmetry with antisymmetric pair (MA).
For the colour part $\theta_{colour}$ one must choose the antisymmetric colour singlet combination.
The three-quark spin states are defined as:
$$\chi_{MS}~=~|((\frac{1}{2},\frac{1}{2})1,\frac{1}{2})\frac{1}{2} \rangle ,\nonumber$$
$$\chi_{MA}~=~|((\frac{1}{2},\frac{1}{2})0,\frac{1}{2})\frac{1}{2} \rangle ,\nonumber$$
$$\chi_{S}~=~|((\frac{1}{2},\frac{1}{2})1,\frac{1}{2})\frac{3}{2} \rangle .\nonumber$$
The antisymmetric combination is absent because there are only two states at disposal for three particles.
Similarly one can define the isospin states $\phi_{MS}, \phi_{MA}, \phi_{S}$.
If the interaction is spin and isospin (flavour) independent, one has to introduce products of $\chi-$ and $\phi-$ states with definite $S_3-$ symmetry. Here we give the explicit forms only for the case that both factors have mixed symmetry, the remaining ones being trivial:
$$\Omega_S~=~\frac{1}{\sqrt{2}}~[\chi_{MA} \phi_{MA} + \chi_{MS} \phi_{MS}],
\nonumber$$
$$\Omega_{MS}~=~\frac{1}{\sqrt{2}}~[\chi_{MA} \phi_{MA} - \chi_{MS} \phi_{MS}],
\nonumber$$
$$\Omega_{MA}~=~\frac{1}{\sqrt{2}}~[\chi_{MA} \phi_{MS} + \chi_{MS} \phi_{MA}],
\nonumber$$
$$\Omega_A~=~\frac{1}{\sqrt{2}}~[\chi_{MA} \phi_{MS} - \chi_{MS} \phi_{MA}].
\nonumber
\label{om}$$
The space wave function is given by $$\psi_{3q}(\vec{\rho},\vec{\lambda})~=
\psi_{\nu\gamma}(x)~~
{Y}_{[{\gamma}]l_{\rho}l_{\lambda}}({\Omega}_{\rho},{\Omega}_{\lambda},\xi),
\label{psi1}$$ where $\gamma= 2n +l_\rho + l_\lambda$; the hyperspherical functions are given by [@baf] $${Y}_{[{\gamma}]l_{\rho}l_{\lambda}}({\Omega}_{\rho},{\Omega}_{\lambda},\xi)~ = ~Y_{l_\rho m_\rho} (\Omega_\rho)~Y_{l_\lambda m_\lambda} (\Omega_\lambda)~^{(2)}P^{l_{\lambda} l_{\rho}}_N(\xi),$$ where $$^{(2)}P^{l_{\lambda} l_{\rho}}_N(\xi)~ = ~C_{n l_\rho l_\lambda} (cos \xi)^{l_\lambda} (sin \xi)^l_{\rho} P_n^{l_\rho+1/2, l_\lambda+1/2}(cos 2\xi),$$ with $$C_{n l_\rho l_\lambda} ~ = ~ \sqrt{\frac{2(2 \gamma +2) \Gamma(\gamma+2-n) \Gamma(n+1)}{\Gamma(n+l_\rho+3/2) \Gamma(n+l_\lambda+3/2}}$$ and $P^{\alpha, \beta}_n(z)$ is a Jacobi polynomial.
The symmetry property of the wave function Eq. (\[psi1\]) is determined by the hyperspherical part only, since the hyperradius $x$ is completely symmetric. In Table \[config\] we report the combinations of the hyperspherical harmonics having definite $S_3-$symmetry.
---------- -- --------- -- ------------------------------------------- -- -------
$\gamma$ $L^P_t$ $(Y_{[\gamma]l_{\rho}l_{\lambda}})_{S_3}$ $S_3$
$0$ $0^+_S$ $Y_{[0]00}$ $S$
$1$ $1^-_M$ $Y_{[1]10}$ $MA$
$ $ $Y_{[1]01}$ $MS$
$2$ $2^+_S$ $\frac{1}{\sqrt{2}}[Y_{[2]20}+Y_{[2]02}]$ $S$
$2^+_M$ $Y_{[2]11}$ $MA$
$\frac{1}{\sqrt{2}}[Y_{[2]20}-Y_{[2]02}]$ $MA$
$1^+_A$ $Y_{[2]11}$ $A$
$0^+_M$ $Y_{[2]11}$ $MA$
$Y_{[2]00}$ $MA$
---------- -- --------- -- ------------------------------------------- -- -------
: Combinations $(Y_{[\gamma]l_{\rho}l_{\lambda}})_{S_3}$ [@sig2] of the hyperspherical harmonics $Y_{[\gamma]l_{\rho}l_{\lambda}}$ that have definite $S_3-$symmetry. For simplicity of notation, in the third column we have omitted the coupling of $l_{\rho}$ and $l_{\lambda}$ to the total orbital angular momentum $L$. Each combination is labelled as $L^P_{t}$, specifying the total orbital angular momentum $L$, the parity $P$ and the $S_3-$symmetry type $t=A, MA, MS, S$. \[config\]
-----------------------------------------------------------------------------------------------------------------------------------------
State $L^P_{S_3}$ S T SU(6) configurations
------- ------------- --------------- --------------- -- --------------------------------------------------------------------------------
$P11$ $0^+_S$ $\frac{1}{2}$ $\frac{1}{2}$ ${\psi}_{00}~Y_{[0]00}
~\Omega_S $
$0^+_S$ $\frac{1}{2}$ $\frac{1}{2}$ ${\psi}_{10}~ Y_{[0]00}
~ \Omega_S$
$0^+_S$ $\frac{1}{2}$ $\frac{1}{2}$ ${\psi}_{20}~ Y_{[0]00}
~ \Omega_S$
$0^+_M$ $\frac{1}{2}$ $\frac{1}{2}$ ${\psi}_{22}~ \frac{1}
{\sqrt{2}}~ [ Y_{[2]00} ~\Omega_{MS} + Y_{[2]11} ~\Omega_{MA}] $
$2^+_M$ $\frac{3}{2}$ $\frac{1}{2}$ ${\psi}_{22} ~\frac{1}
{\sqrt{2}}~ [\frac{1}{\sqrt{2}}~ (Y_{[2]20} - Y_{[2]02 })~\phi_{MS} +
Y_{[2]11} ~\phi_{MA} ]~\chi_S$
$P13$ $2^+_M$ $\frac{1}{2}$ $\frac{1}{2}$ ${\psi}_{22}
~\frac{1}{\sqrt{2}}~[\frac{1}{\sqrt{2}} ~(Y_{[2]20} - Y_{[2]02 })
~\Omega_{MS} + Y_{[2]11} ~\Omega_{MA} ] $
$2^+_M$ $\frac{3}{2}$ $\frac{1}{2}$ ${\psi}_{22}
~ \frac{1}{\sqrt{2}}~[\frac{1}{\sqrt{2}}~ (Y_{[2]20} - Y_{[2]02 })
~\phi_{MS} + Y_{[2]11}~ \phi_{MA} ]~\chi_S$
$0^+_M$ $\frac{3}{2}$ $\frac{1}{2}$ ${\psi}_{22}
~\frac{1}{\sqrt{2}}~ [Y_{[2]00} ~\phi_{MS} + Y_{[2]11} ~\phi_{MA}]~\chi_S$
$2^+_S$ $\frac{1}{2}$ $\frac{1}{2}$ ${\psi}_{22}
~\frac{1}{\sqrt{2}}~ [ Y_{[2]20} + Y_{[2]02 }] ~\Omega_S$
$F15$ $2^+_M$ $\frac{1}{2}$ $\frac{1}{2}$ ${\psi}_{22}
~ \frac{1}{\sqrt{2}}~[\frac{1}{\sqrt{2}}~ (Y_{[2]20} - Y_{[2]02 })
~\Omega_{MS} + Y_{[2]11} ~\Omega_{MA} ] $
$2^+_M$ $\frac{3}{2}$ $\frac{1}{2}$ ${\psi}_{22}
~\frac{1}{\sqrt{2}}~[\frac{1}{\sqrt{2}}~ (Y_{[2]20} - Y_{[2]02 })~\phi_{MS}
+ Y_{[2]11} ~\phi_{MA} ] ~\chi_S$
$2^+_S$ $\frac{1}{2}$ $\frac{1}{2}$ ${\psi}_{22}
~\frac{1}{\sqrt{2}}~ [Y_{[2]20} + Y_{[2]02 }]~ \Omega_S$
$F17$ $2^+_M$ $\frac{3}{2}$ $\frac{1}{2}$ ${\psi}_{22}
~\frac{1}{\sqrt{2}} ~[\frac{1}{\sqrt{2}} ~(Y_{[2]20} - Y_{[2]02 })~\phi_{MS}
+ Y_{[2]11} ~\phi_{MA} ] ~\chi_S$
$P31$ $2^+_S$ $\frac{3}{2}$ $\frac{3}{2}$ ${\psi}_{22}
~\frac{1}{\sqrt{2}}~ [ (Y_{[2]20} + Y_{[2]02 }] ~\chi_S ~\phi_S$
$0^+_M$ $\frac{1}{2}$ $\frac{3}{2}$ ${\psi}_{22}
~\frac{1}{\sqrt{2}}~ [ Y_{[2]00} ~\chi_{MS} + Y_{[2]11} ~\chi_{MA}] ~\phi_S$
$P33$ $0^+_S$ $\frac{3}{2}$ $\frac{3}{2}$ ${\psi}_{00}~Y_{[0]00}
~\chi_S ~\phi_S$
$0^+_S$ $\frac{3}{2}$ $\frac{3}{2}$ ${\psi}_{10}~Y_{[0]00} ~\chi_S
~\phi_S$
$0^+_S$ $\frac{3}{2}$ $\frac{3}{2}$ ${\psi}_{20}~Y_{[0]00} ~\chi_S~
\phi_S$
$2^+_S$ $\frac{3}{2}$ $\frac{3}{2}$ ${\psi}_{22}~\frac{1}{\sqrt{2}}
~ [Y_{[2]20} + Y_{[2]02 }] ~\chi_S ~\phi_S$
$2^+_M$ $\frac{1}{2}$ $\frac{3}{2}$ ${\psi}_{22}~\frac{1}{\sqrt{2}}
~ [\frac{1}{\sqrt{2}} ~(Y_{[2]20} - Y_{[2]02 })~\chi_{MS} + Y_{[2]11}
~\chi_{MA} ]~\phi_S$
$F35$ $2^+_M$ $\frac{1}{2}$ $\frac{3}{2}$ ${\psi}_{22}
~\frac{1}{\sqrt{2}} ~ [\frac{1}{\sqrt{2}}~ (Y_{[2]20} - Y_{[2]02 })~\chi_{MS}
+ Y_{[2]11}~\chi_{MA} ] ~\phi_S$
$2^+_S$ $\frac{3}{2}$ $\frac{3}{2}$ ${\psi}_{22}~\frac{1}{\sqrt{2}}
~ [Y_{[2]20} + Y_{[2]02 }] ~\chi_S ~\phi_S$
$F37$ $2^+_S$ $\frac{3}{2}$ $\frac{3}{2}$ ${\psi}_{22}
~\frac{1}{\sqrt{2}}~ [ Y_{[2]20} + Y_{[2]02 }] ~\chi_S ~\phi_S$
-----------------------------------------------------------------------------------------------------------------------------------------
: Three-quark states with positive parity [@sig2]. The second, third and fourth columns show the angular momentum, parity and $S_3$-symmetry, $L^P_{S_3}$, the spin, $S$, and isospin, $T$. States are shown in the last column and are written in terms of the hyperradial wave functions, $\psi_{\nu \gamma}$, of the hyperspherical harmonics, $(Y_{[\gamma]})_{S_3}$ of Table \[config\] and of the spin and isospin states. \[statpos\]
-----------------------------------------------------------------------------------------------------------------------------------------------
Resonances $L^P_{S_3}$ S T States
------------ -- ------------- --------------- --------------- -- ------------------------------------------------------------------------------
$S11$ $1^-_M$ $\frac{1}{2}$ $\frac{1}{2}$ ${\psi}_{11}
~\frac{1}{\sqrt{2}} ~[Y_{[1]10} ~\Omega_{MA} + Y_{[1]01} ~\Omega_{MS}]$
$1^-_M$ $\frac{1}{2}$ $\frac{1}{2}$ ${\psi}_{21}~\frac{1}{\sqrt{2}}
~[Y_{[1]10} ~\Omega_{MA} + Y_{[1]01}] ~\Omega_{MS}$
$1^-_M$ $\frac{3}{2}$ $\frac{1}{2}$ ${\psi}_{11}~\frac{1}{\sqrt{2}}
~[Y_{[1]10} ~\phi_{MA} + Y_{[1]01} ~\phi_{MS}] ~\chi_S$
$1^-_M$ $\frac{3}{2}$ $\frac{1}{2}$ ${\psi}_{21}~\frac{1}{\sqrt{2}}
~[Y_{[1]10} ~\phi_{MA} + Y_{[1]01} ~\phi_{MS}] ~\chi_S$
$D13$ $1^-_M$ $\frac{1}{2}$ $\frac{1}{2}$ ${\psi}_{11}
~\frac{1}{\sqrt{2}}
~[Y_{[1]10} ~\Omega_{MA} + Y_{[1]01} ~\Omega_{MS}]$
$1^-_M$ $\frac{1}{2}$ $\frac{1}{2}$ ${\psi}_{21}~ \frac{1}{\sqrt{2}}
~ [Y_{[1]10}~ \Omega_{MA} + Y_{[1]01}] ~\Omega_{MS}$
$1^-_M$ $\frac{3}{2}$ $\frac{1}{2}$ ${\psi}_{11}
~\frac{1}{\sqrt{2}}~ [Y_{[1]10} ~\phi_{MA} + Y_{[1]01}~\phi_{MS}] ~\chi_S$
$1^-_M$ $\frac{3}{2}$ $\frac{1}{2}$ ${\psi}_{21}~\frac{1}{\sqrt{2}}
~[Y_{[1]10} ~\phi_{MA} + Y_{[1]01} ~\phi_{MS}] ~\chi_S$
$D15$ $1^-_M$ $\frac{3}{2}$ $\frac{1}{2}$ ${\psi}_{11}
~\frac{1}{\sqrt{2}} ~[Y_{[1]10} ~\phi_{MA} + Y_{[1]01} ~\phi_{MS}] ~\chi_S
$
$1^-_M$ $\frac{3}{2}$ $\frac{1}{2}$ ${\psi}_{21}~\frac{1}{\sqrt{2}}
~[Y_{[1]10} ~\phi_{MA} + Y_{[1]01} ~\phi_{MS}] ~\chi_S$
$S31$ $1^-_M$ $\frac{1}{2}$ $\frac{3}{2}$ ${\psi}_{11}~
\frac{1}{\sqrt{2}}~[Y_{[1]10} ~\chi_{MA} + Y_{[1]01} ~\chi_{MS}] ~\phi_S$
$1^-_M$ $\frac{1}{2}$ $\frac{3}{2}$ ${\psi}_{21}~\frac{1}{\sqrt{2}}
~[Y_{[1]10} ~\chi_{MA} + Y_{[1]01} ~\chi_{MS}] ~\phi_S$
$S33$ $1^-_M$ $\frac{1}{2}$ $\frac{3}{2}$ ${\psi}_{11}
~\frac{1}{\sqrt{2}} ~[Y_{[1]10} ~\chi_{MA} + Y_{[1]01} ~\chi_{MS}] ~\phi_S
$
$1^-_M$ $\frac{1}{2}$ $\frac{3}{2}$ ${\psi}_{21}
~\frac{1}{\sqrt{2}}~ [Y_{[1]10} ~\chi_{MA} + Y_{[1]01} ~\chi_{MS}] ~\phi_S$
-----------------------------------------------------------------------------------------------------------------------------------------------
: Three quark states with negative parity [@sig2]. Notation as in Table \[statpos\] \[statneg\]
In Tables \[statpos\] and \[statneg\], we give the explicit form of the three-quark states with positive and negative parity, respectively. In these Tables the hyperradial wave functions $\psi_{\nu\gamma}$ are solutions of the hyperradial equation Eq. (\[hyrad\]) of the text; their form depends of course on the hypercentral potential.
|
The finite-temperature conductivity of a clean one-dimensional wire [@review] is a fundamental and much studied question. Clearly the “bulk” conductivity of a wire in the [*absence*]{} of a periodic potential is infinite even at finite temperatures $T$. In this case the conductance is independent of the length of the wire and is determined by the contacts only. Surprisingly, much less is known about the conductivity in the presence of Umklapp scattering induced by a periodic potential. There is not even an agreement whether it is finite or infinite at finite temperatures for generic systems [@giamarchi; @millis; @zotos; @zotos2; @everybody]. We shall show that the correct answer emerges when all relevant (weakly violated) conservation laws are taken into account. Those conservation laws are exact at the Fermi surface and are violated by Umklapp terms away from it. We shall study the associated slow modes by means of a [ *memory matrix*]{} formalism able to keep track of their dynamics. It will allow us to calculate reliably the low temperature, low frequency conductivity.
The topology of the Fermi surface of a $1d$ metal determines its low-energy excitations. Two well defined Fermi-points exist at momenta $k=\pm k_F$, allowing us to define left and right moving excitations, to be described by $\Psi_{L/R,\sigma=\uparrow
\downarrow}$. We shall include in the fields momentum modes extending to the edge of the Brillouin zone, usually omitted in treatments that concentrate on physics very close to the Fermi-surface.
The Hamiltonian, including high energy processes, is $$\begin{aligned}
H=H_{LL}+H_{\text{irr}}+\sum_{n,m}^\infty H_{n,m}^U.\end{aligned}$$ $H_{LL}$ is the well-known Luttinger liquid Hamiltonian capturing the low energy behavior[@review], $$\begin{aligned}
H_{LL}&=&v_F \int \left(\Psi_{L
\sigma}^\dagger i \partial_x \Psi_{L \sigma}- \Psi_{R\sigma}^\dagger
i \partial_x \Psi_{R\sigma} \right) + g \int \rho(x)^2 \nonumber
\\ &=& \frac{1}{2} \int \frac{dx}{2 \pi}
\sum_{\nu=\sigma,\rho} v_\nu\left( K_\nu (\partial_x
\theta_\nu)^2+\frac{1}{K_\nu} (\partial_x \phi_\nu)^2 \right) \nonumber\end{aligned}$$ $v_F$ is the Fermi velocity, $g>0$ measures the strength of interactions, $\rho=\rho_L+\rho_R$ is the sum of the left and right moving electron densities. In the second line we wrote the bosonized[@review] version of the Hamiltonian. Here $v_\sigma$, $v_\rho$ are the spin and charge velocities, and the interactions determine the Luttinger parameters $K_\nu$ with $v_\nu K_\nu=v_F$, $v_\rho/K_\rho=v_F+g/\pi $, $v_\sigma/K_\sigma=v_F-g/\pi$.
The high energy processes are captured in the subsequent terms which are formally irrelevant at low energies (we consider only systems away from a Mott transition, i.e. away from half filling). Some of them, however, determine the low-frequency behavior of the conductivity at any finite $T$, since they induce the decay of the conserved modes of $H_{LL}$ (they are “dangerously irrelevant”). We classify these irrelevant terms with the help of two operators which will play the central role in our discussion. The first one is the translation operator $P_T$ of the right- and left-moving fields, the second one, $J_0 =N_R-N_L$, is the difference of the number of right- and left-moving electrons, and is up to $v_F$, the charge current of $H_{LL}$: $$\begin{aligned}
\label{PT}
P_T&=&\sum_\sigma \int dx \left(\Psi_{R
\sigma}^\dagger (-i \partial_x) \Psi_{R \sigma}+ \Psi_{L\sigma}^\dagger
(-i \partial_x) \Psi_{L\sigma}\right) \\ \label{J0}
J_0&=&N_R-N_L= \sum_{\sigma} \int dx \left( \Psi_{R
\sigma}^\dagger \Psi_{R\sigma}-\Psi_{L\sigma}^\dagger
\Psi_{L\sigma} \right)\end{aligned}$$ Both $P_T$ and $J_0$ are conserved by $H_{LL}$; their importance for transport properties is due to the fact that both stay [*approximately*]{} conserved in [*any*]{} one dimensional metal (away from half filling): processes which change $J_0$ are forbidden [*close to the Fermi surface*]{} by momentum conservation. The linear combination $P_0=P_T + k_F J_0$ can be identified with the total momentum of the full Hamiltonian $H$ and is therefore also approximately conserved.
We proceed to the classification of the formally irrelevant terms in the Hamiltonian. This classification allows us to select all those terms (actually few in number) that determine the current dynamics. $H_{\text{irr}}$ includes all terms in $H-H_{LL}$ which commute with both $P_T$ and $J_0$, such as corrections due to the finite band curvature, due to finite-range interactions and similar terms. We will not need their explicit form.
The Umklapp terms $H^U_{n,m}$ ($n,m=0,1,...$) convert $n$ right-movers to left-movers (and vice versa) picking up lattice momentum $m 2 \pi/a=m G$, and do not commute with either $P_T$ or $J_0$. Leading terms are of the form, $$\begin{aligned}
\label{HU}
H^U_{0,m}&\approx& g^U_{0,m} \int e^{i \Delta k_{0,m} x} (\rho_L+\rho_R)^2 + h.c. \\
H^U_{1,m}&\approx& g^U_{1,m} \sum_{\sigma}\int e^{i \Delta k_{1,m} x} \Psi^\dagger_{ R \sigma} \Psi_{ L \sigma}
\rho_{-\sigma}+h.c. \\
H^U_{2,m}&\approx&
g^U_{2,m} \int e^{i \Delta k_{2,m} x} \Psi^\dagger_{ R \uparrow}\Psi^\dagger_{ R \downarrow}
\Psi_{ L \downarrow}\Psi_{ L \uparrow}+h.c. \end{aligned}$$ with momentum transfer $
\Delta k_{n,m}= n 2 k_F-m G$. A process transfering $n>1$ electrons with total spin $ n_s/2$ pointing in the $z$-direction can be neatly expressed as $$\begin{aligned}
\label{boso}
H^U_{n,m}= \frac{ g^U_{n,m,n_s}}{(2 \pi \alpha)^n} \int
e^{i \Delta k_{n,m} x}
e^{ i \sqrt{2} (n \phi_{\rho} + n_s \phi_{\sigma}) }+ h.c.,\end{aligned}$$ $\alpha$ being a cut-off, of the order of the lattice spacing. In fermionic variables the integrand takes the form $ \prod_{j=0}^{n/2-1}
\Psi_{R \uparrow}^\dagger(x+j \alpha)
\Psi_{R \downarrow}^\dagger(x+j \alpha)
\Psi_{L \downarrow}(x+j \alpha)
\Psi_{L \uparrow}(x+j \alpha)$ (for $n_s=0$ and even $n $).
Note, though, that any [*single*]{} term $H^U_{nm}$ conserves a linear combination of $J_0$ and $P_T$, $$\begin{aligned}
\left[ H^U_{nm},\Delta k_{nm} J_0 + 2 n P_T \right]=0.\end{aligned}$$ Indeed, a term of the form (\[boso\]) would appear in a continuum model [*without*]{} Umklapp scattering, but with a Fermi momentum $\tilde{k}_F=\Delta k_{nm}/(2n)$. In such a model, $ \Delta k_{nm} J_0/(2 n)
+ P_T$ is the total momentum of the system and therefore conserved. The importance of this simple but essential conservation law has to our knowledge not been sufficiently realized in previous calculations of the conductivity. Due to this conservation law a single Umklapp term can never induce a finite conductivity! At least two independent Umklapp terms are required to lead to a complete decay of the current. Further, two incommensurate Umklapp terms suffice to generate the rest.
To calculate the conductivity it is necessary to keep track of the nearly conserved quantities and their relation to the current. We will develop a description of the slowest variables using the Mori-Zwanzig memory functional [@forster; @woelfle; @giamarchi]. Approximations within this scheme amount to short-time expansions. In general, the short time decay of a quantity carries little information on its long-time behavior; this, however, is [*not*]{} the case for the slowest variables in the system, where the short time and hydrodynamic behavior coincide.
To set up the formalism [@forster] we define a scalar product $(A|B)$ in the space of [*operators*]{}, $$\begin{aligned}
\left(A(t)|B\right)&\equiv& \frac{1}{\beta} \int_0^\beta d\lambda
\left\langle A(t)^\dagger B(i \lambda) \right\rangle, \end{aligned}$$ where we use the usual Heisenberg picture with $A(t)=e^{i H t} A e^{-i
H t}$. We choose a set “slow” operators $j_1, j_2, ... j_N$ which includes $j_1=J$, the full current operator. Standard arguments [@forster] lead to the electric conductivity, $$\begin{aligned}
\label{sigmaE}
\sigma({\omega},T)&=&\left[\left(\hat{M}({\omega},T)- i {\omega}\right)^{-1}
\hat{\chi}(T) \right]_{11}. \end{aligned}$$ Here $\hat{\chi}_{pq}=\beta (j_p|j_q)$ is the matrix of the static $j_p j_q$ susceptibilities (as usually defined), and $\hat{M}$ is the matrix of memory functions given by the projected correlation functions of time-derivatives of the “slow” operators, $$\begin{aligned}
\label{M}
\hat{M}_{pq}({\omega})=\beta \sum_r
\left(\partial_t j_q \left| Q \frac{i}{{\omega}-QLQ} Q
\right| \partial_t j_r \right) (\hat{\chi}^{-1})_{rp}.\end{aligned}$$ The Liouville “super”-operator, $L$, is defined by $L
A=[H,A]$ and $Q$ is the projection operator on the space perpendicular to the slowly varying variables $j_p$, $$\begin{aligned}
\label{Q}
Q=1-\sum_{pq} |j_q) \beta (\hat{\chi}^{-1})_{qp} (j_p|.\end{aligned}$$ We assumed for simplicity that all $j_p$ have the same signature under time reversal.
The perturbative expansion of the memory matrix $\hat{M}$ is accompanied by factors $1/{\omega}$ guaranteeing it is always valid at short times. It is also valid for small frequencies provided the slowly evolving degrees of freedom are projected out (by the operator $Q$). Unlike the conductivity it is expected to be a smooth function of the coupling constants which can be perturbatively evaluated.
We first consider a situation where some linear combinations of the $j_p$ are conserved by $H$, in which case an infinite conductivity is expected. We introduce ${\cal P}_c$, the projection operator on the space of conserved currents, and carry out the required matrix inversion to find, $$\begin{aligned}
\label{sigmajjj}
\sigma({\omega}\to 0,T>0)&=&
\sigma_{\text{reg}}({\omega},T)
+i \frac{(\hat{\chi} \hat{\chi}_c^{-1} \hat{\chi})_{11}}{{\omega}+i 0},\end{aligned}$$ where $\hat{\chi}_c^{-1} = {\cal P}_c({\cal P}_c \hat{\chi} {\cal
P}_c)^{-1}{\cal P}_c$. Within any simple (short-time) approximation, $\sigma_{\text{reg}}({\omega},T)$ as defined above, is regular (this approximation fails e.g. if some conserved current $\hat{j}$ is not included in $j_1 ...j_N$). Hence the Drude weight $D(T)$ is finite at finite temperatures, $ {\text{Re}}\sigma({\omega}\to 0)=2 \pi
D(T) \delta({\omega})=\pi (\hat{\chi} \hat{\chi}_c^{-1} \hat{\chi})_{11}
\delta({\omega})$. It is determined by the “overlap” of the physical current operator $J$ with the conserved quantities $\chi_{1 s}$, $s$ labeling the conserved currents. Remarkably, our perturbative approximation is in accord with an exact inequality [@zotos2] for the Drude weight, $D(T)\ge \frac{1}{2} (\hat{\chi} \hat{\chi}_c^{-1}
\hat{\chi})_{11}$. Note that $\hat{\chi}$ can be calculated to an arbitrary degree of precision around a Luttinger liquid and that the lower bound can be improved by including more conserved quantities [@zotos2].
Now consider the more realistic situation where the previously conserved currents decay slowly (via Umklapp processes), in which case a finite conductivity is expected. We restrict ourselves to the two-dimensional space spanned by $v_F J_0$ and $P_T$, which we argue have the longest decay rate and dominate the transport. Here we approximate $J \approx v_F
J_0$ to keep the presentation simple. This affects only the high frequency behavior of the conductivity [@millis]. There is a large number of other nearly conserved quantities. For example $H_{LL}+H^U_{21}$, the relevant low-energy model close to half filling, is integrable and therefore is characterized by an [*infinite*]{} number of conservation laws. We can, however, neglect them at low $T$ if our initial model is not integrable, expecting that practically all conservation laws are destroyed by (formally irrelevant) terms [*close to the Fermi surface*]{} leading to decay rates proportional to some power of $T$. This is to be compared to $J_0$ and $P_T$ which commute with [*all*]{} scattering processes at the Fermi surface, leading to exponentially large lifetimes.
We now proceed to calculate the Memory matrix. To leading order in the perturbations we can replace $L$ in (\[M\]) by $L_{LL}=[H_{LL}, .]$ [@LL], since $\partial_t v_F J_0$ and $\partial_t P_T$ are already linear in $g^U_{n,m}$. As $L_{LL} P_T=L_{LL} J_0=0$, there is no contribution from the projection operator $Q$. The memory matrix takes the form, $$\begin{aligned}
\hat{M}&\approx& \sum_{nm} M_{nm}({\omega},T) \left(
\begin{array}{cc}
v_F^2 (2 n)^2 & -2 n v_F \Delta k_{nm} \\
-2 n v_F \Delta k_{nm} & (\Delta k_{nm})^2
\end{array}
\right) \hat{\chi}^{-1} \nonumber
$$ where, $$\begin{aligned}
\hat{\chi}&\approx& \left(
\begin{array}{cc}
2 v_F/\pi & 0 \\
0 & \frac{\pi T^2}{3} \left(\frac{1}{v_\rho^3}+\frac{1}{v_\sigma^3}\right)
\end{array}
\right) \\ M_{nm}&\equiv& (g^U_{nm})^2 M_n(\Delta k_{n,m},{\omega})\equiv \frac{
\langle F;F \rangle^0_{\omega}-\langle F;F \rangle^0_{{\omega}=0}}{i {\omega}}. \nonumber\end{aligned}$$ Here $F=[J_0,H^U_{nm}]/(2 n) $ (for simplicity we drop the indices $n,m$ on F), and $\langle F;F\rangle^0_{{\omega}}$ is the retarded correlation function of $F$ calculated with respect to $H_{LL}$.
The memory function $M_2$ of the $4 k_F-Q$ process $H^U_{21}$ was calculated by Giamarchi [@giamarchi], (not considering the matrix structure of $\hat{M}$ required by the conservation laws.) Higher Umklapps are considered in [@millis]. For $n_s=0$ and even $n$ the memory function due to the term (\[boso\]) can be analytically calculated, $$\begin{aligned}
\label{M2}
M_n(\Delta k,{\omega})=&&\frac{2 \sin 2 \pi K^n_\rho}{\pi^4 \alpha^{2n-2} v_\rho }
\left[\frac{ 2 \pi \alpha T}{v_\rho} \right]^{4 K^n_\rho-2} \frac{1}{i {\omega}}\times \nonumber\\
\times [B(K^n_\rho-&&i S_+,1-2 K^n_\rho) B(K^n_\rho-i S_+,1-2 K^n_\rho) \nonumber \\
- B(K^n_\rho-&&i S_+^0,1-2 K^n_\rho) B(K^n_\rho-i S_+^0,1-2 K^n_\rho)]
\nonumber\\
&&\hspace*{-1cm}\approx
\frac{\alpha^{2-2n}}{ \pi^2 \Gamma^2(2K^n_\rho) v_\rho T}
\left( \frac{ \alpha \Delta k}{2} \right)^{4K_{\rho}^n-2}
e^{-v_\rho \Delta k /(2 T)}
\nonumber \end{aligned}$$ where $K^n_\rho=(n/2)^2K_\rho$, $B(x,y)=\Gamma(x) \Gamma(y)/\Gamma(x+y)$ and $S_{\pm}=({\omega}\pm v_\rho
\Delta k)/(4 \pi T)$, $S_{\pm}^0=S_{\pm}({\omega}=0)$. The last line is valid for ${\omega}=0$ and $T \ll v_\rho \Delta k$.
The origin of the exponential factor is as follows: processes involving momentum transfer $\Delta k$ are associated with initial and final states of energies $v | \Delta k|/2$, which are exponentially suppressed. If only charge degrees of freedom are involved $v=v_\rho$, otherwise $v=\text{min}(v_\sigma,v_\rho)=v_\sigma$ [@LL]. For $T \ll v_{\sigma} \Delta k_{nm} $, $n_s>0$ and ${\omega}=0$, we have, $$\begin{aligned}
\label{MlargeN}
M_n(\Delta k)\sim \frac{\left(\alpha T/v_\rho\right)^{n^2 K_\rho-1}
\left(\alpha \Delta k \right)^{n_s^2 K_\sigma-2}}{
\Gamma^2(n_s^2 K_\sigma/2) v_\sigma^2 \alpha^{2n-3}} e^{-v_{\sigma} \Delta k/(2 T)},\end{aligned}$$ while for $T \gg v_{\rho} \Delta k_{nm}$: $ M_{n}\sim
T^{n^2 K_\rho+n_s^2 K_\sigma-3}$.
Using the above expressions with only one Umklapp term leads to a finite Drude weight (cf eq(\[sigmajjj\])), $$\begin{aligned}
\label{DT}
D(T)\approx \frac{ v_\rho K_\rho}{\pi} \frac{1}{1 +T^2
\frac{2 \pi^2 n^2 K_\rho}{3 (v_\rho \Delta k_{nm})^2}
\left(1+\frac{v_\rho^3}{v_\sigma^3}\right)}.\end{aligned}$$ in accord with the observation that one process $H^U_{nm}$ is not sufficient to degrade the current.
Only in the presence of a second incommensurate process $H^U_{n'm'}$ is the dc conductivity finite, $$\begin{aligned}
\label{sigT}
\sigma(T, \omega=0)=\frac{(\Delta k_{nm})^2/M_{n'm'}+ (\Delta
k_{n'm'})^2/M_{nm}}{\pi^2 (n \Delta k_{n'm'}-n' \Delta k_{nm})^2}\end{aligned}$$ Note that the [*slowest*]{} process determines the low-$T$ conductivity. The frequency and temperature dependence of the conductivity in the case of two competing Umklapp terms is shown in Fig. \[fig2Umklapp\].
The commensurate situation $\Delta k_{nm}=0$ requires extra considerations. Whether the dominant scattering process $H_{nm}$ will completely relax the current $J$ depends according to (\[sigmajjj\]) on the overlap $\chi_{J P_T}$ ($[P_T,H_{nm}]=0$). Using the continuity equation for the charge, $\chi_{J P_T}$ can be related to the deviation $\Delta \rho=2 \Delta n/a$ of the electron density from commensurate filling with the remarkable identity $\chi_{J P_T}=2 \Delta n/a+O(e^{-\beta\epsilon_F})$. In a 3d lattice of 1d wires, $ \Delta n$ is fixed by charge neutrality and is $T$ independent, in a single wire with contacts $ \Delta n$ varies at low $T$ with $\Delta n(T) \sim T^2/(m v^3)$, where the mass $m$ is a measure of the breaking of particle-hole symmetry, e.g. due to a band-curvature $ k^2/2 m$. In this case it is important to replace $\Delta k_{nm}=0$ in Eqn. (\[DT\]) or (\[sigT\]) by $G \Delta n(T)$.
Which of the various scattering processes will eventually dominate at lowest $T$? At intermediate temperatures, certainly low-order (small $n$) scattering events win, being less suppressed by Pauli blocking. At lower temperature the exponential factors in (\[MlargeN\]) prevail and the processes with the smallest $\Delta k_{nm}$ are favored. We first analyze the situation close to a commensurate point $k_F\approx G M_0/(2N_0)$. The [*two*]{} dominant processes are $H^U_{N_0 M_0}$ with $\Delta k_{N_0 M_0}\approx 0$ and $H^U_{N_1 M_1}$ with $\Delta k_{N_1
M_1}=\pm G/ N_0$ (or $N_1 M_0 = \pm 1 \text{ mod } N_0$). The integer $N_1$ of order $N_0$, $N_1= \gamma_1 N_0$, depends strongly on the precise values of $N_0$ and $M_0$. We thus find that the d.c. conductivity at low $T$ is [*largest*]{} close to commensurate points with, $$\begin{aligned}
\label{sigmaKomm}
\sigma(k_F \approx G M_0/(2 N_0))\sim (\Delta n(T))^2 \exp[\beta v G/(2 N_0)]\end{aligned}$$ but $\sigma \sim T^{-N_0^2 K_\rho-(N_0 \text{ mod } 2)^2 K_\sigma+3}$ if the density is exactly commensurate with $|\Delta n(T)|< e^{-\beta G v/(4 N_0)}$.
To estimate the conductivity at a typical “incommensurate” point or at commensurate points at temperatures not too low, we have to balance algebraic and exponential suppression in (\[MlargeN\]) by minimizing $-\beta v G/(2 N) + (\gamma_1 N)^2 K \log[T]$ in a saddle-point approximation to the sum over all Umklapp processes in ${\hat{M}}$. Up to logarithmic corrections we obtain $N_{\text{max}}^3 \sim \beta v G/( \gamma_1)^2$ and therefore for a “typical” incommensurate filling, $$\begin{aligned}
\sigma_{\text{typical}}\sim \exp[c (\beta v G)^{2/3}]\end{aligned}$$ where $c$ is a number depending logarithmically on $T$. At present we cannot rule out that various logarithmic corrections sum up to modify the power law in the exponent. We argue, however, that due to the exponential increase (\[sigmaKomm\]) of $\sigma$ at commensurate fillings with exponents proportional to $1/N_0$, the conductivity at small $T$ at any incommensurate point is smaller than any exponential (but is larger than any power since any single process is exponentially suppressed). In Fig. [\[fractal\]]{} we show schematically the conductivity as a function of filling becoming more and more “fractal-like ” for lower $T$.
Can the effects we predict be seen experimentally? The complicated structures as a function of filling shown in Fig. \[fractal\] are not observable in practice as they occur only at exponentially large conductivities. The $T$-dependence of the conductivity at intermediate temperatures, however, should be accessible, e.g. by comparing the conductivities of clean wires of different length. Perhaps more importantly, it is straightforward to apply our method to a large number of other relevant situation, e.g. close to a Mott transition or in the presence of 3d phonons, as we will discuss in a forthcoming paper.
We thank R. Chitra, A.J. Millis, E. Orignac, A.E. Ruckenstein, S. Sachdev, and P. Wölfle for helpful discussions. Part of this work was supported by the A. v. Humboldt Foundation and NSF grant DMR9632294 (A.R.).
J. Sólyom, Adv. Phys. [**28**]{}, 209 (1979); V.J. Emery in [*Highly Conducting One-Dimensional Solids*]{}, eds. J. Devreese [*et al.*]{} (Plenum, New York, 1979), p. 247. T. Giamarchi, Phys. Rev. B [**44**]{}, 2905 (1991). T. Giamarchi and A.J. Millis, Phys. Rev. B [**[46]{}**]{}, 9325 (1992). S. Fujimoto and N. Kawakami, J. Phys. A [**31**]{}, 465 (1998); X. Zotos, Phys. Rev. Lett. [**82**]{}, 1764 (1998); H. Castella, X. Zotos, and P. Prelov$\check{\text{s}}$ek, Phys. Rev. Lett. [**74**]{}, 972 (1995). X. Zotos, F. Naef, and P. Prelov$\check{\text{s}}$ek, Phys. Rev. B [**55**]{}, 11029 (1997). S. Kirchner [*et al.*]{}, Phys. Rev. B [**59**]{}, 1825 (1999); S. Sachdev and K. Damle, Phys. Rev. Lett. [**78**]{}, 943 (1997); V.V. Ponomarenko and N. Nagaosa, Phys. Rev. Lett. [**79**]{}, 1714 (1997); A.A. Odintsov, Y. Tokura, S. Tarucha, Phys. Rev. B [**56**]{}, 12729 (1997); M. Mori, M. Ogata, H. Fukuyama, J. Phys. Soc. J. [**66**]{}, 3363 (1997). K. Le Hur, cond-mat/0001439. D. Forster, [*Hydrodynamic Fluctuations, Broken Symmetry, and Correlation Functions*]{}, (Benjamin, Massachusetts, 1975). W. Götze and P. Wölfle, Phys. Rev. B [**6**]{}, 1226 (1972). For large $\Delta k_{nm}$ corrections from $H_{\text{irr}}$ like band-curvature terms are important, we neglect them here Their inclusion leads again to exponential suppression in eq(\[MlargeN\]) with modified numerical factors.
|
---
author:
- '[^1] [^2] [^3] [^4]'
bibliography:
- 'MIT\_broadcast\_bibliography.bib'
title: |
Throughput-Optimal Multihop Broadcast on\
Directed Acyclic Wireless Networks
---
[^1]: Part of the paper appeared in the proceedings of INFOCOM, 2015, IEEE.
[^2]: This work was supported by NSF Grant CNS-1217048, ONR Grant N00014-12-1-0064, and ARO MURI Grant W911NF-08-1-0238
[^3]: The work of G. Paschos was done while he was at MIT and affiliated with CERTH-ITI, and it was supported in part by the WiNC project of the Action: Supporting Postdoctoral Researchers, funded by national and Community funds (European Social Fund).
[^4]: The work of C.p.Li was done when he was a Postdoctoral scholar at LIDS, MIT.
|
---
abstract: 'This paper gives an overview of current trends in manual indexing on the Web. Along with a general rise of user generated content there are more and more tagging systems that allow users to annotate digital resources with tags (keywords) and share their annotations with other users. Tagging is frequently seen in contrast to traditional knowledge organization systems or as something completely new. This paper shows that tagging should better be seen as a popular form of manual indexing on the Web. Difference between controlled and free indexing blurs with sufficient feedback mechanisms. A revised typology of tagging systems is presented that includes different user roles and knowledge organization systems with hierarchical relationships and vocabulary control. A detailed bibliography of current research in collaborative tagging is included.'
author:
- '*Jakob Voß*'
bibliography:
- 'isi2007voss.bib'
date: January 2007
title: '[Tagging, Folksonomy & Co – Renaissance of Manual Indexing?]{}[^1]'
---
[ tagging, indexing, knowledge organization, typology]{}
[ Content Analysis and Indexing]{}
[ Information Retrieval, [DL.]{} Digital Libraries]{}
[ Index languages, processes and schemes]{}
Introduction
============
The World Wide Web, a framework originally designed for information management [@BernersLee1989], has long ago become a heterogeneous, exponentially growing mass of connected, digital resources. After first, unsuccessful attempts to classify the Web with traditional, intellectual methods of library and information science, the standard to search the Web is now fulltext indexing – most of all made popular by Google’s PageRank algorithm. The success of such automatic techniques is a reason why “many now working in information retrieval seem completely unaware that procedures other than fully automatic ones have been applied, with some success, to information retrieval for more then 100 years, and that there exist an information retrieval literature beyond that of the computer science community.”[@Lancaster2003] However in the recent years there is a renaissance of manual subject indexing and analysis: Structured metadata is published with techniques like RSS, OAI-PMH, and RDF. OpenSearch[^2] and browser search plugins allow it to aggregate specialised search engines. Last but not least many popular social software systems contain methods to annotate resources with keywords. This type of manual indexing is called *tagging* with index terms referred to as *tags*. Based on [@Marlow2006] this paper presents a revised typology of tagging systems that also includes systems with controlled and structured vocabularies. Section \[TaggingIntroduction\] gives a short introduction to current tagging systems and its research. Afterwards (section \[IndexingProcess\]) theory of subject indexing is pictured with the indexing process, typology of knowledge organization systems, and an unconventional look at vocabulary control. In section \[TaggingTypology\] the typology of tagging systems is presented with conclusion in section \[Conclusion\].
Tagging systems on the rise {#TaggingIntroduction}
===========================
Tagging is referred to with several names: *collaborative tagging*, *social classification*, *social indexing*, *folksonomy* etc. The basic principle is that end users do subject indexing instead of experts only, and the assigned tags are being shown immediately on the Web. The number of websites that support tagging has rapidly increased since 2004. Popular examples are del.icio.us (<http://del.icio.us>), furl (<http://furl.net>), reddit (<http://reddit.com>), and Digg (<http://digg.net>) for bookmarks [@Hammond2005] and flickr (<http://flickr.com>, [@Winget2006]) for photos. Weblog authors usually tag their articles and specialized search engines like Technorati (<http://technorati.com/>) and RawSugar (<http://rawsugar.com>) make use of it. But tagging is not limited to simple keywords only: BibSonomy (<http://bibsonomy.org>, [@Hotho2006Emergent; @Hotho2006Entstehen]), Connotea (<http://connotea.org>, [@Lund2005]), CiteULike (<http://citeulike.org/>), and LibraryThing (<http://librarything.com>) allow users to manage and share bibliographic metadata on the Web (also known as *social reference managing* or *collaborative cataloging*). Additionally to librarian’s subject indexing the University of Pennsylvania Library allows users to tag records in their online catalog since 2005 (<http://tags.library.upenn.edu/>). Other systems to tag bibliographic data are LibraryThing (<http://www.librarything.com>) and Amazon’s tagging feature (<http://amazon.com/gp/tagging/cloud/>). The popular free encyclopedia Wikipedia contains so called categories that are used as hierarchical tags to order the articles by topic [@Voss2006]. Apart from social software there is also a rise of manual indexing in other fields [@Wright2005; @Maislin2005].
The details of tagging vary a lot but all applications are designed to be used as easy and as open as possible. Sometimes the greenness in theory of users and developers let you stumble upon known problems like homonyms an synonyms but on the other hand unloaded trial and error has led to many unconventional and innovative solutions.
Research on Tagging
-------------------
The astonishing popularity of tagging led some even claim that it would overcome classification systems [@Shirky2005], but it is more likely that there is no dichotomy [@Crawford2006] and tagging just stresses certain aspects of subject indexing. Meanwhile serious research about collaborative tagging is growing — hopefully it will not have to redo all the works that has been done in the 20th century. At the 15th World Wide Web Conference there was a *Collaborative Web Tagging Workshop*[^3]. The 17th SIG/CR Classification Research Workshop was about *Social Classification*[^4]. One of the first papers on folksonomies is [@Mathes2004]. Shirky’s paper [@Shirky2005] has reached huge impact. It is probably outdated but still worth to read. A good overview until the beginning of 2006 is given in [@Macgregor2006]. Some papers that deal with specific tagging systems are cited at the beginning of this section at page . Trant and Smith describe the application of tagging in a museum [@Trant2006a; @Trant2006b; @Smith2006]. Other works focus on tagging in enterprises [@Farrell2006; @John2006; @Damianos2006; @Millen2006] or knowledge management [@Wu2006]. Another application is tagging people to find experts [@Bogers2006; @Farrell2006]. Mathematical models of tagging are elaborated in [@Tosic2006; @Lambiotte2005]. The usual model of tagging is a tripartite graph, the nodes being resources, users, and tags [@Lambiotte2005]. Several papers provide statistical analysis of tagging over time and evolution of tagging systems [@Kowatsch2007; @Cattuto2006; @Dubinko2006; @Hotho2006Trend; @Lin2006]. Tagging behaviour is also topic of Kipp and Campbell [@Kipp2006] and Feinberg [@Feinberg2006]. Types of structured and compound tagging are analyzed in [@BarIlan2006; @Tonkin2006; @Guy2006]. Like in traditional scientometrics you can find communities and trends based on tagging data [@Jaeschke2006; @Hotho2006Trend]. Voss [@Voss2006] finds typical distributions among different types of tagging systems and compares tagging systems with traditional classification and thesaurus structures. Tennis [@Tennis2006] uses framework analysis to compare social tagging and subject cataloguing. Tagging is manual indexing instead of automatic indexing. Ironically a focus of research is again on automatic systems that do data mining in tagging data [@Aurnhammer2006; @Hotho2006Emergent; @Hotho2006Information; @Schmitz2006Ontology; @Schmitz2006Mining]. Heymann and Garcia-Molina[@Heymann2006] presented an algorithm to automatically generate hierarchies of tags out of flat, aggregated tagging systems with del.icio.us data. Similar approaches are used by Begelman et al. [@Begelman2006] and Mika et al. [@Mika2005]. Research on tagging mostly comes from computer science and library science — obviously there is a lack of input from psychology, sociology, and cognitive science in general (an exception from philosophy is Campell [@Campbell2006] who applies Husserl’s theory of phenomenology to tagging).
The indexing process {#IndexingProcess}
====================
[r]{}[8cm]{}
The main purpose of subject indexing is to construct a representation of a resource that is being tagged. According to Lancaster [@Lancaster2003 chapter 2] subject indexing involves two steps: conceptual analysis and translation (see figure \[indexingProcess\]). These are intellectually separate although they are not always clearly distinguished. The semiotic triangle can be applied to indexing to demonstrate the distinction between object (resource), concept (what the resource is about), and symbol (set of tags to represent the resource). *Conceptual analysis* involves deciding on what a resource is about and what is relevant in particular. Note that the result of conceptual analysis heavily depends on the needs and interests of users that a resource is tagged for — different people can be interested in different aspects. *Translation* is the process of finding an appropriate set of index terms (tags) that represent the substance of conceptual analysis. Tags can be extracted from the resource or assigned by an indexer. Many studies have shown that high consistency among different indexers is very difficult to achieve and affected by many factors [@Lancaster2003 chapter 3]. One factor is control of the vocabulary that is used for tagging. Synonyms (multiple words and spellings for the same concept) and homonyms/homographs (words with different meanings) are frequent problems in the process of translation. A controlled vocabulary tries to eliminate them by providing a list of preferred and non-preferred terms, often together with definitions and a semantic structure. Controlled vocabularies are subsumed as knowledge organization systems (KOS) [@Zeng2004]. These systems have been studied and developed in library and information science for more then 100 years. Popular examples are the Dewey Decimal Classification, Ranganathan’s faceted classification, and the first thesauri in the 1960s. Beginning with the 1950s library and information science has lost its leading role in the development of information retrieval systems and a rich variety of KOS has come into existence. However it is one of the constant activities of this profession to summarize and evaluate the complexity of attempts to organize the world’s knowledge.
Typology of knowledge organization systems
------------------------------------------
Hodge, Zeng, and Tudhope [@Hodge2000; @Zeng2000; @Tudhope2006] distinguish by growing degree of language control and growing strength of semantic structure: term list, classifications and categories, and relationship groups. Term lists like authority files, glossaries, gazetteers, and dictionaries emphasize lists of terms often with definitions. Classifications and categories like subject headings and classification schemes (also known as taxonomies) emphasize the creation of subject sets. Relationships groups like thesauri, semantic networks, and ontologies emphasize the connections between concepts. Apart from the training of what now may be called *ontology engineers* the theoretical research on knowledge organization systems has had little impact on technical development. Only now common formats are being standardized with SKOS[^5], the microformats movement[^6] and other initiatives. Common formats are a necessary but not sufficient condition for interoperability among knowledge organization systems — an important but also frequently underestimated task [@Zeng2004; @Mayr2006].
Vocabulary control and feedback
-------------------------------
In the process of indexing the controlled vocabulary is used to supply translation via *feedback* (figure \[indexingProcess\]). The indexer searches for index terms supported by the structure of the knowledge organization systems until he finds the best matching tag. Also search is supported by the structure of the knowledge organization systems. Collaborative tagging also provide feedback.
A special kind of tagging system is the category system of Wikipedia. The free encyclopedia is probably the first application of collaborative tagging with a thesaurus [@Voss2006]. The extend of contribution in Wikipedia is distributed very inhomogeneously (more precise it is a power law [@Voss2005]) — this also applies for the category system. Everyone is allowed to change and add categories but most authors only edit the article text instead of tagging articles and even less authors change and add the category system. Furthermore each article is not tagged independently by every user but users have to agree on a single set of categories per article. So tagging in Wikipedia is somewhere between indexing with a controlled vocabulary and free keywords. Most of the time authors just use the categories that exist but they can also switch to editing the vocabulary at any time. The emerging system may look partly chaotic but rather useful. With a comparison of Wikipedia and the AGROVOC[^7] thesaurus Milne et al. [@Milne2006] show that domain-specific thesauri can be enriched and created with Wikipedia’s category and link structure.
We can deduce that the border between free keyword tagging on the one hand and tagging with a controlled vocabulary is less strict. Although most tagging systems do not implement vocabulary control there is almost always some feedback that influences tagging behaviour towards consensus: the Folksonomy emerges [@Mathes2004]. The phenomena is also known as *emergent semantics* or *Wisdom of the crowds* (But you should keep in mind that masses do not always act wise – see Lanier’s critic of ‘Digital Maoism’ [@Lanier2006]).
Typology of tagging systems {#TaggingTypology}
===========================
Based on Marlow’s taxonomy of tagging systems[@Marlow2006] I provide a revised typology. The following key dimensions do not represent simple continuums but basic properties that should be clarified for a given tagging system — so they are presented here as questions.
Tagging Rights
: Who is allowed to tag resources? Can any user tag any resource or are there restrictions? Are restrictions based on resources, tags, or users? Who decides on restrictions? Is there a distinction between tags by different types of users and resources?
Source of Resources
: Do users contribute resources and have resources been created or just supplied by users? Or do users tag resources that are already in the system? Who decides which resources are tagged?
Resource Representation
: What kind of resource is being tagged? How are resources presented while tagging (autopsi principle)?
Tagging Feedback
: How does the interface support tag entry? Do users see other tags assigned to the resource by other users or other resources tagged with the same tags? Does the system suggest tags and if so based on which algorithms? Does the system reject inappropriate tags?
Tag Aggregation
: Can a tag be assigned only once to a resource (set-model) or can the same tag can be assigned multiple times (bag-model with aggregation)?
Vocabulary control
: : Is there a restriction on which tags to use and which tags not to use? Are tags created while tagging or is management of the vocabulary a separated task? Who manages the vocabulary, how frequently is it updated, and how are changes recorded?
Vocabulary Connectivity
: Are tags connected with relations? Are relations associative (authority file), monohierarchical (classification or taxonomy), multihierarchical (thesaurus), or typed (ontology)? Where do the relations come from? Are relations limited to the common vocabulary (precoordination) or can they dynamically be used in tagging (postcoordination with syntactic indexing)?
Resource Connectivity
: How are resources connected to each other with links or grouped hierarchically? Can resources be tagged on different hierarchy levels? How are connections created?
Automatic Tagging
: Is tagging enriched with automatically created tags and relations (for instance file types, automatic expansion of terms etc.)?
The analysis shows that the classic tripartite model of tagging with resources, users, and tags is too simplified to cover the variety of tagging system. Depending on the application you can distinguish different kind of resources, tags, and users. At least you should distinguish four user roles:
1. [Resource Author]{} A person that creates or edits a resource
2. [Resource Collector]{} A person that adds a resource to a tagging system
3. [Indexer or Tagger]{} A person that tags resources
4. [Searcher]{} A person that uses tags to search for resources
In most systems some of the roles overlap and people can fullfill different roles at different times (large libraries may be a counterexample). For instance the author of a private blog combines 1, 2, and 3, a user of del.icio.us combines either 2 and 3 (tagging a new webpage) or 3 and 4 (copying a webpage that someone else has already tagged), and a Wikipedia author combines either 1 and 2 (new articles) or 1 and 3 (existing articles).
Conclusion {#Conclusion}
==========
The popularity of collaborative tagging on the Web has resurged interest in manual indexing. Tagging systems encourage users to manually annotate digital objects with free keywords and share their annotations. Tags are directly assigned by anyone who likes to participate. The instant visibility is motivation and helps to install feedback mechanism. Through feedback the drawbacks of uncontrolled indexing are less dramatic then in previous systems and the border between controlled and free indexing starts to blur. Vocabulary control and relationships between index terms should not be distinctive features of tagging systems and traditional knowledge organization systems but possible properties of manual indexing systems. Further research is needed to find out under which circumstances which features (for instance vocabulary control) are needed and how they influence tagging behaviour and evolution of the tagging system. The typology of tagging systems that was presented in section \[TaggingTypology\] combines all of them. The possibility to allow non-experts to add keywords has made collaborative tagging so popular — but it is nothing fundamentally new. Perhaps the most important feature of tagging systems on the Web is its implementation or how Joseph Busch entitled his keynote speech at the ASIST SIG-CR workshop: “It’s the interface, stupid!” Today’s tagging websites make many traditional knowledge organization systems look like stone age technique: effective but just too uncomfortable. Some of the costly created thesauri and classifications are not even accessible in digital form at all (because of licensing issues grounded in a pre-digital understanding of copyright or because of a lack of technological skills)! But also computer scientists tend to forget that a clever interface to support tagging can be worth much more than any elaborated algorithm. Anyway the art of creating interfaces for developed tagging systems is still in its infancy. Knowledge organization will always need manual input so it is costly to manage — but Wikipedia showed that groups of volunteers can create large knowledge resources if a common goal and the right toolkit exist! And obviously there is not one way of indexing that fits for all applications. Collaborative Tagging is neither the successor of traditional indexing nor a short-dated trend but — like Tennis [@Tennis2006] concludes — a catalyst for improvement and innovation in indexing.
[^1]: Submitted to the 10th International Symposium for Information Science, Cologne.
[^2]: To gain an insight on RSS, OAI-PMH, RDF, and OpenSearch see <http://en.wikipedia.org/>.
[^3]: [
http://www.rawsugar.com/www2006/taggingworkshopschedule.html](
http://www.rawsugar.com/www2006/taggingworkshopschedule.html)
[^4]: <http://www.slais.ubc.ca/users/sigcr/events.html>
[^5]: <http://www.w3.org/2004/02/skos/>
[^6]: <http://microformats.org/>
[^7]: <http://www.fao.org/agrovoc>
|
---
abstract: 'We analyze various prominent quantum repeater protocols in the context of long-distance quantum key distribution. These protocols are the original quantum repeater proposal by Briegel, Dür, Cirac and Zoller, the so-called hybrid quantum repeater using optical coherent states dispersively interacting with atomic spin qubits, and the Duan-Lukin-Cirac-Zoller-type repeater using atomic ensembles together with linear optics and, in its most recent extension, heralded qubit amplifiers. For our analysis, we investigate the most important experimental parameters of every repeater component and find their minimally required values for obtaining a nonzero secret key. Additionally, we examine in detail the impact of device imperfections on the final secret key rate and on the optimal number of rounds of distillation when the entangled states are purified right after their initial distribution.'
author:
- Silvestre Abruzzo
- Sylvia Bratzik
- Nadja K Bernardes
- Hermann Kampermann
- Peter van Loock
- 'Dagmar Bru[ß]{}'
bibliography:
- 'allbib.bib'
title: 'Quantum repeaters and quantum key distribution: analysis of secret key rates'
---
Introduction
============
Quantum communication is one of the most exciting and well developed areas of quantum information. Quantum key distribution (QKD) is a sub-field, where two parties, usually called Alice and Bob, want to establish a secret key. For this purpose, typically, they perform some quantum operations on two-level systems, the qubits, which, for instance, can be realized by using polarized photons. [@ekert1991quantum; @PhysRevA.84.022325; @PhysRevA.84.010304; @PhysRevLett.105.070501; @PhysRevLett.98.230501].
Photons naturally have a long decoherence time and hence could be transmitted over long distances. Nevertheless, recent experiments show that QKD so far is limited to about 150 km [@Scarani:2009], due to losses in the optical-fiber channel. Hence, the concept of quantum relays and repeaters was developed [@briegel_quantum_1998; @RevModPhys.74.145; @gisinrelay; @PhysRevLett.92.047904; @PhysRevA.65.052310]. These aim at entangling qubits over long distances by means of entanglement swapping and entanglement distillation. There exist various proposals for an experimental implementation, such as those based upon atomic ensembles and single-rail entanglement [@duan_long-distance_2001], the hybrid quantum repeater [@van_loock_hybrid_2006], the ion-trap quantum repeater [@PhysRevA.79.042340], repeaters based on deterministic Rydberg gates [@PhysRevA.81.052329; @PhysRevA.81.052311], and repeaters based on nitrogen-vacancy (NV) centers in diamond [@childress_fault-tolerant_2005].
In this paper, we analyze the performance of quantum repeaters within a QKD set-up, for calculating secret key rates as a function of the relevant experimental parameters. Previous investigations on long-distance QKD either consider quantum relays [@gisinrelay; @PhysRevA.65.052310; @scherer_long-distance_2011], which only employ entanglement swapping without using quantum memories or entanglement distillation, or, like the works in [@razavi_quantum_2010; @amirloo_quantum_2010], they exclusively refer to the original Duan-Lukin-Cirac-Zoller (DLCZ) quantum repeater [@duan_long-distance_2001]. Finally, in [@2012arXiv1210.8042L] the authors analyze a variation of the DLCZ protocol [@2007PhRvA.76e0301S] where they consider at most one repeater station. Here, our aim is to quantify the influence of characteristic experimental parameters on the secret key rate for three different repeater schemes, namely the original quantum repeater protocol [@briegel_quantum_1998], the hybrid quantum repeater [@van_loock_hybrid_2006], and a recent variation of the DLCZ-repeater [@minar]. We investigate the minimally required parameters that allow a non-zero secret key rate. In order to reduce the complexity of the full repeater protocol, we consider entanglement distillation only directly after the initial entanglement distribution. Within this scenario, we investigate also the optimal number of distillation rounds for a wide range of parameters. The influence of distillation during later stages of the repeater, as well as the comparison between different distillation protocols, will be studied elsewhere [@Bratzik2012].
This manuscript is organized as follows: In [Sec. \[sec:genfrac\]]{} we present a description of the relevant parameters of a quantum repeater, as well as the main tools for analyzing its performance for QKD. This section should also provide a general framework for analyzing other existing quantum repeater protocols, and for studying the performance and the potential of new protocols. Sections \[sec:briegel\], \[sec:Hybrid\], and \[sec:atomicensenbles\] investigate long-distance QKD protocols for three different quantum repeater schemes; these sections can be read independently. Section \[sec:briegel\] is devoted to the original proposal for a quantum repeater [@briegel_quantum_1998], section \[sec:Hybrid\] analyzes the hybrid quantum repeater [@van_loock_hybrid_2006], and finally, in section \[sec:atomicensenbles\], we investigate quantum repeaters with atomic ensembles [@duan_long-distance_2001]. The conclusion will be given in section \[sec:conclusions\], and more details on the calculations will be presented in the appendix.
General framework {#sec:genfrac}
=================
Quantum repeater
----------------
The purpose of this section is to provide a general framework that describes formally the theoretical analysis of a quantum repeater.
### The protocol
Let $L$ be the distance between the two parties Alice and Bob who wish to share an entangled state. A quantum repeater [@briegel_quantum_1998] consists of a chain of $2^N$ segments of fundamental length $L_0:=L/2^N$ and $2^N-1$ repeater stations which are placed at the intersection points between two segments (see [Fig. \[fig:qr\]]{}). Each repeater station is equipped with quantum memories and local quantum processors to perform entanglement swapping and, in general, also entanglement distillation. In consecutive *nesting levels*, the distances over which the entangled states are shared will be doubled. The parameter $N$ is the *maximal nesting level*.
![Scheme of a generic quantum repeater protocol. We adopt the nested protocol proposed in [@briegel_quantum_1998]. The distance between Alice and Bob is $L$, which is divided in $2^{N}$ segments, each having the length $L_0:=L/2^{N}$. The parameter $n$ describes the different nesting levels, and the value $N$ represents the maximum nesting level. In this paper, we consider quantum repeaters where distillation is performed exclusively before the first entanglement swapping step. The number of distillation rounds is denoted by $k$. \[fig:qr\]](figure1){width="8cm"}
The protocol starts by creating entangled states in all segments, i.e., between two quantum memories over distance $L_0$. After that, if necessary, entanglement distillation is performed. This distillation is a probabilistic process which requires sufficiently many initial pairs shared over distance $L_0$. As a next step, entanglement swapping is performed at the corresponding repeater stations in order to connect two adjacent entangled pairs and thus gradually extend the entanglement. In those protocols where entanglement swapping is a probabilistic process, the whole quantum repeater protocol is performed in a recursive way as shown in [Fig. \[fig:qr\]]{}. Whenever the swapping is deterministic (i.e., it never fails), then all swappings can be executed simultaneously, provided that no further probabilistic entanglement distillation steps are to be incorporated at some intermediate nesting levels for enhancing the fidelities. Recall that in the present work, we do not include such intermediate distillations in order to keep the experimental requirements as low as possible. At the same time it allows us to find analytical rate formulas with no need for numerically optimizing the distillation-versus-swapping scheduling in a fully nested quantum repeater.
### \[subsec:parameters\]Building blocks of the quantum repeater and their imperfections
In this section we describe a model of the imperfections for the main building blocks of a quantum repeater. In an experimental set-up more imperfections than those considered in this model may affect the devices. However, most of them can be incorporated into our model. We point out that if not all possible imperfections are included, the resulting curves for the figure of merit (throughout this paper: the secret key rate) can be interpreted as an upper bound for a given repeater protocol.
#### Quantum channel
Let us consider photons (in form of single- or multi-photon pulses) traveling through optical fibers.
Photon losses are the main source of imperfection. Other imperfections like birefringence are negligible in our context [@RevModPhys.74.145; @sangouard_quantum_2011]. Losses scale exponentially with the length $\ell$, i.e., the transmittivity is given by [@RevModPhys.74.145] $${\eta_{t}\left(\ell\right)}:=10^{-\frac{\alpha_{att} \ell}{10}},
\label{eq:etaT}$$ where $\alpha_{att}$ is the attenuation coefficient given in dB/km. The lowest attenuation is achieved in the telecom wavelength range around 1550 nm and it corresponds to $\alpha_{att}=0.17$ dB/km. This attenuation will also be used throughout the paper. Note that other types of quantum channels, such as free space, can be treated in an equivalent way (see e.g. [@tatarski1961wave]). Further note that besides losses, the effect of the quantum channel can be incorporated into the form of the initial state shared between the connecting repeater stations.
#### Source of entanglement
The purpose of a source is to create entanglement between quantum memories over distance $L_0$. An ideal source produces maximally entangled Bell states (see below) on demand. In practice, however, the created state may not be maximally entangled and may be produced in a probabilistic way. We denote by $\rho_0$ a state shared between two quantum memories over the elementary distance $L_0$ and by $P_0$ the total probability to generate and distribute this state. This probability would contain any finite local state-preparation probabilities before the distribution, the effect of channel losses, and the success probabilities of other processes, such as the conditioning on a desired initial state $\rho_0$ after the state distribution over $L_0$.
For improving the scaling over the total distance $L$ from exponential to sub-exponential, it is necessary to have a heralded creation and storage of $\rho_0$. How this heralding is implemented depends on the particular protocol and it usually involves a form of post-processing, e.g. conditioning the state on a specific pattern of detector clicks. This can also be a finite postselection window of quadrature values in homodyne detection. However, in the present work, the measurements employed in all protocols considered here are either photon-number measurements or Pauli measurements on memory qubits.
#### Detectors
We will consider photon-number resolving detectors (PNRD) which can be described by a positive-operator valued measure (POVM) with elements [@kok_introduction_2010] $$\label{eq:POVMPNRD}
\Pi^{(n)}:={\eta_\mathrm{d}}^{n}\sum_{m=0}^{\infty}{n+m \choose n} (1-{\eta_\mathrm{d}})^m{\left\vert{n+m}\right\rangle}{\left\langle{n+m}\right\vert}.$$ Here, $\Pi^{(n)}$ is the element of the POVM related to the detection of $n$ photons, ${\eta_\mathrm{d}}$ is the efficiency of the detector, and ${\left\vert{n+m}\right\rangle}$ is a state of $(n+m)$-photons. In the POVM above, we have neglected dark counts; we have shown analytically for those protocols considered in this paper that realistic dark counts of the order of $10^{-5}$ are negligible \[see Appendix \[app:general-type\], below Eq. , for the proof\]. Note that our analysis could also be extended to threshold detectors, by replacing the corresponding POVM (see e.g. [@kok_introduction_2010]) in our formulas.
#### Gates
Imperfections of gates also depend on the particular quantum repeater implementation. Such imperfections are e.g. described in [@gilchrist2005distance]. In our analysis, we will characterize them using the gate quality which will be denoted by $p_G$ (see [Eq. ]{} and [Eq. ]{}).
#### Quantum memories
Quantum memories are a crucial part of a quantum repeater. A complete characterization of imperfections of quantum memories is beyond the purpose of this paper (see [@qmemrev] for a recent review). Here we account for memory errors by using a fixed time-independent quantum memory efficiency $\eta_{m}$ when appropriate. This is the probability that a photon is released when a reading signal is applied to the quantum memory, or, more generally, the probability that an initial qubit state is still intact after write-in, storage, and read-out. We discuss the role of $\eta_{m}$ only for the quantum repeater with atomic ensembles (see section \[sec:atomicensenbles\]).
#### Entanglement distillation
As mentioned before, throughout this work we only consider distillation at the beginning of each repeater protocol. Entanglement distillation is a probabilistic process requiring local multi-qubit gates and classical communication. In this paper, we consider the protocol by Deutsch [*et al. *]{}[@deutsch1996quantum]. This protocol performs especially well when there are different types of errors (e.g. bit flips and phase flips). However, depending on the particular form of the initial state and on the particular quantum repeater protocol, other distillation schemes may perform better (see [@Bratzik2012] for a detailed discussion). The Deutsch [*et al. *]{}protocol starts with $2^k$ pairs and after $k$ rounds, it produces one entangled pair with higher fidelity than at the beginning. Every round requires two Controlled Not (CNOT), each performed on two qubits at the same repeater station, and projective measurements with post-selection.
Distillation has two main sources of errors: imperfect quantum gates which no longer permit to achieve the ideal fidelity, as well as imperfections of the quantum memories and the detectors, decreasing the success probability. We denote the success probability in the $i$-th distillation round by $P_{D}[i]$.
We study entanglement distillation for the original quantum repeater protocol (section \[sec:briegel\]) and the hybrid quantum repeater (section \[sec:Hybrid\]). For the quantum repeater with atomic ensembles (section \[sec:atomicensenbles\]), we do not consider any additional distillation on two or more initial memory pairs.
#### Entanglement swapping
In order to extend the initial distances of the shared entanglement, entanglement swapping can be achieved through a Bell measurement performed at the corresponding stations between two adjacent segments. Such a Bell measurement can be, in principle, realized using a CNOT gate and suitable projection measurements on the corresponding quantum memories [@Zukowski1993]. An alternative implementation of the Bell measurement uses photons released from the quantum memories and linear optics [@Pan1998]. The latter technique is probabilistic, but typically much less demanding from an experimental point of view.
We should emphasize that the single-qubit rotation depending on the result of the Bell measurement, as generally needed to complete the entanglement swapping step, is not necessary when the final state is used for QKD applications. In fact, it simply corresponds to suitable bit flip operations on the outcomes of the QKD measurements, i.e., the effect of that single-qubit rotation can be included into the classical post-processing.
Imperfections of entanglement swapping are characterized by the imperfections of the gates (which introduce noise and therefore a decrease in fidelity) and by the imperfections of the measurement process, caused by imperfect quantum memories and imperfect detectors. We denote the probability that entanglement swapping is successful in the $n$-th nesting level by $P_{\rm ES}^{(n)}$.
#### Other imperfections
Other imperfections which are not explicitly considered in this paper but which are likely to be present in a real experiment include imperfections of the interconversion process, fluctuations of the quantum channel, fiber coupling losses and passive losses of optical elements (see [@sangouard_quantum_2011] and reference therein for additional details). These imperfections can be accounted for by a suitable adjustment of the relevant parameters in our model.
### \[subsec:gen\]Generation rate of long-distance entangled pairs
In order to evaluate the performance of a quantum repeater protocol it is necessary to assess how many entangled pairs across distance $L$ can be generated per second.
A relevant unit of time is the *fundamental time* needed to communicate the successful distribution of an elementary entangled pair over distance $L_0$, which is given by: $$\label{eq:fundtime}
T_{0}:=\frac{\beta L_0}{c},$$ where $c=2\cdot10^5$ km/s is the speed of light in the fiber channel (see e.g. [@sangouard_quantum_2011]) and $\beta$ is a factor depending on the type of entanglement distribution. Note that here we have neglected the additional local times needed for preparing and manipulating the physical systems at each repeater station. [Figure \[fig:comm\_time\]]{} shows three different possibilities how to model the initial entanglement distribution. The fundamental time $T_0$ consists of the time to distribute the photonic signals, $T_{dist}$, and the time of acknowledgment, $T_{ack}$, which all together can be different for the three cases shown.
![The fundamental time for different models of entanglement generation and distribution. The source (S) that produces the initial entangled states is either placed in the middle (a), at one side (b), or at both sides (c). In the latter case, photons are emitted from a source and interfere in the middle (see [@Cabrillo1999; @Feng2003]).[]{data-label="fig:comm_time"}](figure2a_2c){width="8cm"}
Throughout the paper, we denote the average number of final entangled pairs produced in the repeater per second by ${R_{\mathrm{REP}}}$. We emphasize that regarding any figures and plots, for each protocol, we are interested in the consumption of time rather than spatial memories. Thus, if one wants to compare different set-ups for the same number of spatial memories, one has to rescale the rates such that the number of memories becomes equal. For example, in order to compare a protocol without distillation with another one with $k$ rounds of distillation, one has to divide the rates for the case with distillation by $2^k$ (as we need two initial pairs to obtain one distilled pair in every round).
In the literature, two different upper bounds on the entanglement generation rate ${R_{\mathrm{REP}}}$ are known. In the case of deterministic entanglement swapping ($P_{ES}^{(n)}=1$) we have [@NadjaHybrid]
$${R_{\mathrm{REP}}}^{\rm det}=\left(T_0 {Z_{N}(P_{L_0}[k])}\right)^{-1},
\label{eq:zn:approx}$$
with $P_{L_0}[i]$ being a recursive probability depending on the rounds of distillation $i$ as follows [@NadjaHybrid] $$\begin{aligned}
P_{L_0}[i=0]&=&P_0,\\
P_{L_0} [i>0]&=&\frac{{P_D}[i]}{Z_{1}(P_{L_0}[i-1])}.
\label{eq:PL0}\end{aligned}$$ We remind the reader that $P_D[i]$ is the success probability in the $i$-th distillation round. Here, $$\label{eq:zn}
Z_{N}(P_0):=\sum_{j=1}^{2^N}{2^N \choose j}\frac{(-1)^{j+1}}{1-(1-P_0)^j}$$ is the average number of attempts to connect $2^N$ pairs, each generated with probability $P_0$.
In the case of probabilistic entanglement swapping, probabilistic entanglement distillation, and $P_0<<1$, we find an upper bound on the entanglement generation rate: $$\label{eq:avgngen}
{R_{\mathrm{REP}}}^{\rm prob}= \frac{1}{T_0}\left(\frac{2}{3 a}\right)^{N+k} P_{0}P_{ES}^{(1)}P_{ES}^{(2)}...P_{ES}^{({N})}\prod_{i=1}^{k}{P_D}[i],$$ with $a\leq\frac{2}{3}P_{L_0}[k] Z_1(P_{L_0}[k])$. Our derivation is given in App. \[sec:app:general\]. For the plots we bound $a$ according to the occuring parameters, typically $a$ is close to one which corresponds to the approximate formula given in [@sangouard_quantum_2011] for the case when there is no distillation.
Equations and should be interpreted as a limiting upper bound on the repeater rate, due to the minimal time needed for communicating the quantum and classical signals. For this minimal time , we consider explicitly only those communication times for initially generating entanglement, but not those for entanglement swapping and entanglement distillation.
Quantum key distribution (QKD) {#sec:QKD}
------------------------------
### The QKD protocol {#subsec:qkd .unnumbered}
In [Fig. \[fig:QKDSetup\]]{} a general quantum key distribution set-up is shown. For long-distance QKD, Alice and Bob will generate entangled pairs using the quantum repeater protocol. For the security analysis of the whole repeater-based QKD scheme, we assume that a potential eavesdropper (Eve) has complete control of the repeater stations, the quantum channels connecting them, and the classical channels used for communicating the measurement outcomes for entanglement swapping and distillation (see figure \[fig:QKDSetup\]). The QKD protocol itself starts with Alice and Bob performing measurements on their shared, long-distance entangled pairs (see figure \[fig:QKDSetup\]). For this purpose, they would both independently choose a certain measurement from a given set of measurement settings. The next step is the classical post-processing and for this an authenticated channel is necessary. First, Alice and Bob discard those measurement outcomes where their choice of the setting did not coincide (sifting), thus obtaining a raw key associated with a *raw key rate*. They proceed by comparing publicly a small subset of outcomes (parameter estimation). From this subset, they can estimate the *quantum bit error rate* (QBER), which corresponds to the fraction of uncorrelated bits. If the QBER is below a certain threshold, they apply an error correction protocol and privacy amplification in order to shrink the eavesdropper’s information about the secret key (for more details, see e.g. [@Renner2008]).
![Scheme of quantum key distribution. The state $\rho_{AB}$ is produced using a quantum repeater. Alice and Bob locally rotate this state in a measurement basis and then they perform the measurement. The detectors are denoted by $d_0^A,d_1^A,d_0^B,d_1^B$ and to each detector click a classical outcome is assigned.[]{data-label="fig:QKDSetup"}](figure3){width="8cm"}
Various QKD protocols exist in the literature. Besides the original QKD protocol by Bennett and Brassard from 1984, the so-called BB84-protocol [@bennett1984quantum], the first QKD protocol based upon entanglement was the Ekert protocol [@ekert1991quantum]. Shortly thereafter the relation of the Ekert protocol to the BB84-protocol was found [@bennett_quantum_1992]. Another protocol which can also be applied in entanglement-based QKD is the six-state protocol [@bruss1998optimal; @6stategisin].
### The quantum bit error rate (QBER)
In order to evaluate the performance of a QKD protocol, it is necessary to determine the quantum bit error rate. This is the fraction of discordant outcomes when Alice and Bob compare a small amount of outcomes taken from a specified measurement basis. This measurement can be modelled by means of four detectors (two on Alice’s side and two on Bob’s side, see figure \[fig:QKDSetup\]) where to each detector click a classical binary outcome is assigned. Particular care is necessary when multi-photon states are measured [@PhysRevA.84.020303; @PhysRevLett.101.093601]. In the following, we give the definition of the QBER for the case of photon-number-resolving detectors and we refer to [@amirloo_quantum_2010] for the definition in the case of threshold detectors. The probability that a particular detection pattern occurs is given by $$\label{eq:detpattern}
P_{jklm}^{(i)}:={\mathrm{tr}}\left(\Pi_{d_0^A}^{(j)}\Pi_{d_1^A}^{(k)}\Pi_{d_0^B}^{(l)}\Pi_{d_1^B}^{(m)}\rho_{AB}^{(i)}\right),$$ where the POVM $\Pi^{(n)}$ has been defined in [Eq. ]{} with a subscript denoting the detectors given in [Fig. \[fig:QKDSetup\]]{}. The superscript $i$ refers to the measurement basis and $\rho_{AB}^{(i)}$ represents the state $\rho_{AB}$ rotated in the basis $i$.
A valid QKD measurement event happens when one detector on Alice’s side and one on Bob’s side click. The probability of this event is given by [@amirloo_quantum_2010] $$P_{\rm click}^{(i)}:= P_{1010}^{(i)}+ P_{0101}^{(i)}+P_{0110}^{(i)}+P_{1001}^{(i)}.$$ The probability that two outcomes do not coincide is given by [@amirloo_quantum_2010] $$P_{\rm err}^{(i)}:= P_{0110}^{(i)}+P_{1001}^{(i)}.$$ Thus, the fraction of discordant bits, i.e., the quantum bit error rate for measurement basis $i$ is [@amirloo_quantum_2010] $$e_{i}:=\frac{P_{\rm err}^{(i)}}{P_{\rm click}^{(i)}}.$$ For the case that $\rho_{AB}$ is a two-qubit state, we find that the QBER does not depend on the efficiency of the detectors, as $P_{\rm click}^{(i)}={\eta_\mathrm{d}}^2$ and $P_{\rm err}^{(i)}\propto{\eta_\mathrm{d}}^2$.
If we assume a genuine two-qubit system[^1] like in the original quantum repeater proposal (see section \[sec:briegel\]) or the hybrid quantum repeater (see section \[sec:Hybrid\]), without loss of generality[^2], the entangled state $\rho_{AB}$ can be considered diagonal in the Bell-basis, i.e., $\rho_{AB}=A{\left\vert{\phi^+}\right\rangle \left\langle{\phi^+}\right\vert}+B{\left\vert{\phi^-}\right\rangle \left\langle{\phi^-}\right\vert}+C{\left\vert{\psi^+}\right\rangle \left\langle{\psi^+}\right\vert}+D{\left\vert{\psi^-}\right\rangle \left\langle{\psi^-}\right\vert}$, with the probabilities $A, B, C, D$, $A+B+C+D=1$, and with the dual-rail[^3] encoded Bell states[^4] ${\left\vert{\phi^{\pm}}\right\rangle}=({\left\vert{1010}\right\rangle}\pm{\left\vert{0101}\right\rangle})/\sqrt{2}$ and ${\left\vert{\psi^{\pm}}\right\rangle}=({\left\vert{1001}\right\rangle}\pm{\left\vert{0110}\right\rangle})/\sqrt{2}$ (we shall use the notation ${\left\vert{\phi^{\pm}}\right\rangle}$ and ${\left\vert{\psi^{\pm}}\right\rangle}$ for the Bell basis in any type of encoding throughout the paper). Then the QBER along the directions $X$, $Y$, and $Z$ corresponds to [@Scarani:2009] $$\label{eq:QBERBELL}
{e_{X}}:=B+D,\quad\quad
{e_{Z}}:=C+D,\quad\quad
{e_{Y}}:=B+C.$$ Throughout the whole paper $X$, $Y$ and $Z$ denote the three Pauli operators acting on the restricted Hilbert space of qubits.
### The secret key rate {#sec:qber}
The figure of merit representing the performance of quantum key distribution is the *secret key rate* ${R_{\rm QKD}}$ which is the product of the *raw key rate* ${R_{\mathrm{raw}}}$ (see above) and the *secret fraction* $r_{\infty}$. Throughout this paper, we will use asymptotic secret key rates. The secret fraction represents the fraction of secure bits that may be extracted from the raw key. Formally, we have $${R_{\rm QKD}}:={R_{\mathrm{raw}}}r_{\infty}={R_{\mathrm{REP}}}P_{\rm click}{R_{\mathrm{sift}}}r_{\infty},
\label{eq:keyrate}$$ where the sifting rate ${R_{\mathrm{sift}}}$ is the fraction of measurements performed in the same basis by Alice and Bob Throughout the whole paper we will use ${R_{\mathrm{sift}}}=1$ which represents the asymptotic bound for ${R_{\mathrm{sift}}}$ when the measurement basis are chosen with biased probability [@Lo2005]. We point out that both ${R_{\mathrm{REP}}}$ and $r_{\infty}$ are functions of the explicit repeater protocol and the involved experimental parameters, as we will discuss in detail later. Our aim is to maximize the overall secret key rate ${R_{\rm QKD}}$. There will be a trade-off between ${R_{\mathrm{REP}}}$ and $r_{\infty}$, as the secret key fraction $r_{\infty}$ is an increasing function of the final fidelity, while the repeater rate ${R_{\mathrm{REP}}}$ typically decreases with increasing final fidelity.
Note that even though for the considered protocol we find upper bounds on the secret key rate, an improved model (e.g. including distillation in later nesting levels or multiplexing[@Collins2007]) could lead to improved key rates.
The secret fraction represents the fraction of secure bits over the total number of measured bits. We adopt the *composable security definition* discussed in [@BenOr:2005fk; @renner05; @muller2009composability]. Here, composable means that the secret key can be used in successive tasks without compromising its security. In the following we calculate secret key rates using the state produced by the quantum repeater protocol.
In the present work, we consider only two QKD protocols, namely the BB84-protocol and the six-state protocol, for which collective and coherent attacks are equivalent [@Renner:2005pi; @Kraus:2005kx] in the limit of a large number of exchanged signals. The unique parameter entering the formula of the secret fraction is the quantum bit error rate (QBER).
In the BB84-protocol only two of the three Pauli matrices are measured. We adopt the asymmetric protocol where the measurement operators are chosen with different probabilities [@Lo2005], because this leads to higher key rates. We call $X$ the basis used for extracting a key, i.e., the basis that will be chosen with a probability of almost one in the measurement process, while $Z$ is the basis used for the estimation of the QBER. Thus, in the asymptotic limit, we have ${R_{\mathrm{sift}}}=1$. The formula for the secret fraction is [@Scarani:2009] $$\label{eq:r:BB84}
r_{\infty}^{\rm BB84}:=1-h(e_{Z})-h(e_{X}),$$ with $h(p):=-p\log_2 p-(1-p)\log_2(1-p)$ being the binary entropy. This formula is an upper bound on the secret fraction, which is only achievable for ideal implementations of the protocol; any realistic, experimental imperfection will decrease this secret key rate.
In the six-state protocol we use all three Pauli matrices. We call $X$ the basis used for extracting a key, which will be chosen with a probability of almost one, and both $Y$ and $Z$ are the bases used for parameter estimation. In this case, the formula for the secret fraction is given by [@Scarani:2009; @Renner2008][^5] $$\begin{aligned}
\label{eq:r:6S}
r^{\rm 6S}_{\infty}:=&1-e_{Z}h\left(\frac{1+(e_{X}-e_{Y})/e_{Z}}{2}\right)\nonumber\\
&-(1-e_{Z})h\left(\frac{1-(e_X+e_Y+e_Z)/2}{1-e_Z}\right)-h(e_{Z}).\end{aligned}$$
Methods
-------
The secret key rate represents the central figure of merit for our investigations. We study the BB84-protocol, because it is most easily implementable and can also be used for protocols, where $\rho_{AB}$ is not a two-qubit state, with help of the squashing model [@PhysRevA.84.020303; @PhysRevLett.101.093601]. Throughout the paper, we also report on results of the six-state protocol if applicable. We evaluate [Eq. ]{} exactly, except for the quantum repeater based on atomic ensembles where we truncate the states and cut off the higher excitations at some maximal number (see footnote \[foot:dlczcalc\] for the details). For the maximization of the secret key rate, we have used the numerical functions provided by Mathematica [@mathematica8].
The original quantum repeater {#sec:briegel}
=============================
In this section, we consider a general class of quantum repeaters in the spirit of the original proposal by Briegel [*et al. *]{}[@briegel_quantum_1998]. We will analyze the requirements for the experimental parameters such that the quantum repeater is useful in conjunction with QKD. The model we consider in this section is applicable whenever two-qubit entanglement is distributed by using qubits encoded into single photons. This is the case, for instance, for quantum repeaters based on ion traps or Rydberg-blockade gates. We emphasize that we do not aim to capture all peculiarities of a specific set-up. Instead, our intention is to present a fairly general analysis that can give an idea of the order of magnitude, which has to be achieved for the relevant experimental parameters. The error-model we consider is the one used in [@briegel_quantum_1998].
The set-up
----------
### Elementary entanglement creation {#elementary-entanglement-creation .unnumbered}
The probability that two adjacent repeater stations (separated by distance $L_0$) share an entangled pair is given by $$\label{eq:briegP0}
P_{0}:={\eta_{t}\left(L_0\right)},$$ where ${\eta_{t}\left(\ell\right)}$, as defined in [Eq. ]{}, is the probability that a photon is not absorbed during the channel transmission. In a specific protocol, $P_0$ may contain an additional multiplicative factor such as the probability that entanglement is heralded or also a source efficiency.
We assume that the state created over distance $L_0$ is a depolarized state of fidelity $F_0$ with respect to ${\left\vert{\phi^+}\right\rangle}$, i.e., $$\begin{aligned}
\label{eq:briegelinitstate}
\rho_0:=&F_0{\left\vert{\phi^+}\right\rangle}{\left\langle{\phi^+}\right\vert}&\nonumber\\
&+\frac{1-F_0}{3}\left({\left\vert{\psi^+}\right\rangle}{\left\langle{\psi^+}\right\vert}+{\left\vert{\psi^-}\right\rangle}{\left\langle{\psi^-}\right\vert}+{\left\vert{\phi^-}\right\rangle}{\left\langle{\phi^-}\right\vert}\right).\end{aligned}$$ The fidelity $F_0$ contains the noise due to an imperfect preparation and the noise in the quantum channel. We have chosen a depolarized state, because this corresponds to a generic noise model and, moreover, any two-qubit mixed quantum state can be brought into this form using local twirling operations [@bennett_mixed-state_1996].
### Imperfect gates {#imperfect-gates .unnumbered}
For the local qubit operations, such as the CNOT gates, we use a generic gate model with depolarizing noise, as considered in [@briegel_quantum_1998]. Thus, we assume that a noisy gate $O_{BC}$ acting upon two qubits $B$ and $C$ can be modeled by $$\label{eq:p2}
O_{BC}(\rho_{BC})=p_{G}O_{BC}^{\mathrm{ideal}}(\rho_{BC})+\frac{1-p_{G}}{4}{\mbox{$1 \hspace{-1.0mm} {\bf l}$}}_{BC},$$ where $O_{BC}^{\mathrm{ideal}}$ is the ideal gate operation and $p_G$ describes the gate quality. Note that, in general, the noisy gates realized in an experiment do not necessarily have this form, however, such a noise model is useful for having an indication as to how good the corresponding gates must be. Other noise models could be analogously incorporated into our analysis. Further, we assume that one-qubit gates are perfect.
### Entanglement distillation {#entanglement-distillation-1 .unnumbered}
We consider entanglement distillation only before the first entanglement swapping steps, right after the initial pair distributions over $L_0$. We employ the Deutsch [*et al. *]{}protocol [@deutsch1996quantum] which indeed has some advantages, as shown in the analysis of [@Bratzik2012]. In App. \[sec:appDistill\], we review this protocol and we also present the corresponding formulas in the presence of imperfections. We point out that when starting with two copies of depolarized states, the distillation protocol will generate an output state which is no longer a depolarized state, but instead a generic Bell diagonal state. Distillation requires two-qubit gates, which we describe using [Eq. ]{}.
### Entanglement swapping {#entanglement-swapping-1 .unnumbered}
The entanglement connections are performed through entanglement swapping by implementing a (noisy) Bell measurement on the photons stored in two local quantum memories. We consider a Bell measurement that is deterministic in the ideal case. It is implemented using a two-qubit gate with gate quality $p_G$ (see [Eq. ]{}). Analogous to the case of distillation, starting with two depolarized states, at the end of the noisy Bell measurement, we will obtain generic Bell diagonal states. Also in this case, it turns out that a successive depolarization decreases the secret key rate and this step is therefore not performed in our scheme.
\[subsec:ImperfG\]Performance in the presence of imperfections
--------------------------------------------------------------
The secret key rate [Eq. ]{} represents our central object of study, as it characterizes the performance of a QKD protocol. It can be written explicitly as a function of the relevant parameters,
$$\begin{aligned}
\label{eq:rqkd:briegel}
{R_{\rm QKD}}^{\rm O}=&&{R_{\mathrm{REP}}}(L_0, N, k, F_0, p_G, {\eta_\mathrm{d}})P_{\rm click}({\eta_\mathrm{d}})\nonumber\\
&&\times{R_{\mathrm{sift}}}r_{\infty}(N, k, F_0, p_G),\end{aligned}$$
where ${R_{\mathrm{REP}}}$ is given by [Eq. ]{} when ${\eta_\mathrm{d}}=1$ (because then $P_{\rm ES}=1$) or by [Eq. ]{} if ${\eta_\mathrm{d}}<1$[^6]. The probability that the QKD measurement is successful is given by $P_{\rm click}={\eta_\mathrm{d}}^2$ and the secret fraction $r_{\infty}$ is given by either [Eq. ]{} or [Eq. ]{}, depending on the type of QKD protocol. For the asymmetric BB84-protocol, we have ${R_{\mathrm{sift}}}=1$ (see [Sec. \[sec:QKD\]]{}). The superscript ${\rm O}$ refers to the original quantum repeater proposal as considered in this section. In order to have a non-zero secret key rate, it is then necessary that the repeater rate, the probability for a valid QKD measurement event, and the secret fraction are each non-zero too. As typically ${R_{\mathrm{REP}}}> 0$, ${R_{\mathrm{sift}}}>0$ and $P_{\rm click} > 0$, for ${R_{\rm QKD}}>0$, it is sufficient to have a non-zero secret fraction, $r_{\infty}>0$. The value of the secret fraction does not depend on the distance, and therefore some properties of this protocol are distance-invariant.
#### Minimally required parameters {#minimally-required-parameters .unnumbered}
In this paragraph, we will discuss the minimal requirements that are necessary to be able to extract a secret key, i.e., we will specify the parameter region where the secret fraction is non-zero. From the discussion in the previous paragraph, we know that this region does not depend on the total distance, but only on the initial fidelity $F_0$, the gate quality $p_G$, the number of segments $2^N$, and the maximal number of distillation rounds $k$. Moreover, note that even if the secret fraction is not zero, the total secret key rate can be very low (see below).
For calculating the minimally required parameters, we start with the initial state in [Eq. ]{}, we distill it $k$ times (see the formulas in App. \[sec:appDistill\]), and then we swap the distilled state $2^N-1$ times ( see the formulas in \[sec:brig:es\]). At the end, a generic Bell diagonal state is obtained. Using [Eq. ]{} one can then calculate the QBER, which is sufficient to calculate the secret fraction.
[Table \[tab:minFid\]]{} and [Tab. \[tab:minpg\]]{} show the minimally required values for $F_0$ and $p_G$ for different maximal nesting levels $N$ (i.e., different numbers of segments $2^N$) and different numbers of rounds of distillation $k$. Throughout these tables, we can see that for the six-state protocol, the minimal fidelity and the minimal gate quality $p_G$ are lower than for the BB84-protocol. Our results confirm the intuition that the larger the number of distillation rounds, the smaller the affordable initial fidelity can be (at the cost of needing higher gate qualities).
[@l\*[15]{}[l]{}]{} &&&&\
\
&BB84&6S&BB84&6S&BB84&6S&BB84&6S\
0 & 0.835 & 0.810 &0.733&0.728& 0.671 & 0.669 &0.620& 0.614\
1 & 0.912 &0.898 & 0.821& 0.818&0.742& 0.740 & 0.669& 0.664\
2 & 0.955& 0.947 & 0.885 & 0.884 &0.801&0.800 & 0.713& 0.709\
3 & 0.977 &0.973 & 0.929 & 0.928 & 0.849 & 0.848 & 0.752& 0.749\
4 & 0.988 & 0.987& 0.957& 0.957& 0.887& 0.887&0.788& 0.785\
5&0.994& 0.993&0.975& 0.975& 0.917& 0.917&0.819&0.818\
6&0.997&0.997& 0.985& 0.985&0.939&0.939&0.847& 0.846\
7&0.999&0.998&0.992&0.992&0.956&0.956&0.872&0.870\
[@l\*[15]{}[l]{}]{} &&&&\
\
&BB84&6S&BB84&6S&BB84&6S&BB84&6S\
0 & - &-&0.800&0.773 & 0.869&0.860 &0.891&0.884\
1 & 0.780&0.748 &0.922&0.910& 0.942 &0.937 & 0.947& 0.942\
2 & 0.920 &0.908 & 0.965& 0.960 &0.973&0.970 & 0.974& 0.972\
3 & 0.965& 0.959 & 0.984& 0.981 & 0.987& 0.986& 0.987& 0.986\
4&0.984& 0.981&0.992&0.991&0.994&0.993&0.994&0.993\
5&0.992&0.991&0.996&0.995&0.997&0.997&0.997&0.997\
6&0.996& 0.995&0.998&0.998&0.999&0.998&0.999&0.998\
7&0.998&0.998&0.999&0.999&0.999&0.999&0.999&0.999\
In [Fig. \[fig:distillationNew\]]{}, the lines represent the values of the initial infidelity and the gate error for a specific ${N}$ that allow for extracting a secret key. As shown in [Fig. \[fig:distillationNew\]]{}, any lower initial fidelity requires a correspondingly higher gate quality and vice versa. Note that above the lines in [Fig. \[fig:distillationNew\]]{} it is not possible to extract a secret key.
![(Color online) Original quantum repeater and the BB84-protocol: Maximal infidelity $(1-F_0)$ as a function of gate error $(1-p_G)$ permitting to extract a secret key for various maximal nesting levels $N$ and numbers of distillation rounds $k$ (Parameter: $L=600$ km).[]{data-label="fig:distillationNew"}](figure4){width="8cm"}
#### The secret key rate {#the-secret-key-rate .unnumbered}
In this section, we will analyze the influence of the imperfections on the secret key rate, see [Eq. ]{}.
![(Color online) Original quantum repeater and the BB84-protocol: Secret key rate [Eq. ]{} versus gate quality $p_G$ for different rounds of distillation $k$. The case $k=0$ leads to a vanishing secret key rate. (Parameters: $F_0=0.9$, $N=2$, $L=600$ km)[]{data-label="fig:rsecvspg"}](figure5){width="8cm"}
In [Fig. \[fig:rsecvspg\]]{} we illustrate the effect of gate imperfections on the secret key rate for different numbers of rounds of distillation and for a fixed distance, initial fidelity, and maximal number of nesting levels. Throughout this whole section, we use $\beta=2$ in [Eq. ]{} for the fundamental time, which corresponds to the case where a source is placed at one side of an elementary segment (see [Fig. \[fig:comm\_time\]]{}). The optimal number of distillation rounds decreases as $p_G$ increases. We see from the figure that $k=2$ is optimal when $p_G=1$. This is due to the fact that from $k=1$ to $k=2$, the raw key rate decreases by $40\%$, but the secret fraction increases by $850\%$. However, from $k=2$ to $k=3$, the raw key rate decreases once again by $40\%$, but now the secret fraction increases only by $141\%$. In this case, the net gain is smaller than 1 and therefore three rounds of distillation do not help to increase the secret key rate compared to the case of two rounds. In other words, what is lost in terms of success probability when having three probabilistic distillation rounds is not added to the secret fraction. For a decreasing $p_G$, more rounds of distillation become optimal. The reason is that when the gates become worse, additional rounds of distillation permit to increase the secret key rate sufficiently much to compensate the decrease of ${R_{\mathrm{REP}}}$.
![(Color online) Original quantum repeater and the BB84-protocol: Number of distillation rounds $k$ that maximizes the secret key rate as a function of gate quality $p_G$ and initial fidelity $F_0$. In the white area, it is no longer possible to extract a secret key. (Parameters: $N=2$, $L=600$ km)[]{data-label="fig:simulateprotocol"}](figure6){width="8cm"}
In [Fig. \[fig:simulateprotocol\]]{} we show the optimal number of rounds of distillation $k$ as a function of the imperfections of the gates and the initial fidelity. It turns out that when the experimental parameters are good enough, then distillation is not necessary at all.
![(Color online) Original quantum repeater and the BB84-protocol: Optimal secret key rate [Eq. ]{} versus distance for different nesting levels, with and without perfect detectors. For each maximal nesting level $N$, we have chosen the optimal number of distillation rounds $k$. A nesting level $N\geq5$ no longer permits to obtain a non-zero secret key rate. (Parameters: $F_0=0.9$ and $p_G=0.995$.)[]{data-label="fig:scrvsL"}](figure7){width="8cm"}
Let us now investigate the secret key rate [Eq. ]{} as a function of the distance $L$ between Alice and Bob. In [Fig. \[fig:scrvsL\]]{} the secret key rate for the optimal number of distillation rounds versus the distance for various nesting levels is shown, for a fixed initial fidelity and gate quality. These curves should be interpreted as upper bounds; when additional imperfections are included, the secret key rate will further decrease. We see that for a distance of more than $400$ km, the value $N=4$ (which corresponds to 16 segments) is optimal. Note that with the initial fidelity and gate quality assumed here, it is no longer possible to extract a secret key for $N=5$.
In many implementations, detectors are far from being perfect. The general expression of the raw key rate including detector efficiencies ${\eta_\mathrm{d}}$ becomes $${R_{\mathrm{raw}}}=\frac{1}{T_0}{R_{\mathrm{sift}}}\left(\frac{2}{3}\right)^{N+k}{\eta_\mathrm{d}}^{2 (k+N+1)} P_0\prod_{i=1}^{k}{P_D}[i],
\label{eq:rrawES}$$ using [Eq. ]{} with the repeater rate $R_{\rm REP}$ given by [Eq. ]{}. The term ${\eta_\mathrm{d}}^{2 k}$ arises from the two-fold detections for the distillation, and similarly, ${\eta_\mathrm{d}}^{2N}$ comes from the entanglement swapping and ${\eta_\mathrm{d}}^2$ from the QKD measurements.
In [Fig. \[fig:scrvsL\]]{} we observe that even if detectors are imperfect, it is advantageous to do the same number of rounds of distillation as for the perfect case. This is due to the fact that the initial fidelity is so low that even with a lower success probability, the gain in the secret fraction produces a net gain greater than 1.
For realistic detectors, the dark count probability is much smaller than their efficiency. We show in App. \[app:general-type\] that, provided that the dark count probability is smaller than $10^{-5}$, dark counts can be neglected. This indeed applies to most modern detectors [@eisaman2011invited].
\[sec:Hybrid\]The hybrid quantum repeater
=========================================
In this section, we will investigate the so-called hybrid quantum repeater (HQR) introduced by van Loock [*et al. *]{}[@van_loock_hybrid_2006] and Ladd [*et al. *]{}[@ladd_hybrid_2006]. In this scheme, the resulting entangled pairs are discrete atomic qubits, but the probe system (also called *qubus*) that mediates the two-qubit entangling interaction is an optical mode in a coherent state. The scheme does not only employ atoms and light at the same time, but it also uses both discrete and continuous quantum variables; hence the name hybrid. The entangled pair is conditionally prepared by suitably measuring the probe state after it has interacted with two atomic qubits located in the two spatially separated cavities at two neighboring repeater stations. Below we shall consider a HQR where the detection is based on an unambiguous state discrimination (USD) scheme [@van_loock_quantum_2008; @azuma]. In this case, arbitrarily high fidelities can be achieved at the expense of low probabilities of success.
The set-up
----------
### Elementary entanglement creation {#elementary-entanglement-creation-1 .unnumbered}
![(Color online) Schematic diagram for the entanglement generation by means of a USD measurement following [@azuma]. The two quantum memories $A$ and $B$ are separated by a distance $L_0$. The part on the left side (an intermediate Alice) prepares a pulse in a coherent state ${\left\vert{\alpha}\right\rangle}_a$ (the subscript refers to the corresponding spatial mode). This pulse first interacts with her qubit $A$ and is then sent to the right side together with the local oscillator pulse (LO). The part on the right side (an intermediate Bob) receives the state ${\left\vert{\sqrt{\eta_t}\alpha}\right\rangle}_{b_1}$ and produces from the LO through beam splitting a second probe pulse ${\left\vert{\sqrt{\eta_t}\alpha}\right\rangle}_{b_2}$ which interacts with his qubit $B$. He further applies a 50:50 beam splitter to the pulses in modes $b_1$ and $b_2$, and a displacement $D(-\sqrt{2\eta_t}\alpha\cos{\theta/2})=e^{-\sqrt{2\eta_{t}}\alpha\cos{\theta/2}(a^{\dagger}-a)}$ to the pulse in mode $b_4$. The entangled state is conditionally generated depending on the results of detectors $D_1$ and $D_2$. The fiber attenuation ${\eta_{t}\left(L_0\right)}$ has been defined in [Eq. ]{}.[]{data-label="fig:hqr"}](figure8){width="8cm"}
Entanglement is shared between two electronic spins (such as $\Lambda$ systems effectively acting as two-level systems) in two distant cavities (separated by $L_0$). The entanglement distribution occurs through the interaction of the coherent-state pulse with both atomic systems. The coherent-state pulse and the cavity are in resonance, but they are detuned from the transition between the ground state and the excited state of the two-level system. This interaction can then be described by the Jaynes-Cummings interaction Hamiltonian in the limit of large detuning, $H_{int}=\hbar\chi Z a^{\dagger}a$, where $\chi$ is the light-atom coupling strength, $a$ ($a^{\dagger}$) is the annihilation (creation) operator of the electromagnetic field mode, and $Z={\left\vert{0}\right\rangle}{\left\langle{0}\right\vert}-{\left\vert{1}\right\rangle}{\left\langle{1}\right\vert}$ is the $Z$ operator for a two-level atom (throughout this section, ${\left\vert{0}\right\rangle}$ and ${\left\vert{1}\right\rangle}$ refer to the two $Z$ Pauli eigenstates of the effective two-level matter system and not to the optical vacuum and one-photon Fock states). After the interaction of the qubus in state ${\left\vert{\alpha}\right\rangle}$ with the first atomic state, which is initially prepared in a superposition, the output state is $U_{int}\left[{\left\vert{\alpha}\right\rangle}({\left\vert{0}\right\rangle}+{\left\vert{1}\right\rangle})/\sqrt{2}\right]=({\left\vert{\alpha e^{-i\theta/2}}\right\rangle}{\left\vert{0}\right\rangle}+{\left\vert{\alpha e^{i\theta/2}}\right\rangle}{\left\vert{1}\right\rangle})/\sqrt{2}$, with $\theta=2\chi t$ an effective light-matter interaction time inside the cavity. The qubus probe pulse is then sent through the lossy fiber channel and interacts with the second atomic qubit also prepared in a superposition. Here we consider the protocol of [@azuma], where linear optical elements and photon detectors are used for the unambiguous discrimination of the phase-rotated coherent states. Different from [@azuma], however, we use imperfect photon-number-resolving detectors (PNRD), as described by [Eq. ]{}, instead of threshold detectors. By performing such a USD measurement on the probe state, as illustrated in [Fig. \[fig:hqr\]]{}, the following entangled state can be conditionally prepared, $$\rho_0:=F_0{\left\vert{\phi^+}\right\rangle}{\left\langle{\phi^+}\right\vert}+(1-F_0){\left\vert{\phi^-}\right\rangle}{\left\langle{\phi^-}\right\vert},
\label{initial.state}$$ where we find $F_0=[1+e^{-2(1+\eta_t(1-2\eta_d))\alpha^2\sin^2(\theta/2)}]/2$ for $\alpha$ real, $\eta_t(L_0)$ is the channel transmission given in [Eq. ]{}, and $\eta_d$ is the detection efficiency (see section \[subsec:parameters\]). Our derivation of the fidelity $F_0$ can be found in App. \[sec:app:gen\]. Note that the form of this state is different from the state considered in section \[sec:briegel\]. It is a mixture of only two Bell states, since the two other (bit flipped) Bell states are filtered out through the USD measurement. The remaining mixedness is due to a phase flip induced by the coupling of the qubus mode with the lossy fiber environment. We find the optimal probability of success to generate an entangled pair in state $\rho_0$ $$\label{initial.p0}
P_0=1-(2F_{0}-1)^\frac{\eta_t\eta_d}{1+\eta_t(1-2\eta_d)},$$ which generalizes the formula for the quantum mechanically optimal USD with perfect detectors, as given in [@van_loock_quantum_2008], to the case of imperfect, photon-number-resolving detectors. We explain our derivation of [Eq. ]{} in App. \[sec:app:gen\][^7].
### \[sec:ESHybrid\]Entanglement swapping {#seceshybridentanglement-swapping .unnumbered}
A two-qubit gate is essential to perform entanglement swapping and entanglement distillation. In the HQR a controlled-Z (CZ) gate operation can be achieved by using dispersive interactions of another coherent-state probe with the two input qubits of the gate. This is similar to the initial entanglement distribution, but this time without any final measurement on the qubus [@van_loock_gate_2008]. Controlled rotations and uncontrolled displacements of the qubus are the essence of this scheme. The controlled rotations are realized through the same dispersive interaction as explained above. In an ideal scheme, after a sequence of controlled rotations and displacements on the qubus, the qubus mode will automatically disentangle from the two qubits and the only effect will be a sign flip on the $|11\rangle$ component of the input two-qubit state (up to single-qubit rotations), corresponding to a CZ gate operation. Thus, this gate implementation can be characterized as measurement-free and deterministic. Using this gate, one can then perform a fully deterministic Bell measurement (i.e., one is able to distinguish between all four Bell states), and consequently, the swapping occurs deterministically (i.e., $P_{ES}\equiv1$).
In a more realistic approach, local losses will cause errors in these gates. Following [@louis], after dissipation, we may consider the more general, noisy two-qubit operation $O_{BC}$ acting upon qubits $B$ and $C$, $$\begin{aligned}
\label{gateerror}
O_{BC}(\rho_{BC})=&O_{BC}^{ideal}\left(p^2_c(x)\rho_{BC}+\right.\\ \nonumber
&\left.p_c(x)(1-p_c(x))(Z^B\rho_{BC}Z^B+Z^C\rho_{BC} Z^C)\right. \\ \nonumber
&\left. +(1-p_c(x))^2 Z^B Z^C\rho_{BC} Z^C Z^B\right),\end{aligned}$$ where $$p_c(x):=\frac{1+e^{-x/2}}{2}$$ is the probability for each qubit to not suffer a $Z$ error, and $x:=\pi\frac{1-p_G^2}{\sqrt{p_G}(1+p_G)}$; here $p_G$ is the local transmission parameter that incorporates photon losses in the local gates.[^8] We derive explicit formulas for entanglement swapping including imperfect two-qubit gates in App. \[sec:app:swap\].
### Entanglement distillation {#entanglement-distillation-2 .unnumbered}
For the distillation, the same two-qubit operation as described above in [Eq. ]{} can be used. It is then interesting to notice that if we start with a state given in [Eq. ]{}, after one round of imperfect distillation, the resulting state is a generic Bell diagonal state. The effect of gate errors in the distillation step is derived in App. \[sec:app:dist\].[^9]
Performance in the presence of imperfections
--------------------------------------------
In the following, we will only consider the BB84-protocol, because it is experimentally less demanding and also, because we found in our simulations that the six-state protocol produces almost the same secret key rates, due to the symmetry of the state in [Eq. ]{}. The secret key rate per second for the hybrid quantum repeater can be written as a function of the relevant parameters:
$$\begin{aligned}
R_{\rm QKD}^{\rm H}=&&R_{\rm REP}^{\rm det}(L_0,N, k, F_0, p_G, {\eta_\mathrm{d}})\nonumber\\
&&\times{R_{\mathrm{sift}}}r_{\infty}^{\rm BB84}(L_0, N, k, F_0, p_G),
\label{KeyH}\end{aligned}$$
where $R_{\rm REP}^{\rm det}$ is the repeater pair-creation rate for deterministic swapping [Eq. ]{} described in section \[subsec:gen\] and $r_{\infty}^{\rm BB84}$ is the secret fraction for the BB84-protocol [Eq. ]{}. For the asymmetric BB84-protocol, we have ${R_{\mathrm{sift}}}=1$ (see [Sec. \[sec:QKD\]]{}). The superscript ${\rm H}$ stands for hybrid quantum repeater. Note that the fundamental time is $T_0=\frac{2 L_0}{c}$, as the qubus is sent from Alice to Bob and then classical communication in the other direction is used (see section \[subsec:gen\] and [Fig. \[fig:comm\_time\]]{}). Further notice that the final projective qubit measurements which are necessary for the QKD protocol are assumed to be perfect. Thus, the secret key rate presented here represents an upper bound and, depending on the particular set-up adopted for these measurements, it should be multiplied by the square of the detector efficiency.
![(Color online) Hybrid quantum repeater with perfect quantum operations ($p_G=1$) and perfect detectors ($\eta_d=1$) (black lines) compared to imperfect quantum operations ($p_G=0.995$) and imperfect detectors ($\eta_d=0.9$) (orange lines): Secret key rate per second [Eq. ]{} as a function of the initial fidelity for $2^3$ segments ($N=3$) and various rounds of distillation $k$. The distance between Alice and Bob is 600 km.[]{data-label="fig:KHybridPerf"}](figure9){width="8cm"}
#### The secret key rate {#the-secret-key-rate-1 .unnumbered}
[Figure \[fig:KHybridPerf\]]{} shows the secret key rate for $2^3$ segments ($N=3$) for various rounds of distillation. We see from the figure that for the hybrid quantum repeater the secret key rate is not a monotonic function of the initial fidelity. The reason is that increasing $F_0$ decreases $P_0$ (see [Eq. ]{}) and vice versa. We find that the optimal initial fidelity, i.e., the fidelity where the secret key rate is maximal, increases as the maximal number of segments increases (see [Table \[tab:optF\]]{}). On the other hand, examining the optimal initial fidelity as a function of the distance, it turns out that it is almost constant for $L>100$ km. Thus, for such distances, it is neither useful nor necessary to produce higher fidelities, because these would not permit to increase the secret key rate.
[@l\*[15]{}[l]{}]{} & & & &\
1&0.898 &0.836 &0.765 &0.705\
2&0.946 &0.876 &0.788 &0.715\
3&0.972 &0.907 &0.812 &0.726\
4&0.986 &0.931 &0.834 &0.741\
We also observe that the maximum of the initial fidelity is quite broad for small $N$, and gets narrower as $N$ increases. If we now consider perfect gates and perfect detectors, we see that by fixing a certain secret key rate, we can reach this value with lower initial fidelities by performing distillation. Furthermore, by distilling the initial entanglement, we can even exceed the optimal secret key rate without distillation by one order of magnitude. However, note that distillation for $k$ rounds requires $2^k$ memories at each side. If we then assume that we choose the protocol with no distillation and perform it in parallel $2^k$ times, i.e., we use the same amount of memories as for the scheme including distillation, the secret key rate without distillation (as shown in [Fig. \[fig:KHybridPerf\]]{}) should be multiplied by $2^k$. As a result, the total secret key rate can then be even higher than that obtained with distillation.
Let us now assess the impact of the gate and detector imperfections on the secret key rate (orange lines) in [Fig. \[fig:KHybridPerf\]]{}. We notice that $p_G$ has a large impact even if it is only changed by a small amount, like here from $p_G=1$ to $p_G=0.995$; the secret key rates drop by one order of magnitude. Imperfect detectors are employed in the creation of entanglement. As we see in [Fig. \[fig:KeyEta\]]{}, imperfect detectors do not affect the secret key rate significantly. As for $N=3$ and $k=0$, improving the detector efficiency from $0.5$ to $1$ leads to a doubling of the secret key rate. We conclude that for the hybrid quantum repeater, the final secret key rates are much more sensitive to the presence of gate errors than to inefficiencies of the detectors. However, recall that in our analysis, we only take into account detector imperfections that occur during the initial USD-based entanglement distribution. For simplicity, any measurements on the memory qubits performed in the local circuits for swapping and distillation are assumed to be perfect, whereas the corresponding two-qubit gates for swapping and distillation are modeled as imperfect quantum operations (see footnote \[footnote9\] for more details).
![(Color online) Hybrid quantum repeater with perfect gates ($p_G=1$): The optimal secret key rate [Eq. ]{} for the BB84-protocol in terms of the detector efficiency $\eta_d$ for the distance $L=600$ km with various numbers of segments $2^N$ and rounds of distillation $k$.[]{data-label="fig:KeyEta"}](figure10){width="8cm"}
#### Minimally required parameters {#minimally-required-parameters-1 .unnumbered}
As we have seen in the previous section, it is also worth finding the minimal parameters for $F_0$ and $p_G$, for which we can extract a secret key. [Figure \[fig:MinFidLocalT\]]{} shows the initial infidelity required for extracting a secret key as a function of the local loss probability $p_G$, which was introduced in [Sec. \[sec:ESHybrid\]]{}.
We obtain also the minimal values of the local transmission probability ${p_{G,{N}}^{\rm min}}$ without distillation (solid lines in [Fig. \[fig:MinFidLocalT\]]{}). If $p_G<{p_{G,{N}}^{\rm min}}$, then it is no longer possible to extract a secret key. As shown in [Fig. \[fig:MinFidLocalT\]]{}, these minimal values (for which the minimal initial fidelity becomes $F_0=1$, without distillation) are ${p_{G,{1}}^{\rm min}}=0.853$ (not shown in the plot), ${p_{G,{2}}^{\rm min}}=0.948$, ${p_{G,{3}}^{\rm min}}=0.977$, and ${p_{G,{4}}^{\rm min}}=0.989$ (not shown in the plot). When including distillation, we can extend the regime of non-zero secret key rate to smaller initial fidelities at the cost of better local transmission probabilities. So there is a trade-off: if we can produce almost perfect Bell pairs, that is initial states with high fidelities $F_0$, we can afford larger gate errors. Conversely, if high-quality gates are available, we may operate the repeater with initial states having a lower fidelity. Note that these results and [Fig. \[fig:MinFidLocalT\]]{} do not depend on the length of each segment in the quantum repeater, but only on the number of segments.
![(Color online) Hybrid quantum repeater with distillation and imperfections: Maximally allowed infidelity $(1-F_0)$ as a function of the local loss probability $(1-p_G)$ for various maximal numbers of segments $2^N$ and rounds of distillation $k$ (distance: $L=600$ km). Above the curves it is no longer possible to extract a secret key. The lines with $k=0$ correspond to entanglement swapping without distillation.[]{data-label="fig:MinFidLocalT"}](figure11){width="8cm"}
![(Color online) Hybrid quantum repeater with imperfect quantum operations ($p_G=0.995$) and imperfect detectors ($\eta_d=0.9$): Optimal secret key rate [Eq. ]{} for the BB84-protocol as a function of the total distance $L$, for various numbers of segments $2^N$ and rounds of distillation $k$. For $N=5$, it is not possible to obtain a secret key when distillation is applied.[]{data-label="fig:ImpKeyVsL"}](figure12){width="8cm"}
In figure \[fig:ImpKeyVsL\] we plotted the optimal secret key rate for a fixed local transmission probability $p_G$ and detector efficiency $\eta_d$ in terms of the total distance $L$. We varied the number of segments $2^N$ and the number of distillation rounds $k$. We observe that a high value of $k$ is not always advantageous: There exists for every $N$ an optimal $k$, for which we obtain the highest key rate. We see, for example, that for $N=1$, the optimal choice is $k=2$, whereas for $N=3$, the optimal $k$ is 3. One can also see that there are distances, where it is advantageous to double the number of segments if one wants to avoid distillation, as, for example, for $N=3$ and $N=4$ at a distance of around 750 km.
Quantum repeaters based on atomic ensembles {#sec:atomicensenbles}
===========================================
The probably most influential proposal for a practical realization of quantum repeaters was made in [@duan_long-distance_2001] and it is known as the Duan-Lukin-Cirac-Zoller (DLCZ)-protocol. These authors suggested to use atomic ensembles as quantum memories and linear optics combined with single-photon detection for entanglement distribution, swapping, and (built-in) distillation. This proposal influenced experiments and theoretical investigations and led to improved protocols based on atomic ensembles and linear optics (see [@sangouard_quantum_2011] for a recent review).
To our knowledge, the most efficient scheme based on atomic ensembles and linear optics was proposed very recently by Minář [*et al. *]{}[@minar]. These authors suggest to use heralded qubit amplifiers [@ralph] to produce entanglement on demand and then to extend it using entanglement swapping based on two-photon detections. The state produced at the end of the protocol no longer contains vacuum components and therefore can be used directly for QKD. This is an improvement over the original DLCZ protocol in which the final long-distance pair is still contaminated by a fairly large vacuum term that accumulates during the imperfect storage and swapping processes.[^10]
In this section, we first review the protocol proposed in [@minar] and then we analyze the role of the parameters and the performance in relation to QKD.
The set-up
----------
The protocol is organized in three logical steps. First, local entanglement is created in a repeater station, then it is distributed, and finally it is extended over the entire distance [@minar].
![Quantum repeater based on atomic ensembles: Set-up for creation of on-demand entanglement (see also [@minar]). The whole set-up is situated at one physical location. A pair source produces the state $\rho_{\rm pair}$. One part of the pair (the mode $g$) is stored in an atomic ensemble and the other part (mode $in$) goes into a linear-optics network. A single-photon source produces the states $\rho_{\rm single}^{H}$ and $\rho_{\rm single}^{V}$ which go through a beam splitter of reflectivity $R$. The output modes of the beam splitter are called $c$ and $out$. The mode $out$ is stored in a quantum memory and the mode $c$ goes into a linear-optics network which is composed of a polarizing beam splitter in the diagonal basis $\pm 45^{\circ}$ (square with a circle inside), two polarizing beam splitters in the rectilinear basis (square with a diagonal line inside), and four detectors.[]{data-label="fig:minarentprod"}](figure13){width="8cm"}
As a probabilistic entangled-pair source we consider spontaneous parametric down-conversion (SPDC) [@PhysRevLett.75.4337] which produces the state (see [@PhysRevA.61.042304] and [@minar])[^11]
$$\label{eq:spdc}
\rho_{\rm pair}:=(1-p)\sum_{m=0}^{\infty} \frac{2^m p^m}{(m!)^2 (m+1)} ({B^{\dagger}})^{m}{\left\vert{0}\right\rangle}{\left\langle{0}\right\vert}B^m,$$
where ${B^{\dagger}}:=({g_{H}^{\dagger}}{in_{H}^{\dagger}}+{g_{V}^{\dagger}}{in_{V}^{\dagger}})/\sqrt{2}$. The operator $g_{i}^{\dagger}$ ($in_{i}^{\dagger}$) denotes a spatial mode with polarization given by $i=H,V$. The *pump parameter* $p$ is related to the probability to have an $n$-photon pulse by $P(n)=p^n(1-p)$.
A probabilistic single-photon source with efficiency $q$ produces states of the form $$\rho_{\rm single}^{i}:=(1-q){\left\vert{0}\right\rangle}{\left\langle{0}\right\vert}+qa_{i}^{\dagger}{\left\vert{0}\right\rangle}{\left\langle{0}\right\vert}a_{i},$$ where $a^{\dagger}_{i}$($a_{i}$) is the creation (annihilation) operator of a photon with polarization $i=H, V$.
We also define by $\gamma_{\rm rep}$ the smallest repetition rate among the repetition rates of the SPDC source and the single-photon sources.
### On-demand entanglement source {#on-demand-entanglement-source .unnumbered}
The protocol that produces local entangled pairs works as follows (see [Fig. \[fig:minarentprod\]]{} and [@minar] for additional details):
1. The state $\rho_{\rm pair}\otimes\rho_{\rm single}^{H}\otimes\rho_{\rm single}^{V}$ is produced.
2. The single photons, which are in the same spatial mode, are sent through a tunable beam splitter of reflectivity $R$ corresponding to the transformation $a_{i}\to \sqrt{R}~c_{i} + \sqrt{1-R}~out_{i}$.
3. The spatial modes $in$ and $c$ are sent through a linear-optics network which is part of the heralded qubit amplifiers, and the following transformations are realized, $$\begin{aligned}
c_{H}\to\frac{d_3+d_4+d_2-d_1}{2},\\ c_{V}\to\frac{d_3+d_4-d_2+d_1}{2},\\
in_{H}\to\frac{d_2+d_1+d_3-d_4}{2},\\ in_{V}\to\frac{d_2+d_1-d_3+d_4}{2},\end{aligned}$$ where $d_1,\; d_2,\; d_3,\; d_4$ are four spatial modes, corresponding to the four detectors.
4. A twofold coincidence detection between $d_1$ and $d_3$ (or $d_1$ and $d_4$ or $d_2$ and $d_3$ or $d_2$ and $d_4$) projects the modes $g$ and $out$ onto an entangled state. These are the heralding events that acknowledge the storage of an entangled pair in the quantum memories $out$ and $g$.The probability of a successful measurement is given by
$$P_0^{s}(p, q, R, {\eta_\mathrm{d}})=4{\mathrm{tr}}\left(\Pi^{(1)}_{d_1}({\eta_\mathrm{d}})\Pi^{(0)}_{d_2}({\eta_\mathrm{d}}) \Pi^{(1)}_{d_3}({\eta_\mathrm{d}})\Pi^{(0)}_{d_4}({\eta_\mathrm{d}}) \rho'_{g, out, d_1, d_2, d_3, d_4}\right),$$
where $\rho'_{g, out, d_1, d_2, d_3, d_4}$ is the total state obtained at the end of step (iii) and the superscript $s$ stands for source. The POVM for the detectors has been defined in [Eq. ]{}. The factor $4$ accounts for the fact that there are four possible twofold coincidences. The resulting state is
$$\label{eq:rho0s}
\rho_{0}^{s}(p, q, R, {\eta_\mathrm{d}})=\frac{4}{P_0^{s}} {\mathrm{tr}}_{d_1, d_2, d_3, d_4}\left(\Pi^{(1)}_{d_1}({\eta_\mathrm{d}})\Pi^{(0)}_{d_2}({\eta_\mathrm{d}}) \Pi^{(1)}_{d_3}({\eta_\mathrm{d}})\Pi^{(0)}_{d_4}({\eta_\mathrm{d}}) \rho'_{g, out, d_1, d_2, d_3, d_4}\right).$$
This is the locally prepared state that will be distributed between the repeater stations. In the ideal case with perfect detectors and perfect single-photon sources, the resulting state (after a suitable rotation) is $\rho_{0}^{s}={\left\vert{\phi^{+}}\right\rangle}{\left\langle{\phi^{+}}\right\vert}$ which can be obtained with probability $P_0^s=pR(1-R)$. In the realistic case, however, additional higher-order excitations are present. In [@minar], the explicit form of $\rho_{0}^{s}$ and $P_{0}^{s}$ can be found for the case when $1>R\gg p$ and $1\gg 1-q$.
Therefore, we have seen that the protocol proposed in [@minar] permits to turn a probabilistic entangled-pair source (SPDC in our case) into an on-demand entangled photon source. In this context *on-demand* means that when a heralding event is obtained then it is known for sure that an entangled quantum state is stored in the quantum memories $out$ and $g$.
### Entanglement distribution and swapping {#entanglement-distribution-and-swapping .unnumbered}
![Quantum repeater based on atomic ensembles: Set-up used for entanglement distribution (swapping) (see [@minar] for additional details). The modes $out$ and $out'$ are released from two quantum memories separated by distance $L_0$ (or located at the same station for the case of swapping) and sent into a linear-optics network consisting of one polarizing beam splitter in the rectilinear basis (square with diagonal line inside), two polarizing beam splitters in the diagonal basis (square with circle inside), and four detectors.[]{data-label="fig:minardistr"}](figure14){width=".3\textwidth"}
Once local entangled states are created, it is necessary to distribute the entanglement over segments of length $L_0$ and then to perform entanglement swapping. Both procedures are achieved in a similar way (see [Fig. \[fig:minardistr\]]{}), as we shall describe in this section. Entanglement distribution is done as follows (see [Fig. \[fig:minardistr\]]{} and [@minar] for additional details):
1. Each of the two adjacent stations create a state of the form $\rho_{0}^{s}$. We call $g$ and $out$ the modes belonging to the first station and $g'$ and $out'$ the modes of the second station.
2. The modes $out$ and $out'$ are read out from the quantum memories and sent through an optical fiber to a central station where a linear-optics network is used in order to perform entanglement swapping. The transformations of the modes are as follows: $$\begin{aligned}
out_{H}\to\frac{d_3+d_4}{\sqrt{2}},\quad\ out_{V}\to\frac{d_1-d_2}{\sqrt{2}},\\
out'_{H}\to\frac{d_1+d_2}{\sqrt{2}},\quad out'_{V}\to\frac{d_3-d_4}{\sqrt{2}},\end{aligned}$$ where $d_1,\; d_2,\; d_3,\; d_4$ are four spatial modes.
3. A twofold coincidence detection between $d_1$ and $d_3$ (or $d_1$ and $d_4$ or $d_2$ and $d_3$ or $d_2$ and $d_4$) projects the modes $out$ and $out'$ onto an entangled state. The probability of this event is given by
$$P_0(p, q, R, {\eta_\mathrm{d}}, \eta_{\rm mtd})=4{\mathrm{tr}}\left(\Pi^{(1)}_{d_1}(\eta_{mtd})\Pi^{(0)}_{d_2}(\eta_{mtd}) \Pi^{(1)}_{d_3}(\eta_{mtd})\Pi^{(0)}_{d_4}(\eta_{mtd}) \rho'_{g, g', d_1, d_2, d_3, d_4}\right),$$
where $\rho'_{g, g', d_1, d_2, d_3, d_4}$ is the total state obtained at the end of step (ii) and $\eta_{mtd}:=\eta_{m}{\eta_{t}\left(\frac{L_0}{2}\right)}{\eta_\mathrm{d}}$, with $\eta_{m}$ being the probability that the quantum memory releases a photon. The factor $4$ accounts for the fact that there are four possible twofold coincidences. The resulting state is
$$\rho_{0, g, g'}=\frac{4}{P_0} {\mathrm{tr}}_{d_1, d_2, d_3, d_4}\left(\Pi^{(1)}_{d_1}(\eta_{mtd})\Pi^{(0)}_{d_2}(\eta_{mtd}) \Pi^{(1)}_{d_3}(\eta_{mtd})\Pi^{(0)}_{d_4}(\eta_{mtd}) \rho'_{g, g', d_1, d_2, d_3, d_4}\right).$$
The state $\rho_{0, g, g'}$ is the entangled state shared between two adjacent stations over distance $L_{0}$. In order to perform entanglement swapping, the same steps as described above are repeated until those two stations separated by distance $L$ are finally connected. Formally, the probability that entanglement swapping is successful in the nesting level $n$ is given by
$$P_{ES}^{(n)}(p, q, R, {\eta_\mathrm{d}}, \eta_{\rm mtd})=4{\mathrm{tr}}\left(\Pi^{(1)}_{d_1}(\eta_{md})\Pi^{(0)}_{d_2}(\eta_{md}) \Pi^{(1)}_{d_3}(\eta_{md})\Pi^{(0)}_{d_4}(\eta_{md}) \rho'_{n-1,g, g', d_1, d_2, d_3, d_4}\right),$$
where $\rho'_{n-1,g, g', d_1, d_2, d_3, d_4}$ is the total state resulting from steps (i) and (ii) described above in this section, and $\eta_{md}:=\eta_{m}{\eta_\mathrm{d}}$.
The swapped state is given by
$$\rho_{k, g, g'}=\frac{4}{P_{ES}^{(i)}} {\mathrm{tr}}_{d_1, d_2, d_3, d_4}\left(\Pi^{(1)}_{d_1}(\eta_{md})\Pi^{(0)}_{d_2}(\eta_{md}) \Pi^{(1)}_{d_3}(\eta_{md})\Pi^{(0)}_{d_4}(\eta_{md}) \rho'_{k-1,g, g', d_1, d_2, d_3, d_4}\right).$$
The state $\rho_{n, g, g'}$ is the state that will be used for quantum key distribution when $n = N$. In a regime where higher-order excitations can be neglected, the state $\rho_{n, g, g'}$ is a maximally entangled Bell state. In [@minar] it is given the expression of the state $\rho_{n, g, g'}$ under the same assumptions on the reflectivity $R$ and the efficiency $q$ of the single-photon sources as discussed regarding $\rho_0^s$ in [Eq. ]{}.
Given the final state $\rho_{AB}:= \rho_{N, g, g'}$ it is possible to calculate $P_{\rm click}$ and the QBER, using the formalism of [Sec. \[sec:qber\]]{} and inserting $\eta_{md}$ for the detector efficiency.
The final secret key rate then reads
$${R_{\rm QKD}}^{\rm AE}={R_{\mathrm{REP}}}(L_0, p, N, {\eta_\mathrm{d}}, \eta_{m},\gamma_{\rm rep}, q)P_{\rm click}(L_0, p, N, {\eta_\mathrm{d}}, \eta_{m}, q){R_{\mathrm{sift}}}r_{\infty}^{\rm BB84}(L_0, p, N, {\eta_\mathrm{d}}, \eta_{m}, q),$$
where ${R_{\mathrm{REP}}}$ is given by [Eq. ]{} with $\beta=1$ for the communication time (see [Fig. \[fig:comm\_time\]]{}c). As for the QKD protocol, we consider the asymmetric BB84-protocol (${R_{\mathrm{sift}}}=1$, see [Sec. \[sec:QKD\]]{}). The superscript ${\rm AE}$ stands for atomic ensembles.
Note that even though for the explicit calculations we used PNRD, the previous formulas hold for any type of measurement.
Performance in the presence of imperfections
--------------------------------------------
As in the previous sections, we shall focus on the secret key rate. The free parameters are the pump parameter $p$ and the reflectivity of the beam splitter $R$. In all plots, we optimize these parameters in such a way that the secret key rate is maximized. As all optimizations have been done numerically, our results may not correspond to the global maximum, but only to a local maximum. In general, we observed that if we treat the secret key rate as a function of $p$ (calculated at the optimal $R$), the maximum of the secret key rate is rather narrow. On the other hand, when calculated as a function of $R$ (at the optimal $p$), this maximum is quite broad.
The most favorable scenario (ideal case) is characterized by perfect detectors (${\eta_\mathrm{d}}=1$), perfect quantum memories ($\eta_{m}=1$), and deterministic single-photon sources ($q=1$) which can emit photons at an arbitrarily high rate ($\gamma_{\rm rep}=\infty$). In this case, the heralded qubit amplifier is assumed to be able to create perfect Bell states and the secret fraction therefore becomes one. The only contribution to the secret key rate is then given by the repeater rate. In [Fig. \[fig:optscrminar\]]{} the optimal secret key rate versus the distance, obtained by maximizing over $p$ and $R$, is shown (see solid lines).
![(Color online) Quantum repeaters based on atomic ensembles: Optimal secret key rate per second versus the distance between Alice and Bob. The secret key rate has been obtained by maximizing over $p$ and $R$. Ideal set-up (solid line) with parameters $\eta_{m}={\eta_\mathrm{d}}=q=1,\gamma_{rep}=\infty$. More realistic set-up (dashed line) with parameters $\eta_{m}=1$, ${\eta_\mathrm{d}}=0.9$, $q=0.96$, $\gamma_{rep}=50$ MHz. []{data-label="fig:optscrminar"}](figure15){width="8cm"}
For the calculation of [Fig. \[fig:optscrminar\]]{}, we have assumed that the creation of local entanglement, i.e., of state $\rho_{0}^{s}$, is so fast that we can neglect the creation time. In the case of SPDC, the repetition rate of the source is related to the pump parameter $p$ and, moreover, the single-photon sources also have finite generation rates that should be taken into account. For this purpose, we introduce the photon-pair preparation time which is given by $T_{0}^{s}=\frac{1}{\gamma_{\rm rep} P_0^{s}}$ [@minar]. The formula for the repeater rate in this case corresponds to [Eq. ]{} with $T_0\to T_0+T_0^s$. As shown in [Fig. \[fig:scroptgamma\]]{}, when ${\eta_\mathrm{d}}=1$ the secret key rate is constant for $\gamma_{rep}>10^7$, however, for realistic detectors with ${\eta_\mathrm{d}}=0.9$, much higher repetition rates are required in order to reach the asymptotic value. Nowadays, SPDC sources reach a rate of about 100 MHz, whereas single-photon sources have a repetition rate of a few MHz [@eisaman2011invited]. Recently, a new single-photon source with repetition rate of 50 MHz has been realized [@lee2011planar]. In the following, we will employ $\gamma_{rep}=50$ MHz.
![(Color online) Quantum repeaters based on atomic ensembles: Optimal secret key rate per second versus the basic repetition rate of the source $\gamma_{\rm rep}$. The secret key rate has been obtained by maximizing over $p$ and $R$. (Parameters: ${\eta_\mathrm{d}}=\eta_{m}=q=1$).[]{data-label="fig:scroptgamma"}](figure16){width="8cm"}
A consequence of imperfect detectors is that multi-photon pulses contribute to the final state. The protocol we are considering here is less robust against detector inefficiencies than the original DLCZ protocol. This is due to the fact that successful entanglement swapping is conditioned on twofold detection as compared to one-photon detection of the DLCZ protocol. However, twofold detections permit to eliminate the vacuum in the memories [@sangouard_quantum_2011], thus increasing the final secret key rate. As shown in [Fig. \[fig:scroptetaD\]]{}, the secret key rate spans four orders of magnitude as ${\eta_\mathrm{d}}$ increases from $0.7$ to $1$. Thus, an improvement of the detector efficiency causes a considerable increase of the secret key rate. For example, for $N=3$, an improvement from ${\eta_\mathrm{d}}=0.85$ to ${\eta_\mathrm{d}}=0.88$ leads to a threefold increase of the secret key rate. Notice that we have considered photon detectors which are able to resolve photon numbers. Photon detectors with an efficiency as high as 95% have been realized [@lita2008counting]. These detectors work at the telecom bandwidth of 1556 nm and they have negligible dark counts. The drawback is that they need to operate at very low temperatures of 100 mK. The reading efficiency of the quantum memory ${\eta_{m}}$ plays a similar role as the detector efficiency. In accordance to [@sangouard_quantum_2011], intrinsic quantum memory efficiencies above 80% have been realized [@PhysRevLett.98.190503]; however, total efficiencies where coupling losses are included are much lower.
![(Color online) Quantum repeaters based on atomic ensembles: Optimal secret key rate per second versus the efficiency of the detectors ${\eta_\mathrm{d}}$. The secret key rate has been obtained by maximizing over $p$ and $R$. (Parameters: $\eta_{m}=q=1$, $\gamma_{\rm rep}=50$ MHz, $L=600$ km). []{data-label="fig:scroptetaD"}](figure17){width="8cm"}
A single-photon source is also characterized by its efficiency, i.e., the probability $q$ to emit a photon. As shown in [Fig. \[fig:scroptq\]]{}, we see that it is necessary to have single-photon sources with high efficiencies, in particular, when detectors are imperfect. The source proposed in [@lee2011planar] reaches $q=0.96$.
![(Color online) Quantum repeaters based on atomic ensembles: Optimal secret key rate per second versus the probability to emit a single photon. The secret key rate has been obtained by maximizing over $p$ and $R$. (Parameters: $\eta_{m}=1, \gamma_{rep}=50$ MHz, $L=600$ km).[]{data-label="fig:scroptq"}](figure18){width="8cm"}
In [Fig. \[fig:optscrminar\]]{} we show the secret key rate as a function of the distance between Alice and Bob for parameters (dashed lines) which are optimistic in the sense that they could be possibly reached in the near future. We observe that with an imperfect set-up and for $N=4$, the realistic secret key rate is by one order of magnitude smaller than the ideal value. This decrease is mainly due to finite detector efficiencies. For $N=4$, the secret key rate scales proportionally to ${\eta_\mathrm{d}}^2{\eta_\mathrm{d}}^2{\eta_\mathrm{d}}^{2\cdot4}{\eta_\mathrm{d}}^2$ (local creation, distribution, entanglement swapping, and QKD measurement). For ${\eta_\mathrm{d}}=0.9$, finite detector efficiencies lead to a decrease of the secret key rate by $78\%$. Regarding the optimal pump parameter $p$, we observe in [Fig. \[fig:scroptp\]]{} that for large distances ($L>600$km) its value is about $0.15\%$. The order of magnitude of this value is in agreement with the results found in [@amirloo_quantum_2010] regarding the original DLCZ protocol and the BB84-protocol.
![(Color online) Quantum repeaters based on atomic ensembles: Optimal value of $p$ versus the distance between Alice and Bob. The corresponding secret key rate is shown in [Fig. \[fig:optscrminar\]]{}. (Parameters: $\eta_{m}=1$, $\eta_D=0.9$, $q=0.96$, $\gamma_{rep}=50$ MHz, $L=600$ km)[]{data-label="fig:scroptp"}](figure19){width="8cm"}
The optimal reflectivity $R$ is given in [Fig. \[fig:scroptR\]]{}. We observe that as $N$ increases, the optimal value of $R$ has a modest increase.
![(Color online) Quantum repeaters based on atomic ensembles: Optimal value of the reflectivity $R$ versus the distance between Alice and Bob. The corresponding secret key rate is shown in [Fig. \[fig:optscrminar\]]{}. (Parameters: $\eta_{m}=1$, $\eta_D=0.9$, $q=0.96$, $\gamma_{rep}=50$ MHz)[]{data-label="fig:scroptR"}](figure20){width="8cm"}
\[sec:conclusions\] Conclusions and Outlook
============================================
Quantum repeaters represent nowadays the most promising and advanced approach to create long-distance entanglement. Quantum key distribution (QKD) is a developed technology which has already reached the market. One of the main limitations of current QKD is that the two parties have a maximal separation of 150 km, due to losses in optical fibers. In this paper, we have studied long-distance QKD by using quantum repeaters.
We have studied three of the main protocols for quantum repeaters, namely, the original protocol, the hybrid quantum repeater, and a variation of the so-called DLCZ protocol. Our analysis differs from previous treatments, in which only final fidelities have been investigated, because we maximize the main figure of merit for QKD – the secret key rate. Such an optimization is non-trivial, since there is a trade-off between the repeater pair-generation rate and the secret fraction: the former typically decreases when the final fidelity grows, whereas the latter increases when the final fidelity becomes larger. Our analysis allows to calculate secret key rates under the assumption of a single repeater chain with at most $2^k$ quantum memories per half station for respectively $k$ distillation rounds occurring strictly before the swappings start. The use of additional memories when parallelizing or even multiplexing several such repeater chains as well as the use of additional quantum error detection or even correction will certainly improve these rates, but also render the experimental realization much more difficult.
The comparison of different protocols is highly subjective, as there are different experimental requirements and difficulties for each of them, therefore here we investigated the main aspects for every protocol separately.
The general type of quantum repeater is a kind of prototype for a quantum repeater based on the original proposal [@briegel_quantum_1998]. We have provided an estimate of the experimental parameters needed to extract a secret key and showed what the role of each parameter is. We have found that the requirement on the initial fidelity is not so strong if distillation is allowed. However, quantum gates need to be very good (errors of the order of $1\%$).
Further, we have studied the hybrid quantum repeater. This protocol permits to perform both the initial entanglement distribution and the entanglement swapping with high efficiencies. The reason is that bright light sources are used for communication and Cavity Quantum Electrodynamics (CQED) interactions are employed for the local quantum gates, making the swapping, in principle, deterministic. Using photon-number resolving detectors, we have derived explicit formulas for the initial fidelity and the probability of success for entanglement distribution. Furthermore, we have found the form of the states after entanglement swapping and entanglement distribution in the presence of gate errors. We have seen that finite detector efficiencies do not play a major role regarding the generation probability. This permits to have high secret key rates in a set-up where it is possible to neglect imperfections of the detectors. By studying imperfect gates we found that excellent gates are necessary (errors of the order of $0.1\%$).
Finally, we have considered repeaters with atomic ensembles and linear optics. There exist many experimental proposals and therefore we have studied the scheme which is believed to be the fastest [@minar]. This scheme uses heralded qubit amplifiers for creating dual-rail encoded entanglement and entanglement swapping based on two-fold detection events. In contrast to the previous two schemes, the Bell measurement used for entanglement swapping is not able to distinguish all four Bell states. We have characterized all common imperfections and we have seen that using present technology, the performance of this type of quantum repeater in terms of secret key rates is only about one order of magnitude different from the corresponding ideal set-up. Thus, this scheme seems robust against most imperfections. These types of repeater schemes, as currently being restricted to linear optics, could still be potentially improved by allowing for additional nonlinear-optics elements. This may render the entanglement swapping steps deterministic, similar to the hybrid quantum repeater using CQED, and thus further enhance the secret key rates.
For the protocols considered here, single-qubit rotations were assumed to be perfect. Obviously, this assumption is not correct in any realistic situation. However, most of these single-qubit rotations can be replaced by simple bit flips of the classical outcomes which are used when the QKD protocol starts. Therefore, we see that in this case, specifically building a quantum repeater for QKD applications permits to relax the requirements on certain operations that otherwise must be satisfied for a more general quantum application, such as distributed quantum computation.
As an outlook our analysis can be extended in various directions: In our work we have considered standard quantum key distribution, in which Alice and Bob trust their measurement devices. To be more realistic, it is possible to relax this assumption and to consider device-independent quantum key distribution (DI-QKD) [@ekert1991quantum; @PhysRevLett.98.230501; @PhysRevLett.105.070501; @PhysRevA.84.010304; @PhysRevA.84.022325]. An analysis of the performance of long-distance DI-QKD can also be done using the methods that we developed in this paper.
A possible continuation of our work is the analysis of multiplexing [@Collins2007; @sangouard_quantum_2011]. It has been shown that this technique has significant advantage in terms of the decoherence time required by the quantum memories. On the other hand it produces only a moderate increase of the repeater rate [@jiang_fast_2007; @sangouard_quantum_2011; @razavi_quantum_2009]. Possible future analyses include the effect on the secret key rate by distilling in all nesting levels [@Bratzik2012] or by optimizing the repeater protocol as done in Refs. [@jiang_optimal_2007; @van2009system]. Moreover, other repeater protocols which are based on quantum error correction codes [@jiang_quantum_2009; @fowler.2010; @nadja.2012] may help to increase the secret key rate.
The authors acknowledge financial support by the German Federal Ministry of Education and Research (BMBF, project QuOReP). The authors would like to thank the organizers and participants of the quantum repeater workshops (project QuOReP) held in Hannover and Bad Honnef in 2011 and 2012. N. K.B. and P. v.L. thank the Emmy Noether Program of the Deutsche Forschungsgemeinschaft for financial support. S. A. thanks J. Minář for enlightening discussions and insightful comments.
\[sec:app:general\]Additional material for the general framework
================================================================
Generation rate with probabilistic entanglement swapping and distillation
-------------------------------------------------------------------------
In this appendix, we give the derivation of [Eq. ]{} in [Sec. \[subsec:parameters\]]{} which describes the generation rate of entangled pairs per time unit $T_0$ with probabilistic entanglement swapping and distillation, i.e.,
$$\label{eq:avgngen2}
{R_{\mathrm{REP}}}^{\rm prob}= \frac{1}{T_0}\left(\frac{2}{3a}\right)^{N+k} P_{0}P_{ES}^{(1)}P_{ES}^{(2)}...P_{ES}^{({N})}\prod_{i=1}^{k}{P_D}[i].$$
In [@sangouard_quantum_2011] the formula has been derived only for the case without distillation and there it reads as follows, $$\label{eq:avgngen3}
{R_{\mathrm{REP}}}^{\rm prob}= \frac{1}{T_0}\left(\frac{2}{3}\right)^{N} P_{0} P_{ES}^{(1)}P_{ES}^{(2)}...P_{ES}^{({N})},$$ where $P_{0}$ is the probability to generate a pair for entanglement swapping. This formula was derived for small $P_0$.
In order to incorporate distillation into [Eq. ]{} we use the definition of the recursive probability $P_{L_0}[k]$ given in [Eq. ]{}, see [@NadjaHybrid]. It describes the generation probability of an entangled pair after $k$ rounds of purification. If we choose an appropriate $a<1$ such that $Z_{1}(x)=\frac{3-2x}{x(2-x)}\geq\frac{3}{2x}a$ , we can rewrite $P_{L_0}[k]$: $$\begin{aligned}
\label{eq:approxPL0}
P_{L_0}[k]&=&\frac{{P_D}[k]}{Z_{1}(P_{L_0}[k-1])}\leq\frac{2}{3a}{P_D}[k]P_{L_0}[k-1]\nonumber \\
&=&\frac{2}{3a}{P_D}[k]\frac{{P_D}[k-1]}{Z_1(P_{L_0}[k-2])}\nonumber\\
&\leq&...\leq\left(\frac{2}{3a}\right)^{k}P_0\prod_{i=1}^{k}{P_D}[i],\end{aligned}$$ where in the last line $P_{L_0}[k]$ is a recursive formula. For deriving [Eq. ]{}, we replace in [Eq. ]{} $P_0$ by $P_{L_0}$ and we use [Eq. ]{}.
For the plots we have $L=600$ km and usually $\eta_d=0.9$ which leads to $P_{L_0}[k]\leq0.037$ and $a\leq0.994$.
Additional material for the original quantum repeater\[app:general-type\]
=========================================================================
Entanglement swapping\[sec:brig:es\]\[app:ES\]
----------------------------------------------
In this appendix we present the formulas of the state after entanglement swapping and the distillation protocol. Moreover, we bound also the role of dark counts in the entanglement swapping probability.
### The protocol {#the-protocol-1 .unnumbered}
We consider the total state $\rho_{ab}\otimes\rho_{cd}$. The entanglement swapping algorithm consists of the following steps:
1. A CNOT is applied on system $b$ as source and $c$ as target.
2. One output system is measured in the computational basis and the other one in the basis $\{{\left\vert{+}\right\rangle}:=\frac{{\left\vert{H}\right\rangle}+{\left\vert{V}\right\rangle}}{\sqrt{2}}, {\left\vert{-}\right\rangle}=\frac{{\left\vert{H}\right\rangle}-{\left\vert{V}\right\rangle}}{\sqrt{2}}\}$, obtained by applying a Hadamard gate.
3. In the standard entanglement swapping algorithm, a single qubit rotation depending on the outcome of the measurement is performed. However, for the purpose of QKD it is not necessary to do this single-qubit rotation[^12]. We propose that Bob collects the results of the Bell measurements, performs the standard QKD measurement and then he can apply a classical bit flip depending on the QKD measurement basis and on the Bell measurement outcomes.
### Formulas in the presence of imperfections {#formulas-in-the-presence-of-imperfections .unnumbered}
We consider a set-up with two detectors $d_1$ and $d_2$. We associate the detection pattern of these two detectors with a two-dimensional Hilbert space, e.g $d_1={\rm click}, d_2={\rm no click}\Rightarrow{\left\vert{H}\right\rangle}={\left\vert{1_{d_1},0_{d_2}}\right\rangle}$ and $d_1={\rm no click}, d_2={\rm click}\Rightarrow{\left\vert{V}\right\rangle}={\left\vert{0_{d_1},1_{d_2}}\right\rangle}$ where $\{{\left\vert{H}\right\rangle}, {\left\vert{V}\right\rangle}\}$ are a basis of a two-dimensional Hilbert space which can be, for example, identified with horizontal and vertical polarizations of a qubit. We discard those events where there are no clicks or when both detectors click. If the detectors are imperfect, we may have an error in the detection of the quantum state. The POVM consists of two elements $\Pi_H\; (\Pi_V)$ which detect mode ${\left\vert{H}\right\rangle} ({\left\vert{V}\right\rangle})$:
$$\begin{aligned}
\label{eq:POVM}
\Pi_H&:=\gamma{\left\vert{H}\right\rangle}{\left\langle{H}\right\vert}+(1-\gamma){\left\vert{V}\right\rangle}{\left\langle{V}\right\vert},\\
\Pi_V&:=\gamma{\left\vert{V}\right\rangle}{\left\langle{V}\right\vert}+(1-\gamma){\left\vert{H}\right\rangle}{\left\langle{H}\right\vert},\end{aligned}$$
with $$\begin{aligned}
\gamma&:=\frac{{\eta_\mathrm{d}}+{{{p_{\mathrm{dark}}}}}(1-{\eta_\mathrm{d}})}{{\eta_\mathrm{d}}+2{{{p_{\mathrm{dark}}}}}(1-{\eta_\mathrm{d}})},\end{aligned}$$ where ${{{p_{\mathrm{dark}}}}}$ is the dark count probability of the detectors and ${\eta_\mathrm{d}}$ is their efficiency[^13].
The POVM above has been used also in [@briegel_quantum_1998; @duer_quantum_1999], however, the connection with the imperfections of the detectors was not made.
If we start with the states $\rho_{ab}=\rho_{cd}=A{\left\vert{\phi^+}\right\rangle \left\langle{\phi^+}\right\vert}+B{\left\vert{\phi^-}\right\rangle \left\langle{\phi^-}\right\vert}+C{\left\vert{\psi^+}\right\rangle \left\langle{\psi^+}\right\vert}+D{\left\vert{\psi^-}\right\rangle \left\langle{\psi^-}\right\vert}$, the resulting state after entanglement swapping between $a$ and $d$ is still a Bell diagonal state with coefficients of the form [@duer_quantum_1998]:
\[eq:esfinstate\]
$$\begin{aligned}
A'=&\frac{1-p_{G}}{4}+p_{G}\left[\gamma^2(A^2+B^2+C^2+D^2) + 2(1-\gamma)^2(AD+BC) +2\gamma(1-\gamma)(A+D)(C+B)\right],\nonumber\\
B'=&\frac{1-p_{G}}{4}+p_{G}\left[2\gamma^2(AB+CD) + 2(1-\gamma)^2(AC+BD)+\gamma(1-\gamma)(A^2 +B^2 +C^2 +D^2 +2AD +2BC)\right],\nonumber\\
C'=&\frac{1-p_{G}}{4}+p_{G}\left[2\gamma^2(AC+BD) + 2(1-\gamma)^2(AB+CD)+\gamma(1-\gamma)(A^2 +B^2 +C^2 +D^2 +2AD +2BC)\right],\nonumber\\
D'=&\frac{1-p_{G}}{4}+p_{G}\left[2\gamma^2(AD+BC) + (1-\gamma)^2(A^2+B^2+C^2+D^2)+2\gamma(1-\gamma)(A+D)(B+C)\right],\end{aligned}$$
and the probability to obtain the state above is equal to $$\label{eq:briegpes}
P_{ES}({\eta_\mathrm{d}}, {{p_{\mathrm{dark}}}}):=\left((1-{{{p_{\mathrm{dark}}}}})({\eta_\mathrm{d}}+2{{{p_{\mathrm{dark}}}}}(1-{\eta_\mathrm{d}}))\right)^2,$$ which can be interpreted as the probability that entanglement swapping is successful[^14]. Note that $P(\eta, 0)=\eta^2$ and $P(1, 0)=1$ as we expect. When we consider dark counts $p_{\rm dark}<10^{-5}$, then these are negligible as $(P_{ES}(0.1,10^{-5})/(P_{ES}(0.1,0)))^N<1.03^N$, so the impact on the secret key rate is minimal. Note that we open the gates only for a short time window, which is the interval of time where we expect the arrival of a photon. The dark count probability ${p_{\mathrm{dark}}}$ represents the probability that in the involved time window the detector gets a dark count.
Distillation
------------
### The protocol {#the-protocol-2 .unnumbered}
We assume that Alice and Bob hold two Bell diagonal states $\rho_{a_1,b_1}$ and $\rho_{a_2,b_2}$. The algorithm is the following:
1. In the computational basis, Alice rotates her particles by $\frac{\pi}{2}$ about the $X$-axis, whereas Bob applies the inverse rotation ($-\frac{\pi}{2}$) on his particles.
2. Then they apply on both sides a CNOT operation, where the states $a_1\; (b_1)$ serve as source and $a_2\; (b_2)$ as target.
3. The states corresponding to the target are measured in the computational basis. If the measurement results coincide, the resulting state $\rho_{a_1, b_1}$ is a purified state; otherwise, the resulting state is discarded. Therefore, this entanglement distillation scheme is probabilistic.
### \[sec:appDistill\]Formulas in the presence of imperfections {#secappdistillformulas-in-the-presence-of-imperfections .unnumbered}
Given a Bell diagonal state with the following coefficients $$\rho_{ab}=A{\left\vert{\phi^+}\right\rangle \left\langle{\phi^+}\right\vert}+B{\left\vert{\phi^-}\right\rangle \left\langle{\phi^-}\right\vert}+C{\left\vert{\psi^+}\right\rangle \left\langle{\psi^+}\right\vert}+D{\left\vert{\psi^-}\right\rangle \left\langle{\psi^-}\right\vert},
\label{eq:bell}$$
the coefficients transform according to the following map [@deutsch1996quantum]: $$\begin{aligned}
A'&=& \frac{1}{{P_D}}\left(A^2+D^2\right),\\
B'&=&\frac{1}{{P_D}}\left(2AD\right),\\
C'&=& \frac{1}{{P_D}}\left(B^2+C^2\right),\\
D'&=&\frac{1}{{P_D}}\left(2BC\right),\end{aligned}$$ where ${P_D}$ is the probability that the measurement outcomes are both the same for Alice and Bob, and thus the probability of successful distillation is: $${P_D}[k]=\left(A_{k-1} +D_{k-1}\right)^2+\left(B_{k-1}+C_{k-1}\right)^2.
\label{eq:pevenD}$$
Including the gate quality $p_G$, these formulas change to [@duer_quantum_1998]:
$${P_D}[k]=\frac{1}{2}\left\{1+p_G^2\left(-1+2A_{k-1} +2D_{k-1}\right)^2\right\}.
\label{eq:pevenDe}$$
with
$$\begin{aligned}
A'&=&\left[1+p_G^2\left( (A-B-C+D) (3 A+B+C+3 D)+4 (A-D)^2\right)\right]/(8 {P_D}),\\
B' &=&\left[1-p_G^2 \left(A^2+2 A (B+C-7 D)+(B+C+D)^2\right)\right]/(8 {P_D}),\\
C' &=&\left[1+p_G^2 \left(4 (B-C)^2-(A-B-C+D) (A+3 (B+C)+D)\right)\right]/(8 {P_D}),\\
D' &=& \left[1-p_G^2 \left(A^2+2 A (B+C+D)+B^2+2 B (D-7 C)+(C+D)^2\right)\right]/(8 {P_D}).\end{aligned}$$
\[sec:app:hybrid\]Additional material for the hybrid quantum repeater
=====================================================================
In this appendix we derive the formula for successful entanglement generation when PNRD are used for the measurements. Moreover, we present the formulas for the states after entanglement swapping and entanglement distillation.
\[sec:app:gen\]Entanglement generation
--------------------------------------
The total state before the detector measurements is described by [@azuma]
$$\begin{aligned}
\rho_{AB,b_3,b_5}=&p\left\{\left[{\left\vert{0}\right\rangle}_{b_3}({\left\vert{00}\right\rangle}_{AB}{\left\vert{\beta}\right\rangle}_{b_5}+{\left\vert{11}\right\rangle}_{AB}{\left\vert{-\beta}\right\rangle}_{b_5})/2+{\left\vert{0}\right\rangle}_{b_5}({\left\vert{01}\right\rangle}_{AB}{\left\vert{-\beta}\right\rangle}_{b_3}+{\left\vert{10}\right\rangle}_{AB}{\left\vert{\beta}\right\rangle}_{b_3})/2\right]\times H.c.\right\}+\nonumber\\
&(1-p)\left\{\left[{\left\vert{0}\right\rangle}_{b_3}({\left\vert{00}\right\rangle}_{AB}{\left\vert{\beta}\right\rangle}_{b_5}-{\left\vert{11}\right\rangle}_{AB}{\left\vert{-\beta}\right\rangle}_{b_5})/2+{\left\vert{0}\right\rangle}_{b_5}({\left\vert{01}\right\rangle}_{AB}{\left\vert{-\beta}\right\rangle}_{b_3}-{\left\vert{10}\right\rangle}_{AB}{\left\vert{\beta}\right\rangle}_{b_3})/2\right]\times H.c.\right\},
\label{app:instate}\end{aligned}$$
where $H.c.$ stays for the Hermitian conjugate of the previous term, $A$ ($B$) represents the qubit at Alice’s (Bob’s) side, $b_3$ is the coherent-state mode arriving at the detector $D_1$, $b_5$ is the coherent-state mode arriving at the detector $D_2$, and $\beta=i\sqrt{2\eta_t}\sin{(\theta/2)}$ (see figure [Eq. ]{}). The probability of error caused by photon losses in the transmission channel is given by $(1-p)$, with $p=(1+e^{-2(1-\eta_t)\alpha^2\sin^2{(\theta/2)}})/2$. It is possible to observe from [Eq. ]{} that whenever Bob detects a click in either one of the detectors $D_1$ or $D_2$, an entangled state has been distributed between qubits $A$ and $B$.
We discuss in the following the case that $D_1$ and $D_2$ are imperfect PNRD (see [Eq. ]{}). When detector $D_1$ does not click and $D_2$ clicks, the resulting state $\rho_{AB}$ is then given by $$\label{eq:rhoabhybridapp}
\rho_{AB}=\frac{\mbox{tr}_{b_3b_5}(\Pi_{b_3}^{(0)}\Pi_{b_5}^{(n)}\rho_{AB,b_3,b_5})}{\mbox{tr}(\Pi_{b_3}^{(0)}\Pi_{b_5}^{(n)}\rho_{AB,b_3,b_5})},$$ with $n>0$. The same result up to local operations can be obtained in the opposite case (a click in detector $D_1$ and no click in detector $D_2$).
Depending on the outcome of the detector, a local operation maybe applied to change the resulting state into the desired state. In this way, if the outcome is an even number, nothing should be done, otherwise a $Z$ operation should be applied. Following this, the resulting state can be written as $$\rho=F_0{\left\vert{\phi^+}\right\rangle}{\left\langle{\phi^+}\right\vert}+(1-F_0){\left\vert{\phi^-}\right\rangle}{\left\langle{\phi^-}\right\vert},\nonumber$$ where $$\begin{aligned}
F_0=&\frac{({\left\langle{00}\right\vert}_{AB}+(-1)^n{\left\langle{11}\right\vert}_{AB})}{\sqrt{2}}\rho_{A,B}\frac{({\left\vert{00}\right\rangle}_{AB}+(-1)^n{\left\vert{11}\right\rangle}_{AB})}{\sqrt{2}}\nonumber\\
=&\frac{1+e^{-2(1+\eta_t(1-2\eta_d))\alpha^2\sin^2(\theta/2)}}{2}.\end{aligned}$$ The probability of success is calculated by adding all successful events, and is given by $$P_0=\sum_{n=1}^{\infty}\mbox{tr}(\Pi_{b_3}^{(0)}\Pi_{b_5}^{(n)}\rho_{AB,b_3,b_5}+\Pi_{b_5}^{(0)}\Pi_{b_3}^{(n)}\rho_{AB,b_3,b_5}).$$
Combining [Eq. ]{} and [Eq. ]{} we obtain [Eq. ]{}.
\[sec:app:swap\]Entanglement swapping
-------------------------------------
The initial states used in the swapping operation are a full rank mixture of the Bell states, $\rho_0:=A{\left\vert{\phi^+}\right\rangle}{\left\langle{\phi^+}\right\vert}+B{\left\vert{\phi^-}\right\rangle}{\left\langle{\phi^-}\right\vert}+C{\left\vert{\psi^+}\right\rangle}{\left\langle{\psi^+}\right\vert}+D{\left\vert{\psi^-}\right\rangle}{\left\langle{\psi^-}\right\vert}$. After the connection, the resulting state will remain in the same form, $A'{\left\vert{\phi^+}\right\rangle}{\left\langle{\phi^+}\right\vert}+B'{\left\vert{\phi^-}\right\rangle}{\left\langle{\phi^-}\right\vert}+C'{\left\vert{\psi^+}\right\rangle}{\left\langle{\psi^+}\right\vert}+D'{\left\vert{\psi^-}\right\rangle}{\left\langle{\psi^-}\right\vert}$, but with new coefficients:
$$\begin{aligned}
A'&=2 B C + 2 A D +
2 [-2 B C + A (B + C - 2 D) + (B + C) D] p_G + (A - B - C + D)^2 p_G^2,\nonumber\\
B'&=2 A C + 2 B D + [A^2 + (B + C)^2 - 4 B D + D^2 +
2 A (-2 C + D)] p_G - (A - B - C + D)^2 p_G^2,\nonumber\\
C'&=2 A B + 2 C D + [A^2 + (B + C)^2 - 4 C D + D^2 +
2 A (-2 B + D)] p_G - (A - B - C + D)^2 p_G^2,\nonumber\\
D'&=A^2 + B^2 + C^2 + D^2 -
2 [A^2 + B^2 + C^2 - A (B + C) -(B + C) D + D^2] p_G + (A - B - C +
D)^2 p_G^2.
\label{coeff}\end{aligned}$$
It is possible to see that $A'+B'+C'+D'=1$, such that even for the case of imperfect connection operations, the swapping occurs deterministically.
\[sec:app:dist\]Entanglement distillation
-----------------------------------------
We calculated also the effect of the gate error in the distillation step. Starting with two copies of states in the form of $\rho_0:=A{\left\vert{\phi^+}\right\rangle}{\left\langle{\phi^+}\right\vert}+B{\left\vert{\phi^-}\right\rangle}{\left\langle{\phi^-}\right\vert}+C{\left\vert{\psi^+}\right\rangle}{\left\langle{\psi^+}\right\vert}+D{\left\vert{\psi^-}\right\rangle}{\left\langle{\psi^-}\right\vert}$, the resulting state after one round of distillation is given by $A'{\left\vert{\phi^+}\right\rangle}{\left\langle{\phi^+}\right\vert}+B'{\left\vert{\phi^-}\right\rangle}{\left\langle{\phi^-}\right\vert}+C'{\left\vert{\psi^+}\right\rangle}{\left\langle{\psi^+}\right\vert}+D'{\left\vert{\psi^-}\right\rangle}{\left\langle{\psi^-}\right\vert}$, where
$$\begin{aligned}
A'&=\frac{1}{{P_D}}\left(D^2 + A^2 [1 + 2 (-1 + p_G) p_G]^2 -
2 A (-1 + p_G) p_G [C + 2 D+ 2 (B - C - 2 D) p_G +2 (-B + C + 2 D) p_G^2]\right.\nonumber\\
&\left.-2 D (-1 + p_G) p_G \{-2 D - 2 (C + D) (-1 + p_G) p_G+ B [1 + 2 (-1 + p_G) p_G]\}\right),\nonumber\\
B'&=\frac{1}{{P_D}}\left[-2 \bm{(}D (-1 + p_G) p_G (C + D + 2 B p_G - 2 C p_G - 2 D p_G - 2 B p_G^2 + 2 C p_G^2 +
2 D p_G^2)+A^2 p_G (-1 + 3 p_G - 4 p_G^2 + 2 p_G^3)\right.\nonumber\\
&\left.-A \{D (1 - 2 p_G + 2 p_G^2)^2 - (-1 + p_G) p_G [-2 C (-1 + p_G) p_G +
B (1 - 2 p_G + 2 p_G^2)]\}\bm{)}\right],\nonumber\\
C'&=\frac{1}{{P_D}}\left(B^2 (1 - 2 p_G + 2 p_G^2)^2 -
2 B (-1 + p_G) p_G [-2 A (-1 + p_G) p_G + D (1 - 2 p_G + 2 p_G^2) +
C (2 - 4 p_G + 4 p_G^2)] \right.\nonumber\\
&\left.+C \{C (1 - 2 p_G + 2 p_G^2)^2-
2 (-1 + p_G) p_G [-2 D (-1 + p_G) p_G+ A (1 - 2 p_G + 2 p_G^2)]\}\right),\nonumber\\
D'&=\frac{1}{{P_D}}\left\{-2 (C (-1 + p_G) p_G (C + D + 2 A p_G - 2 C p_G - 2 D p_G - 2 A p_G^2 + 2 C p_G^2 +
2 D p_G^2) +B^2 p_G (-1 + 3 p_G- 4 p_G^2 + 2 p_G^3)\right. \nonumber\\
&\left.-B \{C (1 - 2 p_G + 2 p_G^2)^2- (-1 + p_G) p_G [-2 D (-1 + p_G) p_G +
A (1 - 2 p_G + 2 p_G^2)]\})\right\},
\label{coeffpur}\end{aligned}$$
${P_D}$ is the distillation probability of success and is given by $$\begin{aligned}
{P_D}=&&(B + C)^2 + (A + D)^2 - 2 (A - B - C + D)^2 p_G\nonumber\\
&& +2 (A - B - C + D)^2 p_G^2.
\label{ppur}\end{aligned}$$ For the case of $p_G=1$, [Eq. ]{} and [Eq. ]{} are in accordance with [@deutsch1996quantum].
[^1]: Note that the states of the DLCZ-type quantum repeaters (see section \[sec:atomicensenbles\]) are only effectively two-qubit states, when higher-order excitations of the atom-light entangled states [@duan_long-distance_2001], or those of the states created through parametric down conversion [@minar], are neglected.
[^2]: As proven in [@Renner:2005pi; @Kraus:2005kx], it is possible to apply an appropriate local twirling operation that transforms an arbitrary two-qubit state into a Bell diagonal state, while the security of the protocol is not compromised.
[^3]: In this paper, by *dual-rail representation* we mean that a single photon can be in a superposition of two optical modes, thus representing a single qubit. By *single-rail representation* we mean that a qubit is implemented using only one single optical mode. See [@kok_introduction_2010] for additional details.
[^4]: The ket ${\left\vert{abcd}\right\rangle}$ is a vector in a Hilbert space of four modes and the values of $a$, $b$, $c$ and $d$ represent the number of excitations in the Fock basis.
[^5]: Note that the formula for the six-state protocol is independent of the choice of basis, when we assume the state of Alice and Bob $\rho_{AB}$ to be Bell diagonal. Then the secret fraction reduces to $r_{\infty}^{6S}=1-S(\rho_E)$ with $S(\rho)$ the von Neumann entropy and $\rho_E$ is the eavesdropper’s state.
[^6]: The supposed link between the effect of imperfect detectors and the determinism of the entanglement swapping here assumes the following. Any incomplete detection patterns that occur in the Bell measurements due to imperfect detectors are considered as inconclusive results and will be discarded. Conversely, with perfect detectors, we assume that we always have complete patterns and thus the Bell state discrimination becomes complete too. Note that this kind of reasoning directly applies to Bell measurements in dual-rail encoding, where the conclusive output patterns always have the same fixed total number for every Bell state (namely two photons leading to two-fold detection events), and so any loss of photons will result in patterns considered inconclusive. In single-rail encoding, the situation is more complicated and patterns considered conclusive may be the result of an imperfect detection.
[^7]: One may also measure the qubus using homodyne detection [@van_loock_hybrid_2006]. However, for this scheme, final fidelities would be limited to $F_0 <0.8$ for $L_0=10$ km [@van_loock_hybrid_2006], whereas by using unambiguous state discrimination, we can tune the parameters for any distance $L_0$, such that the fidelity $F_0$ can be chosen freely and, in particular, made arbitrarily close to unity at the expense of the success probability dropping close to zero [@van_loock_quantum_2008].
[^8]: Note that this error model is considering a CZ gate operation. For a CNOT gate, $Z$ errors can be transformed into $X$ errors.
[^9]: \[footnote9\] Note that we assume perfect qubit measurements for the distillation and the swapping, but imperfect two-qubit gates. In principle, these qubit measurements can be done using a local qubus and homodyne measurement [@van_loock_quantum_2008]. In this case, losses in the qubit measurement can be absorbed into losses of the gates. On the other hand, if we consider imperfect detectors for the qubit measurement then entanglement swapping will succeed with probability given by [Eq. ]{}.
[^10]: Very recently it was shown that in the context of QKD over continuous variables, an effective suppression of channel losses and imperfections can also be achieved via a virtual, heralded amplification on the level of the classical post-processing [@2012arXiv1205.6933F; @2012arXiv1206.0936W]. In this case, it is not even necessary to physically realize a heralded amplifier.
[^11]: \[foot:dlczcalc\]In our calculation, similar to [@minar], we consider only those terms with $m\leq2$. The reason is that the contribution to the total trace of the first three terms is given by $1-p^3$ and therefore for $p<0.1$ the state obtained by considering only the first three terms differs in a negligible way from the full state.
[^12]: Note that this step is different from [@briegel_quantum_1998], where the single-qubit rotations were explicitly included.
[^13]: The coefficient $\gamma$ can be calculated as follows: the POVM for having a click under the assumption of single-photon sources and imperfect detectors is given by $$E^{\rm (click)}={{{p_{\mathrm{dark}}}}}{\left\vert{0}\right\rangle \left\langle{0}\right\vert}+\left(1-(1-{{{p_{\mathrm{dark}}}}})(1-{\eta_\mathrm{d}})\right){\left\vert{1}\right\rangle \left\langle{1}\right\vert}$$ and no click $$E^{\rm(no click)}=(1-{{{p_{\mathrm{dark}}}}}){\left\vert{0}\right\rangle \left\langle{0}\right\vert}+(1-{{{p_{\mathrm{dark}}}}})(1-{\eta_\mathrm{d}}){\left\vert{1}\right\rangle \left\langle{1}\right\vert}.$$ When we say that the detector $a$ clicked, and $b$ did not click and we discard the vacuum events, and those where both detectors clicked, the POVM looks as follows: $$\begin{aligned}
&& E_a^{\rm(click)}\otimes E_b^{\rm (noclick)}\\
&&=\left(1-(1-{{{p_{\mathrm{dark}}}}})(1-{\eta_\mathrm{d}})\right)(1-{{{p_{\mathrm{dark}}}}}){\left\vert{1_a,0_b}\right\rangle \left\langle{1_a,0_b}\right\vert}\\
&&+{{{p_{\mathrm{dark}}}}}(1-{{{p_{\mathrm{dark}}}}})(1-{\eta_\mathrm{d}}) {\left\vert{0_a,1_b}\right\rangle \left\langle{0_a,1_b}\right\vert}.\end{aligned}$$ The trace is $(1-{{{p_{\mathrm{dark}}}}})({\eta_\mathrm{d}}+2{{{p_{\mathrm{dark}}}}}(1-{\eta_\mathrm{d}}))$, which is exactly the probability that we have this measurement. If we normalize this measurement and relate it to the POVM in [Eq. ]{}, we get $\gamma$.
[^14]: This probability was derived by taking the probability of the measurement in the preceding footnote squared, as we need two coincident clicks for the Bell measurement.
|
---
abstract: |
We consider a time discretization of incompressible Navier-Stokes equations with spatial periodic boundary conditions in the vorticity-velocity formulation. The approximation is based on freezing the velocity on time subintervals resulting in linear parabolic equations for vorticity. Probabilistic representations for solutions of these linear equations are given. At each time step, the velocity is expressed via vorticity using a formula corresponding to the Biot–Savart-type law. We show that the approximation is divergent free and of first order. The results are extended to two-dimensional stochastic Navier-Stokes equations with additive noise, where, in particular, we prove the first mean-square convergence order of the vorticity approximation.
**Keywords.** Navier-Stokes equations, vorticity, numerical method, stochastic partial differential equations, mean-square convergence.
**AMS 2000 subject classification.** 65C30, 60H15, 60H35, 35Q30
author:
- 'G.N. Milstein[^1]'
- 'M.V. Tretyakov[^2]'
title: 'Approximation of deterministic and stochastic Navier-Stokes equations in vorticity-velocity formulation'
---
Introduction
============
Navier-Stokes equations (NSE), both deterministic and stochastic, are important for a number of applications and, consequently, development and analysis of numerical methods for simulation of NSE are of significant interest. The theory and applications of deterministic NSE can be found, e.g. in [@CF; @DJT; @MB; @FMRT01; @RT; @T] and for stochastic NSE – e.g. in [Flandoli,MatPhD,RozNS05]{}. The literature on numerics for deterministic NSE is extensive [@FEM; @Peyret; @W] (see also references therein) while the literature on numerics for stochastic NSE is still rather sparse, let us mention [@CP; @BCP10; @BBM14; @Dorsek; @MTsns].
In this paper we consider incompressible NSE in the vorticity-velocity formulation with periodic boundary conditions (see, e.g. [@MB] for the deterministic case and [@HaMa06] for the stochastic one). In the deterministic case we deal with both two-dimensional and three-dimensional NSE, in the stochastic case we are interested in two-dimensional NSE with additive noise. We study their time discretization which is based on freezing the velocity at every time step. Consequently, at every step we just need to solve a system of linear parabolic PDEs. To compute the velocity, we express it via the vorticity field, i.e. we derive a periodic version of Biot-Savart’s law (see e.g. [@MB p. 50 and p. 71]). The constructed approximations of both vorticity and velocity are divergent free. We prove convergence theorems for the suggested approximation. The second part of the paper deals with NSE with additive noise.
The paper is organised as follows. We introduce function spaces required, recall the Helmholtz-Hodge decomposition, and derive the periodic version of Biot-Savart’s law in Section \[Secpre\]. In the deterministic case we suggest to use probabilistic representations together with ideas of weak-sense numerical integration of SDEs for solving the system of linear parabolic PDEs at every step of the time discretization. In Section [SecPrb]{}, we present probabilistic representations appropriate for this task which are based on [@M] (see also [@MSS; @MT1]). The numerical method is proposed and analysed in Section \[sec:approx\], where in particular its first order convergence in $L^{2}$-norm is proved. Then the ideas of Section \[sec:approx\] are transferred over to the case of NSE with additive noise in Section \[sec:sns\], including a proof of first-order mean-square convergence of the time-discretization of the stochastic NSE in the vorticity-velocity formulation. Ideas used in the proof are of potential interest for convergence analysis of numerical methods for a wider class of semilinear SPDEs.
Preliminaries\[Secpre\]
=======================
In the first part of this paper (Sections \[SecPrb\] and \[sec:approx\]), we consider the two- and three-dimensional deterministic incompressible NSE for velocity $v$ and pressure $q$ with space periodic conditions: $$\begin{aligned}
\frac{\partial v}{\partial s}+(v,\nabla )v+\nabla q-\frac{\sigma ^{2}}{2}%
\Delta v=F, \label{NS1} \\
\mathop{\rm div}v=0. \label{NS2}\end{aligned}$$In (\[NS1\])-(\[NS2\]) we have $0<s\leq T,$ $x\in \mathbf{R}^{n},$ $%
v\in \mathbf{R}^{n},$ $F$ $\in \mathbf{R}^{n},$ $n=2,3,$ $q$ is a scalar. The velocity vector $v=(v^{1},\ldots ,v^{n})^{\top }$ satisfies initial conditions$$v(0,x)=\varphi (x) \label{NS3}$$and spatial periodic conditions$$v(s,x+Le_{i})=v(s,x),\ i=1,\ldots ,n,\ 0\leq s\leq T. \label{NS4}$$Here $\mathop{\rm div}\varphi =0,$ $\{e_{i}\}$ is the canonical basis in $%
\mathbf{R}^{n},$ and $L>0$ is the period. For simplicity in writing, the periods in all the directions are taken to be equal. The function $F=F(s,x)$ and pressure $q=q(s,x)$ are assumed to be spatial periodic as well.
In what follows we will consider the deterministic NSE with negative direction of time which is convenient for probabilistic representations considered in Section \[SecPrb\]. By an appropriate change of the time variable and functions, the NSE (\[NS1\])-(\[NS2\]) with positive direction of time can be rewritten in the form: $$\begin{aligned}
\frac{\partial u}{\partial t}+\frac{\sigma ^{2}}{2}\Delta u-(u,\nabla
)u-\nabla p+f=0, \label{ns1p} \\
\mathop{\rm div}u=0, \label{ns2p}\end{aligned}$$where $0\leq t<T,$ $x\in \mathbf{R}^{n},$ $u\in \mathbf{R}^{n},$ $f$ $\in
\mathbf{R}^{n},$ $n=2,3,$ the pressure $p$ is a scalar. The velocity vector $u=(u^{1},\ldots ,u^{n})^{\top }$ satisfies the terminal condition$$u(T,x)=\varphi (x) \label{ns3p}$$and spatial periodic conditions$$u(t,x+Le_{i})=u(t,x),\ i=1,\ldots ,n,\ 0\leq t\leq T. \label{ns4p}$$Throughout the paper we will assume that this problem has a unique, sufficiently smooth classical solution. In the two-dimensional case ($n=2)$ the corresponding theory is available, e.g. in [@FMRT01; @MB]. In the remaining part of this preliminary section we recall the required function spaces [@DJT; @RT; @T] and write the NSE in vorticity formulation.
Function spaces and the Helmholtz-Hodge decomposition
-----------------------------------------------------
Let $\{e_{i}\}$ be the canonical basis in $\mathbf{R}^{n}.$ We shall consider spatial periodic $n$-vector functions $u(x)=(u^{1}(x),\ldots
,u^{n}(x))^{\top }$ in $\mathbf{R}^{n}:$ $u(x+Le_{i})=u(x),\ i=1,\ldots ,n,$ where $L>0$ is the period in $i$th direction. Denote by $Q=(0,L]^{n}$ the cube of the period. We denote by $\mathbf{L}^{2}(Q)$ the Hilbert space of functions on $Q$ with the scalar product and the norm$$(u,v)=\int_{Q}\sum_{i=1}^{n}u^{i}(x)v^{i}(x)dx,\ \Vert u\Vert =(u,u)^{1/2}.$$ We keep the notation $|\cdot|$ for the absolute value of numbers and for the length of $n$-dimensional vectors, for example,$$|u(x)|=[(u^{1}(x))^{2}+\cdots+(u^{n}(x))^{2}]^{1/2}.$$
We denote by $\mathbf{H}_{p}^{m}(Q),\ m=0,1,\ldots,$ the Sobolev space of functions which are in $\mathbf{L}^{2}(Q),$ together with all their derivatives of order less than or equal to $m,$ and which are periodic functions with the period $Q.$ The space $\mathbf{H}_{p}^{m}(Q)$ is a Hilbert space with the scalar product and the norm$$(u,v)_{m}=\int_{Q}\sum_{i=1}^{n}\sum_{[\alpha^{i}]\leq
m}D^{\alpha^{i}}u^{i}(x)D^{\alpha^{i}}v^{i}(x)dx,\ \Vert
u\Vert_{m}=[(u,u)_{m}]^{1/2},$$ where $\alpha^{i}=(\alpha_{1}^{i},\ldots,\alpha_{n}^{i}),\
\alpha_{j}^{i}\in\{0,\ldots,m\},\
[\alpha^{i}]=\alpha_{1}^{i}+\cdots+\alpha_{n}^{i},$ and$$D^{\alpha^{i}}=D_{1}^{\alpha_{1}^{i}}\cdots D_{n}^{\alpha_{n}^{i}}=\frac{%
\partial^{\lbrack\alpha^{i}]}}{\partial(x^{1})^{\alpha_{1}^{i}}\cdots%
\partial(x^{n})^{\alpha_{n}^{i}}}\ ,\ i=1,\ldots,n.$$ Note that $\mathbf{H}_{p}^{0}(Q)=\mathbf{L}^{2}(Q).$
Introduce the Hilbert subspaces of $\mathbf{H}_{p}^{m}(Q):$$$\begin{aligned}
\mathbf{V}_{p}^{m}& = \{v:\ v\in \mathbf{H}_{p}^{m}(Q),\ \mathop{\rm div}%
v=0\},\ m>0, \\
\mathbf{V}_{p}^{0}& = \text{the closure of } \mathbf{V}_{p}^{m},\ m>0 \text{
in }\mathbf{L}^{2}(Q).\end{aligned}$$
Denote by $P$ the orthogonal projection in $\mathbf{H}_{p}^{m}(Q)$ onto $%
\mathbf{V}_{p}^{m}$ (we omit $m$ in the notation $P$ here). The operator $P$ is often called the Leray projection. Due to the Helmholtz-Hodge decomposition, any function $u\in\mathbf{H}_{p}^{m}(Q)$ can be represented as $$u=Pu+\nabla g,\ \mathop{\rm div}Pu=0,$$ where $g=g(x)$ is a scalar $Q$-periodic function such that $\nabla g\in%
\mathbf{H}_{p}^{m}(Q).$ It is natural to introduce the notation $P^{\bot
}u:=\nabla g$ and hence write $$u=Pu+P^{\bot}u$$ with $$P^{\bot}u\in(\mathbf{V}_{p}^{m})^{\bot}=\{v:\ v\in\mathbf{H}_{p}^{m}(Q),\
v=\nabla g\}.$$
Let$$\begin{aligned}
u(x)=\sum_{\mathbf{n}\in \mathbf{Z}^{n}}u_{\mathbf{n}}e^{i(2\pi /L)(\mathbf{n%
},x)},\ g(x)=\sum_{\mathbf{n}\in \mathbf{Z}^{n}}g_{\mathbf{n}}e^{i(2\pi /L)(%
\mathbf{n},x)},\ g_{\mathbf{0}}=0, \label{N00} \\
Pu(x)=\sum_{\mathbf{n}\in \mathbf{Z}^{n}}(Pu)_{\mathbf{n}}e^{i(2\pi /L)(%
\mathbf{n},x)},\ P^{\bot }u(x)=\nabla g(x)=\sum_{\mathbf{n}\in \mathbf{Z}%
^{n}}(P^{\bot }u)_{\mathbf{n}}e^{i(2\pi /L)(\mathbf{n},x)} \notag\end{aligned}$$be the Fourier expansions of $u,$ $g,$ $Pu,$ and $P^{\bot }u=\nabla g.$ Here $u_{\mathbf{n}},$ $(Pu)_{\mathbf{n}},\ $and $(P^{\bot }u)_{\mathbf{n}%
}=(\nabla g)_{\mathbf{n}}$ are $n$-dimensional vectors and $g_{\mathbf{n}}$ are scalars. We note that $g_{\mathbf{0}}$ can be any real number but for definiteness we set $g_{\mathbf{0}}=0$ without loss of generality [FMRT01]{}. The coefficients $(Pu)_{\mathbf{n}},\ (P^{\bot }u)_{\mathbf{n}}$, and $g_{\mathbf{n}}$ can be easily expressed in terms of $u_{\mathbf{n}}:$ $$\begin{aligned}
(Pu)_{\mathbf{n}}& =u_{\mathbf{n}}-\frac{u_{\mathbf{n}}^{\top }\mathbf{n}}{|%
\mathbf{n}|^{2}}\mathbf{n,\ }(P^{\bot }u)_{\mathbf{n}}=i\frac{2\pi }{L}g_{%
\mathbf{n}}\mathbf{n=}\frac{u_{\mathbf{n}}^{\top }\mathbf{n}}{|\mathbf{n}%
|^{2}}\mathbf{n,\ } \label{N01} \\
g_{\mathbf{n}}& =-i\frac{L}{2\pi }\frac{u_{\mathbf{n}}^{\top }\mathbf{n}}{|%
\mathbf{n}|^{2}},\ \mathbf{n\neq 0,\ }g_{\mathbf{0}}=0. \notag\end{aligned}$$
We have$$\nabla e^{i(2\pi /L)(\mathbf{n},x)}=\mathbf{n}e^{i(2\pi /L)(\mathbf{n}%
,x)}\cdot i\frac{2\pi }{L},$$hence $u_{\mathbf{n}}e^{i(2\pi /L)(\mathbf{n},x)}\in \mathbf{V}_{p}^{m}$ if and only if $(u_{\mathbf{n}},\mathbf{n)}=0.$ We obtain from here that the orthogonal basis of the subspace $(\mathbf{V}_{p}^{m})^{\bot }$ consists of $%
\mathbf{n}e^{i(2\pi /L)(\mathbf{n},x)},\ \mathbf{n}\in \mathbf{Z}^{n},\
\mathbf{n\neq 0}$; and an orthogonal basis of $\mathbf{V}_{p}^{m}$ consists of $_{k}u_{\mathbf{n}}e^{i(2\pi /L)(\mathbf{n},x)},$ $k=1,\ldots ,n-1,\
\mathbf{n}\in \mathbf{Z}^{n}$, where under $\mathbf{n\neq 0}$ the vectors $%
_{k}u_{\mathbf{n}}$ are orthogonal to $\mathbf{n:}$** **$\mathbf{(}%
_{k}u_{\mathbf{n}},\mathbf{n)}=0,\ k=1,\ldots ,n-1,$ and they are orthogonal among themselves: $\mathbf{(}_{k}u_{\mathbf{n}},\ _{m}u_{\mathbf{n}}\mathbf{)%
}=0,$ $k,m=1,\ldots ,n-1,$ $m\neq k,$ and finally, for $\mathbf{n=0,}$ the vectors $_{k}u_{\mathbf{0}},\ k=1,\ldots ,n,$ are orthogonal. In particular, in the two-dimensional case $(n=2),$ these bases are, correspondingly (for $%
\mathbf{n}\neq 0$): $$\left[
\begin{array}{c}
n_{1} \\
n_{2}%
\end{array}%
\right] e^{i(2\pi /L)(\mathbf{n},x)} \text{ \ \ and \ }\ \left[
\begin{array}{c}
-n_{2} \\
n_{1}%
\end{array}%
\right] e^{i(2\pi /L)(\mathbf{n},x)},\ \mathbf{n}=(n_{1},n_{2})^{\top }.
\label{N02}$$
We shall consider the case of zero space average (see e.g. [@FMRT01]), i.e. when $$\int_{Q}u(x)=0. \label{av}$$In this case the Fourier series expansion for $u(x)$ does not contain the constant term and $\sum_{\mathbf{n}\in \mathbf{Z}^{n}}$ in (\[N00\]) can be replaced by $\sum_{\mathbf{n}\in \mathbf{Z}^{n},\mathbf{n}\neq 0},$ which in what follows we will write as simply $\sum .$
We recall Parseval’s identity $$\Vert u\Vert ^{2}=\int_{Q}|u(x)|^{2}dx=L^{n}\sum |u_{\mathbf{n}}|^{2},\ \
n=2,3. \label{eq:pars}$$We also note the following two relationships. Since the vector field $u=u(x)$ is real valued, we have $$u_{-\mathbf{n}}=\bar{u}_{\mathbf{n}},\ \ \mathbf{n}\in \mathbf{Z}^{n},\
\mathbf{n}\neq 0,$$where $\bar{u}_{\mathbf{n}}$ denotes the complex conjugate of $u_{\mathbf{n}%
}.$ The divergence-free condition reads $$u_{\mathbf{n}}^{\top }\mathbf{n}=(u_{\mathbf{n}},\mathbf{n}%
)=\sum_{k=1}^{n}u_{\mathbf{n}}^{k}\mathbf{n}^{k}=0,\ \ n=2,3.$$
We will need the following estimate for the tri-linear form (see [@CF p. 50, eq. (6.10)] or [@T p. 12, eq. (2.29)]): $$|((v,\nabla )u,g)|\leq K\Vert v\Vert _{m_{1}}\Vert u\Vert _{m_{2}+1}\Vert
g\Vert _{m_{3}}, \label{A2}$$where $K>0$ is a constant, $m_{1},\ m_{2}$ and $m_{3}$ are such that $%
m_{1}+m_{2}+m_{3}\geq n/2$ and $(m_{1},m_{2},m_{3})\neq (0,0,n/2),$ $%
(0,n/2,0),$ $(n/2,0,0),$ and $u,$ $v,$ $g$ are arbitrary functions from the corresponding spaces. Further, we recall the standard interpolation inequality for Sobolev spaces (see e.g. [@T p. 11]): $$\Vert u\Vert _{m}\leq \Vert u\Vert _{m_{1}}^{1-l}\Vert u\Vert _{m_{2}}^{l},
\label{A4i}$$where $m=(1-l)m_{1}+lm_{2},\ m_{1},$ $m_{2}\geq 0,$ $l\in (0,1),$ and $u\in
\mathbf{H}_{p}^{\max (m_{1},m_{2})}(Q).$ For any $c>0,$ we get from ([A4i]{}) and Young’s inequality: $$\Vert u\Vert _{m}^{2}\leq \Vert u\Vert _{m_{1}}^{2-2l}\Vert u\Vert
_{m_{2}}^{2l}\leq (1-l)c\Vert u\Vert _{m_{1}}^{2}+lc^{1-\frac{1}{l}}\Vert
u\Vert _{m_{2}}^{2}. \label{A4}$$Let us take $m_{1}=1,$ $m_{2}=0$ and $m_{3}=1/2$ in (\[A2\]), then, using (\[A4\]), we get (see also [@HaMa06 p. 1028, eq. (A6)]) for any $%
c_{1}>0$ and $c_{2}>0$: $$\begin{aligned}
|((v,\nabla )u,g)| &\leq &K\Vert v\Vert _{1}\Vert u\Vert _{1}\Vert g\Vert
_{1/2}\leq \frac{K^{2}}{4c_{1}}\Vert g\Vert _{1/2}^{2}+c_{1}\Vert v\Vert
_{1}^{2}\Vert u\Vert _{1}^{2} \label{A6} \\
&\leq &\frac{K^{2}}{4c_{1}}\Vert g\Vert _{1/2}^{2}+c_{1}\Vert v\Vert
_{1}^{2}\Vert u\Vert _{1}^{2} \notag \\
&\leq &c_{2}\Vert g\Vert _{1}^{2}+\frac{K^{4}}{64c_{1}^{2}c_{2}}\Vert g\Vert
^{2}+c_{1}\Vert v\Vert _{1}^{2}\Vert u\Vert _{1}^{2}, \notag\end{aligned}$$where we used (\[A4\]) with $m=1/2,$ $m_{1}=1,$ $m_{2}=0$ and $l=1/2$ (fractional $\mathbf{H}_{p}^{m}(Q)$ spaces are defined in the usual way via the Fourier series expansions, see e.g. [@T pp. 7-8]).
We also recall (see e.g. [@FMRT01 p. 20 eq. (4.14) ]) Poincare’s inequality for functions $u\in \mathbf{H}_{p}^{1}(Q)$ satisfying (\[av\]): $$||u||\leq \alpha ||\nabla u|| \label{Poin}$$for some constant $\alpha >0$ which depends only on the shape of $Q$ and on the period $L.$ We note that here and in what follows: when $u(x)$ is a vector, $\nabla u(x)$ means the matrix with elements $\partial
u^{i}/\partial x^{j}$ and $||\nabla u||$ means $L^{2}$-norm of the Frobenius norm of the matrix $\nabla u(x).$
Equation for vorticity
----------------------
Introduce the vorticity $\omega :$$$\omega =\left[
\begin{array}{c}
\omega ^{1} \\
\omega ^{2} \\
\omega ^{3}%
\end{array}%
\right] :=\mathop{\rm curl}u=\mathop{\rm curl}\left[
\begin{array}{c}
u^{1} \\
u^{2} \\
u^{3}%
\end{array}%
\right] =\left[
\begin{array}{ccc}
i & j & k \\
\frac{\partial }{\partial x^{1}} & \frac{\partial }{\partial x^{2}} & \frac{%
\partial }{\partial x^{3}} \\
u^{1} & u^{2} & u^{3}%
\end{array}%
\right] =\left[
\begin{array}{c}
\frac{\partial u^{3}}{\partial x^{2}}-\frac{\partial u^{2}}{\partial x^{3}}
\\
\frac{\partial u^{1}}{\partial x^{3}}-\frac{\partial u^{3}}{\partial x^{1}}
\\
\frac{\partial u^{2}}{\partial x^{1}}-\frac{\partial u^{1}}{\partial x^{2}}%
\end{array}%
\right] . \label{vor1}$$We note that (\[vor1\]) implies $\mathop{\rm div}\omega =0.$
Taking the $\mathop{\rm curl}$ of equation (\[ns1p\]) gives the evolution equation for the vorticity $\omega =\mathop{\rm curl}u:$ $$\frac{\partial \omega }{\partial t}-(u,\nabla )\omega +(\omega ,\nabla )u+%
\frac{\sigma ^{2}}{2}\Delta \omega +g=0, \label{V1}$$where $g=\mathop{\rm curl}f.$ From (\[ns3p\])-(\[ns4p\]), we get $$\omega (T,x)=\mathop{\rm curl}\varphi (x):=\phi (x)$$and spatial periodic conditions$$\omega (t,x+Le_{i})=\omega (t,x),\ i=1,\ldots ,n,\ 0\leq t\leq T.$$
Analogously to (\[N00\]), we write the Fourier expansion for $\omega :$ $$\omega (t,x)=\sum \omega _{\mathbf{n}}(t)e^{i(2\pi /L)(\mathbf{n},x)},
\label{vor2}$$where $$\begin{aligned}
\omega _{\mathbf{n}}(t) &=&\frac{1}{L^{n}}\left( \omega (t,\cdot ),e^{i(2\pi
/L)(\mathbf{n},\cdot )}\right) \\
&=&\frac{1}{L^{n}}\int_{Q}\omega (t,x)e^{-i(2\pi /L)(\mathbf{n},x)}dx,\ \
\mathbf{n\in Z}^{n},\ \mathbf{n}\neq 0.\end{aligned}$$
Substituting the Fourier expansions for $\omega $ and $u$ in (\[vor1\]), we obtain $$\omega =\left[
\begin{array}{c}
\sum \omega _{\mathbf{n}}^{1}(t)e^{i(2\pi /L)(\mathbf{n},x)} \\
\sum \omega _{\mathbf{n}}^{2}(t)e^{i(2\pi /L)(\mathbf{n},x)} \\
\sum \omega _{\mathbf{n}}^{3}(t)e^{i(2\pi /L)(\mathbf{n},x)}%
\end{array}%
\right] =i\frac{2\pi }{L}\left[
\begin{array}{c}
\sum \left( u_{\mathbf{n}}^{3}(t)\mathbf{n}^{2}-u_{\mathbf{n}}^{2}(t)\mathbf{%
n}^{3}\right) e^{i(2\pi /L)(\mathbf{n},x)} \\
\sum \left( u_{\mathbf{n}}^{1}(t)\mathbf{n}^{3}-u_{\mathbf{n}}^{3}(t)\mathbf{%
n}^{1}\right) e^{i(2\pi /L)(\mathbf{n},x)} \\
\sum \left( u_{\mathbf{n}}^{2}(t)\mathbf{n}^{1}-u_{\mathbf{n}}^{1}(t)\mathbf{%
n}^{2}\right) e^{i(2\pi /L)(\mathbf{n},x)}%
\end{array}%
\right] . \label{vor3}$$The equality (\[vor3\]) gives for any $\mathbf{n}\neq 0$ the equations with respect to $u_{\mathbf{n}}^{k},$ $k=1,2,3:$ $$\begin{aligned}
\mathbf{n}^{2}u_{\mathbf{n}}^{3}-\mathbf{n}^{3}u_{\mathbf{n}}^{2} &=&-\frac{%
iL}{2\pi }\omega _{\mathbf{n}}^{1} \label{vor4} \\
\mathbf{n}^{3}u_{\mathbf{n}}^{1}-\mathbf{n}^{1}u_{\mathbf{n}}^{3} &=&-\frac{%
iL}{2\pi }\omega _{\mathbf{n}}^{2} \notag \\
\mathbf{n}^{1}u_{\mathbf{n}}^{2}-\mathbf{n}^{2}u_{\mathbf{n}}^{1} &=&-\frac{%
iL}{2\pi }\omega _{\mathbf{n}}^{3}\ . \notag\end{aligned}$$Thanks to $\mathop{\rm div}u=0$ and $\mathop{\rm div}\omega =0,$ we also have for any $\mathbf{n:}$ $$\begin{aligned}
\mathbf{n}^{1}u_{\mathbf{n}}^{1}+\mathbf{n}^{2}u_{\mathbf{n}}^{2}+\mathbf{n}%
^{3}u_{\mathbf{n}}^{3} &=&0 \label{vor5} \\
\mathbf{n}_{\mathbf{n}}^{1}\omega _{\mathbf{n}}^{1}+\mathbf{n}^{2}\omega _{%
\mathbf{n}}^{2}+\mathbf{n}^{3}\omega _{\mathbf{n}}^{3} &=&0\ . \label{vor6}\end{aligned}$$Due to the property (\[vor6\]), the system (\[vor4\])-(\[vor5\]) with respect to $u_{\mathbf{n}}^{k}$ is compatible. It is not difficult to prove directly that the solution of this system is unique. This also follows from the two observations that the vector field $u_{\mathbf{n}}e^{i(2\pi /L)(%
\mathbf{n},x)}$ is solenoidal because it is divergence-free and that in the case of $\omega _{\mathbf{n}}^{k}=0,$ $k=1,2,3,$ the field is irrotational, i.e. potential. But if a vector field is simultaneously solenoidal and potential, it is trivial, i.e. $u_{\mathbf{n}}=0.$ Thus, the homogeneous system corresponding to (\[vor4\])-(\[vor5\]) has the trivial solution only, and hence the solution to the system (\[vor4\])-(\[vor5\]) exists and it is unique.
Our nearest goal consists in solving this system, i.e., in expressing $u$ via $\omega .$ We observe that $$\begin{aligned}
\mathop{\rm curl}u &=&\omega , \label{vor7} \\
\mathop{\rm div}u &=&0. \label{vor8}\end{aligned}$$
\[prop:curl\] For a sufficiently smooth $\psi ,$ let $\mathop{\rm div}%
\psi =0.$ Then$$\mathop{\rm curl}[-\func{curl}\psi ]=\Delta \psi . \label{V5}$$
**Proof**. The first component of the vector $%
\mathop{\rm
curl}[-\func{curl}\psi ]$ is equal to$$\begin{aligned}
&&-\frac{\partial ^{2}\psi ^{2}}{\partial x^{1}\partial x^{2}}+\frac{%
\partial ^{2}\psi ^{1}}{(\partial x^{2})^{2}}+\frac{\partial ^{2}\psi ^{1}}{%
(\partial x^{3})^{2}}-\frac{\partial ^{2}\psi ^{3}}{\partial x^{1}\partial
x^{3}} \\
&=&\frac{\partial }{\partial x^{1}}(-\frac{\partial \psi ^{2}}{\partial x^{2}%
}-\frac{\partial \psi ^{3}}{\partial x^{3}})+\frac{\partial ^{2}\psi ^{1}}{%
(\partial x^{2})^{2}}+\frac{\partial ^{2}\psi ^{1}}{(\partial x^{3})^{2}}.\end{aligned}$$Because of the condition $\mathop{\rm div}\psi =0,$ this component is equal to $\Delta \psi ^{1}.$ Analogously, the second and third components are equal to $\Delta \psi ^{2}$ and $\Delta \psi ^{3},$ correspondingly. The proposition is proved.
Let us look for $u$ in the form$$u=-\mathop{\rm curl}\psi , \label{V7}$$where $\mathop{\rm div}\psi =0.$ Due to (\[vor7\])-(\[V5\]), we now have to solve$$\Delta \psi =\omega ,\ \omega =\sum \omega _{\mathbf{n}}e^{i(2\pi /L)(%
\mathbf{n},x)}. \label{V8}$$Equation (\[V8\]) is solvable (uniquely, if we assume $\psi _{\mathbf{0}%
}=0 $):$$\psi =\sum \psi _{\mathbf{n}}e^{i(2\pi /L)(\mathbf{n},x)},\ \psi _{\mathbf{n}%
}=-\frac{\omega _{\mathbf{n}}L^{2}}{4\pi ^{2}|\mathbf{n}|^{2}},$$i.e., $$\psi _{\mathbf{n}}^{j}=-\frac{\omega _{\mathbf{n}}^{j}L^{2}}{4\pi ^{2}|%
\mathbf{n}|^{2}},\ j=1,2,3.$$Hence, using (\[V7\]), we have$$u=-\mathop{\rm curl}\psi =\frac{Li}{2\pi }\left[
\begin{array}{c}
\sum \frac{1}{|\mathbf{n}|^{2}}e^{i(2\pi /L)(\mathbf{n},x)}(\omega _{\mathbf{%
n}}^{3}\mathbf{n}^{2}-\omega _{\mathbf{n}}^{2}\mathbf{n}^{3}) \\
\sum \frac{1}{|\mathbf{n}|^{2}}e^{i(2\pi /L)(\mathbf{n},x)}(\omega _{\mathbf{%
n}}^{1}\mathbf{n}^{3}-\omega _{\mathbf{n}}^{3}\mathbf{n}^{1}) \\
\sum \frac{1}{|\mathbf{n}|^{2}}e^{i(2\pi /L)(\mathbf{n},x)}(\omega _{\mathbf{%
n}}^{2}\mathbf{n}^{1}-\omega _{\mathbf{n}}^{1}\mathbf{n}^{2})%
\end{array}%
\right] :=U\omega , \label{V9}$$where $U$ is a linear operator. It is not difficult to verify that the equality $\mathop{\rm div}u=0$ (see (\[vor8\])) holds for $u$ from ([V9]{}) under arbitrary $\omega $. However, the equality (\[vor7\]) is not fulfilled by $u$ from (\[V9\]) for arbitrary $\omega $. But the considered $\omega $ is not arbitrary, it is divergent free and we show below that for a divergent-free $\omega $ the equality (\[vor7\]) is satisfied by $u$ from (\[V9\]).
For $u$ from (\[V9\]), we get$$\mathop{\rm curl}u=-\left[
\begin{array}{c}
\sum \frac{1}{|\mathbf{n}|^{2}}e^{i(2\pi /L)(\mathbf{n},x)}[-\omega _{%
\mathbf{n}}^{1}((\mathbf{n}^{2})^{2}+(\mathbf{n}^{3})^{2})+\mathbf{n}^{1}(%
\mathbf{n}^{2}\omega _{\mathbf{n}}^{2}+\mathbf{n}^{3}\omega _{\mathbf{n}%
}^{3})] \\
\sum \frac{1}{|\mathbf{n}|^{2}}e^{i(2\pi /L)(\mathbf{n},x)}[-\omega _{%
\mathbf{n}}^{2}((\mathbf{n}^{1})^{2}+(\mathbf{n}^{3})^{2})+\mathbf{n}^{2}(%
\mathbf{n}^{1}\omega _{\mathbf{n}}^{1}+\mathbf{n}^{3}\omega _{\mathbf{n}%
}^{3})] \\
\sum \frac{1}{|\mathbf{n}|^{2}}e^{i(2\pi /L)(\mathbf{n},x)}[-\omega _{%
\mathbf{n}}^{3}((\mathbf{n}^{1})^{2}+(\mathbf{n}^{2})^{2})+\mathbf{n}^{3}(%
\mathbf{n}^{1}\omega _{\mathbf{n}}^{1}+\mathbf{n}^{2}\omega _{\mathbf{n}%
}^{2})]%
\end{array}%
\right] .$$Recall that (\[vor6\]) holds for a divergent-free $\omega .$ Then, using (\[vor6\]), we have$$\begin{aligned}
\mathbf{n}^{1}(\mathbf{n}^{2}\omega _{\mathbf{n}}^{2}+\mathbf{n}^{3}\omega _{%
\mathbf{n}}^{3}) &=&-(\mathbf{n}^{1})^{2}\omega _{\mathbf{n}}^{1} \\
\mathbf{n}^{2}(\mathbf{n}^{1}\omega _{\mathbf{n}}^{1}+\mathbf{n}^{3}\omega _{%
\mathbf{n}}^{3}) &=&-(\mathbf{n}^{2})^{2}\omega _{\mathbf{n}}^{2} \\
\mathbf{n}^{3}(\mathbf{n}^{1}\omega _{\mathbf{n}}^{1}+\mathbf{n}^{2}\omega _{%
\mathbf{n}}^{2}) &=&-(\mathbf{n}^{3})^{2}\omega _{\mathbf{n}}^{3}\end{aligned}$$and therefore$$\mathop{\rm curl}u=\left[
\begin{array}{c}
\sum \frac{1}{|\mathbf{n}|^{2}}e^{i(2\pi /L)(\mathbf{n},x)}\omega _{\mathbf{n%
}}^{1}|\mathbf{n}|^{2} \\
\sum \frac{1}{|\mathbf{n}|^{2}}e^{i(2\pi /L)(\mathbf{n},x)}\omega _{\mathbf{n%
}}^{2}|\mathbf{n}|^{2} \\
\sum \frac{1}{|\mathbf{n}|^{2}}e^{i(2\pi /L)(\mathbf{n},x)}\omega _{\mathbf{n%
}}^{3}|\mathbf{n}|^{2}%
\end{array}%
\right] ,$$i.e., $\mathop{\rm curl}u=\omega .$ Thus, the following theorem is proved.
\[thm:nsvor\]The velocity field $u$ (with $u_{\mathbf{0}}=0$) is determined explicitly through vorticity field $\omega =\mathop{\rm curl}v$ by formula $(\ref{V9})$. The closed-form equation for $\omega $ is given by$$\frac{\partial \omega }{\partial t}-(U\omega ,\nabla )\omega +(\omega
,\nabla )U\omega +\frac{\sigma ^{2}}{2}\Delta \omega +g=0, \label{V11}$$where $U$ is from $(\ref{V9})$.
This theorem is related to analogous results for periodic $2D$ flows (see [@MB p. 50]) and for flows in the whole space (see [@MB p. 71]).
Let us mention the corresponding formulas in the 2D case, for which we have$$u(x)=\left[
\begin{array}{c}
u^{1}(x^{1},x^{2}) \\
u^{2}(x^{1},x^{2}) \\
0%
\end{array}%
\right] =\sum \left[
\begin{array}{c}
u_{\mathbf{n}}^{1} \\
u_{\mathbf{n}}^{2} \\
0%
\end{array}%
\right] e^{i(2\pi /L)(\mathbf{n}^{1}x^{1}+\mathbf{n}^{2}x^{2})},$$i.e., $u^{1}(x)$ and $u^{2}(x)$ are independent of $x^{3}$ and $u^{3}=0.$ Hence$$\omega =\left[
\begin{array}{c}
\omega ^{1} \\
\omega ^{2} \\
\omega ^{3}%
\end{array}%
\right] =\left[
\begin{array}{c}
0 \\
0 \\
\frac{\partial u^{2}}{\partial x^{1}}-\frac{\partial u^{1}}{\partial x^{2}}%
\end{array}%
\right] .$$We shall denote the scalar $\omega ^{3}(x)$ as $\omega (x)=\frac{\partial
u^{2}}{\partial x^{1}}(x)-\frac{\partial u^{1}}{\partial x^{2}}(x)$ and the two dimensional vector $(u^{1}(x^{1},x^{2}),u^{2}(x^{1},x^{2}))^{\intercal }$ as $u(x).$ This does not lead to confusion. We have $\mathop{\rm div}u(x)=%
\frac{\partial u^{1}}{\partial x^{1}}(x)+\frac{\partial u^{2}}{\partial x^{2}%
}(x)=0 $, i.e., $u_{\mathbf{n}}^{1}\mathbf{n}^{1}+u_{\mathbf{n}}^{2}\mathbf{n%
}^{2}=0 $ for any $\mathbf{n}=(\mathbf{n}^{1},\mathbf{n}^{2}),$ and$$\omega (x)=\sum \omega _{\mathbf{n}}e^{i(2\pi /L)(\mathbf{n}^{1}x^{1}+%
\mathbf{n}^{2}x^{2})},$$where$$\omega _{\mathbf{n}}=i\frac{2\pi }{L}(u_{\mathbf{n}}^{2}\mathbf{n}^{1}-u_{%
\mathbf{n}}^{1}\mathbf{n}^{2}).$$Due to (\[V9\]), $u$ is expressed through $\omega :$$$u(x)=\frac{Li}{2\pi }\left[
\begin{array}{c}
\sum \frac{1}{|\mathbf{n}|^{2}}e^{i(2\pi /L)(\mathbf{n},x)}\omega _{\mathbf{n%
}}\mathbf{n}^{2} \\
-\sum \frac{1}{|\mathbf{n}|^{2}}e^{i(2\pi /L)(\mathbf{n},x)}\omega _{\mathbf{%
n}}\mathbf{n}^{1}%
\end{array}%
\right] :=U\omega . \label{Mnew1}$$Clearly, $(\omega ,\nabla )u=0$ in the $2D$ case. Hence (\[V1\]) takes the form$$\frac{\partial \omega }{\partial t}-u^{1}(t,x)\frac{\partial \omega }{%
\partial x^{1}}(t,x)-u^{2}(t,x)\frac{\partial \omega }{\partial x^{2}}(t,x)+%
\frac{\sigma ^{2}}{2}\Delta \omega (t,x)+g(t,x^{1},x^{2})=0. \label{M6}$$
\[ex:21\] Consider the Stokes equation with negative direction of time: $$\frac{\partial u}{\partial t}+\frac{\sigma ^{2}}{2}\Delta u-\nabla p+f=0
\label{vor15}$$with the conditions $(\ref{ns2p})$-$(\ref{ns4p})$. In this case the vorticity $\omega (t,x)$ satisfies the equation $$\frac{\partial \omega }{\partial t}+\frac{\sigma ^{2}}{2}\Delta \omega +g=0,
\label{vor16}$$where $g(t,x)=\sum g_{\mathbf{n}}e^{i(2\pi /L)(\mathbf{n},x)}.$ Substituting the Fourier expansions for $\omega $ and $g$ in $(\ref{vor16})$, we get$$\sum \omega _{\mathbf{n}}^{\prime }(t)e^{i(2\pi /L)(\mathbf{n},x)}+\frac{%
\sigma ^{2}}{2}\sum \omega _{\mathbf{n}}(t)\left( -\frac{4\pi ^{2}}{L^{2}}%
\right) |\mathbf{n}|^{2}e^{i(2\pi /L)(\mathbf{n},x)}+\sum g_{\mathbf{n}%
}e^{i(2\pi /L)(\mathbf{n},x)}=0,$$whence $$\frac{d\omega _{\mathbf{n}}^{k}}{dt}-\frac{2\pi ^{2}\sigma ^{2}}{L^{2}}|%
\mathbf{n}|^{2}\omega _{\mathbf{n}}^{k}+g_{\mathbf{n}}^{k}=0,\ \ \omega _{%
\mathbf{n}}^{k}(T)=\phi _{\mathbf{n}}^{k},$$where $\phi _{\mathbf{n}}$ are the Fourier coefficients for $\phi :=%
\mathop{\rm curl}\varphi .$ Hence $$\omega _{\mathbf{n}}^{k}(t)=\phi _{\mathbf{n}}^{k}\exp \left( \frac{2\pi
^{2}\sigma ^{2}}{L^{2}}|\mathbf{n}|^{2}(t-T)\right) +\int_{t}^{T}\exp \left(
\frac{2\pi ^{2}\sigma ^{2}}{L^{2}}|\mathbf{n}|^{2}(t-s)\right) g_{\mathbf{n}%
}^{k}(s)ds.$$
In future we will need the following estimates. One can obtain from (\[V9\]) that for $m\geq 1$$$||u||_{m}=||U\omega ||_{m}\leq K||\omega ||_{m-1} \label{A3}$$for some $K>0.$ Further, we note that $$||\omega ||_{1}^{2}=||\omega ||^{2}+||\nabla \omega ||^{2} \label{A31}$$and then by (\[Poin\])$$||\omega ||_{1}^{2}\leq K||\nabla \omega ||^{2} \label{A32}$$for some $K>0$. Using (\[A6\]), (\[A3\]), (\[A31\]) and (\[A32\]), we get for $\omega ,$ $v$, $g$ from appropriate spaces and arbitrary $%
c_{1}>0 $ and $c_{2}>0:$ $$\begin{aligned}
|((U\omega ,\nabla )v,g)| &\leq &c_{2}\Vert g\Vert _{1}^{2}+\frac{K^{4}}{%
64c_{1}^{2}c_{2}}\Vert g\Vert ^{2}+c_{1}\Vert U\omega \Vert _{1}^{2}\Vert
v\Vert _{1}^{2} \label{A66} \\
&\leq &c_{2}||\nabla g||^{2}+(c_{2}+\frac{K^{4}}{64c_{1}^{2}c_{2}})\Vert
g\Vert ^{2}+Kc_{1}\Vert \omega \Vert ^{2}\Vert \nabla v\Vert ^{2} \notag \\
&=&c_{2}||\nabla g||^{2}+K\Vert g\Vert ^{2}+c_{3}\Vert \omega \Vert
^{2}\Vert \nabla v\Vert ^{2}, \notag\end{aligned}$$where in the third line $c_{3}>0$ is an arbitrary constant and $K>0$ is some constant dependent on $c_{2}$ and $c_{3}$ (it differs from $K>0$ in the first and second line but this should not cause any confusion).
Probabilistic representations of solutions to linear systems of parabolic equations with application to vorticity equations\[SecPrb\]
=====================================================================================================================================
In this section we derive probabilistic representations for systems of parabolic equations based on the approach developed in [@M]. They can be used for constructing probabilistic methods for NSE in vorticity-velocity formulation (\[V1\]) (see probabilistic numerical methods for semilinear PDEs in e.g. [@M1; @MT1] and for NSE in velocity formulation in e.g. [MTns13]{}). For this purpose, it is useful to have a wide class of such probabilistic representations, and, in addition to [@M], we also exploit ideas from [@MSS; @MT1]. Note that we obtain more general probabilistic representations than in [@BFR].
The basic probabilistic representation
--------------------------------------
We consider the following Cauchy problem for system of parabolic equations$$\begin{aligned}
\frac{\partial u^{k}}{\partial t}+\frac{1}{2}\sum_{i,j=1}^{n}\left[
\sum_{r=1}^{l}\sigma _{r}\sigma _{r}^{\top }\right] ^{ij}\frac{\partial
^{2}u^{k}}{\partial x^{i}\partial x^{j}}+\sum_{i=1}^{n}\sum_{j=1}^{m}\left(
\sum_{r=1}^{l}[\sigma _{r}]^{i}[\vartheta _{r}]^{jk}\right) \frac{\partial
u^{j}}{\partial x^{i}} \label{PR1} \\
+\sum_{i=1}^{n}a^{i}\frac{\partial u^{k}}{\partial x^{i}}+[B^{\top
}u]^{k}+f^{k}=0,\ k=1,\ldots ,m, \notag\end{aligned}$$$$u(T,x)=\varphi (x). \label{PR2}$$
Introduce the system of SDEs$$\begin{aligned}
dX &=&a(s,X)ds+\sum_{r=1}^{l}\sigma _{r}(s,X)dw_{r}(s),\ X(t)=x, \label{PR3}
\\
dY &=&B(s,X)Yds+\sum_{r=1}^{l}\vartheta _{r}(s,X)Ydw_{r}(s),\ Y(t)=y.
\label{PR4}\end{aligned}$$
In (\[PR1\])-(\[PR4\]), $0\leq t\leq s\leq T;\ x$ and $X$ are column-vectors of dimension $n;\ y$ and $Y$ are column-vectors of dimension $%
m;\ w_{r},\ r=1,\ldots ,l,$ are independent standard Wiener processes; $%
a(s,x)$ and $\sigma _{r}(s,x)$ are column-vectors of dimension $n;\ B(s,x)$ and $\vartheta _{r}(s,x)$ are $m\times m$ - matrices; $u(s,x),\ f(s,x),$ and $\varphi (x)$ are column-vector of dimension $m$ with components $u^{k},\
f^{k},\ \varphi ^{k},\ k=1,\ldots ,m.$ We assume that there exist a sufficiently smooth solution of the problem (\[PR1\])-(\[PR2\]) and a unique solution of the problem (\[PR3\])-(\[PR4\]).
Introduce the process$$\xi _{t,x,y}(s)=\int_{t}^{s}f^{\top }(s^{\prime },X_{t,x}(s^{\prime
}))Y_{t,x,y}(s^{\prime })ds^{\prime }+u^{\top }(s,X_{t,x}(s))Y_{t,x,y}(s).
\label{PR5}$$Using Ito’s formula, we get$$d\xi =\sum_{k=1}^{m}f^{k}Y^{k}ds+d(u^{\top }Y), \label{PR05}$$$$\begin{aligned}
d(u^{\top }Y) &=&\sum_{k=1}^{m}d[u^{k}Y^{k}]=\sum_{k=1}^{m}\frac{\partial
u^{k}}{\partial s}Y^{k}ds+\sum_{k=1}^{m}\sum_{i=1}^{n}\frac{\partial u^{k}}{%
\partial x^{i}}[a^{i}ds+\sum_{r=1}^{l}\sigma _{r}^{i}dw_{r}(s)]Y^{k}
\label{PR6} \\
&&+\frac{1}{2}\sum_{k=1}^{m}\sum_{i,j=1}^{n}\frac{\partial ^{2}u^{k}}{%
\partial x^{i}\partial x^{j}}\sum_{r=1}^{l}\sigma _{r}^{i}\sigma
_{r}^{j}ds\cdot Y^{k}+\sum_{k=1}^{m}u^{k}[BY]^{k}ds \notag \\
&&+\sum_{k=1}^{m}u^{k}\left[ \sum_{r=1}^{l}\vartheta _{r}Ydw_{r}(s)\right]
^{k}+\sum_{k=1}^{m}\sum_{i=1}^{n}\frac{\partial u^{k}}{\partial x^{i}}%
dX^{i}dY^{k}. \notag\end{aligned}$$Further,$$\sum_{k=1}^{m}u^{k}[BY]^{k}ds=\sum_{k=1}^{m}[B^{\top }u]^{k}Y^{k},
\label{PR7}$$$$\begin{aligned}
\sum_{k=1}^{m}\sum_{i=1}^{n}\frac{\partial u^{k}}{\partial x^{i}}%
dX^{i}dY^{k}=\sum_{k=1}^{m}\sum_{i=1}^{n}\frac{\partial u^{k}}{\partial x^{i}%
}\sum_{r=1}^{l}\sigma _{r}^{i}dw_{r}(s)\sum_{r^{\prime
}=1}^{l}\sum_{j=1}^{m}\vartheta _{r^{\prime }}^{kj}Y^{j}dw_{r^{\prime }}(s)
\label{PR8} \\
=\sum_{k=1}^{m}\sum_{i=1}^{n}\frac{\partial u^{k}}{\partial x^{i}}%
\sum_{r=1}^{l}\sum_{j=1}^{m}\sigma _{r}^{i}\vartheta
_{r}^{kj}Y^{j}ds=\sum_{k=1}^{m}(\sum_{i=1}^{n}\sum_{j=1}^{m}\sum_{r=1}^{l}%
\sigma _{r}^{i}\vartheta _{r}^{jk}\frac{\partial u^{j}}{\partial x^{i}}%
)Y^{k}ds. \notag\end{aligned}$$In (\[PR05\])-(\[PR8\]) all the coefficients and functions have $s,$ $%
X_{t,x}(s)$ as their arguments.
Substituting (\[PR6\])-(\[PR8\]) in (\[PR05\]) and taking into account that $u$ is a solution of (\[PR1\]), we get$$d\xi =\sum_{r=1}^{l}\sum_{k=1}^{m}(u^{k}(\vartheta _{r}Y)^{k}+\sum_{i=1}^{n}%
\frac{\partial u^{k}}{\partial x^{i}}\sigma _{r}^{i}Y^{k})dw_{r}(s).
\label{PR9}$$It is known that if $$E\int_{t}^{T}\sum_{r=1}^{l}\left[ \sum_{k=1}^{m}(u^{k}(\vartheta
_{r}Y)^{k}+\sum_{i=1}^{n}\frac{\partial u^{k}}{\partial x^{i}}\sigma
_{r}^{i}Y^{k})\right] ^{2}ds<\infty \label{PR10}$$then$$E\int_{t}^{T}\sum_{r=1}^{l}\sum_{k=1}^{m}(u^{k}(\vartheta
_{r}Y)^{k}+\sum_{i=1}^{n}\frac{\partial u^{k}}{\partial x^{i}}\sigma
_{r}^{i}Y^{k})dw_{r}(s)=E[\xi _{t,x,y}(T)-\xi _{t,x,y}(t)]=0. \label{PR11}$$At the same time (see (\[PR5\]))$$\xi _{t,x,y}(T)-\xi _{t,x,y}(t)=\int_{t}^{T}f^{\top }(s^{\prime
},X_{t,x}(s^{\prime }))Y_{t,x,y}(s^{\prime })ds^{\prime }+\varphi ^{\top
}(X_{t,x}(T))Y_{t,x,y}(T)-u^{\top }(t,x)y. \label{PR12}$$Hence$$u^{\top }(t,x)y=E\left[ \int_{t}^{T}f^{\top }(s^{\prime },X_{t,x}(s^{\prime
}))Y_{t,x,y}(s^{\prime })ds^{\prime }+\varphi ^{\top
}(X_{t,x}(T))Y_{t,x,y}(T)\right] . \label{PR13}$$
So, we have obtained that under certain conditions ensuring existence of a sufficiently smooth solution of (\[PR1\])-(\[PR2\]), existence and uniqueness of solution of (\[PR3\])-(\[PR4\]), and boundedness ([PR10]{}), the probabilistic representation of the solution to the problem (\[PR1\])-(\[PR2\]) is given by formula (\[PR13\]).
A family of probabilistic representations
-----------------------------------------
Now we restrict ourselves to the case $\vartheta _{r}=0,\ r=1,\ldots ,l$, i.e. (see (\[PR1\])-(\[PR4\])):$$\frac{\partial u^{k}}{\partial t}+\frac{1}{2}\sum_{i,j=1}^{n}\left[
\sum_{r=1}^{l}\sigma _{r}\sigma _{r}^{\top }\right] ^{ij}\frac{\partial
^{2}u^{k}}{\partial x^{i}\partial x^{j}}+\sum_{i=1}^{n}a^{i}\frac{\partial
u^{k}}{\partial x^{i}}+[B^{\top }u]^{k}+f^{k}=0,\ k=1,\ldots ,m,
\label{PR14}$$$$u(T,x)=\varphi (x), \label{PR15}$$and$$\begin{aligned}
dX &=&a(s,X)ds+\sum_{r=1}^{l}\sigma _{r}(s,X)dw_{r}(s),\ X(t)=x,
\label{PR16} \\
dY &=&B(s,X)Yds,\ Y(t)=y. \label{PR17}\end{aligned}$$
Introduce the system $$dX=a(s,X)ds-\sum_{r=1}^{l}\mu _{r}(s,X)\sigma
_{r}(s,X)ds+\sum_{r=1}^{l}\sigma _{r}(s,X)dw_{r}(s),\ X(t)=x, \label{PR18}$$$$dY=B(s,X)Yds,\ Y(t)=y, \label{PR19}$$$$dQ=\sum_{r=1}^{l}\mu _{r}(s,X)Qdw_{r}(s),\ Q(t)=1, \label{PR20}$$$$dZ=Qf^{\top }(s,X)Yds+Q\sum_{r=1}^{l}F_{r}^{\top }(s,X)Ydw_{r}(s),\ Z(t)=0.
\label{PR21}$$In (\[PR18\])-(\[PR21\]) $\mu _{r},\ Q,$ and $Z$ are scalars; $\ F_{r},\
r=1,\ldots ,l,$ are column-vectors of dimension $m;\ \mu _{r}$ and $F_{r}$ are arbitrary functions, however, with good analytical properties.
Introduce the process$$\eta _{t,x,y}(s)=Q_{t,x,y,1}(s)u^{\top
}(s,X_{t,x}(s))Y_{t,x,y}(s)+Z_{t,x,y,1,0}(s). \label{PR22}$$Using Ito’s formula and taking into account that $u$ is a solution of ([PR14]{}), we get that under arbitrary $\mu _{r}$ and $F_{r}$$$d\eta =\sum_{r=1}^{l}Q\mathbb{\cdot (}\sum_{i=1}^{n}\frac{\partial u^{\top }%
}{\partial x^{i}}\sigma _{r}^{i}+\mu _{r}u^{\top }+F_{r}^{\top })Ydw_{r}(s).
\label{PR23}$$
If $$E\int_{t}^{T}\sum_{r=1}^{l}\left[ Q\mathbb{\cdot (}\sum_{i=1}^{n}\frac{%
\partial u^{\top }}{\partial x^{i}}\sigma _{r}^{i}+\mu _{r}u^{\top
}+F_{r}^{\top })Y\right] ^{2}ds<\infty \label{PR24}$$then$$E\int_{t}^{T}\sum_{r=1}^{l}Q\mathbb{\cdot (}\sum_{i=1}^{n}\frac{\partial
u^{\top }}{\partial x^{i}}\sigma _{r}^{i}+\mu _{r}u^{\top }+F_{r}^{\top
})Ydw_{r}(s)=E(\eta _{t,x,y}(T)-\eta _{t,x,y}(t))=0. \label{PR25}$$We have$$\begin{aligned}
\eta _{t,x,y}(T)-\eta _{t,x,y}(t) &=&Q(T)\varphi ^{\top
}(X_{t,x}(T))Y(T)-u(t,x)y \label{PR26} \\
&&+\int_{t}^{T}Q_{t,x,y,1}(s^{\prime })f^{\top }(s^{\prime
},X_{t,x}(s^{\prime }))Y_{t,x,y}(s^{\prime })ds^{\prime } \notag \\
&&+\int_{t}^{T}Q_{t,x,y,1}(s^{\prime })\sum_{r=1}^{l}F_{r}^{\top }(s^{\prime
},X_{t,x}(s^{\prime }))Y_{t,x,y}(s^{\prime })dw_{r}(s^{\prime }). \notag\end{aligned}$$Under the natural assumption$$E\left[ \int_{t}^{T}Q_{t,x,y,1}(s^{\prime })\sum_{r=1}^{l}F_{r}^{\top
}(s^{\prime },X_{t,x}(s^{\prime }))Y_{t,x,y}(s^{\prime })dw_{r}(s^{\prime })%
\right] =0,$$using (\[PR25\]) and (\[PR26\]), we obtain the family of probabilistic representations for the solution of (\[PR14\])-(\[PR15\]): $$u^{\top }(t,x)y=E\left[ Q_{t,x,y,1}(T)\varphi ^{\top
}(X_{t,x}(T))Y_{t,x,y}(T)+\int_{t}^{T}Q_{t,x,y,1}(s^{\prime })f^{\top
}(s^{\prime },X_{t,x}(s^{\prime }))Y_{t,x,y}(s^{\prime })ds^{\prime }\right]
, \label{PR27}$$where the expressions under sign $E$ depend on a choice of $\mu _{r}$ and $%
F_{r}.$ We see that the expectation of $\eta _{t,x,y}(T)$ in the right hand side of (\[PR27\]) is equal to $u(t,x)y$ and it is independent of a choice of $\mu _{r}$ and $F_{r}.$ At the same time, the variance $Var[\eta
_{t,x,y}(T)]$ does depend on $\mu _{r}$ and $F_{r}.$
Probabilistic representations for the vorticity
-----------------------------------------------
System (\[V1\]) has the form of (\[PR14\]) with $m=3,\ n=3,$$$\begin{aligned}
\sigma _{1} &=&\left[
\begin{array}{c}
\sigma \\
0 \\
0%
\end{array}%
\right] ,\ \sigma _{2}=\left[
\begin{array}{c}
0 \\
\sigma \\
0%
\end{array}%
\right] ,\ \sigma _{3}=\left[
\begin{array}{c}
0 \\
0 \\
\sigma%
\end{array}%
\right] , \\
a(t,x) &=&-\left[
\begin{array}{c}
u^{1}(t,x) \\
u^{2}(t,x) \\
u^{3}(t,x)%
\end{array}%
\right] ,\ B^{\top }=\left[
\begin{array}{ccc}
\partial u^{1}/\partial x^{1} & \partial u^{1}/\partial x^{2} & \partial
u^{1}/\partial x^{3} \\
\partial u^{2}/\partial x^{1} & \partial u^{2}/\partial x^{2} & \partial
u^{2}/\partial x^{3} \\
\partial u^{3}/\partial x^{1} & \partial u^{3}/\partial x^{2} & \partial
u^{3}/\partial x^{3}%
\end{array}%
\right] ,\end{aligned}$$with $\omega $ instead of $u$ and $g$ instead of $f,\ \sigma $ is a positive constant, i.e., the more detailed writing of (\[V1\]) has the form: $$\begin{aligned}
\frac{\partial \omega ^{k}}{\partial t}+\frac{1}{2}\sigma ^{2}\Delta \omega
^{k}-\sum_{i=1}^{3}u^{i}(t,x)\frac{\partial \omega ^{k}}{\partial x^{i}}%
+\sum_{i=1}^{3}\frac{\partial u^{k}}{\partial x^{i}}(t,x)\omega
^{i}+g^{k}(t,x)=0, \label{V13} \\
\omega ^{k}(T,x)=\phi ^{k}(x),\ k=1,2,3. \label{V14}\end{aligned}$$
Let us put $\mu _{r}(s,x)=-a^{r}(s,x),\ F^{r}(s,x)=0$ in the family of representations (\[PR18\])-(\[PR21\]), (\[PR27\]) for the problem ([V13]{})-(\[V14\]). We get$$\begin{aligned}
dX^{i} &=&\sigma dw_{i}(s),\ X^{i}(t)=x^{i},\ i=1,2,3, \label{V15} \\
dY^{i} &=&\sum_{j=1}^{3}\frac{\partial u^{j}}{\partial x^{i}}(s,X)Y^{j}ds,\
Y^{i}(t)=y^{i},\ i=1,2,3, \label{V16} \\
dQ &=&-Q\sum_{j=1}^{3}u^{j}(s,X)dw_{j}(s),\ Q(t)=1, \label{V17} \\
dZ &=&Q\sum_{j=1}^{3}g^{j}(s,X)Y^{j}ds,\ Z(t)=0, \label{V18} \\
\omega ^{\top }(t,x)y &=&E\left[ Q_{t,x,y,1}(T)\phi ^{\top
}(X_{t,x}(T))Y_{t,x,y}(T)+Z_{t,x,y,1,0}(T)\right] . \label{V19}\end{aligned}$$The components $\omega ^{1},\ \omega ^{2},~\omega ^{3}$ of $\omega $ are obtained from (\[V19\]) under $y$ equal subsequently to $(1,0,0)^{\top },\
(0,1,0)^{\top },\ (0,0,1)^{\top }.$
\[ex:31\](*The Monte Carlo calculation of the Fourier coefficients*) Due to (\[vor2\]), we have $$\begin{aligned}
\omega _{\mathbf{n}}^{j}(t) &=&\frac{1}{L^{3}}\left( \omega
^{j}(t,x),e^{i(2\pi /L)(\mathbf{n},x)}\right) \label{ex31_1} \\
&=&\frac{1}{L^{3}}\int_{Q}\omega ^{j}(t,x)e^{-i(2\pi /L)(\mathbf{n},x)}dx,\
\ \mathbf{n\in Z}^{3},\ \mathbf{n}\neq 0,\ j=1,2,3. \notag\end{aligned}$$Let $\xi $ be a random variable uniformly distributed on $Q.$ Then ([ex31\_1]{}) can be written as $$\omega _{\mathbf{n}}^{j}(t)=E\left[ \omega ^{j}(t,\xi )e^{-i(2\pi /L)(%
\mathbf{n},\xi )}\right] ,$$where the expectation can be approximated using the Monte Carlo technique and hence $$\omega _{\mathbf{n}}^{j}(t)\doteq \frac{1}{M}\sum_{m=1}^{M}\omega ^{j}(t,\xi
^{(m)})e^{-i(2\pi /L)(\mathbf{n},\xi ^{(m)})}$$with $\xi ^{(m)}$ being independent realizations of $\xi .$ In turn, every $%
\omega (t,\xi ^{(m)})$ can be computed by applying the Monte Carlo technique and weak-sense approximation of SDEs to the representation (\[PR27\]), (\[V15\])-(\[V19\]).
Approximation method based on vorticity\[sec:approx\]
=====================================================
Let us introduce a uniform partition of the time interval $[0,T]$: $%
0=t_{0}<t_{1}<\cdots <t_{N}=T$ and the time step $h=T/N$ (we restrict ourselves to the uniform time discretization for simplicity only). In this section we derive an approximation for the vorticity (Section [sec:approxcon]{}) and study its properties (divergence free property in Section \[sec:approxdiv\], one-step error in Section \[sec:approxerr\], and global convergence in Section \[sec:conv\]).
Construction of the method\[sec:approxcon\]
-------------------------------------------
Let $\omega (t_{k+1},x),\ k=0,\ldots ,N-1,$ be known exactly. Then $%
u(t_{k+1},x)$ can be calculated exactly due to (\[V9\]): $%
u(t_{k+1},x)=U\omega (t_{k+1},x).$ The formula (\[V19\]),$$\omega ^{\top }(t_{k},x)y=E\left[ Q_{t_{k},x,y,1}(t_{k+1})\omega ^{\top
}(t_{k+1},X_{t_{k},x}(t_{k+1}))Y_{t_{k},x,y}(t_{k+1})+Z_{t_{k},x,y,1,0}(t_{k+1})%
\right] , \label{M1}$$gives the value of the solution of (\[V1\]) at $t_{k}$ assuming that $%
u(t,x),\ t_{k}\leq t<t_{k+1},$ is known exactly. We note that knowing this $%
u(t,x)$ is necessary for equations (\[V16\])-(\[V17\]).
Let us replace the unknown $u(t,x)$ in (\[V13\]) by the function$$\hat{u}(t,x):=u(t_{k+1},x):=\hat{u}(x),\ t_{k}\leq t<t_{k+1}.
\label{eq:freeze}$$As an approximation of $\omega (t,x)$ on $\ t_{k}\leq t\leq t_{k+1},$ we take $\tilde{\omega}(t,x)$ satisfying the system $$\begin{aligned}
\frac{\partial \tilde{\omega}^{i}}{\partial t}+\frac{\sigma ^{2}}{2}\Delta
\tilde{\omega}^{i}-\sum_{j=1}^{3}\hat{u}^{j}(x)\frac{\partial \tilde{\omega}%
^{i}}{\partial x^{j}}+\sum_{i=1}^{3}\frac{\partial \hat{u}^{i}}{\partial
x^{j}}(x)\tilde{\omega}^{j}+g(t,x) &=&0,\ t_{k}\leq t<t_{k+1}, \label{M2} \\
\tilde{\omega}^{i}(t_{k+1},x) &=&\omega ^{i}(t_{k+1},x),\ \ \tilde{\omega}%
^{i}(t_{k+1},x+Le_{j})=\tilde{\omega}^{i}(t_{k+1},x),\ j=1,2,3, \label{M3}
\\
i &=&1,2,3. \notag\end{aligned}$$We observe that (\[M2\])-(\[M3\]) can be also obtained from (\[V13\]) by freezing the velocity $u(t,x)$ on every time step according to ([eq:freeze]{}).
Now we propose the method for solving the problem (\[ns1p\])-(\[ns4p\]) with negative direction of time. On the first step of the method we set $$\tilde{\omega}(t_{N},x)=\mathop{\rm curl}u(t_{N},x)=\phi (x)=\mathop{\rm
curl}\varphi (x)$$and $$\hat{u}(x)=\hat{u}(t,x)=u(t_{N},x)=\varphi (x),\ \ t_{N-1}\leq t\leq t_{N}.$$Then we solve the system (\[M2\])-(\[M3\]) on* *$[t_{N-1},t_{N}]$ to obtain $\tilde{\omega}(t,x)$ and to construct $$\hat{u}(t_{N-1},x)=U\tilde{\omega}(t_{N-1},x).$$On the second step we solve (\[M2\])-(\[M3\]) on* *$%
[t_{N-2},t_{N-1})$ having $\tilde{\omega}(t_{N-1},x)$ and setting $\hat{u}%
(t,x)=\hat{u}(x)=\hat{u}(t_{N-1},x)$ for $t_{N-2}\leq t<t_{N-1}.$ As a result, we obtain $\tilde{\omega}(t,x)$ on $[t_{N-2},t_{N-1})$ and $\hat{u}%
(t_{N-2},x)=U\tilde{\omega}(t_{N-2},x),$ and so on. Proceeding in this way, we obtain on the $N$-th step the approximation $\tilde{\omega}(t,x)$ on $%
[t_{0},t_{1})$ for $\omega (t,x)$ having $\tilde{\omega}(t_{1},x)$ and $\hat{%
u}(x)=\hat{u}(t_{1},x)=U\tilde{\omega}(t_{1},x)$ and setting $\hat{u}(t,x)=%
\hat{u}(x)=\hat{u}(t_{1},x)$ for $t_{0}\leq t<t_{1}.$ Finally, $\hat{u}%
(t_{0},x)=U\tilde{\omega}(t_{0},x).$
It is also useful to introduce $$\tilde{u}(t,x):=U\tilde{\omega}(t,x),\ \ t_{0}\leq t\leq t_{N}.
\label{u_tilde}$$In contrast to $\hat{u},$ the function $\tilde{u}$ is continuous in $t.$ These functions coincide at $t=t_{k},$ $k=0,\ldots ,N.$
At each step of this method one has to solve the system (\[M2\])-(\[M3\]). In contrast to the system (\[ns1p\])-(\[ns4p\]), the system (\[M2\])-(\[M3\]) does not have the divergence-free condition and it is linear. Then the solution of (\[M2\])-(\[M3\]) can be found using probabilistic representations. We pay attention to the fact that in the vorticity formulation of the NSE the pressure term disappears.
In order to realise the approximation process described above, it is sufficient that on every time interval $[t_{k},t_{k+1}],$ $k=N-1,N-2,\ldots
,1,0,$ there exists a solution of the linear parabolic system (\[M2\])-(\[M3\]) (we denote such a solution $\tilde{\omega}_{k}(t,x))$ which satisfies the condition $$\tilde{\omega}_{k}(t_{k+1},x)=\left\{
\begin{array}{c}
\mathop{\rm curl}\varphi (x),\ k=N-1, \\
\tilde{\omega}_{k+1}(t_{k+1},x),\ k=N-2,\ldots ,0,%
\end{array}%
\right. \label{vor45}$$and has the time-independent $\hat{u}(x)$ within each interval $%
[t_{k},t_{k+1})$ $\ $defined as$$\hat{u}(x):=\hat{u}_{k}(x)=U\tilde{\omega}_{k}(t_{k+1},x),\ t_{k}\leq
t<t_{k+1}. \label{vor46}$$Clearly, $\hat{u}(x)$ used in (\[M2\]) are different on the time intervals $[t_{k},t_{k+1})$.
The divergence-free property of the method\[sec:approxdiv\]
-----------------------------------------------------------
The evolution equation (\[V1\]) for vorticity has the form$$\frac{\partial \omega }{\partial t}=\mathop{\rm curl}[\ldots ].$$Due to this fact, any solution of (\[V1\]) with $\mathop{\rm div}\omega
(t_{k},x)=0$ is divergence free for $t\leq t_{k}:$ $\mathop{\rm div}\omega
(t,x)=0, $ $t\leq t_{k}.$ Indeed, this property can be seen after applying the operator $\mathop{\rm div}$ to (\[V1\]) and taking into account the equality $\mathop{\rm div}\mathop{\rm curl}[\ldots ]=0.$
A very important property of the proposed method is that the constructed approximation $\tilde{\omega}_{k}(t,x)$ is also divergent free.
\[thm:divfree\]The solution $\tilde{\omega}(t,x),\ t_{k}\leq t\leq
t_{k+1},$ of $(\ref{M2})$-$(\ref{M3})$ is divergent free.
**Proof**. Let us take $\mathop{\rm div}$ of the equation (\[M2\]). In (\[M2\]) we have that $\hat{u}(t,x)=\hat{u}%
(x)=u(t_{k+1},x),\ t_{k}\leq t<t_{k+1},$ and $\hat{u}(x)$ is divergent free: $\mathop{\rm div}\hat{u}=0$. Besides, $\mathop{\rm div}g=0$. We have$$-(\hat{u},\nabla )\tilde{\omega}=-(\hat{u}^{1}\frac{\partial }{\partial x^{1}%
}+\hat{u}^{2}\frac{\partial }{\partial x^{2}}+\hat{u}^{3}\frac{\partial }{%
\partial x^{3}})\tilde{\omega}=-\left[
\begin{array}{c}
\hat{u}^{1}\frac{\partial \tilde{\omega}^{1}}{\partial x^{1}}+\hat{u}^{2}%
\frac{\partial \tilde{\omega}^{1}}{\partial x^{2}}+\hat{u}^{3}\frac{\partial
\tilde{\omega}^{1}}{\partial x^{3}} \\
\hat{u}^{1}\frac{\partial \tilde{\omega}^{2}}{\partial x^{1}}+\hat{u}^{2}%
\frac{\partial \tilde{\omega}^{2}}{\partial x^{2}}+\hat{u}^{3}\frac{\partial
\tilde{\omega}^{2}}{\partial x^{3}} \\
\hat{u}^{1}\frac{\partial \tilde{\omega}^{3}}{\partial x^{1}}+\hat{u}^{2}%
\frac{\partial \tilde{\omega}^{3}}{\partial x^{2}}+\hat{u}^{3}\frac{\partial
\tilde{\omega}^{3}}{\partial x^{3}}%
\end{array}%
\right] ,$$$$\begin{aligned}
\mathop{\rm div}[-(\hat{u},\nabla )\tilde{\omega}]=-(\frac{\partial \hat{u}%
^{1}}{\partial x^{1}}\frac{\partial \tilde{\omega}^{1}}{\partial x^{1}}+%
\frac{\partial \hat{u}^{2}}{\partial x^{1}}\frac{\partial \tilde{\omega}^{1}%
}{\partial x^{2}}+\frac{\partial \hat{u}^{3}}{\partial x^{1}}\frac{\partial
\tilde{\omega}^{1}}{\partial x^{3}} \\
\frac{\partial \hat{u}^{1}}{\partial x^{2}}\frac{\partial \tilde{\omega}^{2}%
}{\partial x^{1}}+\frac{\partial \hat{u}^{2}}{\partial x^{2}}\frac{\partial
\tilde{\omega}^{2}}{\partial x^{2}}+\frac{\partial \hat{u}^{3}}{\partial
x^{2}}\frac{\partial \tilde{\omega}^{2}}{\partial x^{3}}+\frac{\partial \hat{%
u}^{1}}{\partial x^{3}}\frac{\partial \tilde{\omega}^{3}}{\partial x^{1}}+%
\frac{\partial \hat{u}^{2}}{\partial x^{3}}\frac{\partial \tilde{\omega}^{3}%
}{\partial x^{2}}+\frac{\partial \hat{u}^{3}}{\partial x^{3}}\frac{\partial
\tilde{\omega}^{3}}{\partial x^{3}}) \\
-(\hat{u}^{1}\frac{\partial }{\partial x^{1}}\mathop{\rm div}\tilde{\omega}+%
\hat{u}^{2}\frac{\partial }{\partial x^{2}}\mathop{\rm div}\tilde{\omega}+%
\hat{u}^{3}\frac{\partial }{\partial x^{3}}\mathop{\rm div}\tilde{\omega}).\end{aligned}$$Analogously,$$\begin{aligned}
\mathop{\rm div}[(\tilde{\omega},\nabla )\hat{u}]=\frac{\partial \tilde{%
\omega}^{1}}{\partial x^{1}}\frac{\partial \hat{u}^{1}}{\partial x^{1}}+%
\frac{\partial \tilde{\omega}^{2}}{\partial x^{1}}\frac{\partial \hat{u}^{1}%
}{\partial x^{2}}+\frac{\partial \tilde{\omega}^{3}}{\partial x^{1}}\frac{%
\partial \hat{u}^{1}}{\partial x^{3}} \\
\frac{\partial \tilde{\omega}^{1}}{\partial x^{2}}\frac{\partial \hat{u}^{2}%
}{\partial x^{1}}+\frac{\partial \tilde{\omega}^{2}}{\partial x^{2}}\frac{%
\partial \hat{u}^{2}}{\partial x^{2}}+\frac{\partial \tilde{\omega}^{3}}{%
\partial x^{2}}\frac{\partial \hat{u}^{2}}{\partial x^{3}}+\frac{\partial
\tilde{\omega}^{1}}{\partial x^{3}}\frac{\partial \hat{u}^{3}}{\partial x^{1}%
}+\frac{\partial \tilde{\omega}^{2}}{\partial x^{3}}\frac{\partial \hat{u}%
^{3}}{\partial x^{2}}+\frac{\partial \tilde{\omega}^{3}}{\partial x^{3}}%
\frac{\partial \hat{u}^{3}}{\partial x^{3}} \\
+(\tilde{\omega}^{1}\frac{\partial }{\partial x^{1}}\mathop{\rm div}\hat{u}+%
\tilde{\omega}^{2}\frac{\partial }{\partial x^{2}}\mathop{\rm div}\hat{u}+%
\tilde{\omega}^{3}\frac{\partial }{\partial x^{3}}\mathop{\rm div}\hat{u}).\end{aligned}$$Since $\mathop{\rm div}\hat{u}=0,$ we get$$\begin{aligned}
&&\mathop{\rm div}[-(\hat{u},\nabla )\tilde{\omega}]+\mathop{\rm
div}[(\tilde{\omega},\nabla )\hat{u}] \\
&=&-(\hat{u}^{1}\frac{\partial }{\partial x^{1}}\mathop{\rm div}\tilde{\omega%
}+\hat{u}^{2}\frac{\partial }{\partial x^{2}}\mathop{\rm div}\tilde{\omega}+%
\hat{u}^{3}\frac{\partial }{\partial x^{3}}\mathop{\rm div}\tilde{\omega}).\end{aligned}$$Hence, taking $\mathop{\rm div}$ of (\[M2\]) gives the following equation for $\mathop{\rm div}\tilde{\omega}:$$$\begin{aligned}
\frac{\partial \mathop{\rm div}\tilde{\omega}}{\partial t}-(\hat{u}^{1}\frac{%
\partial }{\partial x^{1}}\mathop{\rm div}\tilde{\omega}+\hat{u}^{2}\frac{%
\partial }{\partial x^{2}}\mathop{\rm div}\tilde{\omega}+\hat{u}^{3}\frac{%
\partial }{\partial x^{3}}\mathop{\rm div}\tilde{\omega})+\frac{\sigma ^{2}}{%
2}\Delta \mathop{\rm div}\tilde{\omega}=0, \label{M4} \\
t_{k}\leq t<t_{k+1},\ \ \mathop{\rm div}\tilde{\omega}(t_{k+1},x)=0.
\label{M5}\end{aligned}$$From here, due to uniqueness of solution to the problem (\[M4\])-(\[M5\]), we obtain$$\mathop{\rm div}\tilde{\omega}(t,x)=0,\ t_{k}\leq t\leq t_{k+1},\ x\in
\mathbf{R}^{3}.$$Theorem \[thm:divfree\] is proved.
The one-step error of the method\[sec:approxerr\]
-------------------------------------------------
For estimating the local error (the one-step error) in the 2D case, together with the solution $\omega (t,x),\ t_{k}\leq t\leq t_{k+1},$ of (\[M6\]), we consider the approximation $\tilde{\omega}(t,x),\ $which satisfies the equation$$\frac{\partial \tilde{\omega}}{\partial t}-\hat{u}^{1}(x)\frac{\partial
\tilde{\omega}}{\partial x^{1}}(t,x)-\hat{u}^{2}(x)\frac{\partial \tilde{%
\omega}}{\partial x^{2}}(t,x)+\frac{\sigma ^{2}}{2}\Delta \tilde{\omega}%
(t,x)+g(t,x^{1},x^{2})=0 \label{M7}$$and the Cauchy condition$$\tilde{\omega}(t_{k+1},x)=\omega (t_{k+1},x). \label{M07}$$The difference $$\delta _{\omega }(t,x):=\omega (t,x)-\tilde{\omega}(t,x),$$which is the one step error, is a solution to the problem$$\begin{gathered}
\frac{\partial \delta _{\omega }}{\partial t}+\frac{\sigma ^{2}}{2}\Delta
\delta _{\omega }-u^{1}\frac{\partial \delta _{\omega }}{\partial x^{1}}%
-u^{2}\frac{\partial \delta _{\omega }}{\partial x^{2}}-(u^{1}-\hat{u}^{1})%
\frac{\partial \tilde{\omega}}{\partial x^{1}}-(u^{2}-\hat{u}^{2})\frac{%
\partial \tilde{\omega}}{\partial x^{2}}=0, \label{M8} \\
\delta _{\omega }(t_{k+1},x)=0. \label{M9}\end{gathered}$$
\[thm:onestep2D\] The one-step error of $\tilde{\omega}(t,x),\ t_{k}\leq
t\leq t_{k+1},$ which solves $(\ref{M2})$-$(\ref{M3})$ is of second order with respect to $h:$$$|\delta _{\omega }(t,x)|\leq Kh^{2},\ t_{k}\leq t\leq t_{k+1},\ x\in \mathbf{%
R}^{2}. \label{M10}$$
**Proof**. Let us write the probabilistic representation of the form (\[V15\])-(\[V19\]) for the solution to problem (\[M8\])-([M9]{}):$$dX^{i}=\sigma dw_{i}(s),\ X^{i}(t)=x^{i},\ i=1,2, \label{M11}$$$$dQ=-Q(u^{1}dw_{1}+u^{2}dw_{2}),\ Q(t)=1, \label{M12}$$$$dZ=-Q((u^{1}-\hat{u}^{1})\frac{\partial \tilde{\omega}}{\partial x^{1}}%
+(u^{2}-\hat{u}^{2})\frac{\partial \tilde{\omega}}{\partial x^{2}})ds,\
Z(t)=0, \label{M13}$$$$\delta _{\omega }(t,x)=-E\int_{t}^{t_{k+1}}Q((u^{1}-\hat{u}^{1})\frac{%
\partial \tilde{\omega}}{\partial x^{1}}+(u^{2}-\hat{u}^{2})\frac{\partial
\tilde{\omega}}{\partial x^{2}})ds. \label{M14}$$Using boundedness of $\partial \tilde{\omega}/\partial x^{i},\ i=1,2,$ and the inequalities$$|u^{i}(s,X_{t,x}(s))-\hat{u}%
^{i}(X_{t,x}(s))|=|u^{i}(s,X_{t,x}(s))-u^{i}(t_{k+1},X_{t,x}(s))|\leq Ch,$$for $t_{k}\leq s\leq t_{k+1},$ we get$$|\delta _{\omega }(t,x)|\leq \int_{t}^{t_{k+1}}E|Q|ds\cdot Kh.$$But $Q>0$ and$$E|Q|=EQ=1,$$whence (\[M10\]) follows. Theorem \[thm:onestep2D\] is proved.
Introduce the one-step error for $\tilde{u}(t,x)$ from (\[u\_tilde\]): $$\delta _{u}(t,x):=u(t,x)-U\tilde{\omega}(t,x)=U\delta _{\omega }(t,x),
\label{delta_u}$$where $\tilde{\omega}(t,x),\ t_{k}\leq t\leq t_{k+1},$ is the solution of (\[M2\])-(\[M3\]).
\[Cor2D\]The one-step error of $\tilde{u}(t,x)$ from (\[u\_tilde\]) is of second order with respect to $h$ in $L^{2}$-norm:$$||\delta _{u}(t,x)||_{L^{2}}\leq Kh^{2},\ t_{k}\leq t\leq t_{k+1}.
\label{Cor2D1}$$
**Proof**. Let the Fourier coefficients for $\delta _{\omega
}(t,x)$ be $(\delta _{\omega }(t,\cdot ))_{\mathbf{n}},$ i.e. $$\delta _{\omega }(t,x)=\sum (\delta _{\omega }(t,\cdot ))_{\mathbf{n}%
}e^{i(2\pi /L)(\mathbf{n},x)}.$$Hence (cf. (\[Mnew1\]))$$\delta _{u}(t,x)=\frac{iL}{2\pi }\sum \frac{1}{|\mathbf{n}|^{2}}e^{i(2\pi
/L)(\mathbf{n},x)}(\delta _{\omega }(t,\cdot ))_{\mathbf{n}}\left[
\begin{array}{c}
n^{2} \\
-n^{1}%
\end{array}%
\right] ,$$i.e., the Fourier coefficients for $\delta _{u}(t,x)$ are $$(\delta _{u}(t,\cdot ))_{\mathbf{n}}=\frac{iL}{2\pi }\frac{1}{|\mathbf{n}%
|^{2}}(\delta _{\omega }(t,\cdot ))_{\mathbf{n}}\left[
\begin{array}{c}
n^{2} \\
-n^{1}%
\end{array}%
\right] .$$Then, by Parseval’s identity (\[eq:pars\]), we have $$\begin{aligned}
||\delta _{u}(t,\cdot )||_{L^{2}} &=&\int_{Q}|\delta
_{u}(t,x)|^{2}dx=L^{2}\sum |(\delta _{u}(t,\cdot ))_{\mathbf{n}}|^{2}
\label{extra} \\
&=&\frac{L^{4}}{4\pi ^{2}}\sum |(\delta _{\omega }(t,\cdot ))_{\mathbf{n}%
}|^{2}\frac{\left( n^{1}\right) ^{2}+\left( n^{2}\right) ^{2}}{|\mathbf{n}%
|^{4}} \notag \\
&=&\frac{L^{4}}{4\pi ^{2}}\sum \frac{|(\delta _{\omega }(t,\cdot ))_{\mathbf{%
n}}|^{2}}{|\mathbf{n}|^{2}}\leq \frac{L^{4}}{4\pi ^{2}}\sum |(\delta
_{\omega }(t,\cdot ))_{\mathbf{n}}|^{2} \notag \\
&=&\frac{L^{2}}{4\pi ^{2}}\int_{Q}|\delta _{\omega }(t,x)|^{2}dx\leq \frac{%
L^{2}}{4\pi ^{2}}\max_{x}|\delta _{\omega }(t,x)|^{2}, \notag\end{aligned}$$which together with (\[M10\]) implies (\[Cor2D1\]). Corollary \[Cor2D\] is proved.
The result of Theorem \[thm:onestep2D\] is carried over to the 3D case without any substantial changes in the proof. In the 3D case the difference $%
\delta _{\omega }(t,x):=\omega (t,x)-\tilde{\omega}(t,x)$ is a solution to the problem$$\begin{gathered}
\frac{\partial \delta _{\omega }}{\partial t}+\frac{\sigma ^{2}}{2}\Delta
\delta _{\omega }-\sum_{i=1}^{3}u^{i}\frac{\partial \delta _{\omega }}{%
\partial x^{i}}+\sum_{i=1}^{3}\frac{\partial u}{\partial x^{i}}\delta
_{\omega }^{i}-\sum_{i=1}^{3}(u^{i}-\hat{u}^{i})\frac{\partial \tilde{\omega}%
}{\partial x^{i}}+\sum_{i=1}^{3}(\frac{\partial u}{\partial x^{i}}-\frac{%
\partial \hat{u}}{\partial x^{i}})\tilde{\omega}^{i}=0, \label{M15} \\
\delta _{\omega }(t_{k+1},x)=0. \label{M16}\end{gathered}$$
\[thm:onestep3D\]The one-step error of $\tilde{\omega}(t,x),\ t_{k}\leq
t\leq t_{k+1},$ which solves $(\ref{M2})$-$(\ref{M3}),$ is of second order with respect to $h:$$$|\delta _{\omega }(t,x)|\leq Kh^{2},\ t_{k}\leq t\leq t_{k+1},\ x\in \mathbf{%
R}^{3}. \label{M017}$$
**Proof**. We apply the probabilistic representation ([V15]{})-(\[V19\]) to the solution of (\[M15\])-(\[M16\]):$$dX^{i}=\sigma dw_{i}(s),\ X^{i}(t)=x^{i},\ i=1,2,3, \label{M17}$$$$dY^{i}=\sum_{j=1}^{3}\frac{\partial u^{j}}{\partial x^{i}}Y^{j}ds,\
Y^{i}(t)=y^{i},\ i=1,2,3, \label{M18}$$$$dQ=-Q\sum_{j=1}^{3}u^{j}(s,X)dw_{j}(s),\ Q(t)=1, \label{M19}$$$$dZ=Q\left[ \sum_{j=1}^{3}\sum_{i=1}^{3}(\frac{\partial u^{j}}{\partial x^{i}}%
-\frac{\partial \hat{u}^{j}}{\partial x^{i}})\tilde{\omega}%
^{i}Y^{j}-\sum_{j=1}^{3}\sum_{i=1}^{3}(u^{i}-\hat{u}^{i})\frac{\partial
\tilde{\omega}^{j}}{\partial x^{i}}Y^{j}\right] ds,\ Z(t)=0, \label{M20}$$$$\begin{aligned}
\delta _{\omega }^{\top }(t,x)y=EZ_{t,x,y,1,0}(t_{k+1}) \label{M21} \\
=E\int_{t}^{t_{k+1}}Q_{t,x,y,1}(s)\left[ \sum_{j=1}^{3}\sum_{i=1}^{3}(\frac{%
\partial u^{j}}{\partial x^{i}}(s,X_{t,x}(s))-\frac{\partial \hat{u}^{j}}{%
\partial x^{i}}(X_{t,x}(s)))\tilde{\omega}^{i}(s,X_{t,x}(s))Y^{j}(s)\right.
\notag \\
-\left. \sum_{j=1}^{3}\sum_{i=1}^{3}(u^{i}(s,X_{t,x}(s))-\hat{u}%
^{i}(X_{t,x}(s)))\frac{\partial \tilde{\omega}^{j}}{\partial x^{i}}%
(s,X_{t,x}(s))Y^{j}(s)\right] ds. \notag\end{aligned}$$Using boundedness of $\tilde{\omega}^{i},\ \partial \tilde{\omega}%
^{j}/\partial x^{i},\ Y^{i}(s),\ i,j=1,2,3,$ the inequalities$$|u^{i}(s,X_{t,x}(s))-\hat{u}%
^{i}(X_{t,x}(s))|=|u^{i}(s,X_{t,x}(s))-u^{i}(t_{k+1},X_{t,x}(s))|\leq Ch,$$$$|\frac{\partial u^{j}}{\partial x^{i}}(s,X_{t,x}(s))-\frac{\partial \hat{u}%
^{j}}{\partial x^{i}}(X_{t,x}(s))|=|\frac{\partial u^{j}}{\partial x^{i}}%
(s,X_{t,x}(s))-\frac{\partial u^{j}}{\partial x^{i}}(t_{k+1},X_{t,x}(s))|%
\leq Ch,$$for $t_{k}\leq s<t_{k+1},$ and the properties $Q>0,\ E|Q|=EQ=1,$ we get ([M017]{}). Theorem \[thm:onestep3D\] is proved.
We note that the one-step error estimate (\[Cor2D1\]) for $\tilde{u}$ from Corollary \[Cor2D\] is also valid in the three-dimensional case.
Convergence theorems\[sec:conv\]
--------------------------------
In this section we first consider the global error for the approximation $%
\tilde{u}(t,x)$ from (\[u\_tilde\]), i.e., we are interested in estimating the difference $$D_{\tilde{u}}:=u(t_{0},x)-\tilde{u}(t_{0},x),$$where $u(t_{0},x)$ is the solution of the NSE (\[ns1p\])-(\[ns4p\]).
Let us introduce the auxiliary functions $_{k}u(t,x)$ on the time intervals $%
[t_{0},t_{k}],$ $k=1,\ldots ,N:$ $$_{k}u(t,x):=u(t,x;t_{k},\tilde{u}(t_{k},\cdot )),\ t_{0}\leq t\leq t_{k},
\label{eq:k_u}$$where $u(t,x;t_{k},\tilde{u}(t_{k},\cdot ))$ denotes the solution of the NSE (\[ns1p\])-(\[ns4p\]) with the terminal condition $\varphi (\cdot )=%
\tilde{u}(t_{k},\cdot )$ prescribed at $T=t_{k}$. To prove the convergence theorem, we assume that all the functions $_{k}u(t,x)$ are bounded together with their derivatives up to some order.
Since $\tilde{u}(t_{N},x)=u(t_{N},x),$ we have $_{N}u(t,x)=u(t,x),$ $%
t_{0}\leq t\leq t_{N}.$ Also, note that $\tilde{u}(t_{0},x)=\
_{0}u(t_{0},x). $ Then we can re-write the global error as$$D_{\tilde{u}}=\sum_{k=0}^{N-1}\left( \ _{k+1}u(t_{0},x)-\
_{k}u(t_{0},x)\right) . \label{c01}$$We have $$\begin{aligned}
_{k+1}u(t_{0},x) &=&u(t_{0},x;t_{k+1},\tilde{u}(t_{k+1},\cdot
))=u(t_{0},x;t_{k},u(t_{k},\cdot ;t_{k+1},\tilde{u}(t_{k+1},\cdot ))),
\label{c02} \\
_{k}u(t_{0},x) &=&u(t_{0},x;t_{k},\tilde{u}(t_{k},\cdot )). \notag\end{aligned}$$Note that the difference $$_{k}\delta _{u}(t_{k},x)=u(t_{k},x;t_{k+1},\tilde{u}(t_{k+1},\cdot ))-\tilde{%
u}(t_{k},x)$$is a one-step error (see (\[delta\_u\])), which $L^{2}$-estimate is of order $h^{2}$ according to Corollary \[Cor2D\]. We remark that $%
_{k+1}u(t_{0},x)-\ _{k}u(t_{0},x)$ is the propagation error which is due to the error in the terminal condition propagated along the trajectory of the NSE solution.
To estimate the propagation error, we are making use of the basic energy estimate from [@MB p. 89], where it is proved in the whole space, but it can be derived for the periodic case as well. In our case this energy estimate takes the form$$\sup_{t_{0}\leq t\leq t_{k}}||\ _{k+1}u(t_{0},\cdot )-\ _{k}u(t_{0},\cdot
)||_{L^{2}}\leq C||\ _{k+1}u(t_{k},\cdot )-\ _{k}u(t_{k},\cdot )||_{L^{2}},
\label{c03}$$where the constant $C>0$ depends on the function $_{k}u(t,x).$
Due to (\[Cor2D1\]) and (\[c03\]), we obtain $$||\ _{k+1}u(t_{0},\cdot )-\ _{k}u(t_{0},\cdot )||_{L^{2}}\leq Kh^{2},
\label{c04}$$where $K>0$ combines the constant $K$ from (\[Cor2D1\]) and $C$ from ([c03]{}). From (\[c04\]) and (\[c01\]), we get $$||D_{\tilde{u}}||_{L^{2}}\leq Kh.$$Thus, we have proved the following theorem.
\[thm:conv\] The approximation $\tilde{u}(t,x)$ from $(\ref{u_tilde})$ for the solution of the NSE $(\ref{ns1p})$-$(\ref{ns4p})$ is of first order in $h.$
We note that the proof of Theorem \[thm:conv\] tacitly used an assumption of existence, uniqueness and regularity of solutions of the NSE problems involved in the error estimates. Such an assumption is natural to make in the work aimed at deriving approximations and we do not consider here how one can prove such properties of $_{k}u(t,x)$ from (\[eq:k\_u\]).
Now we analyse the global error of $\tilde{\omega}_{k}(t,x).$
\[thm:conv2\] The approximation $\tilde{\omega}_{k}(t,x)$ $($see $(\ref%
{u_tilde}),$ $(\ref{vor45}))$ for the solution of the NSE $(\ref{ns1p})$-$(%
\ref{ns4p})$ converges with order 1 in $L^{2}$-norm.
**Proof**. Let $D_{\tilde{\omega}}(t,x;k)$ be the global error for $\tilde{\omega}$ on the interval $[t_{k,}t_{k+1}],$ i.e.$$D_{\tilde{\omega}}(t,x;k):=\omega (t,x)-\tilde{\omega}_{k}(t,x),$$and $D_{\hat{u}}(t,x;k)$ be the global error for $\hat{u}_{k}$ on the interval $[t_{k,}t_{k+1}),$ i.e.$$D_{\hat{u}}(t,x;k):=u(t,x)-\hat{u}_{k}(x).$$We have analogously to (\[M8\])-(\[M9\]):$$\begin{gathered}
-\frac{\partial D_{\tilde{\omega}}(t,x;k)}{\partial t}=\frac{\sigma ^{2}}{2}%
\Delta D_{\tilde{\omega}}(t,x;k)-(u(t,x),\nabla )D_{\tilde{\omega}}(t,x;k)
\notag \\
-(D_{\hat{u}}(t,x;k),\nabla )\tilde{\omega}_{k}(t,x),\ \ t_{k}\leq
t<t_{k+1},\ \ k=N-1,\ldots ,0, \label{D1} \\
D_{\tilde{\omega}}(t_{N},x;N-1)=0, \label{D2} \\
D_{\tilde{\omega}}(t_{k+1},x;k)=D_{\tilde{\omega}}(t_{k+1},x;k+1),\ \ \
k=N-2,\ldots ,0. \label{D3}\end{gathered}$$Then $$\begin{gathered}
-\frac{1}{2}\frac{d||D_{\tilde{\omega}}(t,\cdot ;k)||^{2}}{dt}=-\frac{\sigma
^{2}}{2}||\nabla D_{\tilde{\omega}}||^{2}-((u,\nabla )D_{\tilde{\omega}},D_{%
\tilde{\omega}})-((D_{\hat{u}}(t,\cdot ;k),\nabla )\tilde{\omega}_{k},D_{%
\tilde{\omega}}), \label{sp1} \\
t_{k}\leq t<t_{k+1}. \notag\end{gathered}$$Since $u$ is divergence free, we get$$((u,\nabla )D_{\tilde{\omega}},D_{\tilde{\omega}})=0. \label{sp11}$$Note that (see (\[vor46\])): $$\begin{aligned}
D_{\hat{u}}(t,x;k) &=&u(t,x)-\hat{u}_{k}(x)=U\omega (t,x)-U\tilde{\omega}%
_{k}(t_{k+1},x) \\
&=&UD_{\tilde{\omega}}(t,x;k)+U\tilde{\omega}_{k}(t,x)-U\tilde{\omega}%
_{k}(t_{k+1},x).\end{aligned}$$Then, using (\[A66\]) with $c_{2}=\sigma ^{2}/2$, we obtain for some $K>0$:$$\begin{aligned}
|((D_{\hat{u}}(t,\cdot ;k),\nabla )\tilde{\omega}_{k},D_{\tilde{\omega}})|
&=&|((U\left( D_{\tilde{\omega}}+\tilde{\omega}_{k}(t,\cdot )-\tilde{\omega}%
_{k}(t_{k+1},\cdot )\right) ,\nabla )\tilde{\omega}_{k},D_{\tilde{\omega}})|
\label{sp2} \\
&\leq &\frac{\sigma ^{2}}{2}||\nabla D_{\tilde{\omega}}||^{2}+K||D_{\tilde{%
\omega}}||^{2} \notag \\
&&+K||\nabla \tilde{\omega}_{k}||^{2}||D_{\tilde{\omega}}+\tilde{\omega}%
_{k}(t,\cdot )-\tilde{\omega}_{k}(t_{k+1},\cdot )||^{2} \notag \\
&\leq &\frac{\sigma ^{2}}{2}||\nabla D_{\tilde{\omega}}||^{2}+K||D_{\tilde{%
\omega}}||^{2}+K||\nabla \tilde{\omega}_{k}||^{2}||D_{\tilde{\omega}}||^{2}
\notag \\
&&+K||\nabla \tilde{\omega}_{k}||^{2}||\tilde{\omega}_{k}(t,\cdot )-\tilde{%
\omega}_{k}(t_{k+1},\cdot )||^{2}. \notag\end{aligned}$$Using boundedness of $||\frac{d}{dt}\tilde{\omega}_{k}(t,\cdot )||^{2},$ we get $$||\tilde{\omega}_{k}(t,\cdot )-\tilde{\omega}_{k}(t_{k+1},\cdot )||^{2}\leq
Kh^{2},$$which together with boundedness of $||\nabla \tilde{\omega}_{k}||^{2}$ implies $$|((D_{\hat{u}}(t,\cdot ;k),\nabla )\tilde{\omega}_{k},D_{\tilde{\omega}%
})|\leq \frac{\sigma ^{2}}{2}||\nabla D_{\tilde{\omega}}||^{2}+K||D_{\tilde{%
\omega}}||^{2}+Kh^{2}. \label{sp22}$$It follows from (\[sp1\]), (\[sp11\]) and (\[sp22\]) that $$-\frac{d\left( ||D_{\tilde{\omega}}(t,\cdot ;k)||^{2}+h^{2}\right) }{||D_{%
\tilde{\omega}}(t,\cdot ;k)||^{2}+h^{2}}\leq 2Kdt,\ t_{k}\leq t<t_{k+1}.$$Then $$||D_{\tilde{\omega}}(t_{k},\cdot ;k)||^{2}+h^{2}\leq e^{2Kh}\left( ||D_{%
\tilde{\omega}}(t_{k+1},\cdot ,k)||^{2}+h^{2}\right) .$$From here and due to (\[D3\]), we get $$||D_{\tilde{\omega}}(t_{k},\cdot ;k-1)||^{2}\leq e^{2Kh}||D_{\tilde{\omega}%
}(t_{k+1},\cdot ,k)||^{2}+\left( e^{2Kh}-1\right) h^{2},\ k=N-1,\ldots ,1.$$Denoting $R_{k}:=||D_{\tilde{\omega}}(t_{k+1},\cdot ;k)||^{2},$ $%
k=N-1,\ldots ,0,$ we obtain (see (\[D2\])): $$\begin{aligned}
R_{k-1} &\leq &e^{2Kh}R_{k}+\left( e^{2Kh}-1\right) h^{2},\ k=N-1,\ldots ,1,
\\
R_{N-1} &=&0,\end{aligned}$$and using the discrete Gronwall lemma (see e.g. [@MT1 p. 7]), we arrive at $R_{0}=||D_{\tilde{\omega}}(t_{1},\cdot ;0)||^{2}\leq Kh^{2}.$ Theorem \[thm:conv2\] is proved.
Stochastic Navier-Stokes equations\[sec:sns\]
=============================================
In this section we carry over the results of Section \[sec:approx\] for the deterministic NSE to two-dimensional NSE with additive noise. After introducing the stochastic NSE in velocity-vorticity formulation, we prove two auxiliary lemmas (Section \[sec:sns1\]) about its solution; we consider a one-step approximation of vorticity and its properties (Section \[sec:sns2\]); we introduce the numerical method for vorticity and prove boundedness of its moments in Section \[sec:sns3\]; and, finally, we prove first-order mean-square convergence of the method in Section \[sec:sns4\]. The global convergence proof contains ideas, which can potentially be exploited in analysis of numerical methods for a wider class of semilinear SPDEs.
Let $(\Omega ,\mathcal{F},P)$ be a probability space and $(w(t),\mathcal{F}%
_{t}^{w})=((w_{1}(t),\ldots ,w_{q}(t))^{\top },\mathcal{F}_{t})$ be a $q$-dimensional standard Wiener process, where $\mathcal{F}_{t},\ 0\leq t\leq
T, $ is an increasing family of $\sigma $-subalgebras of $\mathcal{F}$ induced by $w(t).$ We consider the system of stochastic Navier-Stokes equations (SNSE) with additive noise for velocity $v$ and pressure $p$ in a viscous incompressible flow:$$\begin{aligned}
dv(t) &=&\left[ \frac{\sigma ^{2}}{2}\Delta v-(v,\nabla )v-\nabla p+f(t,x)%
\right] dt+\sum_{r=1}^{q}\gamma _{r}(t,x)dw_{r}(t), \label{sns1} \\
\ \ 0 &<&t\leq T,\ x\in \mathbf{R}^{2}, \notag \\
\mathop{\rm div}v &=&0, \label{sns3}\end{aligned}$$with spatial periodic conditions$$\begin{aligned}
v(t,x+Le_{i}) &=&v(t,x),\ p(t,x+Le_{i})=p(t,x), \label{sns4} \\
0 &\leq &t\leq T,\ \ i=1,2, \notag\end{aligned}$$and the initial condition$$v(0,x)=\varphi (x). \label{sns5}$$In (\[sns1\])-(\[sns4\]), $v,$ $f,$ and $\gamma _{r}$ are two-dimensional functions;$\ p$ is a scalar; $\{e_{i}\}$ is the canonical basis in $\mathbf{R}^{2}$ and $L>0$ is the period. The functions $f=f(t,x)$ and $\gamma _{r}(t,x)$ are assumed to be spatial periodic as well. Further, we require that $\gamma _{r}(t,x)$ are divergence free: $$\mathop{\rm div}\gamma _{r}(t,x)=0,\ r=1,\ldots ,q. \label{sns6}$$For simplicity of proofs, we assume that the number of noises $q$ is finite but it can be shown that the theoretical results of this section are also valid when $q$ is infinite if $\Vert \gamma _{r}(t,x)\Vert _{m}$ for some $%
m\geq 0$ decay exponentially fast with increase of $r.$
**Assumption 5.1.** *We assume that the coefficients* $%
f(t,x)$ *and* $\gamma _{r}(s,x),$ $r=1,\ldots ,q,$ *belong to* $\mathbf{H}_{p}^{m+1}(Q)$ *and the initial condition* $\varphi (x)$ *belongs to* $\mathbf{H}_{p}^{m+2}(Q)$ *for some* $m\geq 0$.
Under this assumption the problem (\[sns1\])-(\[sns5\]) has a unique solution $v(t,x),\ p(t,x),$ $(t,x)\in \lbrack 0,T]\times R^{2},$ so that for some $m\geq 0$ and $l\geq 2$ [@MatPhD; @Mat02]:$$E\Vert v(t,\cdot )\Vert _{m+2}^{l}\leq K, \label{momv}$$where $K>0$ may depend on $l$, $m,\ T,$ $f(t,x),$ $\gamma _{r}(t,x),$ and $%
\varphi (x).$ The solution $v(t,x),\ p(t,x),$ $(t,x)\in \lbrack 0,T]\times
\mathbf{R}^{2},$ to (\[sns1\])-(\[sns5\]) is $\mathcal{F}_{t}$-adaptive, $v(t,\cdot )\in \mathbf{V}_{p}^{m+2}$ and $\nabla p(t,\cdot )\in (\mathbf{V}%
_{p}^{m+2})^{\bot }$ for every $t\in \lbrack 0,T]$ and $\mathbf{\omega }\in
\Omega .$ We note that if we were interested in variational solutions of (\[sns1\])-(\[sns5\]) then it is more natural to put $m\geq -1$ in Assumption 5.1; but here our focus is on the vorticity formulation and then it is natural to require more, $m\geq 0$.
The vorticity formulation of the problem (\[sns1\])-(\[sns5\]) has the form $$d\omega =\left[ \frac{\sigma ^{2}}{2}\Delta \omega -(v,\nabla )\omega +g(t,x)%
\right] dt+\sum_{r=1}^{q}\mu _{r}(t,x)dw_{r}(t), \label{sns7}$$where $g=\mathop{\rm curl}f$ and $\mu _{r}=\mathop{\rm curl}\gamma _{r}.$ The vorticity satisfies the initial and periodic boundary conditions $$\omega (0,x)=\mathop{\rm curl}\varphi (x):=\phi (x) \label{sns8}$$and spatial periodic conditions$$\omega (t,x+Le_{i})=\omega (t,x),\ i=1,2,\ 0\leq t\leq T. \label{sns9}$$We note that $\omega (t,x)$ is a one-dimensional function here. Using the linear operator $U$ from (\[Mnew1\]), we can re-write (\[sns7\]) as $$d\omega =\left[ \frac{\sigma ^{2}}{2}\Delta \omega -(U\omega ,\nabla )\omega
+g(t,x)\right] dt+\sum_{r=1}^{q}\mu _{r}(t,x)dw_{r}(t). \label{sns10}$$
Similarly to the solution $v(t,x)$ of (\[sns1\])-(\[sns5\]), the solution $\omega (t,x)$ to the vorticity problem (\[sns7\])-(\[sns9\]) under Assumption 5.1 is so that for some $m\geq 0$ and $p\geq 2$:$$E\Vert \omega (t,\cdot )\Vert _{m+1}^{p}\leq K, \label{sns100}$$where $K>0$ depends on $p$, $m,$ $g,$ $\mu _{r},$ and $\phi .$ Note that under Assumption 5.1 the coefficients $g(t,x)$ and $\mu _{r}(s,x),$ $%
r=1,\ldots ,q,$ belong to $\mathbf{H}_{p}^{m}(Q)$ and the initial condition $%
\phi (x)$ belongs to $\mathbf{H}_{p}^{m+1}(Q).$ As it is clear from the context, we are dealing here with solutions understood in the strong sense probabilistically and PDE-wise in the variational sense.
Two technical lemmas\[sec:sns1\]
--------------------------------
For proving convergence of the numerical method in Section \[sec:sns4\], we need two further properties of the solution $\omega (t,x)$ which are formulated in the next two lemmas.
It is convenient to introduce the notation for the solution $\omega (t,x)$ of the problem (\[sns7\])-(\[sns9\]) which reflects its dependence on the initial condition $\phi (x)$ prescribed at time $s\leq t$: $$\omega (t,x)=\omega (t,x;s,\phi ).$$
Let us prove a technical lemma which is related to Lemmas 4.10(1) and A.1 from [@HaMa06].
\[LemMat\]Let Assumption 5.1 hold with $m=0.$ There exist constants $%
\beta _{0}>0$ and $\alpha >0$ such that for any $\beta \in (0,\beta _{0}]$ and $0\leq t\leq t+h\leq T:$ $$\begin{aligned}
&&E\exp \left( \beta \left[ ||\omega (t+h,\cdot ;t,\phi )||^{2}-||\phi ||^{2}%
\right] +\beta \frac{\sigma ^{2}}{4}\int_{t}^{t+h}||\nabla \omega (s,\cdot
;t,\phi )||^{2}ds\right) \label{sns30x} \\
&\leq &\exp \left( \beta \int_{t}^{t+h}\left( \frac{2}{\alpha \sigma ^{2}}%
||g(s,\cdot )||^{2}+\sum_{r=1}^{q}||\mu _{r}(s,\cdot )||^{2}\right)
ds\right) . \notag\end{aligned}$$
**Proof**. By the Ito formula, integration by parts and using $\mathop{\rm div}v(t,x)=0$, we obtain $$\begin{gathered}
\frac{1}{2}d||\omega (s,\cdot )||^{2}=\left[ -\frac{\sigma ^{2}}{2}||\nabla
\omega (s,\cdot )||^{2}+(g(s,\cdot ),\omega (s,\cdot ))+\frac{1}{2}%
\sum_{r=1}^{q}||\mu _{r}(s,\cdot )||^{2}\right] ds \label{sns302} \\
+\sum_{r=1}^{q}(\mu _{r}(s,\cdot ),\omega (s,\cdot ))dw_{r}(s),\ t<s\leq t+h,
\notag \\
||\omega (t,\cdot )||^{2}=||\phi ||^{2}. \notag\end{gathered}$$Using the elementary inequality, we get for any $\alpha >0:$ $$\begin{aligned}
&&\frac{1}{2}d||\omega (s,\cdot )||^{2} \label{sns304} \\
&\leq &\left[ -\frac{\sigma ^{2}}{2}||\nabla \omega (s,\cdot )||^{2}+\frac{1%
}{\alpha \sigma ^{2}}||g(s,\cdot )||^{2}+\frac{\sigma ^{2}}{4}\alpha
||\omega (s,\cdot )||^{2}+\frac{1}{2}\sum_{r=1}^{q}||\mu _{r}(s,\cdot )||^{2}%
\right] ds \notag \\
&&+\sum_{r=1}^{q}(\mu _{r}(s,\cdot ),\omega (s,\cdot ))dw_{r}(s). \notag\end{aligned}$$By Poincaire’s inequality (\[Poin\]), for some $\alpha >0,$ we have $$||\nabla \omega (t,\cdot )||^{2}\geq \alpha ||\omega (t,\cdot )||^{2}.
\label{sns303}$$By (\[sns303\]), we obtain $$\begin{aligned}
d||\omega (s,\cdot )||^{2} &\leq &\left[ -\frac{\sigma ^{2}}{4}||\nabla
\omega (s,\cdot )||^{2}-\frac{\sigma ^{2}}{4}\alpha ||\omega (s,\cdot
)||^{2}+\frac{2}{\alpha \sigma ^{2}}||g(s,\cdot )||^{2}\right.
\label{sns3045} \\
&&\left. +\sum_{r=1}^{q}||\mu _{r}(s,\cdot )||^{2}\right] ds+2%
\sum_{r=1}^{q}(\mu _{r}(s,\cdot ),\omega (s,\cdot ))dw_{r}(s), \notag\end{aligned}$$then for any $c>0$$$\begin{aligned}
&&c||\omega (t+h,\cdot )||^{2}-c||\phi ||^{2}+c\frac{\sigma ^{2}}{4}%
\int_{t}^{t+h}||\nabla \omega (s,\cdot )||^{2}ds \label{sns306} \\
&&-c\int_{t}^{t+h}\left( \frac{2}{\alpha \sigma ^{2}}||g(s,\cdot
)||^{2}+\sum_{r=1}^{q}||\mu _{r}(s,\cdot )||^{2}\right) ds \notag \\
&\leq &2c\int_{t}^{t+h}\sum_{r=1}^{q}(\mu _{r}(s,\cdot ),\omega (s,\cdot
))dw_{r}(s)-\alpha \frac{\sigma ^{2}}{4}c\int_{t}^{t+h}||\omega (s,\cdot
)||^{2}ds. \notag\end{aligned}$$Let $$M(t,t^{\prime }):=2c\int_{t}^{t^{\prime }}\sum_{r=1}^{q}(\mu _{r}(s,\cdot
),\omega (s,\cdot ))dw_{r}(s)$$which is a continuous $L^{2}$-martingale with quadratic variation $$<M>(t,t^{\prime }):=4c^{2}\int_{t}^{t^{\prime }}\sum_{r=1}^{q}(\mu
_{r}(s,\cdot ),\omega (s,\cdot ))^{2}ds.$$There exists a constant $\beta _{0}>0$ (independent of $h$ and $c)$ so that for all $\beta \in (0,\beta _{0}]:$$$\alpha \frac{\sigma ^{2}}{4}c\int_{t}^{t^{\prime }}||\omega (s,\cdot
)||^{2}ds\geq \frac{\beta }{2c}<M>(t,t^{\prime }).$$Hence $$\begin{aligned}
&&c||\omega (t+h,\cdot )||^{2}-c||\phi ||^{2}+c\frac{\sigma ^{2}}{4}%
\int_{t}^{t+h}||\nabla \omega (s,\cdot )||^{2}ds \label{sns305} \\
&&-c\int_{t}^{t+h}\left( \frac{2}{\alpha \sigma ^{2}}||g(s,\cdot
)||^{2}+\sum_{r=1}^{q}||\mu _{r}(s,\cdot )||^{2}\right) ds \notag \\
&\leq &M(t,t+h)-\frac{\beta }{2c}<M>(t,t+h). \notag\end{aligned}$$For $c=\beta ,$ the right-hand side of (\[sns305\]) is logarithm of a local exponential martingale and therefore $$\begin{gathered}
E\exp \left[ \beta ||\omega (t+h\wedge \tau _{n},\cdot )||^{2}-\beta ||\phi
||^{2}+\beta \frac{\sigma ^{2}}{4}\int_{t}^{t+h\wedge \tau _{n}}||\nabla
\omega (s,\cdot )||^{2}ds\right. \\
\left. -\beta \int_{t}^{t+h\wedge \tau _{n}}\left( \frac{2}{\alpha \sigma
^{2}}||g(s,\cdot )||^{2}+\sum_{r=1}^{q}||\mu _{r}(s,\cdot )||^{2}\right) ds%
\right] \leq 1,\end{gathered}$$where $\tau _{n}=\inf \{s>t:<M>(t,s)\geq n\}$ for a natural number $n.$ Tending $n$ to infinity, we arrive at (\[sns30x\]). Lemma \[LemMat\] is proved.
Note that it follows from (\[sns30x\]) that $$\begin{aligned}
&&E\exp \left( \beta \frac{\sigma ^{2}}{4}\int_{t}^{t+h}||\nabla \omega
(s,\cdot ;t,\phi )||^{2}ds\right) \label{sns30xx} \\
&\leq &\exp \left( \beta ||\phi ||^{2}+\beta \int_{t}^{t+h}\left( \frac{2}{%
\alpha \sigma ^{2}}||g(s,\cdot )||^{2}+\sum_{r=1}^{q}||\mu _{r}(s,\cdot
)||^{2}\right) ds\right) . \notag\end{aligned}$$We also pay attention that the prove of Lemma \[LemMat\] is not relying on smallness of the time step $h$ and, after replacing $t$ with $0$ and $t+h$ with $T,$ the result remains valid: $$\begin{aligned}
&&E\exp \left( \beta \frac{\sigma ^{2}}{4}\int_{0}^{T}||\nabla \omega
(s,\cdot ;0,\phi )||^{2}ds\right) \label{sns30xxg} \\
&\leq &\exp \left( \beta ||\phi ||^{2}+\beta \int_{0}^{T}\left( \frac{2}{%
\alpha \sigma ^{2}}||g(s,\cdot )||^{2}+\sum_{r=1}^{q}||\mu _{r}(s,\cdot
)||^{2}\right) ds\right) . \notag\end{aligned}$$
We now prove the next lemma which gives us dependence of the solution $%
\omega (s,x;t,\phi )$ on the initial data.
\[Lem2\]Let Assumption 5.1 hold with $m=2$ and $\phi _{i}(t,x),$ $i=1,2,$ be $\mathcal{F}_{t}$-measurable processes satisfying (\[sns100\]) with $%
m=2 $. There exists a constant $c_{0}>0$ such that for every $c\in (0,c_{0})$ there is a sufficiently small $h>0$ so that we have for $t\leq s\leq t+h:$$$\omega (s,x;t,\phi _{1}(t,\cdot ))-\omega (s,x;t,\phi _{2}(t,\cdot ))=\phi
_{1}(t,x)-\phi _{2}(t,x)+\eta (s,x) \label{sns30}$$for which $$\begin{aligned}
&&||\omega (s,\cdot ;t,\phi _{1})-\omega (s,\cdot ;t,\phi _{2})||^{2}
\label{sns31} \\
&\leq &||\phi _{1}(t,\cdot )-\phi _{2}(t,\cdot )||^{2}\exp \left(
K(s-t)+c\int_{t}^{s}||\nabla \omega (s^{\prime },\cdot ;t,\phi _{1}(t,\cdot
))||^{2}ds^{\prime }\right) \mathbf{,} \notag\end{aligned}$$where $K>0$ is a constant.
The process $\eta (s)$ satisfies the following estimate $$||\eta (s,\cdot )||^{2}\leq (s-t)||\phi _{1}(t,\cdot )-\phi _{2}(t,\cdot
)||^{2}+C(s,\mathbf{\omega })(s-t)^{3}, \label{sns32}$$where $C(s,\mathbf{\omega })>0$ is an $\mathcal{F}_{s}$-adapted process with bounded moments of a sufficiently high order.
**Proof**. Let $$\theta (s,x):=\omega (s,x;t,\phi _{1})-\omega (s,x;t,\phi _{2})$$We have$$\begin{aligned}
d\theta (s,x) &=&\left[ \frac{\sigma ^{2}}{2}\Delta \theta -(U\theta ,\nabla
)\omega (s,\cdot ;t,\phi _{1})-(U\omega (s,\cdot ;t,\phi _{2}),\nabla
)\theta \right] ds,\ t<s\leq t+h, \\
\theta (t,x) &=&\phi _{1}(t,x)-\phi _{2}(t,x).\end{aligned}$$Then $$\begin{gathered}
\frac{1}{2}d||\theta (s,\cdot )||^{2}=\left[ -\frac{\sigma ^{2}}{2}||\nabla
\theta (s,\cdot )||^{2}-((U\theta (s,\cdot ),\nabla )\omega (s,\cdot ;t,\phi
_{1}),\theta (s,\cdot ))\right] ds,\ t<s\leq t+h,\ \ \label{sns33} \\
||\theta (t,\cdot )||^{2}=||\phi _{1}(t,\cdot )-\phi _{2}(t,\cdot )||^{2}.
\notag\end{gathered}$$Using the inequality (\[A66\]) with $c_{2}=\sigma ^{2}/4$, we have that there exists $K>0$ such that for any $c>0$ $$\begin{aligned}
2|((U\theta (s,\cdot ),\nabla )\omega (s,\cdot ;t,\phi _{1}),\theta (s,\cdot
))| &\leq &\frac{\sigma ^{2}}{2}||\nabla \theta (s,\cdot )||^{2}+K||\theta
(s,\cdot )||^{2} \label{sns34} \\
&&+c||\nabla \omega (s,\cdot ;t,\phi _{1})||^{2}||\theta (s,\cdot )||^{2}
\notag\end{aligned}$$and hence$$d||\theta (s,\cdot )||^{2}\leq \left[ K+c||\nabla \omega (s,\cdot ;t,\phi
_{1})||^{2}\right] ||\theta (s,\cdot )||^{2}ds,\ t<s\leq t+h,$$which implies $$||\theta (s,\cdot )||^{2}\leq ||\phi _{1}(t,\cdot )-\phi _{2}(t,\cdot
)||^{2}\exp \left( K(s-t)+c\int_{t}^{s}||\nabla \omega (s^{\prime },\cdot
;t,\phi _{1}(t,\cdot ))||^{2}ds^{\prime }\right) . \label{sns35}$$Thus we have proved the inequality (\[sns31\]).
Let us now prove (\[sns32\]). We have $$\eta (s,x)=\int_{t}^{s}\left[ \frac{\sigma ^{2}}{2}\Delta \theta -(U\theta
,\nabla )\omega (s^{\prime },\cdot ;t,\phi _{1})-(U\omega (s^{\prime },\cdot
;t,\phi _{2}),\nabla )\theta \right] ds^{\prime },\ t<s\leq t+h,$$which together with (\[momv\]) and (\[sns100\]) implies that $$||\eta (s,x)||\leq C(s,\mathbf{\omega })(s-t), \label{sns37}$$where $C(s,\mathbf{\omega })>0$ is an $\mathcal{F}_{s}$-adapted process with bounded moments of a sufficiently high order. It is not difficult to see that the inequality (\[sns37\]) is also valid for $||\nabla \eta (s,x)||$ and $||\Delta \eta (s,x)||:$$$||\nabla \eta (s,x)||\leq C(s,\mathbf{\omega })(s-t),\ \ ||\Delta \eta
(s,x)||\leq C(s,\mathbf{\omega })(s-t). \label{sns377}$$We have $$d||\eta (s^{\prime },x)||^{2}=\left[ (\sigma ^{2}\Delta \theta ,\eta
)-2((U\theta ,\nabla )\omega (s^{\prime },\cdot ;t,\phi _{1}),\eta
)-2((U\omega (s^{\prime },\cdot ;t,\phi _{2}),\nabla )\theta ,\eta )\right]
ds^{\prime }.$$Using integration by parts, (\[sns31\]), and (\[sns377\]) (we also recall that $s^{\prime }-t\leq h$ which is sufficiently small), we get $$|(\sigma ^{2}\Delta \theta ,\eta )|=\sigma ^{2}|(\theta ,\Delta \eta )|\leq
\sigma ^{2}||\theta ||||\Delta \eta ||\leq C(s^{\prime },\mathbf{\omega }%
)(s^{\prime }-t)||\phi _{1}(t,\cdot )-\phi _{2}(t,\cdot )||.$$By (\[A2\]) with $m_{1}=1,$ $m_{2}=0$, and $m_{3}=1$, (\[A3\]), ([sns100]{}), (\[sns31\]), (\[sns37\]) and (\[sns377\])$,$ we obtain $$\begin{aligned}
|2((U\theta ,\nabla )\omega (s,\cdot ;t,\phi _{1}),\eta )| &\leq &K||U\theta
||_{1}||\omega ||_{1}||\eta ||_{1}\leq K||\theta ||||\omega ||_{1}||\eta
||_{1} \\
&\leq &C(s^{\prime },\mathbf{\omega })(s^{\prime }-t)||\phi _{1}(t,\cdot
)-\phi _{2}(t,\cdot )||.\end{aligned}$$And by (\[A2\]) with $m_{1}=1,$ $m_{2}=1$, and $m_{3}=0$, (\[A3\]), ([sns100]{}), (\[sns31\]), (\[sns37\]) and (\[sns377\]), we arrive at $$\begin{aligned}
|2((U\omega (s^{\prime },\cdot ;t,\phi _{2}),\nabla )\theta ,\eta )|
&=&2|((U\omega (s^{\prime },\cdot ;t,\phi _{2}),\nabla )\eta ,\theta )|\leq
K||\omega ||||\eta ||_{2}||\theta || \\
&\leq &C(s^{\prime },\mathbf{\omega })(s^{\prime }-t)||\phi _{1}(t,\cdot
)-\phi _{2}(t,\cdot )||.\end{aligned}$$Then we have $$\begin{aligned}
d||\eta (s^{\prime },x)||^{2} &\leq &C(s^{\prime },\mathbf{\omega }%
)(s^{\prime }-t)||\phi _{1}(t,\cdot )-\phi _{2}(t,\cdot )||ds^{\prime } \\
&\leq &||\phi _{1}(t,\cdot )-\phi _{2}(t,\cdot )||^{2}ds^{\prime }+\frac{%
C^{2}(s^{\prime },\mathbf{\omega })}{4}(s^{\prime }-t)^{2}ds^{\prime }\end{aligned}$$from which (\[sns32\]) follows. Lemma \[Lem2\] is proved.
One-step approximation\[sec:sns2\]
----------------------------------
Similarly to derivation of the approximation for the deterministic NSE in Section \[sec:approx\], we can approximate the stochastic NSE (\[sns7\])-(\[sns9\]) by freezing the velocity as in (\[eq:freeze\]): $$v(t,x)\approx \hat{v}(t,x):=v(t_{k},x):=\hat{v}(x),\ t_{k}<t\leq t_{k+1},
\label{eq:freeze2}$$and obtain an approximation $\tilde{\omega}(t,x)$ of $\omega (t,x)$ on $\
t_{k}\leq t\leq t_{k+1},$ as follows $$\begin{aligned}
d\tilde{\omega}=\left[ \frac{\sigma ^{2}}{2}\Delta \tilde{\omega}-(\hat{v}%
,\nabla )\tilde{\omega}+g(t,x)\right] dt+\sum_{r=1}^{q}\mu
_{r}(t,x)dw_{r}(t),\ t_{k}<t\leq t_{k+1}, \label{sns11} \\
\tilde{\omega}(t_{k},x)=\omega (t_{k},x),\ \ \tilde{\omega}(t_{k},x+Le_{j})=%
\tilde{\omega}(t_{k},x),\ j=1,2. \label{sns12}\end{aligned}$$
It is not difficult to see that the local error $\delta _{\omega }(t,x)=%
\tilde{\omega}(t,x)-\omega (t,x),$ $t_{k}\leq t\leq t_{k+1},$ for the approximation $\tilde{\omega}(t,x)$ of the solution $\omega (t,x)$ of the stochastic NSE (\[sns7\])-(\[sns9\]) satisfies the problem of the same form as (\[M8\])-(\[M9\]) but with positive direction of time: $$\begin{aligned}
d\delta _{\omega }=\left[ \frac{\sigma ^{2}}{2}\Delta \delta _{\omega
}-(v,\nabla )\delta _{\omega }-((v-\hat{v}),\nabla )\tilde{\omega}\right] dt,
\label{sns14} \\
\delta _{\omega }(t_{k},x)=0. \label{sns15}\end{aligned}$$
We note that the main difference of (\[sns14\])-(\[sns15\]) with ([M8]{})-(\[M9\]) is that the functions in (\[sns14\]) are random and non-smooth in time, they have the same regularity in time as Wiener processes.
Moments of $||\tilde{\omega}||_{3}$ (and hence of $||\delta _{\omega
}||_{3}) $ up to a sufficiently high order are bounded under Assumption 5.1 with $m=2$: for $t_{k}<t\leq t_{k+1}$ and $p\geq 1:$$$E\Vert \tilde{\omega}(t,\cdot )\Vert _{3}^{2p}\leq K, \label{sns159}$$where $K>0$ is a constant, which can be proved by arguments similar to boundedness of the global approximation (see Theorems \[thm:sns2\] and [thm:sns2nn]{}) but not considered here.
To obtain bounds for the one-step error $\delta _{\omega },$ we first prove the following lemma.
\[Lem1\] Let Assumption 5.1 hold with $m=1.$ For $v(t,x)$ from $(\ref%
{sns1})$-$(\ref{sns5})$, $\hat{v}(x)$ from $(\ref{eq:freeze2})$, and $\tilde{%
\omega}(t,x)$ from $(\ref{sns11})$-$(\ref{sns12})$, we have for $t_{k}<t\leq
t_{k+1}$ and sufficiently small $h>0:$ $$\begin{aligned}
||E[((v-\hat{v}),\nabla )\tilde{\omega}|\mathcal{F}_{t_{k}}]|| &\leq
&C(t_{k},\mathbf{\omega })h, \label{sns16} \\
\left( E||v-\hat{v}||^{2p}\right) ^{1/2p} &\leq &Kh^{1/2},\ p\geq 1,
\label{sns17}\end{aligned}$$where $C(t_{k},\mathbf{\omega })>0$ is an $\mathcal{F}_{t_{k}}$-measurable random variable with moments of a sufficiently high order bounded by a constant independent of $h$ and $K>0$ is a constant independent of $h.$
**Proof**. From (\[sns1\]) and (\[eq:freeze2\]), we have for $t_{k}<t\leq t_{k+1}:$$$v(t,x)-\hat{v}(x)=\int_{t_{k}}^{t}\left[ \frac{\sigma ^{2}}{2}\Delta
v-(v,\nabla )v-\nabla p+f(s,x)\right] ds+\int_{t_{k}}^{t}\sum_{r=1}^{q}%
\gamma _{r}(s,x)dw_{r}(s). \label{sns177}$$Then it is not difficult to obtain the estimate (\[sns17\]) using ([momv]{}) and the assumptions on $f$ and $\gamma _{r}.$
From (\[sns177\]) and (\[sns11\]), we have $$\begin{aligned}
((v-\hat{v}),\nabla )\tilde{\omega} &=&\left( \int_{t_{k}}^{t}\left[ \frac{%
\sigma ^{2}}{2}\Delta v-(v,\nabla )v-\nabla p+f(s,x)\right] ds,\nabla
\right) \tilde{\omega}(t,x) \\
&&+\left( \int_{t_{k}}^{t}\sum_{r=1}^{q}\gamma _{r}(s,x)dw_{r}(s),\nabla
\right) \\
&&\left\{ \tilde{\omega}(t_{k},x)+\int_{t_{k}}^{t}\left[ \frac{\sigma ^{2}}{2%
}\Delta \tilde{\omega}-(\hat{v},\nabla )\tilde{\omega}+g(s,x)\right]
ds+\int_{t_{k}}^{t}\sum_{r=1}^{q}\mu _{r}(s,x)dw_{r}(s)\right\}\end{aligned}$$from which it is not difficult to see that the inequality (\[sns16\]) holds. Lemma \[Lem1\] is proved.
Now we proceed to proving estimates for the one-step error of $\tilde{\omega}%
(t,x).$
\[thm:sns1\]Let Assumption 5.1 hold with $m=2$. The one-step error of $%
\tilde{\omega}(t,x),\ t_{k}\leq t\leq t_{k+1},$ which solves $(\ref{sns11})$-$(\ref{sns12}),$ has the following bounds for $t_{k}\leq t\leq t_{k+1}$ and sufficiently small $h>0:$$$\begin{aligned}
||E[\delta _{\omega }(t,\cdot )|\mathcal{F}_{t_{k}}]|| &\leq &C(t_{k},%
\mathbf{\omega })h^{2},\ \label{sns18} \\
\left( E||\delta _{\omega }(t,\cdot )||^{2}\right) ^{1/2} &\leq &Kh^{3/2},\
\label{sns19}\end{aligned}$$where $C(t_{k},\mathbf{\omega })>0$ is an $\mathcal{F}_{t_{k}}$-measurable random variable with moments of a sufficiently high order bounded by a constant independent of $h$ and $K>0$ is a constant independent of $h.$
**Proof**. Taking scalar product of (\[sns14\]) and $\delta
_{\omega }(t,x),$ using integration by parts and the property $%
\mathop{\rm
div}v(t,x)=0$, we get $$\begin{aligned}
\frac{1}{2}d||\delta _{\omega }(t,\cdot )||^{2} &=&\frac{\sigma ^{2}}{2}%
(\Delta \delta _{\omega }(t,\cdot ),\delta _{\omega }(t,\cdot ))dt-(\left[
(v(t,\cdot ),\nabla )\delta _{\omega }(t,\cdot )\right] ,\delta _{\omega
}(t,\cdot ))dt \label{sns200} \\
&&-(\left[ ((v(t,\cdot )-\hat{v}(\cdot )),\nabla )\tilde{\omega}(t,\cdot )%
\right] ,\delta _{\omega }(t,\cdot ))dt \notag \\
&=&-\frac{\sigma ^{2}}{2}||\nabla \delta _{\omega }(t,\cdot )||^{2}dt-(\left[
((v(t,\cdot )-\hat{v}(\cdot )),\nabla )\tilde{\omega}(t,\cdot )\right]
,\delta _{\omega }(t,\cdot ))dt. \notag\end{aligned}$$Then $$\frac{1}{2}dE||\delta _{\omega }(t,\cdot )||^{2}=-\frac{\sigma ^{2}}{2}%
E||\nabla \delta _{\omega }(t,\cdot )||^{2}dt-E(\left[ ((v(t,\cdot )-\hat{v}%
(\cdot )),\nabla )\tilde{\omega}(t,\cdot )\right] ,\delta _{\omega }(t,\cdot
))dt. \label{sns20}$$For the last term in (\[sns20\]), we get $$\begin{aligned}
&&|E(\left[ ((v(t,\cdot )-\hat{v}(\cdot )),\nabla )\tilde{\omega}(t,\cdot )%
\right] ,\delta _{\omega }(t,\cdot ))| \label{sns201} \\
&\leq &KE||v(t,\cdot )-\hat{v}(\cdot )||\cdot ||\tilde{\omega}(t,\cdot
)||_{3}\cdot ||\delta _{\omega }(t,\cdot )|| \notag \\
&\leq &K\left( E||v(t,\cdot )-\hat{v}(\cdot )||^{2}\cdot ||\tilde{\omega}%
(t,\cdot )||_{3}^{2}\right) ^{1/2}\left( E||\delta _{\omega }(t,\cdot
)||^{2}\right) ^{1/2} \notag \\
&\leq &K\left( E||v(t,\cdot )-\hat{v}(\cdot )||^{4}\right) ^{1/4}\left( E||%
\tilde{\omega}(t,\cdot )||_{3}^{4}\right) ^{1/4}\left( E||\delta _{\omega
}(t,\cdot )||^{2}\right) ^{1/2} \notag \\
&\leq &Kh^{1/2}\left( E||\delta _{\omega }(t,\cdot )||^{2}\right) ^{1/2},
\notag\end{aligned}$$where for the first line we used the inequality (\[A2\]) with $m_{1}=0,$ $%
m_{2}=2,$ $m_{3}=0$; we applied the Cauchy-Bunyakovski inequality twice to arrive at the pre-last line; and we used the error estimate (\[sns17\]) with $p=2$ and boundedness of the moment $E||\tilde{\omega}(t,\cdot
)||_{3}^{4}$ (see (\[sns159\])) to obtain the last line.
Thus $$\frac{1}{2}dE||\delta _{\omega }(t,\cdot )||^{2}\leq Kh^{1/2}\left(
E||\delta _{\omega }(t,\cdot )||^{2}\right) ^{1/2}dt,$$and since $\delta _{\omega }(t_{k},x)=0,$ we arrive at $$\int_{t_{k}}^{t}\frac{1}{2}\frac{dE||\delta _{\omega }(s,\cdot )||^{2}}{%
\left( E||\delta _{\omega }(s,\cdot )||^{2}\right) ^{1/2}}=\left( E||\delta
_{\omega }(t,\cdot )||^{2}\right) ^{1/2}\leq Kh^{3/2}$$confirming (\[sns19\]).
Now we are to prove (\[sns18\]). Using (\[sns14\]), we write the equation for $dE[\delta _{\omega }(t,x)|\mathcal{F}_{t_{k}}]$ and, after taking scalar product of the components of this equation and $E[\delta
_{\omega }(t,x)|\mathcal{F}_{t_{k}}]$ and doing integration by parts, we arrive $$\begin{aligned}
&&\frac{1}{2}d||E[\delta _{\omega }(t,\cdot )|\mathcal{F}_{t_{k}}]||^{2}
\label{sns21} \\
&=&-\frac{\sigma ^{2}}{2}||\nabla E[\delta _{\omega }(t,\cdot )|\mathcal{F}%
_{t_{k}}]||^{2}dt-(E\left[ (v(t,\cdot ),\nabla )\delta _{\omega }(t,\cdot )|%
\mathcal{F}_{t_{k}}\right] ,E[\delta _{\omega }(t,\cdot )|\mathcal{F}%
_{t_{k}}])dt\ \notag \\
&&-(E\left[ ((v(t,\cdot )-\hat{v}(\cdot )),\nabla )\tilde{\omega}(t,\cdot )|%
\mathcal{F}_{t_{k}}\right] ,E[\delta _{\omega }(t,\cdot )|\mathcal{F}%
_{t_{k}}])dt. \notag\end{aligned}$$By (\[sns16\]), we get for the third term in (\[sns21\]): $$\begin{aligned}
&&|(E\left[ ((v(t,\cdot )-\hat{v}(\cdot )),\nabla )\tilde{\omega}(t,\cdot )|%
\mathcal{F}_{t_{k}}\right] ,E[\delta _{\omega }(t,\cdot )|\mathcal{F}%
_{t_{k}}])| \label{sns22} \\
&\leq &||E\left[ ((v(t,\cdot )-\hat{v}(\cdot )),\nabla )\tilde{\omega}%
(t,\cdot )|\mathcal{F}_{t_{k}}\right] ||\cdot ||E[\delta _{\omega }(t,\cdot
)|\mathcal{F}_{t_{k}}]|| \notag \\
&\leq &C(t_{k},\mathbf{\omega })h||E[\delta _{\omega }(t,\cdot )|\mathcal{F}%
_{t_{k}}]||, \notag\end{aligned}$$where $C(t_{k},\mathbf{\omega })>0$ is an $\mathcal{F}_{t_{k}}$-measurable random variable which has moments up to a sufficiently high order and does not depend on $h.$
By simple maniplations and using (\[A2\]) $m_{1}=2,$ $m_{2}=0,$ $m_{3}=0$ as well as (\[A32\]), the Cauchy-Bunyakovski inequality, (\[sns100\]), a conditional version of (\[sns19\]), and (\[sns159\]), we obtain for the second term in (\[sns21\]):$$\begin{aligned}
&&|(E\left[ (v(t,\cdot ),\nabla )\delta _{\omega }(t,\cdot )|\mathcal{F}%
_{t_{k}}\right] ,E[\delta _{\omega }(t,\cdot )|\mathcal{F}_{t_{k}}])|
\label{sns23} \\
&=&|E\left[ ((v(t,\cdot ),\nabla )\delta _{\omega }(t,\cdot ),E[\delta
_{\omega }(t,\cdot )|\mathcal{F}_{t_{k}}])|\mathcal{F}_{t_{k}}\right] |
\notag \\
&\leq &E\left[ |((v(t,\cdot ),\nabla )\delta _{\omega }(t,\cdot ),E[\delta
_{\omega }(t,\cdot )|\mathcal{F}_{t_{k}}])|\ |\mathcal{F}_{t_{k}}\right]
\notag \\
&=&E\left[ |((v(t,\cdot ),\nabla )E[\delta _{\omega }(t,\cdot )|\mathcal{F}%
_{t_{k}}],\delta _{\omega }(t,\cdot ))|\ |\mathcal{F}_{t_{k}}\right] \notag
\\
&\leq &KE\left[ ||v(t,\cdot )||_{2}\cdot ||E[\delta _{\omega }(t,\cdot )|%
\mathcal{F}_{t_{k}}]||_{1}\cdot ||[\delta _{\omega }(t,\cdot )||\ |\mathcal{F%
}_{t_{k}}\right] \notag \\
&=&K||E[\delta _{\omega }(t,\cdot )|\mathcal{F}_{t_{k}}]||_{1}E\left[
||v(t,\cdot )||_{2}\cdot ||[\delta _{\omega }(t,\cdot )||\ |\mathcal{F}%
_{t_{k}}\right] \notag \\
&\leq &K||\nabla E[\delta _{\omega }(t,\cdot )|\mathcal{F}_{t_{k}}]||E\left[
||\omega (t,\cdot )||_{1}\cdot ||[\delta _{\omega }(t,\cdot )||\ |\mathcal{F}%
_{t_{k}}\right] \notag \\
&\leq &K||\nabla E[\delta _{\omega }(t,\cdot )|\mathcal{F}_{t_{k}}]||(E\left[
||\omega (t,\cdot )||_{1}^{2}|\mathcal{F}_{t_{k}}\right] )^{1/2}(E\left[
||\delta _{\omega }(t,\cdot )||^{2}|\mathcal{F}_{t_{k}}\right] )^{1/2}
\notag \\
&\leq &C(t_{k},\mathbf{\omega })h^{3/2}||\nabla E[\delta _{\omega }(t,\cdot
)|\mathcal{F}_{t_{k}}]||, \notag\end{aligned}$$where $C(t_{k},\mathbf{\omega })>0$ is an $\mathcal{F}_{t_{k}}$-measurable random variable which has moments up to a sufficiently high order and does not depend on $h.$
Combining (\[sns21\])-(\[sns23\]), we arrive at $$\begin{aligned}
\frac{1}{2}d||E[\delta _{\omega }(t,\cdot )|\mathcal{F}_{t_{k}}]||^{2} &\leq
&-\frac{\sigma ^{2}}{2}||\nabla E[\delta _{\omega }(t,\cdot )|\mathcal{F}%
_{t_{k}}]||^{2}dt+ \\
&&+C(t_{k},\mathbf{\omega })h^{3/2}||\nabla E[\delta _{\omega }(t,\cdot )|%
\mathcal{F}_{t_{k}}]||dt+C(t_{k},\mathbf{\omega })h||E[\delta _{\omega
}(t,\cdot )|\mathcal{F}_{t_{k}}]||dt \\
&=&-\frac{1}{2}\left( \sigma ||\nabla E[\delta _{\omega }(t,\cdot )|\mathcal{%
F}_{t_{k}}]||-\frac{C(t_{k},\mathbf{\omega })}{\sigma }h^{3/2}\right) ^{2}dt+%
\frac{C^{2}(t_{k},\mathbf{\omega })}{2\sigma ^{2}}h^{3}dt \\
&&+C(t_{k},\mathbf{\omega })h||E[\delta _{\omega }(t,\cdot )|\mathcal{F}%
_{t_{k}}]||dt \\
&\leq &\frac{C^{2}(t_{k},\mathbf{\omega })}{2\sigma ^{2}}h^{3}dt+C(t_{k},%
\mathbf{\omega })h||E[\delta _{\omega }(t,\cdot )|\mathcal{F}_{t_{k}}]||dt.\end{aligned}$$Then, for some $\mathcal{F}_{t_{k}}$-measurable independent of $h$ random variable $C(t_{k},\mathbf{\omega })>0,$ we have $$d||E[\delta _{\omega }(t,\cdot )|\mathcal{F}_{t_{k}}]||^{2}\leq C(t_{k},%
\mathbf{\omega })h^{3}dt+\frac{1}{h}||E[\delta _{\omega }(t,\cdot )|\mathcal{%
F}_{t_{k}}]||^{2}dt,$$from which (\[sns18\]) follows taking into account that $||E[\delta
_{\omega }(t_{k},\cdot )|\mathcal{F}_{t_{k}}]||=0.$ Theorem \[thm:sns1\] is proved.
As in the deterministic case, we define $$\tilde{v}(t,x):=U\tilde{\omega}(t,x),\ \ t_{k}\leq t\leq t_{k+1},
\label{v_tilde}$$where the operator $U$ is from (\[Mnew1\]).
Using the idea of the proof of Corollary \[Cor2D\], it is not difficult to prove the following corollary to Theorem \[thm:sns1\].
\[Corsns\]The one-step error $\delta _{v}(t,x):=v(t,x)-\tilde{v}%
(t,x)=U\delta _{\omega }(t,x)$ of $v(t,x),\ t_{k}\leq t\leq t_{k+1},$ has the following bounds for $t_{k}\leq t\leq t_{k+1}:$$$\begin{aligned}
||E\delta _{v}(t,\cdot )|| &\leq &Kh^{2},\ \label{sns25} \\
\left( E||\delta _{v}(t,\cdot )||^{2}\right) ^{1/2} &\leq &Kh^{3/2},\
\label{sns26}\end{aligned}$$where $K>0$ is independent of $h.$
The method\[sec:sns3\]
----------------------
Analogously, how it was done in the deterministic case (see Section [sec:approxcon]{}), we can construct the global approximation for the stochastic NSE (\[sns7\])-(\[sns9\]) based on the one-step approximation (\[sns11\])-(\[sns12\]). On the first step of the method we set $$\tilde{\omega}(t_{0},x)=\mathop{\rm curl}v(t_{0},x)=\phi (x)=\mathop{\rm
curl}\varphi (x)$$and $$\hat{v}(x)=\hat{v}(t,x)=u(t_{0},x)=\varphi (x),\ \ 0=t_{0}\leq t\leq t_{1}.$$Then we solve the linear SPDE (\[sns11\])-(\[sns12\]) on* *$%
[t_{0},t_{1}]$ to obtain $\tilde{\omega}(t,x)$ and to construct $$\hat{v}(t_{1},x)=U\tilde{\omega}(t_{1},x).$$On the second step we solve (\[sns11\])-(\[sns12\]) on* *$%
[t_{1},t_{2}]$ having $\tilde{\omega}(t_{1},x)$ and setting $\hat{v}(t,x)=%
\hat{v}(x)=\hat{v}(t_{1},x)$ for $t_{1}<t\leq t_{2}.$ As a result, we obtain $\tilde{\omega}(t,x)$ on $[t_{1},t_{2}]$ and $\hat{v}(t_{2},x)=U\tilde{\omega%
}(t_{2},x),$ and so on. Proceeding in this way, we obtain on the $N$-th step the approximation $\tilde{\omega}(t,x)$ on $[t_{N-1},t_{N}]$ for $\omega
(t,x)$ having $\tilde{\omega}(t_{N-1},x)$ and $\hat{v}(x)=\hat{v}%
(t_{N-1},x)=U\tilde{\omega}(t_{N-1},x)$ and setting $\hat{v}(t,x)=\hat{v}(x)=%
\hat{v}(t_{N-1},x)$ for $t_{N-1}<t\leq t_{N}.$ Finally, $\hat{v}(T,x)=U%
\tilde{\omega}(T,x).$
In order to realise the approximation process described above, it is sufficient that on every time interval $[t_{k},t_{k+1}],$ $k=0,\ldots ,N-1,$ there exists a solution of the linear SPDE (\[sns11\])-(\[sns12\]), we denote such a solution $\tilde{\omega}_{k}(t,x)$ which satisfies the condition $$\tilde{\omega}_{k}(t_{k},x)=\left\{
\begin{array}{c}
\mathop{\rm curl}\varphi (x),\ k=0, \\
\tilde{\omega}_{k-1}(t_{k},x),\ k=1,\ldots ,N,%
\end{array}%
\right. \label{omega_k}$$and has the time-independent $\hat{u}(x)$ within each interval $%
(t_{k},t_{k+1}]$ $\ $defined as$$\hat{v}(x):=\hat{v}_{k}(x)=U\tilde{\omega}_{k}(t_{k},x),\ t_{k}<t\leq
t_{k+1}. \label{v_k}$$Clearly, $\hat{v}(x)$ used in (\[sns11\]) are different on the time intervals $(t_{k},t_{k+1}]$.
Before considering global errors of the approximation in Section [sec:sns4]{}, we now prove boundedness of the approximation’s moments.
\[thm:sns2\]Let Assumption 5.1 hold with $m=0$. The moments of the global approximation $\tilde{\omega}_{k}(t_{k},x)$ and $\hat{v}_{k}(x)$ are uniformly bounded in $h$ and $k:$ $$\begin{aligned}
E||\tilde{\omega}_{k}(t_{k+1},\cdot )||^{2p} &\leq &||\phi (\cdot
)||^{2p}+K\int_{0}^{t_{k+1}}\left( ||g(s,\cdot )||^{2p}+\sum_{r=1}^{q}||\mu
_{r}(s,\cdot )||^{2p}\right) ds, \label{sns27} \\
E||\hat{v}_{k}(\cdot )||^{2p} &\leq &KE||\tilde{\omega}_{k}(t_{k},\cdot
)||^{2p}, \label{sns28}\end{aligned}$$where $K>0$ is independent of $h$ and $t_{k}$ but depends on $p.$
**Proof**. For every sufficiently large integer $n$, define the stopping time$$\tau _{n}=\inf \{0<t\leq T:||\tilde{\omega}(t,\cdot )||^{2}\geq n\}.$$
Using the Ito formula, doing integration by parts and taking into account that $\hat{v}_{k}(x)$ is divergence free, we obtain$$\begin{gathered}
d||\tilde{\omega}_{k}(t,\cdot )||^{2p}=2p||\tilde{\omega}_{k}(t,\cdot
)||^{2(p-1)} \label{sns292} \\
\cdot \left[ -\frac{\sigma ^{2}}{2}||\nabla \tilde{\omega}_{k}(t,\cdot
)||^{2}+(g(t,\cdot ),\tilde{\omega}_{k}(t,\cdot ))+\frac{2p-1}{2}%
\sum_{r=1}^{q}||\mu _{r}(t,\cdot )||^{2}\right] dt \notag \\
+2p||\tilde{\omega}_{k}(t,\cdot )||^{2(p-1)}\sum_{r=1}^{q}(\mu _{r}(t,\cdot
),\tilde{\omega}_{k}(t,\cdot ))dw_{r}(t),\ t_{k}\wedge \tau _{n}\leq t\leq
t_{k+1}\wedge \tau _{n}, \notag \\
||\tilde{\omega}_{k}(t_{k},\cdot )||^{2p}=E||\tilde{\omega}%
_{k-1}(t_{k},\cdot )||^{2p}. \notag\end{gathered}$$We have$$\begin{gathered}
dE||\tilde{\omega}_{k}(t,\cdot )||^{2p} \\
=2p\left[ -\frac{\sigma ^{2}}{2}E\left( ||\tilde{\omega}_{k}(t,\cdot
)||^{2(p-1)}||\nabla \tilde{\omega}_{k}(t,\cdot )||^{2}\right) +E||\tilde{%
\omega}_{k}(t,\cdot )||^{2(p-1)}(g(t,\cdot ),\tilde{\omega}_{k}(t,\cdot
))\right. \\
\left. +\frac{2p-1}{2}E||\tilde{\omega}_{k}(t,\cdot
)||^{2(p-1)}\sum_{r=1}^{q}||\mu _{r}(t,\cdot )||^{2}\right] dt,\ t_{k}\wedge
\tau _{n}\leq t\leq t_{k+1}\wedge \tau _{n}, \\
E||\tilde{\omega}_{k}(t_{k},\cdot )||^{2}=E||\tilde{\omega}%
_{k-1}(t_{k},\cdot )||^{2}.\end{gathered}$$By Poincare’s inequality (\[Poin\]) and doing simple re-arrangements, we arrive at$$\begin{gathered}
dE||\tilde{\omega}_{k}(t,\cdot )||^{2p} \\
\leq 2p\left[ -\alpha \frac{\sigma ^{2}}{2}E||\tilde{\omega}_{k}(t,\cdot
)||^{2p}+\alpha \frac{\sigma ^{2}}{4}E||\tilde{\omega}_{k}(t,\cdot )||^{2p}+%
\frac{1}{\alpha \sigma ^{2}}||g(t,\cdot )||^{2}\right. \\
\left. +\alpha \frac{\sigma ^{2}}{4}E||\tilde{\omega}_{k}(t,\cdot )||^{2p}+%
\frac{2\left[ (2p-1)(p-1)\right] ^{p}}{\left( \alpha \sigma ^{2}\right)
^{p-1}(p-1)}\left[ \sum_{r=1}^{q}||\mu _{r}(t,\cdot )||^{2}\right] ^{p}%
\right] dt.\end{gathered}$$We note that the constant $\alpha >0$ in the above expression is due to Poincare’s inequality (\[Poin\]) and it is, of course, independent of $h$ and $k.$ Hence $$dE||\tilde{\omega}_{k}(t,\cdot )||^{2p}\leq K\left[ ||g(t,\cdot )||^{2}+%
\left[ \sum_{r=1}^{q}||\mu _{r}(t,\cdot )||^{2}\right] ^{p}\right] dt,
\notag$$where the constant $K>0$ depends on $p$ but independent of $h$ and $k.$ The previous inequality implies$$\begin{aligned}
E||\tilde{\omega}_{k}(t_{k+1}\wedge \tau _{n},\cdot )||^{2p} &\leq &E||%
\tilde{\omega}_{k-1}(t_{k},\cdot )||^{2p} \\
&&+KE\int_{t_{k}\wedge \tau _{n}}^{t_{k+1}\wedge \tau _{n}}\left(
||g(s,\cdot )||^{2}+\left[ \sum_{r=1}^{q}||\mu _{r}(s,\cdot )||^{2}\right]
^{p}\right) ds \\
&\leq &E||\phi (\cdot )||^{2p}+E\int_{0}^{t_{k+1}\wedge \tau _{n}}\left(
K||g(s,\cdot )||^{2}+\sum_{r=1}^{q}||\mu _{r}(s,\cdot )||^{2}\right) ds,\end{aligned}$$and letting $n\rightarrow \infty $ we arrive at (\[sns27\]). The estimate (\[sns28\]) is evident (see e.g. (\[Mnew1\])). Theorem \[thm:sns2\] is proved.
It is note difficult to see that repeating the proof of Lemma \[LemMat\] word by word, we immediately get that the exponential moment for $||\tilde{%
\omega}_{k}(t_{k+1},\cdot )||^{2}$ is bounded, more precisely the estimate of the form (\[sns30x\]) holds for $||\tilde{\omega}_{k}(t_{k+1},\cdot
)||^{2}$ under Assumption 5.1 with $m=0.$
Now we consider uniform bounds for moments of higher Sobolev norms of $%
\tilde{\omega}_{k}.$
\[thm:sns2nn\]Let Assumption 5.1 hold with $m>0.$ Then $$E||\tilde{\omega}_{k}(t_{k+1},\cdot )||_{m}^{2p}\leq ||\phi (\cdot
)||_{m}^{2p}+Kt_{k+1}, \label{snd1}$$where $K>0$ is independent of $h$ and $t_{k}$.
**Proof**. The proof is by induction. To this end, we assume that moments $E||\tilde{\omega}_{k}(t,\cdot )||_{m-1}^{2p}$ are bounded (uniformly in $k$ and $h)$ and for sufficiently large $p\geq 1$ (note that Theorem \[thm:sns2\] guarantees their boundedness for $m=0).$
We will be adapting recipes from [@T Section 4.1]. Let the operator $%
\Lambda $ be such that $\Lambda ^{2}=-\Delta .$ We have for an integer $%
m\geq 1$ (cf. [@T p. 29] and also [@MatPhD Section 3.4]): $$\begin{gathered}
d||\tilde{\omega}_{k}(t,\cdot )||_{m}^{2p}=2p||\tilde{\omega}_{k}(t,\cdot
)||_{m}^{2(p-1)}\left[ -\frac{\sigma ^{2}}{2}||\tilde{\omega}_{k}(t,\cdot
)||_{m+1}^{2}+(\tilde{\omega}_{k}(t,\cdot ),g(t,\cdot ))_{m}\right.
\label{qqq} \\
\left. -((\hat{v}_{k}(\cdot ),\nabla )\tilde{\omega}_{k}(t,\cdot ),\Lambda
^{2m}\tilde{\omega}_{k}(t,\cdot ))+\frac{2p-1}{2}\sum_{r=1}^{q}||\mu
_{r}(t,\cdot )||_{m}^{2}\right] dt \notag \\
+2p||\tilde{\omega}_{k}(t,\cdot )||_{m}^{2(p-1)}\sum_{r=1}^{q}(\mu
_{r}(t,\cdot ),\tilde{\omega}_{k}(t,\cdot ))_{m}dw_{r}(t), \notag \\
\ t_{k}\wedge \tau _{n}\leq t\leq t_{k+1}\wedge \tau _{n}, \notag \\
||\tilde{\omega}_{k}(t_{k},\cdot )||_{m}^{2p}=||\tilde{\omega}%
_{k-1}(t_{k},\cdot )||_{m}^{2p}. \notag\end{gathered}$$Here $\tau _{n}$ is as in Theorem \[thm:sns2\].
Let us analyze terms in the right-hand side of (\[qqq\]). We have (e.g. see [@T Eq. (4.4)]):$$\begin{aligned}
|(\tilde{\omega}_{k}(t,\cdot ),g(t,\cdot ))_{m}| &\leq &||g(t,\cdot
)||_{m-1}||\tilde{\omega}_{k}(t,\cdot )||_{m+1} \\
&\leq &\frac{4}{\sigma ^{2}}||g(t,\cdot )||_{m-1}^{2}+\frac{\sigma ^{2}}{16}%
||\tilde{\omega}_{k}(t,\cdot )||_{m+1}^{2}\end{aligned}$$and $$K||\tilde{\omega}_{k}(t,\cdot )||_{m}^{2(p-1)}||g(t,\cdot )||_{m-1}^{2}\leq
\frac{K^{p}}{p}\left( \frac{16p}{\alpha \sigma ^{2}(p-1)}\right)
^{p-1}||g(t,\cdot )||_{m-1}^{2p}+\alpha \frac{\sigma ^{2}}{16}||\tilde{\omega%
}_{k}(t,\cdot )||_{m}^{2p},$$where as before the constant $\alpha >0$ is due to Poincare’s inequality (\[Poin\]). Also, for some $K>0$ dependent on $p:$$$p(2p-1)||\tilde{\omega}_{k}(t,\cdot )||_{m}^{2(p-1)}\sum_{r=1}^{q}||\mu
_{r}(t,\cdot )||_{m}^{2}\leq K\left( \sum_{r=1}^{q}||\mu _{r}(t,\cdot
)||_{m}^{2}\right) ^{p}+\alpha \frac{\sigma ^{2}}{8}||\tilde{\omega}%
_{k}(t,\cdot )||_{m}^{2p}.$$Hence, we can write for some $K>0:$$$\begin{gathered}
d||\tilde{\omega}_{k}(t,\cdot )||_{m}^{2p}\leq \left[ -\frac{\sigma ^{2}}{4}%
p\left\{ ||\tilde{\omega}_{k}(t,\cdot )||_{m}^{2(p-1)}||\tilde{\omega}%
_{k}(t,\cdot )||_{m+1}^{2}+\alpha ||\tilde{\omega}_{k}(t,\cdot
)||_{m}^{2p}\right\} \right. \\
+K||g(t,\cdot )||_{m-1}^{2p}-2p\left\{ ||\tilde{\omega}_{k}(t,\cdot
)||_{m}^{2(p-1)}((\hat{v}_{k}(\cdot ),\nabla )\tilde{\omega}_{k}(t,\cdot
),\Lambda ^{2m}\tilde{\omega}_{k}(t,\cdot ))\right\} \\
\left. +K\left( \sum_{r=1}^{q}||\mu _{r}(t,\cdot )||_{m}^{2}\right) ^{p}
\right] dt \\
+2p||\tilde{\omega}_{k}(t,\cdot )||_{m}^{2(p-1)}\sum_{r=1}^{q}(\mu
_{r}(t,\cdot ),\tilde{\omega}_{k}(t,\cdot ))_{m}dw_{r}(t) \\
\ t_{k}\wedge \tau _{n}\leq t\leq t_{k+1}\wedge \tau _{n},\ \ ||\tilde{\omega%
}_{k}(t_{k},\cdot )||_{m}^{2p}=||\tilde{\omega}_{k-1}(t_{k},\cdot
)||_{m}^{2p}.\end{gathered}$$Let us now estimate the trilinear-form: $$\begin{gathered}
|((\hat{v}_{k}(\cdot ),\nabla )\tilde{\omega}_{k}(t,\cdot ),\Lambda ^{2m}%
\tilde{\omega}_{k}(t,\cdot ))|\leq K\sum_{l=1}^{m}||\hat{v}_{k}(\cdot
)||_{l}||\tilde{\omega}_{k}(t,\cdot )||_{m-l+3/2}||\tilde{\omega}%
_{k}(t,\cdot )||_{m+1} \\
\leq K||\tilde{\omega}_{k}(t_{k},\cdot )||_{m-1}||\tilde{\omega}_{k}(t,\cdot
)||_{m+1/2}||\tilde{\omega}_{k}(t,\cdot )||_{m+1} \\
\leq K||\tilde{\omega}_{k}(t,\cdot )||_{m-1}^{5/4}||\tilde{\omega}%
_{k}(t,\cdot )||_{m+1}^{7/4} \\
\leq \frac{K^{8}}{8}\left( \frac{56}{\sigma ^{2}}\right) ^{7}||\tilde{\omega}%
_{k}(t_{k},\cdot )||_{m-1}^{10}+\frac{\sigma ^{2}}{8}||\tilde{\omega}%
_{k}(t,\cdot )||_{m+1}^{2},\end{gathered}$$where for the first line we used the recipe from [@T pp. 29-30] and the inequality (\[A2\]); for the second line we used (\[A3\]) and that $%
||u(\cdot )||_{m_{1}}\leq ||u(\cdot )||_{m_{2}}$ for $m_{2}\geq m_{1}$; the third line is obtained using (\[A4i\]); and the fourth line follows from Young’s inequality. Note that the constants $K>0$ in the first and second lines are different.
Further, $$\begin{aligned}
&&K||\tilde{\omega}_{k}(t,\cdot )||_{m-1}^{10}||\tilde{\omega}_{k}(t,\cdot
)||_{m}^{2(p-1)} \\
&\leq &\frac{K^{p}}{p}\left( \frac{8p}{\alpha \sigma ^{2}(p-1)}\right)
^{p-1}||\tilde{\omega}_{k}(t_{k},\cdot )||_{m-1}^{10p}+\alpha \frac{\sigma
^{2}}{8}||\tilde{\omega}_{k}(t,\cdot )||_{m}^{2p}.\end{aligned}$$Thus, for some $K>0$$$\begin{gathered}
dE||\tilde{\omega}_{k}(t,\cdot )||_{m}^{2p}\leq \\
+K\left[ ||g(t,\cdot )||_{m-1}^{2p}+E||\tilde{\omega}_{k}(t_{k},\cdot
)||_{m-1}^{10p}+\left( \sum_{r=1}^{q}||\mu _{r}(t,\cdot )||_{m}^{2}\right)
^{p}\right] dt, \\
\ t_{k}\wedge \tau _{n}\leq t\leq t_{k+1}\wedge \tau _{n},\ \ E||\tilde{%
\omega}_{k}(t_{k},\cdot )||_{m}^{2p}=E||\tilde{\omega}_{k-1}(t_{k},\cdot
)||_{m}^{2p}.\end{gathered}$$By the Cauchy-Bunyakovskii inequality and the induction assumption at the start of the proof, we get$$E||\tilde{\omega}_{k}(t_{k},\cdot )||_{m-1}^{10p}\leq K$$with a constant $K>0$ independent of $h$ and $k.$ Hence $$\begin{gathered}
dE||\tilde{\omega}_{k}(t,\cdot )||_{m}^{2p}\leq Kdt, \\
\ t_{k}\wedge \tau _{n}\leq t\leq t_{k+1}\wedge \tau _{n},\ \ E||\tilde{%
\omega}_{k}(t_{k},\cdot )||_{m}^{2p}=E||\tilde{\omega}_{k-1}(t_{k},\cdot
)||_{m}^{2p}\end{gathered}$$and $$E||\tilde{\omega}_{k}(t_{k+1}\wedge \tau _{n},\cdot )||_{m}^{2p}\leq E||%
\tilde{\omega}_{k-1}(t_{k},\cdot )||_{m}^{2p}+Kh,$$from which (\[snd1\]) follows by the standard arguments. Theorem [thm:sns2nn]{} is proved.
Mean-square convergence theorem\[sec:sns4\]
-------------------------------------------
To prove the global convergence of $\tilde{\omega}_{k}(t_{k},\cdot ),$ we use the idea of the proof of the fundamental theorem of mean-square convergence for SDEs [@Mil87] (see also [@MT1 Section 1.1]).
\[thm:sns3\]Let Assumption 5.1 hold with $m=2.$ The global approximation $\tilde{\omega}_{k}(t_{k+1},x)$ for the problem $(\ref{sns7})$-$(\ref{sns9})$ has the first mean-square order accuracy.
**Proof**. We note that in the proof we shall again use letters $K$ and $C(\cdot ,\mathbf{\omega })$ to denote various deterministic constants and random variables, respectively, which are independent of $h$ and $k,$ and $K$ is also independent of $h$ and $k$; their values may change from line to line.
We have $$\begin{aligned}
R(t_{k+1},x) &:&=\omega (t_{k+1},x;0,\phi )-\tilde{\omega}%
_{k}(t_{k+1},x;0,\phi ) \label{Ba27} \\
&=&\omega (t_{k+1},x;t_{k},\omega (t_{k},\cdot ))-\tilde{\omega}%
_{k}(t_{k+1},x;t_{k},\tilde{\omega}_{k}(t_{k},\cdot )) \notag \\
&=&(\omega (t_{k+1},x;t_{k},\omega (t_{k},\cdot ))-\omega (t_{k+1},x;t_{k},%
\tilde{\omega}_{k}(t_{k},\cdot ))) \notag \\
&&+(\omega (t_{k+1},x;t_{k},\tilde{\omega}_{k}(t_{k},\cdot ))-\tilde{\omega}%
_{k}(t_{k+1},x;t_{k},\tilde{\omega}_{k}(t_{k},\cdot )))\,, \notag\end{aligned}$$where $\cdot $ reflects function dependence of solutions on the initial conditions. The first difference in the right-hand side of (\[Ba27\]) is the error of the solution arising due to the error in the initial data at time $t_{k},$ accumulated at the $k$-th step. The second difference is the one-step error at the $(k+1)$-step: $$\delta _{\omega }(t_{k+1},x):=\omega (t_{k+1},x;t_{k},\tilde{\omega}%
_{k}(t_{k},\cdot ))-\tilde{\omega}_{k}(t_{k+1},x;t_{k},\tilde{\omega}%
_{k}(t_{k},\cdot )) \label{Ba30}$$for which estimates are given in Theorem \[thm:sns1\] taking into account that Theorems \[thm:sns2\] and \[thm:sns2nn\] guarantees boundedness of moments of $||\tilde{\omega}_{k}(t_{k},\cdot )||_{3}$ under the conditions of this theorem. Taking the $L^{2}$-norm of both sides of (\[Ba27\]), we obtain $$\begin{gathered}
||R(t_{k+1},\cdot )||^{2}=||\omega (t_{k+1},\cdot ;t_{k},\omega (t_{k},\cdot
))-\omega (t_{k+1},\cdot ;t_{k},\tilde{\omega}_{k}(t_{k},\cdot ))||^{2}\
\label{Ba28} \\
+||\delta _{\omega }(t_{k+1},\cdot )||^{2}+2(\omega (t_{k+1},\cdot
;t_{k},\omega (t_{k},\cdot ))-\omega (t_{k+1},\cdot ;t_{k},\tilde{\omega}%
_{k}(t_{k},\cdot )),\delta _{\omega }(t_{k+1},\cdot )), \notag\end{gathered}$$where the first $\cdot $ in each $\omega $ or $\tilde{\omega}_{k}$ reflects that we took $L^{2}$-norm.
Using (\[sns19\]) from Theorem \[thm:sns1\] together with Theorems [thm:sns2]{} and \[thm:sns2nn\], we obtain for the second term in (\[Ba28\]): $$||\delta _{\omega }(t_{k+1},\cdot )||^{2}\leq C(t_{k+1},\mathbf{\omega }%
)h^{3}, \label{Ba288}$$where $C(t_{k+1},\mathbf{\omega })>0$ is an $\mathcal{F}_{t_{k+1}}$-measurable with bounded second moment.
By (\[sns31\]) from Lemma \[Lem2\] together with Theorems \[thm:sns2\] and \[thm:sns2nn\], we get for the first term in (\[Ba28\]):$$\begin{aligned}
&&||\omega (t_{k+1},x;t_{k},\omega (t_{k},\cdot ))-\omega (t_{k+1},x;t_{k},%
\tilde{\omega}_{k}(t_{k},\cdot ))||^{2} \label{Ba29} \\
&\leq &||R(t_{k},\cdot )||^{2}\exp \left( Kh+c\int_{t_{k}}^{t_{k+1}}||\nabla
\omega (s^{\prime },\cdot ;t_{k},\omega (t_{k},\cdot ))||^{2}ds^{\prime
}\right) . \notag\end{aligned}$$The difference $\omega (t_{k+1},x;t_{k},\omega (t_{k},\cdot ))-\omega
(t_{k+1},x;t_{k},\tilde{\omega}_{k}(t_{k},\cdot ))$ in the last summand in (\[Ba28\]) can be treated using (\[sns30\]) from Lemma \[Lem2\]: $$\omega (t_{k+1},x;t_{k},\omega (t_{k},\cdot ))-\omega (t_{k+1},x;t_{k},%
\tilde{\omega}_{k}(t_{k},\cdot ))=R(t_{k},x)+\eta (t_{k},x)\,.$$Using a conditional version of (\[sns19\]) from Theorem \[thm:sns1\] and (\[sns32\]) from Lemma \[Lem2\] together with Theorems \[thm:sns2\] and \[thm:sns2nn\], we get $$\begin{aligned}
|((\eta (t_{k},\cdot ),\delta _{\omega }(t_{k+1},\cdot ))| &\leq &||\eta
(t_{k},\cdot )||||\delta _{\omega }(t_{k+1},\cdot )||\leq ||\eta
(t_{k},\cdot )||^{2}+\frac{1}{4}||\delta _{\omega }(t_{k+1},\cdot )||^{2}\ \
\ \ \ \label{Ba32} \\
&\leq &h||R(t_{k},\cdot )||^{2}+C(t_{k+1},\mathbf{\omega })h^{3}, \notag\end{aligned}$$where $C(t_{k+1},\mathbf{\omega })>0$ is an $\mathcal{F}_{t_{k+1}}$-measurable with bounded second moment.
Combining the above, we arrive at $$\begin{aligned}
||R(t_{k+1},\cdot )||^{2} &\leq &|R(t_{k},\cdot )||^{2}\exp \left(
Kh+c\int_{t_{k}}^{t_{k+1}}||\nabla \omega (s^{\prime },\cdot ;t_{k},\omega
(t_{k},\cdot ))||^{2}ds^{\prime }\right) \label{bax1} \\
&&+h||R(t_{k},\cdot )||^{2}+(R(t_{k},\cdot ),\delta _{\omega }(t_{k+1},\cdot
))+C(t_{k+1},\mathbf{\omega })h^{3}. \notag\end{aligned}$$
Since $||R(0,\cdot )||=0,$ summing (\[bax1\]) from $k=0$ to $n$, we get $$\begin{aligned}
&&||R(t_{n+1},\cdot )||^{2} \\
&\leq &\sum_{k=1}^{n}||R(t_{k},\cdot )||^{2}\left[ \exp \left(
Kh+c\int_{t_{k}}^{t_{k+1}}||\nabla \omega (s^{\prime },\cdot ;t_{k},\omega
(t_{k},\cdot ))||^{2}ds^{\prime }\right) -1+h\right] \\
&&+h^{3}\sum_{k=0}^{n}C(t_{k+1},\mathbf{\omega })+\sum_{k=1}^{n}(R(t_{k},%
\cdot ),\delta _{\omega }(t_{k+1},\cdot )) \\
&\leq &\sum_{k=1}^{n}||R(t_{k},\cdot )||^{2}\left[ \exp \left(
Kh+c\int_{t_{k}}^{t_{k+1}}||\nabla \omega (s^{\prime },\cdot ;t_{k},\omega
(t_{k},\cdot ))||^{2}ds^{\prime }\right) -1\right] \\
&&+h^{3}\sum_{k=0}^{n}C(t_{k+1},\mathbf{\omega })+\sum_{k=1}^{n}(R(t_{k},%
\cdot ),\delta _{\omega }(t_{k+1},\cdot )).\end{aligned}$$From which, by a version of Gronwall’s lemma (see, e.g. [@Gronw; @Kruse18]), we obtain $$\begin{aligned}
||R(t_{n+1},\cdot )||^{2} &\leq &F_{n}+\sum_{k=1}^{n}F_{k-1} \label{bax3} \\
&&\cdot \left[ \exp \left( Kh+c\int_{t_{k}}^{t_{k+1}}||\nabla \omega
(s^{\prime },\cdot ;t_{k},\omega (t_{k},\cdot ))||^{2}ds^{\prime }\right) -1%
\right] \notag \\
&&\cdot \mathop{\displaystyle \prod }\limits_{j=k+1}^{n}\exp \left(
Kh+c\int_{t_{j}}^{t_{j+1}}||\nabla \omega (s^{\prime },\cdot ;t_{k},\omega
(t_{k},\cdot ))||^{2}ds^{\prime }\right) , \notag\end{aligned}$$where $$F_{k}:=h^{3}\sum_{j=0}^{k}C(t_{j+1},\mathbf{\omega })+%
\sum_{j=1}^{k}(R(t_{j},\cdot ),\delta _{\omega }(t_{j+1},\cdot )).$$We have $$\begin{aligned}
&&||R(t_{n+1},\cdot )||^{2}\leq F_{n} \label{bax31} \\
&&+\sum_{k=1}^{n}F_{k-1}\cdot \left[ \exp \left(
Kh+c\int_{t_{k}}^{t_{k+1}}||\nabla \omega (s^{\prime },\cdot ;t_{k},\omega
(t_{k},\cdot ))||^{2}ds^{\prime }\right) -1\right] \notag \\
&&\cdot \exp \left( K(t_{n+1}-t_{k+1})+c\int_{t_{k+1}}^{t_{n+1}}||\nabla
\omega (s^{\prime },\cdot ;t_{k+1},\omega (t_{k+1},\cdot ))||^{2}ds^{\prime
}\right) \notag \\
&=&F_{n}+\sum_{k=1}^{n}F_{k-1}\cdot \left[ \exp \left(
K(t_{n+1}-t_{k})+c\int_{t_{k}}^{t_{n+1}}||\nabla \omega (s^{\prime },\cdot
;t_{k},\omega (t_{k},\cdot ))||^{2}ds^{\prime }\right) \right. \notag \\
&&\left. -\exp \left( K(t_{n+1}-t_{k+1})+c\int_{t_{k+1}}^{t_{n+1}}||\nabla
\omega (s^{\prime },\cdot ;t_{k+1},\omega (t_{k+1},\cdot ))||^{2}ds^{\prime
}\right) \right] \notag \\
&=&\sum_{k=1}^{n}\left( F_{k}-F_{k-1}\right) \exp \left(
K(t_{n+1}-t_{k+1})+c\int_{t_{k+1}}^{t_{n+1}}||\nabla \omega (s^{\prime
},\cdot ;t_{k+1},\omega (t_{k+1},\cdot ))||^{2}ds^{\prime }\right) \notag
\\
&&+h^{3}C(t_{1},\mathbf{\omega })\exp \left(
K(t_{n+1}-t_{1})+c\int_{t_{1}}^{t_{n+1}}||\nabla \omega (s^{\prime },\cdot
;t_{1},\omega (t_{1},\cdot ))||^{2}ds^{\prime }\right) . \notag\end{aligned}$$
For the last term in the right-hand side of (\[bax31\]), we obtain using the Cauchy-Bunyakovsky inequality and Lemma \[LemMat\]:$$E\left\{ h^{3}C(t_{1},\mathbf{\omega })\exp \left(
K(t_{n+1}-t_{1})+c\int_{t_{k+1}}^{t_{n+1}}||\nabla \omega (s^{\prime },\cdot
;0,\phi (\cdot ))||^{2}ds^{\prime }\right) \right\} \leq Kh^{3}.
\label{bax5}$$
Consider now the first term in the right-hand side of (\[bax31\]). We have $$\begin{aligned}
&&\left( F_{k}-F_{k-1}\right) \exp \left(
K(t_{n+1}-t_{k+1})+c\int_{t_{k+1}}^{t_{n+1}}||\nabla \omega (s^{\prime
},\cdot ;t_{k+1},\omega (t_{k+1},\cdot ))||^{2}ds^{\prime }\right) \ \ \ \ \
\label{bax55} \\
&=&\exp \left( K(t_{n+1}-t_{k+1})+c\int_{t_{k+1}}^{t_{n+1}}||\nabla \omega
(s^{\prime },\cdot ;t_{k+1},\omega (t_{k+1},\cdot ))||^{2}ds^{\prime
}\right) \notag \\
&&\times \left[ h^{3}C(t_{k+1},\mathbf{\omega })+(R(t_{k},\cdot ),\delta
_{\omega }(t_{k+1},\cdot ))\right] . \notag\end{aligned}$$Expectation of the first term from the right-hand side of (\[bax55\]) is estimated by $Kh^{3}$ as in (\[bax5\]). Let us now consider the second term.
By the martingale representation theorem and Lemma \[LemMat\], we can obtain $$\begin{aligned}
&&E\left[ \left. \exp \left(
K(t_{n+1}-t_{k+1})+c\int_{t_{k+1}}^{t_{n+1}}||\nabla \omega (s^{\prime
},\cdot ;t_{k+1},\omega (t_{k+1},\cdot ))||^{2}ds^{\prime }\right)
\right\vert \mathcal{F}_{t_{k+1}}\right] \ \ \ \ \ \ \label{snsmrt} \\
&=&E\left[ \exp \left( K(t_{n+1}-t_{k+1})+c\int_{t_{k+1}}^{t_{n+1}}||\nabla
\omega (s^{\prime },\cdot ;t_{k+1},\omega (t_{k+1},\cdot ))||^{2}ds^{\prime
}\right) \right] \notag \\
&&+\sum_{r=1}^{q}\int_{0}^{t_{k+1}}\lambda _{r}(s)dw_{r}(s), \notag\end{aligned}$$where $\lambda _{r}(s)$ are $\mathcal{F}_{s}$-adapted square-integrable stochastic processes.
Using (\[sns18\]) and a conditional version of (\[sns19\]) from Theorem \[thm:sns1\] together with Theorems \[thm:sns2\] and \[thm:sns2nn\] and also using Lemma \[LemMat\] and (\[snsmrt\]), we arrive at $$\begin{aligned}
&&\left\vert E\left\{ \exp \left(
K(t_{n+1}-t_{k+1})+c\int_{t_{k+1}}^{t_{n+1}}||\nabla \omega (s^{\prime
},\cdot ;t_{k+1},\omega (t_{k+1},\cdot ))||^{2}ds^{\prime }\right) \right.
\right. \\
&&\left. \left. \Bigl.\overset{\ }{\ \ }(R(t_{k},\cdot ),\delta _{\omega
}(t_{k+1},\cdot ))\Bigr\vert\mathcal{F}_{t_{k}}\right\} \right\vert \\
&=&\left\vert \left( R(t_{k},\cdot ),E\left\{ \exp \left(
K(t_{n+1}-t_{k+1})+c\int_{t_{k+1}}^{t_{n+1}}||\nabla \omega (s^{\prime
},\cdot ;t_{k+1},\omega (t_{k+1},\cdot ))||^{2}ds^{\prime }\right) \right.
\right. \right. \\
&&\left. \left. \left. \Biggl.\overset{\ }{\ \ }\delta _{\omega
}(t_{k+1},\cdot )\Biggr\vert\mathcal{F}_{t_{k}}\right\} \right) \right\vert
\\
&\leq &||R(t_{k},\cdot )||\left\Vert E\left\{ \exp \left(
K(t_{n+1}-t_{k+1})+c\int_{t_{k+1}}^{t_{n+1}}||\nabla \omega (s^{\prime
},\cdot ;t_{k+1},\omega (t_{k+1},\cdot ))||^{2}ds^{\prime }\right) \right.
\right. \\
&&\left. \left. \left. \Biggl.\delta _{\omega }(t_{k+1},\cdot )\Biggr\vert%
\mathcal{F}_{t_{k}}\right\} \right) \right\Vert\end{aligned}$$$$\begin{aligned}
&=&||R(t_{k},\cdot )|| \\
&&\cdot \left\Vert E\left\{ \left( E\left[ \exp \left(
K(t_{n+1}-t_{k+1})+c\int_{t_{k+1}}^{t_{n+1}}||\nabla \omega (s^{\prime
},\cdot ;t_{k+1},\omega (t_{k+1},\cdot ))||^{2}ds^{\prime }\right) \right]
\right. \right. \right. \\
&&\left. \left. \left. \left. +\sum_{r=1}^{q}\int_{0}^{t_{k+1}}\lambda
_{r}(s)dw_{r}\right) \delta _{\omega }(t_{k+1},\cdot )\right\vert \mathcal{F}%
_{t_{k}}\right\} \right\Vert \\
&\leq &||R(t_{k},\cdot )||E\left[ \exp \left(
K(t_{n+1}-t_{k+1})+c\int_{t_{k+1}}^{t_{n+1}}||\nabla \omega (s^{\prime
},\cdot ;t_{k+1},\omega (t_{k+1},\cdot ))||^{2}ds^{\prime }\right) \right]
\\
&&\cdot \left\Vert E\left\{ \left. \delta _{\omega }(t_{k+1},\cdot
)\right\vert \mathcal{F}_{t_{k}}\right\} \right\Vert +||R(t_{k},\cdot
)||\left\Vert E\left\{ \left. \delta _{\omega }(t_{k+1},\cdot
)\sum_{r=1}^{q}\int_{0}^{t_{k+1}}\lambda _{r}(s)dw_{r}\right\vert \mathcal{F}%
_{t_{k}}\right\} \right\Vert\end{aligned}$$$$\begin{aligned}
&\leq &||R(t_{k},\cdot )||C(t_{k},\mathbf{\omega })h^{2}+||R(t_{k},\cdot
)||\left\Vert \sum_{r=1}^{q}\int_{0}^{t_{k}}\lambda _{r}(s)dw_{r}E\left\{
\left. \delta _{\omega }(t_{k+1},\cdot )\right\vert \mathcal{F}%
_{t_{k}}\right\} \right\Vert \\
&&+||R(t_{k},\cdot )||\left( E\left[ ||\left. \delta _{\omega
}(t_{k+1},\cdot )||^{2}\right\vert \mathcal{F}_{t_{k}}\right] \right) ^{1/2}
\\
&&\cdot \left( \sum_{r=1}^{q}\left( E\left[ \left. \left\{
\int_{t_{k}}^{t_{k+1}}\lambda _{r}(s)dw_{r}\right\} ^{2}\right\vert \mathcal{%
F}_{t_{k}}\right] \right) ^{1/2}\right) \\
&\leq &||R(t_{k},\cdot )||C(t_{k},\mathbf{\omega })h^{2}\leq
h||R(t_{k},\cdot )||^{2}+\frac{h^{3}}{4}C^{2}(t_{k},\mathbf{\omega }).\end{aligned}$$Therefore, $$\begin{aligned}
&&E\left\{ \left( F_{k}-F_{k-1}\right) \exp \left(
K(t_{n+1}-t_{k+1})+c\int_{t_{k+1}}^{t_{n+1}}||\nabla \omega (s^{\prime
},\cdot ;t_{k+1},\omega (t_{k+1},\cdot ))||^{2}ds^{\prime }\right) \right\}
\ \ \ \ \ \ \ \ \ \ \ \label{bax99} \\
&\leq &hE||R(t_{k},\cdot )||^{2}+Kh^{3}. \notag\end{aligned}$$From (\[bax31\]), (\[bax5\]), and (\[bax99\]), we obtain $$E||R(t_{n+1},\cdot )||^{2}\leq h\sum_{k=1}^{n}E||R(t_{k},\cdot
)||^{2}+Kh^{2},$$from which it follows by a version of Gronwall’s lemma that $$E||R(t_{n+1},\cdot )||^{2}\leq Kh^{2}$$as required. Theorem \[thm:sns3\] is proved.
Various approaches can be used to turn the method, introduced at the start of Section \[sec:sns3\] for the problem $(\ref{sns7})$-$(\ref{sns9})$, into a numerical algorithm. To obtain a constructive numerical algorithm, we need to approximate the linear SPDE $(\ref{sns11})$-$(\ref{sns12})$ at every step. To this end, for instance, we can discretize this SPDE in space using the spectral method based on the Fourier expansion and use a finite difference for time discretization (see such an algorithm in the deterministic setting in e.g. [@Peyret]). Alternatively, we can apply the method based on averaging characteristics to $(\ref{sns11})$-$(\ref%
{sns12})$ [@spde]. We leave construction, analysis and testing of such algorithms for a future work.
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[^1]: Ural Federal University, Lenin Str. 51, 620083 Ekaterinburg, Russia;
[^2]: School of Mathematical Sciences, University of Nottingham, University Park, Nottingham, NG7 2RD, UK, email: Michael.Tretyakov@nottingham.ac.uk
|
---
abstract: 'We measured infrared surface brightness fluctuation (SBF) distances to an isotropically-distributed sample of 16 distant galaxies with redshifts reaching 10,000 [kms$^{-1}$]{} using the near-IR camera and multi-object spectrometer (NICMOS) on the [*Hubble Space Telescope*]{} (HST). The excellent spatial resolution, very low background, and brightness of the IR fluctuations yielded the most distant SBF measurements to date. Twelve nearby galaxies were also observed and used to calibrate the F160W (1.6 ) SBF distance scale. Of these, three have Cepheid variable star distances measured with HST and eleven have optical $I$-band SBF distance measurements. A distance modulus of 18.5 mag to the Large Magellanic Cloud was adopted for this calibration. We present the F160W SBF Hubble diagram and find a Hubble constant [$H_{\rm 0}$]{}$\,{=}\,76\pm 1.3$ (1-$\sigma$ statistical) $\pm 6$ (systematic) [kms$^{-1}$Mpc$^{-1}$]{}. This result is insensitive to the velocity model used to correct for local bulk motions. Restricting the fit to the six most distant galaxies yields the smallest value of [$H_{\rm 0}$]{}$\,{=}\,72\pm 2.3$ [kms$^{-1}$Mpc$^{-1}$]{} consistent with the data. This 6% decrease in the Hubble constant is consistent with the hypothesis that the Local Group inhabits an under-dense region of the universe, but is also consistent with the best-fit value of [$H_{\rm 0}$]{}$\,{=}\,76$ [kms$^{-1}$Mpc$^{-1}$]{} at the 1.5-sigma level.'
author:
- 'Joseph B. Jensen'
- 'John L. Tonry'
- 'Rodger I. Thompson'
- 'Edward A. Ajhar and Tod R. Lauer'
- 'Marcia J. Rieke'
- Marc Postman
- 'Michael C. Liu'
title: The Infrared Surface Brightness Fluctuation Hubble Constant
---
Introduction
============
The Hubble constant, [$H_{\rm 0}$]{}, is the most fundamental of the cosmological parameters. Yet in spite of its key role in our understanding of the universe, an accurate determination of its value eluded researchers for decades. It has only been within the last few years that the promise of knowing [$H_{\rm 0}$]{} to better than 10% has been realized (Mould et al. 2000). The [*Hubble Space Telescope*]{} (HST) has occupied a key role in resolving the debate over the Hubble constant by enabling distance measurements not previously possible from the ground. With HST’s spatial resolution, Cepheid variable stars have been detected in galaxies as distant as 20 Mpc. Cepheid distances to a variety of galaxies, including some in the important Virgo and Fornax clusters, have provided new calibrations of many secondary distance indicators, including type-Ia supernovae (Gibson et al. 2000, Parodi et al. 2000), fundamental plane (Kelson et al. 2000), Tully-Fisher (Sakai et al. 2000), planetary nebulae, globular clusters, and surface brightness fluctuations (Ferrarese et al. 2000a). Uniform HST Cepheid distances were collected by Ferrarese et al. (2000b).
Surface brightness fluctuations have emerged as an accurate and reliable distance indicator (Tonry et al. 1997, Blakeslee et al. 1999). HST has made it possible to not only better calibrate SBFs by providing Cepheid distances to calibration galaxies, but also allowed detection of fluctuations in half a dozen galaxies at much greater distances than possible from the ground (Lauer et al. 1998, Pahre et al. 1999, Thomson et al. 1997). Two additional low signal-to-noise ratio (S/N) measurements in the Coma cluster (Thomson et al. 1997, Jensen et al. 1999), were the most distant SBF measurements until the current NICMOS project.
Surface brightness fluctuations have a much larger amplitude in the near-IR than at optical wavelengths. The Near Infrared Camera and Multi-object Spectrograph (NICMOS) on the HST provides the combination of low background and high spatial resolution needed to measure IR SBFs beyond 100 Mpc for the first time. The purpose of this study was to calibrate the F160W SBF distance scale and to measure [$H_{\rm 0}$]{} beyond the effects of local flows. Reaching distances twice as large as previous SBF studies for a sample uniformly distributed on the sky provided immunity to many of the difficulties that plague all attempts to measure [$H_{\rm 0}$]{} within 50 Mpc.
In the next section we describe the selection of the calibration and distant galaxy samples observed. In Section \[reduxsection\] we discuss the procedures used to acquire and reduce the data. Section \[sbfsection\] describes the methods used to determine the SBF amplitude. Section \[calsection\] discusses the empirical calibration of the F160W SBF distance scale and the comparison to stellar population models. Section \[hubblesection\] presents the IR SBF Hubble diagram. Finally, we discuss the relationship of our measurement to others which find lower values of [$H_{\rm 0}$]{} and conclude with a summary section.
Sample Selection \[samplesection\]
==================================
As part of our program to measure distances to redshifts of 10,000 [kms$^{-1}$]{}, we observed a set of nearby galaxies in the Leo, Virgo, and Fornax clusters. These observations support an empirical distance calibration determined both using Cepheid variable star distances and the extensive $I$-band SBF distance survey (Tonry et al. 1997). The calibration galaxies cover a similar color range as the distant galaxies used to measure [$H_{\rm 0}$]{} ($I$-band SBF brightnesses show a systematic dependence on galaxy [$(V{-}I)$]{} color). In addition to our calibration data, we discovered that several other NICMOS programs included F160W observations of nearby galaxies suitable for SBF analysis that could be used to augment our calibration. The most useful of these are the IR SBF survey of the Fornax cluster (NICMOS program 7458, J. R. Graham et al.) and the programs which targeted galaxies previously observed using WFPC-2 for the purpose of measuring Cepheid distances. We acquired raw NICMOS data from the HST archive and reduced it using the procedures presented in this paper to guarantee a completely consistent calibration.
[llcrrrrr]{} IC 2006 & Fornax & 7458 & 237.51 & $-$50.39 & 0.048 & 0.006 & 256\
NGC 1380 & Fornax & 7458 & 235.93 & $-$54.06 & 0.075 & 0.010 & 256\
NGC 1381 & Fornax & 7458 & 236.47 & $-$54.04 & 0.058 & 0.008 & 256\
NGC 1387 & Fornax & 7458 & 236.82 & $-$53.95 & 0.055 & 0.007 & 256\
NGC 1399 & Fornax & 7453 & 236.72 & $-$53.63 & 0.058 & 0.008 & 384\
NGC 1404 & Fornax & 7453 & 236.95 & $-$53.55 & 0.049 & 0.006 & 384\
NGC 3031 & M 81 & 7331 & 142.09 & $+$40.90 & 0.347 & 0.046 & 384\
NGC 3351 & Leo I & 7330 & 233.95 & $+$56.37 & 0.120 & 0.016 & 640\
NGC 3379 & Leo I & 7453 & 233.49 & $+$57.63 & 0.105 & 0.014 & 384\
NGC 4406 & Virgo & 7453 & 279.08 & $+$74.63 & 0.128 & 0.017 & 384\
NGC 4472 & Virgo & 7453 & 286.92 & $+$70.20 & 0.096 & 0.012 & 384\
NGC 4536 & Virgo & 7331 & 292.95 & $+$64.73 & 0.079 & 0.013 & 384\
NGC 4636 & Virgo & 7886 & 297.75 & $+$65.47 & 0.124 & 0.016 & 640\
NGC 4725 & & 7330 & 295.08 & $+$88.36 & 0.051 & 0.007 & 320\
NGC 708 & Abell 262 & 7453 & 136.57 &$-$25.09 & 0.379 & 0.050 & 960\
NGC 3311 & Abell 1060 & 7820 & 269.60 &$+$26.49 & 0.343 & 0.046 & 2560\
IC 4296 & Abell 3565 & 7453 & 313.54 &$+$27.97 & 0.265 & 0.035 & 1920\
NGC 7014 & Abell 3742 & 7453 & 352.53 &$-$42.35 & 0.142 & 0.019 & 1600\
NGC 4709 & Centaurus & 7453 & 302.66 &$+$21.49 & 0.512 & 0.068 & 1600\
NGC 5193 & (Abell 3560)& 7453 & 312.59 & $+$28.88 & 0.242 & 0.032 & 1920\
PGC 015524& Abell 496 & 7453 & 209.58 &$-$36.49 & 0.602 & 0.079 & 5760\
NGC 2832 & Abell 779 & 7453 & 191.09 &$+$44.39 & 0.073 & 0.010 & 1920\
IC 4051 & Abell 1656(a) & 7820 & 56.22 &$+$87.72 & 0.046 & 0.006 & 2560\
NGC 4874 & Abell 1656(b) & 7820 & 58.06 &$+$88.01 & 0.037 & 0.005 & 2560\
NGC 6166 & Abell 2199 & 7453 & 62.93 &$+$43.69 & 0.050 & 0.007 & 4160\
NGC 7768 & Abell 2666 & 7453 & 106.71 &$-$33.81 & 0.167 & 0.022 & 1600\
NGC 2235 & Abell 3389 & 7453 & 274.67 &$-$27.43 & 0.330 & 0.044 & 2048\
IC 4374 & Abell 3581 & 7453 & 323.14 &$+$32.85 & 0.263 & 0.035 & 1280\
IC 4931 & Abell 3656 & 7453 & 1.92 &$-$29.46 & 0.306 & 0.040 & 1920\
NGC 4073 & & 7820 & 276.91 &$+$62.37 & 0.117 & 0.016 & 2560\
Because we were able to analyze F160W NICMOS data for galaxies with Cepheid distances, we were not required to assume that the ellipticals and spirals in a given cluster are all at a common distance. For at least a few Cepheid-bearing spirals, IR SBF measurements are possible in their bulges. Using a calibration based solely on galaxies with distances measured both using Cepheid variables and SBFs removes the added uncertainty in the calibration arising from the size and distribution of galaxies in the clusters (Tonry et al. 2000, hereafter SBF-II).
In addition to the relatively local calibrators, we also targeted five galaxies with $I$-band SBF distances previously measured using WFPC-2: the four from Lauer et al. (1998) and Ajhar et al. (1997), and NGC 4709 in the Centaurus cluster (Optical SBF team, private communication). These intermediate-distance galaxies provide overlap between our local calibration and the distant galaxies from which we determine [$H_{\rm 0}$]{}. NGC 3311 in the Hydra cluster (NICMOS program 7820, D. Geisler et al.) was added to the intermediate-distance set, although it does not have an $I$-band SBF distance measurement.
The main focus of this study is to measure distances to the set of 16 galaxies (including the six intermediate-distance galaxies) that extend out to redshifts of 10,000 [kms$^{-1}$]{}. The most distant galaxies are uniformly distributed on the sky to provide a robust determination of [$H_{\rm 0}$]{} and minimize sensitivity to bulk streaming motions in the local universe. The sample and observational data are presented in Table \[sampletable\]; the positions of the galaxies on the sky are shown in Figure \[supergal\] in galactic coordinates.
The results presented in this paper are derived from data taken as part of six separate NICMOS programs. In some cases the observers in these other programs were careful to ensure that their data would be useful for SBF measurements. This program demonstrates the value of the HST archive.
Observations and Data Reduction \[reduxsection\]
================================================
All the data were acquired using the background-minimizing F160W filter (1.6 , similar to the standard $H$ filter) and the NIC2 camera read out in the MULTIACCUM mode. NIC2 has a plate scale of 0.075 arcsec per pixel which gives a field of view 19.2 arcsec across. Data were reduced using modified versions of IDL routines developed by the NICMOS team. During each exposure, the NIC2 array was read non-destructively several times, and intermediate images from the MULTIACCUM sequence were created by subtracting the initial read from the intermediate reads. A dark current image from the NICMOS team’s library was then subtracted and pixels exhibiting non-linear response were identified. The differences between intermediate reads were used to identify pixels affected by cosmic rays, which were recognizable as a change in the rate of accumulation of flux in a pixel and could be corrected using the unaffected sub-images. Remaining cosmic rays were fixed when the individual MULTIACCUM images were combined. The next step was to construct the full exposure by multiplying the fitted slope of flux accumulation in each pixel by the total exposure time, divide by the flat field, and mask bad pixels. The combined MULTIACCUM images were then registered to the nearest pixel and added together; integer-pixel registration does not introduce correlations in the noise between pixels that change the spatial power spectrum of the noise. The SBF analysis assumes that the noise is uncorrelated between pixels.
Raw NICMOS images frequently have slightly different bias levels in each quadrant. In the final coadded images, the background level mismatches between quadrants produce horizontal and vertical discontinuities that affect the measurement of the SBF spatial power spectrum. To remove the offsets, we first processed each individual image and subtracted a smooth fit to the galaxy. The differences between residual background levels in narrow regions on either side of the boundaries were measured. Overall offsets were then computed to effectively add zero flux to the overall image background while minimizing the differences across boundaries. Offsets were applied to the images prior to dividing by the flat field. The final coadded images are much smoother and do not suffer from discontinuities in the background.
Once the relative bias levels between quadrants was removed, the overall bias level remained uncertain. Any such background not removed prior to dividing by the flat field image carries the power spectrum of the flat field into the final spatial power spectrum. The NIC2 flat field has significant structure, making it necessary to address the possibility that residual bias adds power to the measured SBF power spectrum. To measure the influence of residual bias levels on the SBF measurement, we constructed an image composed of scaled copies of that flat field added with the offsets of the dither pattern. The resulting image was then scaled to form a “residual bias image” and added to or subtracted from the final galaxy image prior to SBF analysis. The SBF analysis was repeated for different scale factors, corresponding to the likely range of residual bias values. The most likely residual bias level was determined by trial and error: if a residual bias correction was similar to the level of the inter-quadrant bias adjustment, resulted in lower fluctuation amplitudes, and made the fit to the power spectrum better over a larger range of wavenumber, then it was adopted. If adding a scaled residual bias image led to a worse fit to the spatial power spectrum, or increased the fluctuation amplitude, then no correction was adopted. In many cases applying a residual bias correction did not make the power spectrum fit better or worse, and no correction was adopted. The influence of residual bias on the final SBF measurement was included in the uncertainty by noting the change in the fluctuation magnitude resulting from a range of applied residual bias levels.
Some raw NIC2 images were also affected by interference from the operation of the other cameras. Because NIC1 and NIC3 were not operated in precisely the same mode as NIC2, the cameras were not being reset and read at the same time. Interference between cameras resulted in dark and light horizontal lines in the raw images that adversely affect the SBF power spectrum. To remove the lines, we first identified the affected rows in each individual image with a smooth galaxy profile removed. These rows were masked before the final image was constructed.
A few cosmic rays were energetic enough to leave a residual ghost that persisted for several minutes in the subsequent MULTIACCUM sequences. We identified these occasional residual cosmic rays and masked them as well. These, along with any cosmic rays that escaped detection in the MULTIACCUM sequence, were fixed using valid data for the same location on the sky from the other images in the dither sequence. Each individual exposure (a complete MULTIACCUM sequence) was dithered by 1.5 arcsec, or 20 pixels. When the final summed images were created, we also used the spatial information in the dither sequence to fix the lines caused by read out interference. The final images are smooth and clean, free from almost all the defects inherent in the raw images.
Even when great care was taken to remove cosmic rays and detector artifacts, one type of persistent problem proved to be difficult to remove from our data. When the HST passed through the South Atlantic Anomaly (SAA), the NICMOS arrays were completely saturated with cosmic rays. The arrays were turned off during these passages. However, some of the time the arrays were restarted too soon after passage through the SAA, and the number of hard cosmic ray hits was very high. The persistent images from these cosmic rays are obvious in the first MULTIACCUM sequences taken after passing through the SAA, and slowly decay through several subsequent exposures.
The background of residual cosmic rays is best described as a “wormy” pattern, with small, sometimes elongated patches of several pixels having significantly higher signal than surrounding regions. An example of this wormy background is shown for one quadrant in subsequent MULTIACCUM images in Figure \[worms\], in which the galaxy profile has been subtracted to show the background. The “worms” make up the splotchy background (left panel) and are distributed fairly uniformly over the array because they correspond to the locations of cosmic ray hits. Worms are a serious concern for SBF measurements, as they are not confined to a small number of pixels. Their spatial power spectrum, while not exactly matching the PSF power spectrum, has significant power on the spatial scales used to fit the SBF power spectrum. If not removed, the power in worms can dominate the stellar fluctuations. The low level persistence of the wormy background pattern can bias the SBF measurement because worms add power to the fluctuation power spectrum, even when they are no longer obvious in the images.
To deal with the worms, we started by examining the galaxy-subtracted residual images from individual MULTIACCUM sequences (e.g., Fig. \[worms\]) and we excluded the badly affected images. The remaining question, then, is to what extent the rest of the images were affected. The power spectra from sequential exposures showed the total power decaying to an asymptotic value, although it was difficult to know if the contribution from worms at the asymptotic power level was zero or not. Because the worms were not convolved with the PSF, it was sometimes possible to identify wormy images from power spectra that deviated systematically from the PSF power spectrum. A wormy image has more power at high wavenumbers (small scales) and less power at low wavenumbers than the PSF. To estimate the maximum contribution from residual worminess, we examined the behavior of the SBF signal as a function of the distance from the center of the galaxy. The stellar SBF signal scales with the galaxy surface brightness, while any background fluctuation power from cosmic ray image persistence is uniform. The result of background worminess was a fluctuation power that increased with the area of the region being analyzed. We measured the SBF power in three or four apertures centered on the galaxy nucleus, and, assuming the stellar population of the galaxy is reasonably uniform (ie., the intrinsic stellar fluctuation magnitude does not change drastically with radius), we applied a correction proportional to area to make the fluctuation measurements in the different annuli equal, if possible. The details of these corrections are discussed further in the appendix.
SBF Measurements \[sbfsection\]
===============================
We followed the same basic procedures for measuring SBF amplitudes outlined in detail in Jensen et al. (1998). We first fitted and subtracted a smooth fit to the galaxy profile. Fluctuations can be seen in the three examples shown in Figure \[images\]. Objects in the galaxy-subtracted image were identified, their brightnesses and number densities measured, and a luminosity function generated for globular clusters (GCs) and background galaxies (see Jensen et al. 1998 for details of the luminosity function fits). Objects down to the completeness limit were masked, and the luminosity function integrated beyond the completeness limit to determine the contribution to the SBFs from undetected globular clusters and galaxies. The SBF analysis is insensitive to the exact values assumed for the globular cluster luminosity function width and peak magnitude or the galaxy luminosity function slope when the brightest of these populations are well-measured and masked. Residual large-scale variations in the background resulting from incomplete galaxy subtraction were fitted and removed as well; low wavenumbers ($k\,{<}\,20$) were ignored in fitting the power spectrum because the spectrum at low wavenumbers was modified by the background subtraction. The fitting parameters for the galaxy profile and large-scale background were tuned to produce the cleanest power spectrum possible. The uncertainties resulting from the galaxy and background fits were determined and added in quadrature with the other sources of uncertainty. Dusty regions near the centers of a few of the galaxies were masked.
The SBF spatial power spectrum normalized by the mean galaxy surface brightness was fitted with the sum of a white-noise component $P_1$ and the expectation power spectrum $E(k)$ scaled by the fluctuation power $P_0$. $E(k)$ is a combination of the normalized PSF power spectrum and the mask used to excise point sources and select the radial region of the galaxy being analyzed. The data were fitted with the function $$P(k) = P_0 E(k) + P_1.$$
The fluctuation power $P_0$ must be corrected for undetected point sources and residual wormy background. These are represented by $P_r$ and $P_g$, respectively. The power in stellar SBFs is therefore $$P_{\rm fluc} = P_0 - P_r - P_g.$$ $P_{\rm fluc}$ is simply a flux, and has units of electrons per pixel per integration time. Fluctuation powers and the relative sizes of the $P_r$ and $P_g$ corrections are listed in Table \[powertable\]. $P_{\rm fluc}$ can be transformed into an apparent fluctuation magnitude and corrected for galactic extinction: $$\overline m = -2.5\,\log(P_{\rm fluc}) + m_1 - A_H$$ where $m_1$ is the magnitude of a source yielding 1 e$^-$ per total integration time on the Vega system. We adopted the photometric zero point for NIC2 and the F160W filter measured by the NICMOS team of $m_1 = 23.566 \pm 0.02$ mag, the brightness of a source which gives 1 e$^-$s$^{-1}$. The gain is 5.4 e$^-$ per ADU. In this paper we have chosen to adopt the extinction values of Schlegel et al. (1998, hereafter SFD), which are converted to the $H$-band extinction values assuming $A_B=4.315\,E(B{-}V)$ and $A_H=0.132\,A_B$ (SFD).
[lcrcccrcc]{} Abell 262 & 2.4–4.8 &$ 8.8{\pm}0.3 $& 0.22 & 0.00 & 0.34 & 1.2 & 5.8 & d,w\
Abell 496 & 2.4–4.8 &$11.4{\pm}0.8 $& 0.17 & 0.29 & 0.11 & 2.2 & 6.8 &\
Abell 779 & 2.4–4.8 &$ 7.3{\pm}0.6 $& 0.20 & 0.28 & 0.08 & 2.2 & 4.6 & p\
Abell 1060 & 2.4–4.8 &$20.7{\pm}0.5 $& 0.08 & 0.09 & 0.00 & 12.1 & 18.7 & (d)\
Abell 1656(a)& 2.4–4.8 &$15.7{\pm}0.6 $& 0.15 & 0.41 & 0.16 & 1.4 & 6.8 & GC\
Abell 1656(b)& 4.8–9.6 &$11.7{\pm}0.8 $& 0.21 & 0.15 & 0.22 & 1.5 & 7.4 &\
Abell 2199 & 2.4–4.8 &$ 9.0{\pm}0.3 $& 0.22 & 0.35 & 0.00 & 2.9 & 5.9 & (d)\
Abell 2666 & 2.4–4.8 &$ 5.5{\pm}0.3 $& 0.33 & 0.00 & 0.53 & 0.6 & 2.6 & w\
Abell 3389 & 2.4–4.8 &$ 8.2{\pm}0.2 $& 0.24 & 0.33 & 0.26 & 0.8 & 3.3 & w\
Abell 3565 & 4.8–9.6 &$17.3{\pm}1.0 $& 0.12 & 0.06 & 0.07 & 4.7 & 15.1 & (d)\
Abell 3581 & 2.4–4.8 &$ 8.4{\pm}0.9 $& 0.26 & 0.18 & 0.42 & 0.6 & 3.4 & (d),w\
Abell 3656 & 2.4–4.8 &$ 8.0{\pm}0.6 $& 0.24 & 0.10 & 0.10 & 2.4 & 7.0 & p\
Abell 3742 & 4.8–9.6 &$12.1{\pm}1.4 $& 0.22 & 0.10 & 0.05 & 3.1 & 10.3 & drift\
NGC 4073 & 2.4–4.8 &$13.1{\pm}0.9 $& 0.15 & 0.31 & 0.25 & 1.1 & 5.8 & w\
NGC 4709 & 2.4–4.8 &$20.2{\pm}0.8 $& 0.10 & 0.08 & 0.12 & 3.8 & 16.3 & w\
NGC 5193 & 2.4–4.8 &$21.4{\pm}2.2 $& 0.09 & 0.07 & 0.00 & 10.8 & 19.9 & (d)\
Because the stellar SBF pattern is convolved with the diffraction pattern of the telescope and instrument, we require a good measurement of the point spread function (PSF) for each observation. If the reference PSF shape used does not match the PSF of the data, the fit will be poor. If the PSF is not properly normalized, the photometry will not be correct. To ensure a good SBF measurement, we attempted to image a bright star concurrently with each galaxy observation to serve as a high-S/N PSF reference. The PSF star measurements were short, unguided exposures, and in some cases the PSF was blurred slightly by telescope drift. Others were unusable because of close companions that were undetectable without the excellent resolution of the HST. Still another turned out to be a compact galaxy. In the end, we acquired 10 good PSF measurements over the course of our program (spanning approximately 1 year). For each SBF measurement, we chose the PSF taken closest in time to the galaxy observation.
The uncertainty in the SBF measurement resulting from variations in the PSF was determined by using all 10 PSF stars to measure the fluctuation magnitude for a galaxy. The standard deviation in each case was added in quadrature with the other sources of uncertainty, and was typically between 4% and 6%. The time between individual PSF measurements was much longer than the “breathing” timescale of the telescope, so the variation between PSFs was random. In general, the PSF fits to the SBF data were excellent. While the PSF shows diffraction rings and spots that are quite different from the typical smooth PSF observed from the ground, the PSF power spectrum fits the galaxy data very well, both in the tight Airy core (the broad, high-wavenumber component) and in the wings (the steeper component at low wavenumbers).
We compared our snapshot PSF measurements to 16 measurements of four stars made by the NICMOS team. Fluctuation measurements of our most distant galaxy (Abell 496) were made using the library PSFs as a test case. The library PSF results agreed perfectly with measurements made using our snapshot PSFs, and showed a somewhat smaller scatter (3.5%). The smaller dispersion can be attributed to the fact that the library PSF measurements were made on fewer stars and while the telescope guiding was enabled.
The signal-to-noise ratio of an SBF measurement is best quantified as $$\xi = P_{\rm fluc} / (P_1 + P_g).$$ Jensen et al. (1998) showed that $\xi$ is a good figure of merit for IR SBF measurements. Values of $\xi$ less than unity indicate measurements that are unreliable. The higher $\xi$, the better the SBF measurement. Galaxies with $\xi{<}1$ are necessarily those for which the correction for globular clusters ($P_r$) or residual cosmic rays ($P_g$) are large. $P_0/P_1$, while sometimes used as a measure of SBF S/N, significantly overestimates the true S/N because $P_0$ contains the contributions from these other sources of variance.
In several cases, the relative contributions of stellar SBFs ($P_{\rm fluc}$), background worminess ($P_g$), and globular clusters ($P_r$) to the power spectrum were difficult to untangle. Table \[powertable\] lists the powers measured (in electrons per total integration time) for each galaxy and the relative levels of the $P_1$, $P_r$, and $P_g$ contributions. The fluctuation S/N ratio ($\xi$) is listed for each galaxy, and the power spectrum for each annulus listed in Table \[powertable\] is plotted in Figure \[powerfig\]. Fluctuation measurements were made in three annuli for each galaxy, and the results compared. The inner annulus spanned a radial region from 1.2 to 2.4 arcsec, the middle annulus from 2.4 to 4.8 arcsec, and the outer annulus from 4.8 to 9.6 arcsec. In the appendix we discuss the SBF measurements for each intermediate and distant galaxy individually.
Calibration of the F160W SBF Distance Scale \[calsection\]
==========================================================
Absolute Fluctuation Magnitudes
-------------------------------
Apparent fluctuation magnitudes for the nearby calibrator galaxies were combined with previously measured distance moduli to empirically determine the absolute brightness of F160W fluctuations [$\overline M_{\rm F160W}$]{}. Most of the calibration galaxies are giant ellipticals, and we adopt the distances from the $I$-band SBF survey for them (Optical SBF team, private communication). The $I$-band SBF distances were calibrated using Cepheid distances to a handful of spiral galaxies for which $I$-band SBF analysis was possible in the bulges (SBF-II). Thus the $I$-band SBF distances used to calibrate the F160W distance scale are based on SBF and Cepheid distances to individual galaxies, and do not assume common distances for different galaxies within a cluster or group. Optical SBF distance moduli and F160W fluctuation magnitudes are listed in Table \[caltable\].
[lcccccc]{} IC 2006 & $26.58\pm0.05$ & $1.183\pm0.018$ & $31.59\pm0.29$ & $-5.01\pm0.29$ & &\
NGC 1380 & $26.40\pm0.05$ & $1.197\pm0.019$ & $31.32\pm0.18$ & $-4.92\pm0.18$ & &\
NGC 1381 & $26.52\pm0.10$ & $1.189\pm0.018$ & $31.28\pm0.21$ & $-4.76\pm0.23$ & &\
NGC 1387 & $26.0\phn\pm0.7\phn$ & $1.208\pm0.047$ & $31.54\pm0.26$ & $-5.6\phn\pm0.8\phn$ & &\
NGC 1399 & $26.76\pm0.04$ & $1.227\pm0.016$ & $31.50\pm0.16$ & $-4.74\pm0.15$ & &\
NGC 1404 & $26.66\pm0.08$ & $1.224\pm0.016$ & $31.61\pm0.19$ & $-4.95\pm0.19$ & &\
NGC 3031 & $22.96\pm0.05$ & $1.187\pm0.011$ & $27.96\pm0.26$ & $-5.00\pm0.26$ & $27.80\pm0.08$ & $-4.84\pm0.09$\
NGC 3351 & $25.16\pm0.07$ & $1.225\pm0.014$ & & & $30.01\pm0.08$ & $-4.85\pm0.10$\
NGC 3379 & $25.23\pm0.08$ & $1.193\pm0.015$ & $30.12\pm0.11$ & $-4.89\pm0.13$ & &\
NGC 4406 & $26.23\pm0.06$ & $1.167\pm0.008$ & $31.17\pm0.14$ & $-4.94\pm0.14$ & &\
NGC 4472 & $26.23\pm0.04$ & $1.218\pm0.011$ & $31.06\pm0.10$ & $-4.83\pm0.09$ & &\
NGC 4536 & $25.43\pm0.12$ & $1.20\phn\pm0.07\phn$ & & & $30.95\pm0.07$ & $-5.52\pm0.14$\
NGC 4636 & $26.07\pm0.08$ & $1.233\pm0.012$ & $30.83\pm0.13$ & $-4.76\pm0.15$ & &\
NGC 4725 & $25.69\pm0.10$ & $1.209\pm0.023$ & $30.61\pm0.34$ & $-4.92\pm0.35$ & $30.57\pm0.08$ & $-4.88\pm0.12$\
Abell 262 & $29.06$ & $1.275\pm0.015$ & $33.99\pm0.20$ & $-4.96$ & &\
Abell 3565 & $28.79$ & $1.199\pm0.015$ & $33.69\pm0.16$ & $-4.92$ & &\
Abell 3742 & $29.03$ & $1.248\pm0.015$ & $34.00\pm0.15$ & $-5.00$ & &\
NGC 4709 & $28.48$ & $1.221\pm0.015$ & $33.04\pm0.17$ & $-4.58$ & &\
NGC 5193 & $28.49$ & $1.208\pm0.015$ & $33.51\pm0.15$ & $-5.04$ & &\
Four Cepheid-bearing galaxies were observed as part of other NICMOS programs, which allowed us to bypass the $I$-band SBF calibration altogether. We determined reliable [$\overline M_{\rm F160W}$]{} values for three of the galaxies using HST Cepheid distances; the fluctuation measurement in NGC 4536 was contaminated by clumpy dust, and it was excluded from the calibration. Cepheid distances are also compiled in Table \[caltable\].
The measured apparent fluctuation magnitudes are very robust. Because slewing between targets and acquiring guide stars takes a significant fraction of an HST orbit, only one or two calibration galaxies could be observed in one orbit. Each integration was at least 256 s, far longer than the minimum time needed to measure SBFs with NICMOS at distances less than 20 Mpc. As a result, the fluctuations were very strong in the calibration images, and the corrections for undetected globular clusters and worms were insignificant. The S/N ratios of the calibration measurements were $\xi{=}15$ to 50.
Absolute fluctuation magnitudes are plotted as a function of [$(V{-}I)$]{} color in Figure \[calibration\]. NGC 1387 and NGC 4536 are excessively dusty, and the extra spatial power from the clumpy dust leads to the anomalously bright fluctuation magnitudes measured (they lie outside the range plotted in Figure \[calibration\]). The dust is easily seen in the NICMOS images, and these two galaxies were rejected from further consideration based on morphology, not their bright fluctuation magnitudes. They were excluded from the calibration fits. The top panel of Figure \[calibration\] shows [$\overline M_{\rm F160W}$]{} derived using Cepheid distances, and the lower panel shows those calculated using $I$-band SBF distances.
The best fits for the larger $I$-band SBF calibration and the direct Cepheid calibration are practically identical. A weighted fit (including the uncertainties both in [$\overline M_{\rm F160W}$]{} and [$(V{-}I)$]{}) has no significant slope in [$\overline M_{\rm F160W}$]{} for galaxies redder than [$(V{-}I)$]{}${>}1.16$. We therefore adopt a uniform [$\overline M_{\rm F160W}$]{} calibration for galaxies in this color range, and note that none of the distant galaxies are likely to have colors bluer than [$(V{-}I)$]{}${=}1.16$ (as described in the next section). The $I$-band SBF distances give $$\overline M_{\rm F160W} = -4.86\pm0.05\,{\rm mag}$$ with an rms scatter of 0.08 mag. The calibration derived using only the three Cepheid measurements is indistinguishable ($-4.85\pm0.06$ mag). The small scatter in values of [$\overline M_{\rm F160W}$]{} for this color range is remarkable, and emphasizes the potential IR SBFs have as a precision distance indicator and probe of stellar populations.
Five additional galaxies from the intermediate-distance sample are plotted in Figure \[calibration\] with open symbols. These galaxies have $I$-band SBF distances measured using HST. They were not included in the calibration fit; instead, their F160W SBF distances were derived using the calibration and they were included in the computation of [$H_{\rm 0}$]{}. If they had been used as calibrators, the calibration would have been 0.015 mag brighter, which is entirely consistent given the standard deviation of 0.05 mag observed in the [$\overline M_{\rm F160W}$]{} fit. The intermediate set is presented in Figure \[calibration\] to demonstrate the overlap between our calibration sample and the distant galaxies from which [$H_{\rm 0}$]{} is derived. While both intermediate-distance galaxies redder than [$(V{-}I)_{\rm0}$]{}$\,{=}\,1.24$ have brighter than average absolute fluctuation magnitudes, we cannot assume that redder galaxies have intrinsically brighter fluctuations. In fact, the results of Jensen et al. (2000) suggest that [$\overline M_{\rm F160W}$]{} gets fainter with increasing [$(V{-}I)$]{}. Different stellar population models (Sec. \[modelsection\]) provide contradictory predictions for fluctuation magnitudes in galaxies redder than [$(V{-}I)$]{}$\,{=}\,1.24$. At this point, we take the conservative approach and adopt a uniform calibration for all the distant galaxies, relying on the overlap (albeit with significant scatter) between the calibrators and the intermediate set. The consistency of [$\overline M_{\rm F160W}$]{} values shown in Figure \[calibration\] suggests that there are probably no significant stellar population differences between the distant brightest cluster galaxies, the nearby ellipticals, and the bulges of the Cepheid-bearing spirals that produce large variations in the F160W absolute fluctuation magnitudes.
The two versions of the calibration presented here are not independent; both rely on many of the same Cepheid calibrators and are subject to the same systematic uncertainties of the Cepheid distance scale. These significant uncertainties are very much the topic of current debate, and include the issues of the distance to the Large Magellanic Cloud (Mould et al. 2000), metallicity corrections to the Cepheid distance scale (Kennicutt et al. 1998; Ferrarese et al. 2000a), and blending of images in the most distant Cepheid measurements (Ferrarese et al. 1998, 2000c; Gibson, Maloney, & Sakai 2000; Stanek & Udalski 2000; Mochejska et al. 2000). This study adopts a distance modulus to the LMC of 18.50 mag. The Cepheid distances adopted are those of Ferrarese et al. (2000b), without the metallicity correction described in Kennicutt et al. (2000).
[$(V{-}I)$]{} Colors
--------------------
As in the optical $I$-band, [$\overline M_{\rm F160W}$]{} shows a dependence on [$(V{-}I)$]{} color such that bluer ellipticals have intrinsically brighter fluctuations. Stellar population models predict a breaking of the age and metallicity degeneracy in the near-IR, and the observed slope of [$\overline M_{\rm F160W}$]{} with [$(V{-}I)$]{} reveals differences between old, metal-poor populations and young, metal-rich galaxies. A sample of NICMOS SBF measurements in galaxies spanning a wide range in [$(V{-}I)$]{} is presented in a companion paper (Jensen et al. 2000) in which stellar population issues are explored. The slope in [$\overline M_{\rm F160W}$]{} with color among the redder ellipticals [$(V{-}I)$]{}${>}1.16$ is insignificant (Fig. \[calibration\]). [$(V{-}I)$]{} colors have been measured for 7 of the 16 distant galaxies in our sample (Lauer et al. 1998), and all are significantly redder than [$(V{-}I)$]{}$\,{=}\,1.16$. Estimates of the [$(V{-}I)$]{} colors for the rest of the distant sample were made by finding the best-fitting relationship between [$(V{-}I)$]{}and ($B{-}R$) for the 7 galaxies for which both colors are known, and then applying the relationship to the ($B{-}R$) data taken from Lauer & Postman (1995). All of the estimated [$(V{-}I)$]{} colors are significantly redder than 1.16 as well. The mean estimated [$(V{-}I)$]{} is 1.246 mag with a standard deviation of 0.027 (averaging all 14 galaxies with known ($B{-}R$) colors); the mean for the 7 galaxies with measured [$(V{-}I)$]{} colors is 1.255 mag. Measured or estimated [$(V{-}I)_{\rm0}$]{} colors listed in Table \[distancetable\] have been corrected for extinction and redshift. We therefore feel secure adopting the calibration determined for galaxies redder than 1.16 for the distant sample. Distances to bluer galaxies will require a reliable [$(V{-}I)_{\rm0}$]{} color measurement and the full color–[$\overline M_{\rm F160W}$]{} relation to be presented in Jensen et al. (2000).
[lccccccccr]{} A262 & $29.06{\pm}0.08$ & $+0.06$ & $-0.36$ & $1.275{\pm}0.015$ & 0.027 & $33.89{\pm}0.10$ & 60 & 4618 & 128\
A496 & $30.80{\pm}0.09$ & $+0.13$ & $-0.18$ & (1.21) & 0.052 & $35.61{\pm}0.11$ & 132 & 9799 & 147\
A779 & $30.10{\pm}0.09$ & $+0.00$ & $-0.14$ & (1.28) & 0.036 & $34.92{\pm}0.11$ & 97 & 7089 & 59\
A1060 & $28.86{\pm}0.07$ & $+0.26$ & $-0.01$ & (1.28) & 0.020 & $33.70{\pm}0.08$ & 55 & 4061 & 102\
A1656(a)& $30.00{\pm}0.12$ & $+0.16$ & $-0.34$ & $1.297{\pm}0.037$ & 0.039 & $34.82{\pm}0.13$ & 92 & 7245 & 377\
A1656(b)& $29.91{\pm}0.08$ & $+0.00$ & $-0.32$ & $1.295{\pm}0.037$ & 0.039 & $34.73{\pm}0.10$ & 88 & 7244 & 377\
A2199 & $30.68{\pm}0.11$ & $+0.00$ & $-0.64$ & & 0.049 & $35.49{\pm}0.12$ & 125 & 8935 & 121\
A2666 & $30.50{\pm}0.12$ & $+0.69$ & $-0.20$ & (1.25) & 0.044 & $35.32{\pm}0.13$ & 116 & 7888 & 30\
A3389 & $30.49{\pm}0.10$ & $+1.14$ & $-0.54$ & (1.24) & 0.042 & $35.31{\pm}0.12$ & 115 & 8105 & 39\
A3565 & $28.79{\pm}0.08$ & $+0.07$ & $-0.02$ & $1.199{\pm}0.015$ & 0.019 & $33.63{\pm}0.09$ & 53 & 4142 & 15\
A3581 & $29.97{\pm}0.08$ & $+0.24$ & $-0.77$ & (1.27) & 0.035 & $34.80{\pm}0.09$ & 91 & 6778 & 29\
A3656 & $29.62{\pm}0.07$ & $+0.13$ & $-0.07$ & (1.24) & 0.031 & $34.45{\pm}0.09$ & 78 & 5607 & 18\
A3742 & $29.03{\pm}0.11$ & $+0.05$ & $-0.16$ & $1.248{\pm}0.015$ & 0.026 & $33.86{\pm}0.12$ & 59 & 4801 & 20\
N4073 & $30.16{\pm}0.12$ & $+0.36$ & $-0.67$ & & 0.032 & $34.99{\pm}0.13$ & 99 & 6306 & 1\
N4709 & $28.48{\pm}0.07$ & $+0.00$ & $-0.14$ & $1.221{\pm}0.015$ & 0.017 & $33.32{\pm}0.08$ & 46 & 4905 & 1\
N5193 & $28.49{\pm}0.06$ & $+0.36$ & $-0.15$ & $1.208{\pm}0.015$ & 0.019 & $33.33{\pm}0.08$ & 46 & 3920 & 1\
Comparison with Single-Burst Stellar Population Models \[modelsection\]
-----------------------------------------------------------------------
If the intrinsic luminosity of the brightest stars in a population is known, fluctuation distances can be determined directly without an empirical calibration based on another distance indicator. Stellar population models can be used to compute theoretical absolute fluctuation magnitudes by determining the second moment of the luminosity function for an ensemble of stars of a particular age and metallicity. In practice, we adopt the empirical calibration because of the uncertainties involved in modeling populations and because of the variations in the ages and metallicities of real galaxies. Nevertheless, it is useful to compare the empirical calibration to the theoretical predictions of stellar population models.
In Figure \[models\] we plotted the same calibration data points shown in Figure \[calibration\] over three different sets of models. The top panel shows the recent model predictions of Liu, Charlot, & Graham (2000) for the F160W filter. These are the same models used to compute the redshift corrections $k(z)$ to our fluctuation magnitudes. The models plotted in the middle panel were taken from Blakeslee, Vazdekis, & Ajhar (2000). These models were computed for the $H$-band filter and shifted to F160W using the relation: $$\overline M_{\rm F160W} = \overline M_H +0.1(\overline M_J - \overline M_K)$$ (J. Blakeslee, private communication, and Stephens et al. 2000). Finally, Worthey’s (1994) models are plotted in the bottom panel for the F160W filter (G. Worthey, private communication). In all three sets of models, dashed lines indicate isochrones ranging from approximately 5 Gyr at the top to 17 Gyr on the bottom. The fine dotted lines indicate models of constant metallicity. In the Liu et al. case, the SBF measurements straddle the solar-metallicity line, and the next line to the left is \[Fe/H\]${=}\,{-}0.4$. In the center panel, the points are closest to the \[Fe/H\]${=}\,0.2$ models of Blakeslee et al. (2000) and reach down to the solar metallicity line. The transformation to the F160W scale, and hence the vertical position of the models, is somewhat uncertain however. In the bottom panel, the points are closest to the \[Fe/H\]${=}\,{-}0.25$ line; the next metallicity line down is \[Fe/H\]${=}\,{-}0.5$. The calibration data presented here cover a limited range in color, and appear consistent with stellar populations near solar metallicity (between -0.25 and 0.25) and potentially covering a wide range of ages. A detailed comparison of the models with a NICMOS data set covering a much wider range of [$(V{-}I)$]{} colors will be presented in Jensen et al. (2000).
The F160W SBF Hubble Diagram\[hubblesection\]
=============================================
Distances and Uncertainties
---------------------------
To determine the distance to each galaxy, we adopted [$\overline M_{\rm F160W}$]{}${=}\,-4.86\pm0.05$ and computed the $k(z)$-corrected distance modulus: $$(m{-}M) = \overline m_{\rm F160W} - \overline M_{\rm F160W}-k(z).$$ Redshift corrections $k(z)$ to fluctuation magnitudes in the F160W-band were taken from the Liu et al. (2000) models for metallicities between \[Fe/H\]${=}\,{-}0.4$ to 0.0 and old stellar populations. $k(z)$ corrections are listed in Table \[distancetable\]. We compared these $k(z)$ corrections to those determined using Worthey’s models for solar metallicity (G. Worthey, private communication) and found small differences of order ${\lesssim}0.01$ mag. At distances of 10,000 [kms$^{-1}$]{} and less, the magnitude of the $k(z)$ corrections are insensitive to the details of the stellar population models.
[lr]{} PSF normalization and fitting& 0.06 mag\
Sky subtraction& 0.01 mag\
Globular cluster and background galaxy removal& 0.07 mag\
Galaxy profile subtraction& 0.02 mag\
Bias subtraction& 0.01 mag\
Wormy background correction (see text)& ${\sim}0.08$ mag\
[$\overline M_{\rm F160W}$]{} calibration & 0.05 mag\
CMB Velocities& 200 [kms$^{-1}$]{}\
NICMOS photometric zero point& 0.02 mag\
Cepheid distance calibration& 0.16 mag\
The uncertainties in the fluctuation magnitudes in Table \[distancetable\] are the contributions from PSF fitting, sky subtraction, bias removal, and galaxy subtraction, all added in quadrature. The uncertainties in the distance moduli include the uncertainty in [$\overline M_{\rm F160W}$]{} of 0.05 mag. Typical values for the individual uncertainties are listed in Table \[errorbudget\]. We also determined the range of fluctuation magnitudes permitted given the level of worminess in the background and the agreement between individual annuli. Maximum and minimum values derived from the residual background corrections are listed separately from the other sources of uncertainty in Table \[distancetable\].
We treated the different uncertainties as if they were independent, but acknowledge the fact that there are subtle correlations between sources of uncertainty that are difficult to quantify. For example, the procedure that is used to fit and subtract the smooth galaxy profile is affected by errors in sky subtraction. While relationships between sources of uncertainty exist, they are insignificant to the results of this study. Examination of Table \[errorbudget\] shows that the significant sources of uncertainty in the distance measurement are the PSF fit, globular cluster correction, and the intrinsic scatter in the [$\overline M_{\rm F160W}$]{} calibration. The first is due mainly to variations in the drift and focus of the telescope and brightness of the PSF stars. The second is principally a function of the depth of the observation and size of the globular cluster population. The cosmic scatter in [$\overline M_{\rm F160W}$]{} is a result of variations in the stellar populations of galaxies. These uncertainties are independent and may safely be added in quadrature. Furthermore, in many cases even these uncertainties are secondary to the larger uncertainty in the correction for worminess in the background, which is a function of time since the last SAA passage.
Measuring the uncertainty due to residual background patterns was difficult; to make an estimate, we explored the range of correction that is permitted by the data by subtracting various levels of uniformly distributed residual spatial power, and thereby found the maximum and minimum fluctuation magnitudes allowed. The most likely fluctuation magnitude for each galaxy was determined taking into account the details described in the notes in the appendix. Rather than assume a Gaussian distribution of errors about the most probable value, we chose to adopt a probability distribution that increases linearly from zero at the maximum and minimum allowed values to the most likely value and is normalized appropriately. The probability function is not symmetrical about the most likely value because the measurement is not usually midway between the maximum and minimum allowed values. We convolved this skewed saw-tooth distribution function with the normal probability distribution of the other sources of uncertainty to get the probability distribution function that was used to determine [$H_{\rm 0}$]{}.
Some systematic errors listed in Table \[errorbudget\] affect all our measurements equally, and are not included in the uncertainties in Table \[distancetable\]. The first of these is the 0.02 mag uncertainty in the photometric zero point of the F160W filter in the NIC2 camera. The other systematic errors we inherit from the Cepheid distances adopted, either directly or via the $I$-band SBF calibration. The systematic uncertainty in the Cepheid distance scale of 0.16 mag includes the 0.13 mag uncertainty in the distance to the LMC and the 0.02 mag uncertainty in the zero point of the period-luminosity relationship for Cepheid variables. The systematic photometric uncertainty in the WFPC-2 measurements contributes another 0.09 mag to the Cepheid distances. A detailed discussion of these uncertainties can be found in Ferrarese et al. (2000a). Adding all sources of systematic uncertainty in quadrature gives 0.16 mag. An additional systematic uncertainty from the $I$-band SBF distance scale is not included because the $I$-band distances are only used to link the Cepheid calibration to the distant galaxies of our sample. The $I$-band SBF systematic uncertainties are the same as those already discussed, and it would not be correct to include them twice. The 0.01 mag agreement between the $I$-band SBF and the direct Cepheid calibrations confirms that no additional systematic error is incurred by adopting the $I$-band SBF distances for the calibration.
Radial Velocities
-----------------
The heliocentric velocity for each cluster or galaxy was initially measured or collected from the literature by Postman & Lauer (1995, and references therein). The uncertainties on the individual redshift measurements were typically 60 [kms$^{-1}$]{}. New data now available provide velocities to additional cluster members and have been included in this study. Radial velocities are compiled in Table \[distancetable\], along with the number of individual galaxy redshifts that were averaged to get the cluster velocity. The details of how galaxies were selected for inclusion are described by Postman & Lauer (1995). The mean uncertainty in the mean cluster redshift is 184 [kms$^{-1}$]{} for the Postman & Lauer sample. NGC 4709 is listed in Table \[distancetable\] with its own radial velocity; it is a member of the high-velocity (4500 [kms$^{-1}$]{}) component of the Centaurus cluster, and hence has a significant peculiar velocity. NGC 5193 is also listed with its own redshift; it was previously thought to be the cD galaxy in Abell 3560, but Willmer et al. (1999) found that it is in fact a foreground galaxy. NGC 4073 is not associated with a cluster; its heliocentric velocity was taken from Beers et al. (1995). The heliocentric velocities were converted to the reference frame that is at rest with respect to the cosmic microwave background (CMB) radiation. The CMB dipole adopted was that measured by Lineweaver et al. (1996).
The Model Velocity Field and [$H_{\rm 0}$]{}\[flowsection\]
-----------------------------------------------------------
Measurements of the Hubble constant within 50 Mpc must take peculiar velocities into account because they can be a significant fraction of the Hubble velocity. In fact, one of the differences between the Hubble constants measured by Tonry et al. (SBF-II) and Ferrarese et al. (2000a) using the [*same*]{} Cepheid calibrators and the [*same*]{} SBF measurements (albeit with a slightly different calibration) was the result of different assumptions about the local velocity field. We have chosen our distant sample to be distributed in such a way as to minimize sensitivity to local peculiar velocities (Fig. \[supergal\]). By far the greatest immunity to peculiar velocities comes from reaching much greater distances than previously possible. At 130 Mpc, we expect peculiar velocities to be approximately 3% of the Hubble velocity. This insensitivity to peculiar velocities and isotropic distribution of the distant sample produced the most accurate SBF measurement of [$H_{\rm 0}$]{} to date.
We followed the SBF-II maximum-likelihood procedure for computing [$H_{\rm 0}$]{}. We first constructed a model velocity field, which included a 187 [kms$^{-1}$]{} cosmic thermal velocity dispersion. The quadrupole term adopted in SBF-II was not included. Various dipole terms (resulting from the peculiar velocity of the Local Group in the CMB frame) were tried, and a comparison is presented below.
At the position of each galaxy as defined by the F160W SBF distance, the most likely velocity was determined from the velocity model. A number of points were then chosen radially spanning the range of possible distances given the uncertainties in the measured distances. At each point, the joint likelihood of a given combination of distance and velocity measurements was computed, and the likelihood integrated across the radial range in distance. The Hubble constant is a free parameter of the velocity model, and this procedure was repeated to find the value of [$H_{\rm 0}$]{} which maximizes the likelihood of all the distance and velocity measurements together. This procedure used the distance probability distribution function constructed by convolving the normal Gaussian uncertainties with the saw-tooth probability distribution between the maximum and minimum [$\overline m_{\rm F160W}$]{} values. In practice, we attempted to [*minimize*]{} the negative likelihood statistic ${\cal N}$ (see SBF-II for details). The value of $\chi^2$ determined using the maximum likelihood technique and our non-Gaussian probability distributions is not necessarily minimized when ${\cal N}$ is minimized; however, the difference between values of ${\cal N}$ for different input parameters to the velocity model is equivalent to a difference in $\chi^2$. In Table \[flowstable\] we compared the likelihood of various models by indicating the difference in $\chi^2$ relative to the baseline model that ignores all peculiar velocities except the motion of the Local Group in the CMB frame.
Several velocity models were used to determine the sensitivity of the [$H_{\rm 0}$]{} measurement to the input parameters of the models. Results for these tests are listed in Table \[flowstable\]. NGC 4709 was excluded from all fits because the velocity field of the complex Centaurus cluster was not included in the velocity model. The models tried were constructed as follows:
[clclcrl]{} (1) & CMB only& && $76.1{\pm}1.3$ & 0.0 & Lineweaver et al. 1996\
(2) & Virgo, GA & dipole& (306,43) & $205{\pm}83$ & $77.1{\pm}1.6$ &$-$0.2 & SBF-II\
(3) & Willick & Batra dipole& (274,67) & $243$ & $75.6{\pm}1.3$ & 0.2 & Willick & Batra 2000\
(4) & Giovanelli et al. dipole& (295,28) & $151{\pm}120$ & $75.5{\pm}1.3$ & 0.4 & Giovanelli et al. 1998\
(5) & Virgo, GA & dipole& (355,56) & $409{\pm}335$ & $76.9{\pm}1.5$ & 1.4 & SBF-II+free dipole\
(6) & Lauer & Postman dipole& (343,52) & $689{\pm}178$ & $73.8{\pm}1.5$ & 9.7 & Lauer & Postman 1994\
(7) & Model (2) + Bubble& (306,43) & $205{\pm}83$ & $72.3{\pm}2.3$ & & Zehavi et al. 1998\
\(1) The first model does not include any local attractors or peculiar velocities beyond that of the Local Group in the CMB frame. In the CMB frame, [$H_{\rm 0}$]{}${=}76.1$ [kms$^{-1}$Mpc$^{-1}$]{}. We adopt the CMB model as the baseline and compare other models by computing the change in $\chi^2$ relative to this case.
\(2) Adding the contributions from the Virgo and GA mass concentrations and dipole as prescribed by SBF-II increases [$H_{\rm 0}$]{} to 77.1. The slight decrease in $\chi^2$ is not significant. The quadrupole term suggested by SBF-II was not included because it is inappropriate for the distances of the galaxies in our sample (including it would increase $\chi^2$ by 25!).
(3$-$4) For model 3 we used the dipole determined by Willick & Batra (2000). Model 4 includes the dipole measured by Giovanelli et al. (1998) using Tully-Fisher measurements to many clusters out to redshifts of 9000 [kms$^{-1}$]{}. The likelihood of these two dipole models is essentially the same as the best-fitting SBF-II models and the CMB-only baseline model. The Hubble constant implied by these models is approximately 75.5 [kms$^{-1}$Mpc$^{-1}$]{}. The largest difference in [$H_{\rm 0}$]{} between models 1 to 4, which have essentially the same likelihood, is only 1.6 [kms$^{-1}$Mpc$^{-1}$]{}.
\(5) Like model 2, the fifth model used the mass distribution suggested by SBF-II, but allowed the maximum likelihood procedure to determine the most likely dipole velocity in addition to [$H_{\rm 0}$]{}. Despite having more freedom to fit the data with three additional degrees of freedom, the fit is worse and and $\Delta\chi^2$ is larger. The dipole determined is large, but barely larger than the uncertainty. We have sampled the velocity field with only 16 points spanning a range in distance from 50 to 150 Mpc, and the sample was chosen to minimize sensitivity to streaming motions that could bias the measurement of [$H_{\rm 0}$]{}. Towards this end we were successful; the small variation in [$H_{\rm 0}$]{} between the various models confirms this conclusion. On the other hand, to reliably measure the bulk motion of the galaxies in the local universe, distances would need to be measured to a much larger sample of galaxies within the redshift interval of interest (7000 to 10,000 [kms$^{-1}$]{} in this study).
\(6) The one dipole presented here that fails to fit our data very well is the large dipole velocity of 689 [kms$^{-1}$]{}measured by Lauer & Postman (1994). Using the Lauer & Postman dipole reduces [$H_{\rm 0}$]{} by approximately 3 [kms$^{-1}$Mpc$^{-1}$]{}, but $\chi^2$ is significantly larger. It is, however, closer to the free dipole of model 5 than the other model dipoles considered.
Based on the full data set, we conclude that $$H_{0} = 76 \pm 1.3\ ({\rm random}) \pm 6\ ({\rm systematic})\
{\rm km\,s}^{-1}\,{\rm Mpc}^{-1}.$$ The 1-$\sigma$ random uncertainty formally includes all sources of uncertainty in the distance measurement, including the non-Gaussian uncertainty from the residual background correction. Although $\chi^2$ per degree of freedom is not minimized for our non-Gaussian probability distribution, its value of 1.0 for model 1 indicates that the adopted uncertainties are reasonable. The Gaussian 1-$\sigma$ error bars are plotted with thick lines in the Hubble diagram (Fig. \[hubblediagram\]). The full non-Gaussian ranges allowed by the various corrections to the SBF distances are indicated by the lighter lines underneath each point. The Hubble diagram is shown using the CMB velocities (model 1), and our best-fit value of [$H_{\rm 0}$]{}${=}76$ [kms$^{-1}$Mpc$^{-1}$]{} is indicated by the dashed line.
A second line in Figure \[hubblediagram\] indicates the decrease in [$H_{\rm 0}$]{}beyond 70 h$^{-1}$ Mpc (${\sim}7000$ [kms$^{-1}$]{}) suggested by Zehavi et al. (1998). Their “Hubble bubble” model hypothesizes that a locally under-dense region of the Universe gives rise to an expansion rate about 6% higher within 70 h$^{-1}$ Mpc. The SBF analysis was repeated using only the six most distant galaxies and the model 3 (SBF-II) velocity model (model 7 in Table \[flowstable\]). The six galaxies were chosen to minimize [$H_{\rm 0}$]{} and provide the best match to the decrease in [$H_{\rm 0}$]{} predicted by Zehavi et al. (1998). The result, 72.3 [kms$^{-1}$Mpc$^{-1}$]{}, reproduces nearly perfectly the predicted decrease in [$H_{\rm 0}$]{}.
The lower panel in Figure \[hubblediagram\] shows the Hubble ratio $v_{\rm CMB}/d$ for each galaxy. The error bars are shown for the Gaussian component of the distance error, and do not include the uncertainty range from the residual background correction. The curved lines behind each point show the full range of possible distances if a different background correction $P_g$ were adopted. The longest arcs are necessarily those points with the worst worminess, the largest corrections, and the lowest S/N ratios. The best-fit value of [$H_{\rm 0}$]{}$\,{=}\,76$ [kms$^{-1}$Mpc$^{-1}$]{} is indicated by the horizontal line. Once again, the Zehavi et al. (1998) predicted decrease in [$H_{\rm 0}$]{} is shown.
We also explored the sensitivity of our results to the low-S/N observations. Several of the measurements are quite poor, and should arguably be excluded from the fits. Excluding all galaxies obviously contaminated by worms (Abell 262, 2666, 3389, 3581, 3742 and NGC 4073) gives [$H_{\rm 0}$]{}$\,{=}\,77.4\pm1.7$ [kms$^{-1}$Mpc$^{-1}$]{} for model 2, which is nearly the same value determined using the entire data set of 77.1. If we exclude only the three galaxies with $\xi{<}1$ (Abell 2666, Abell 3389, and Abell 3581), then we find that [$H_{\rm 0}$]{}$\,{=}\,78.0\pm1.6$. It is clear that the worst measurements are not systematically biasing the measurement of [$H_{\rm 0}$]{}. This is not surprising, as the maximum likelihood code takes into account the large range of possible distances for these galaxies. The fact that there is no systematic offset shows that the $P_g$ corrections are applied uniformly and that the residual background is not systematically over or under-subtracted. When the six galaxies that have dust lanes or disks are exluded (Abell 262, 1060, 2199, 3565, 3581, and NGC 5193), [$H_{\rm 0}$]{}$\,{=}\,75.7\pm1.5$ [kms$^{-1}$Mpc$^{-1}$]{}. This value is only 1.4 [kms$^{-1}$Mpc$^{-1}$]{} smaller than 77.1, and suggests that clumpy dust in some galaxies does not introduce a significant bias to the distance measurements. If there were a significant bias in the distance measurements due to residual cosmic rays, clumpy dust, improper bias removal, undetected globular clusters, or any of the other sources of variance discussed above, the result would be increasingly underestimated distances as redshifts increase. In fact, the opposite trend is observed: the highest redshift galaxies have somewhat larger measured distances than expected, as predicted by the Zehavi et al. (1998) Hubble bubble model.
Can [$H_{\rm 0}$]{} be 65 [kms$^{-1}$Mpc$^{-1}$]{}?\[whatifsection\]
====================================================================
Several groups have recently reported measurements of the Hubble constant derived from HST Cepheid distance calibrations of various secondary distance indicators. Our best-fit measurement of [$H_{\rm 0}$]{}$\,{=}\,76\,\pm\,1.3\,\pm\,6$ [kms$^{-1}$Mpc$^{-1}$]{} is in good agreement (better than 1$\sigma$) with several, including optical SBFs (SBF-II; Lauer et al. 1998), Cepheid distances alone (Willick & Batra 2000), fundamental plane distances (Kelson et al. 2000), and Tully Fisher distances (Sakai et al. 2000). The SBF Hubble constant as calibrated by Ferrarese et al. (2000a) differs from ours at the 1.5-$\sigma$ level for the same reasons it differs from SBF-II: a slightly different SBF calibration and different peculiar velocities were adopted for the four clusters measured by Lauer et al. (see Ferrarese et al. for a discussion). Our measurement of [$H_{\rm 0}$]{} differs significantly from the results based on type-Ia supernovae. Gibson et al. (2000) report [$H_{\rm 0}$]{}$\,{=}\,68\pm2\pm5$ [kms$^{-1}$Mpc$^{-1}$]{}, 12% lower than our value. Parodi et al. (2000) found [$H_{\rm 0}$]{}$\,{=}\,58.5\pm6.3$ [kms$^{-1}$Mpc$^{-1}$]{} (90% confidence level), which is 24% smaller. These two supernovae measurements cannot be directly compared, however, because of a significant difference (0.18 mag) between the calibration adopted by Parodi et al. and that used by the HST Key Project team (ie., Gibson et al.) The Key Project Cepheid calibration was adopted for this study.
Is it possible that [$H_{\rm 0}$]{}$\,{=}\,65$ [kms$^{-1}$Mpc$^{-1}$]{}, and that we have overestimated it by 15% (or more)? Roughly one third of this difference disappears if the Zehavi et al. (1998) Hubble bubble model is correct. If galaxies nearer than ${\sim}7000$ [kms$^{-1}$]{} must be disregarded because of their enhanced velocities away from a locally under-dense region, then our measurement of [$H_{\rm 0}$]{} can be as low as 72 [kms$^{-1}$Mpc$^{-1}$]{}. Our results do not demand that this be the case, however, and are still consistent with observations that refute the Hubble bubble hypothesis (Giovanelli et al. 1999, Lahav 2000). The fits including all the data, assuming a smooth Hubble flow, are equally good because the nearer meausurements tend to be most reliable.
Extinction from clumpy dust in the individual galaxies could add to the fluctuation power and bias the SBF measurements to shorter distances. No sign of dust was seen in the images of the galaxies used to determine the calibration of [$\overline M_{\rm F160W}$]{} (aside from NGC 1387 and NGC 4536, which were excluded). Six of the distant galaxies do have obvious dust lanes (Table \[powertable\]). Only one (Abell 262) has extensive dust, and we used the optical WFPC-2 image of Lauer et al. (1998) to mask the dusty regions. The other five have dust lanes or disks that are concentrated near the centers of the galaxies, which were masked. There is no evidence for extended patches of dust with sizes comparable to the PSF and smooth on larger scales. The distances for these are not systematically smaller than the others at the same redshifts, nor are their colors redder on average. The Hubble constant measured with the six dusty galaxies excluded was not significantly smaller, and it seems unlikely that all the distant galaxies would have uniformly distributed clumpy dust that would be unrecognizable in our images.
Besides the potential 6% reduction in [$H_{\rm 0}$]{} beyond ${\sim}100$ Mpc suggested by the six most distant measurements, are there systematic problems with the F160W SBF measurements that could explain another ${\sim}$10% (or more) difference between our results and the conclusions of the supernovae measurements? There are potentially three sources of extra power in the power spectrum that are not convolved with the diffraction pattern of the telescope, but do have power on the spatial scales over which we fit the SBF power spectrum. The first of these is the residual bias scaled by the flat field image. We addressed this possibility by explicitly subtracting a dithered bias$\times$flat image as described in Section \[reduxsection\]. The uncertainty in [$H_{\rm 0}$]{} resulting from errors in bias subtraction was measured and found to be less than 1% (Table \[errorbudget\]).
The second potential contributor to the power spectrum is the residual wormy background. We carefully excluded wormy images and subtracted an estimate of the residual power as described in Section \[reduxsection\]. If the correction for worms were systematically underestimated, then our measurement of [$H_{\rm 0}$]{} would be too large. In the previous section we showed that excluding the galaxies contaminated by worms had no significant effect on the measurement of [$H_{\rm 0}$]{}. Excluding the lowest S/N measurements also had no significant effect. The range of [$H_{\rm 0}$]{} values seen during these tests was less than 1 [kms$^{-1}$Mpc$^{-1}$]{}. This suggests that the corrections were applied uniformly.
The third potential contributor to the power spectrum is the residual structure in the background from subtraction of the model galaxy profile. To avoid any bias because of the somewhat arbitrary fit of the left-over large-scale structure in the galaxy, we excluded wavenumbers smaller than 20 from our analysis. The mean uncertainty in the distance modulus from galaxy and smooth background subtraction was 0.02 mag. Residual galactic structure could explain a 1% bias in our [$H_{\rm 0}$]{} measurement, but not a large systematic error.
If the fluctuation powers we measure are reliable, is it possible that other sources of systematic error could cause us to overestimate [$H_{\rm 0}$]{} by 10 to 15%? Perhaps the most obvious candidate for this kind of systematic error would be the calibration of the F160W absolute fluctuation magnitude. We used both Cepheid and $I$-band SBF distances to determine the calibration. Although not completely independent, the two calibrations are remarkably consistent (0.01 mag). The agreement between the $I$-band SBF and direct Cepheid calibration supports the conclusion that there is no significant difference in the fluctuation amplitudes between early and late-type galaxies. The applicability of the calibration to the more distant galaxies is demonstrated by the overlap with the intermediate-distance sample. To explain a 15% difference in [$H_{\rm 0}$]{}, [$\overline M_{\rm F160W}$]{} would have to be brighter by 0.3 mag, or [$\overline M_{\rm F160W}$]{}$\,{=}-5.16$. Examination of Figure \[calibration\] shows that a calibration as bright as $-$5.16 is inconsistent with the data.
Significant systematic errors in the Cepheid distance scale are relevant to the measurement of the true value of the Hubble constant, but cannot explain the difference between our measurement and that determined using type-Ia supernovae because we adopted the same Cepheid calibration as the other groups listed at the beginning of this section. A systematic error in the distance to the LMC (for example) will affect our measurement of [$H_{\rm 0}$]{} in exactly the same way as the other measurements.
Is it possible that the mundane choice of Galactic extinction corrections could result in a systematic calibration error at the 15% level? By observing in the near-IR, our sensitivity to errors in the extinction are significantly reduced. The largest correction in our sample is 0.08 mag (Table \[sampletable\]). Most of the calibrator galaxies have IR extinction corrections of order 0.01 mag. If extinction has been underestimated, the true fluctuation magnitudes will be brighter than we have estimated and the distances smaller. Increasing extinction corrections makes [$H_{\rm 0}$]{} larger. On the other hand, extinction cannot have been overestimated by very much because the corrections are already very close to zero. The only other way to get distance measurements that are systematically underestimated by 15% would be for extinction corrections to all three Cepheid calibrators used in this paper and all the Cepheid calibrators used by Tonry et al. (SBF-II) to be overestimated by 0.3 mag, but not those used to calibrate the supernova distance scale. It seems unlikely that Galactic extinction could be the cause of so large a systematic error.
One reason that previous measurements of [$H_{\rm 0}$]{} have disagreed with each other has been the choice of velocities used (Ferrarese et al. 2000a, Mould et al. 2000). When [$H_{\rm 0}$]{} is measured on scales where the peculiar motions of individual galaxies are a significant fraction of the Hubble velocity, the value of [$H_{\rm 0}$]{} will depend quite sensitively on the velocity adjustments made for infall into local mass concentrations. Our measurement of [$H_{\rm 0}$]{} reaches well into the Hubble flow and is distributed uniformly on the sky, and is therefore very insensitive to the choice of velocity model and the peculiar velocities of individual galaxies and clusters (as described in Section \[flowsection\]).
Finally, we cannot rule out the possibility that modest systematic errors affect both F160W SBF and type-Ia supernovae distance measurement techniques in such a way to create the difference between the measurements.
Summary
=======
We measured accurate IR SBF distances to a collection of 16 uniformly-distributed distant galaxies for the purpose of measuring the Hubble constant well beyond the influence of local peculiar velocities. These NICMOS measurements mark the first time SBFs have been measured in galaxies out to redshifts of 10,000 [kms$^{-1}$]{}, clearly demonstrating the advantages of measuring SBFs in the near-IR with excellent spatial resolution and low background. The calibration of the F160W SBF distance scale presented here was based on SBF measurements of galaxies in which Cepheid variable stars were detected in the same galaxy. Using a maximum-likelihood technique to account both for the influence of local mass concentrations on the velocity field and the non-Gaussian uncertainties on our SBF distance measurements yields a Hubble contsant of [$H_{\rm 0}$]{}$\,{=}\,76\pm1.3$ [kms$^{-1}$Mpc$^{-1}$]{} (1-$\sigma$ statistical uncertainty) with an additional systematic uncertainty of 6 [kms$^{-1}$Mpc$^{-1}$]{}, primarily the result of uncertainty in the distance to the LMC. The small statistical uncertainty in [$H_{\rm 0}$]{} is a result of the fact that our measurement is very insensitive to peculiar velocities, stellar population variations, extinction corrections, and photometric errors. Arbitrarily excluding all but the six most distant galaxies from the fit results in a 6% decrease in [$H_{\rm 0}$]{}, consistent with the hypothesis that the Local Group is located in an under-dense region of the universe.
This study benefitted greatly from NICMOS data collected as part of several programs, and we thank those who worked to acquire that data. In particular, we are grateful to those who helped ensure that the data would be appropriate for SBF analysis and assisted with the data reductions (D. Geisler, J. Elias, J. R. Graham, and S. Charlot). We are endebted to the NICMOS GTO team for their hard work in building, calibrating, and providing software for NICMOS. We greatly appreciated the helpful comments of R. Weymann, J. Blakeslee, and L. Ferrarese. The calibration presented here made use of data collected by the Optical SBF team (J. Tonry, J. Blakeslee, E. Ajhar, and A. Dressler), and we thank them for providing color photometry and $I$-band SBF distances to the calibration galaxies. Finally, we wish to thank G. Worthey, S. Charlot, and A. Vazdekis for constructing stellar population models and providing appropriate $k(z)$-corrections.
This research was supported in part by NASA grant GO-07453.0196A. The NICMOS GTO team was supported by NASA grant NAG 5-3042. J. Jensen acknowledges the support of the Gemini Observatory, which is operated by the Association for Research in Astronomy, Inc., under a cooperative agreement with the National Science Foundation on behalf of the Gemini partnership: the National Science Foundation (United States), the Particle Physics and Astronomy Research Council (United Kingdom), the National Research Council (Canada), CONICYT (Chile), the Australian Research Council (Australia), CNPq (Brazil) and CONICET (Argentina).
Appendix: Notes
===============
Abell 262 (NGC 708): The central galaxy in Abell 262 is littered with dust. We used the high-resolution $I$-band WFPC-2 images (Lauer et al. 1998) to identify dusty regions and create a mask for our NICMOS image. In addition to the copious dust, we had to exclude exposures because of worminess in the background. The uncertainty in fluctuation magnitude is relatively large because of the dustiness and wormy background corrections, even though Abell 262 is among the closest of the clusters we observed.
Abell 496 (PGC 015524): This cluster is the most distant in our sample, and we allocated 3 orbits to ensure a good SBF measurement. Of the 20 individual exposures, only the last two were found to be wormy. The other 18 are unaffected. The S/N is good and the fluctuation measurement is reliable.
Abell 779 (NGC 2832): Aside from a little dither-pattern noise in the power spectrum, the results for Abell 779 are quite good. Pattern noise is an array of spots in the spatial power spectrum with a periodicity corresponding to the 20-pixel offset of the dither pattern. Detector artifacts (e.g., vertical bands or mismatches in the background level at quadrant boundaries) were sometimes incompletely removed by the image reduction procedures and cause pattern noise. Pattern noise is only significant in the power spectrum of the outermost annulus.
Abell 1060 (NGC 3311): The central galaxy in the Hydra cluster was one of four galaxies presented here that were observed by D. Geisler, J. Elias and E. Ajhar as part of NICMOS program 7820. The images were reduced for SBF analysis using the software and procedures described in this paper. NGC 3311 has some dust in the central region that was masked; the SBF fit in the outer regions is nearly perfect and the S/N ratio is very high.
Abell 1656(a) (IC 4051): Two galaxies in the Coma cluster were observed by D. Geisler et al. IC 4051 has an unusually large population of globular clusters (Baum et al. 1997). We found that many are much brighter than expected for a galaxy at this distance. We modified the luminosity fitting parameters and estimated the contribution from unresolved GCs and subtracted it, but a relatively large uncertainty in the GC contribution to the SBF power remains. The fit to the SBF power spectrum is good.
Abell 1656(b) (NGC 4874): The power spectrum for NGC 4874 is clean and the fit is very good. NGC 4874 has a normal globular cluster population (Harris et al. 2000).
Abell 2199 (NGC 6166): The central galaxy in Abell 2199 has dust lanes within 3 arcsec of the center. The dust lanes were masked prior to performing the SBF analysis, but measurements in the innermost aperture are suspect. The SBF analysis did not include the region between NGC 6166 and two nearby companions, where the fit to the galaxy profile is not very good. The fit to the power spectum in the intermediate annulus was excellent, but the outer two apertures disagree at a level (0.64 mag) that cannot be corrected properly by adopting a value of $P_g$ that scales with area. We adopted the measurement in the intermediate aperture and a relatively large range of permitted fluctuation magnitudes.
Abell 2666 (NGC 7768): NGC 7768 has surprisingly few globular clusters, which is consistent with the measurements of Harris, Pritchet, & McClure (1995) and Blakeslee, Tonry, & Metzger (1997). Our attempts to fit a luminosity function to a half-dozen objects failed to produce a reasonable correction for undetected globular clusters. The SBF analysis proceeded without a GC correction, and we adopted an uncertainty that is larger than the other galaxies that reflects our lack of knowledge of the GC luminosity function. The only way this measurement could be significantly biased by undetected GCs is if the GC luminosity function is skewed to the faint end and contains practically no GCs on the bright side of the peak. The Abell 2666 observation is also contaminated by a wormy background. One of the six exposures is excluded, and the worst regions masked in the two subsequent exposures. There is always a tradeoff between including frames that increase the SBF signal but also contain the decaying wormy background. In this case, a good fit to the power spectrum was achieved, but a significant correction for the background must be applied to make the outer two apertures agree. $P_g$ is further enhanced by the presence of undetected globular clusters and background galaxies that could not be handled with the usual procedure of fitting luminosity functions due to the paucity of bright objects in this field. $\xi{<}1$ for this galaxy and the possible range of fluctuation magnitudes is therefore quite large.
Abell 3389 (NGC 2235): We observed the central galaxy in Abell 3389 in the continuous viewing zone to achieve a longer total integration time for this galaxy. Unfortunately, the longer MULTIACCUM sequences used to avoid frequent NICMOS buffer dumps had many more persistent cosmic rays in each image and a significantly wormy background. We abandoned half of our images, and the remaining ones must be corrected for residual worminess. As a result, $P_g$ is significant and $\xi$ is less than unity. The fluctuation power increases significantly with aperture area, and the globular cluster and worminess corrections are large. The uncertainties reflect the fact that the fluctuation magnitude is poorly constrained.
Abell 3565 (IC 4296): IC 4296 has a compact dust ring close to the nucleus, but no sign of dust outside of a radius of 1.5 arcsec. Very small residual spatial variance corrections ($P_g$) bring the annuli into nearly perfect agreement. The S/N of this measurement is high and the GC correction small.
Abell 3581 (IC 4374): the observations of IC 4374 were strongly affected by worminess in the background. Two of six exposures were excluded from the final image, and residual worms were masked in three of the remaining four. The potential for bias is strong in this case, and a significant $P_g$ correction for background power was applied, resulting in a $\xi{<}1$ and a large range of allowed fluctuation magnitudes. Furthermore, the central regions contain a dust lane, which we masked.
Abell 3656 (IC 4931): The only problem that arose in the analysis of IC 4931 is the presence of dither pattern noise in the background. This problem is only significant in the largest annulus; the inner two agree nicely.
Abell 3742 (NGC 7014): The HST failed to lock onto the guide stars for the observations of NGC 7014. Because some drift occurred during the MULTIACCUM sequences, our IDL procedures interpreted the changing flux in each pixel as cosmic rays. To overcome this problem, we were forced to abandon the temporal cosmic ray rejection and rely on the spatial information alone. The galaxy fitting routine also had trouble because of the smeared image. The residual image shows extra background structure close to the center where the galaxy fit is worst. In this case, we included a small correction to the fluctuation power that scales as the galaxy brightness (rather than by the area of the aperture, as with the worminess corrections applied to some of the other galaxies). Furthermore, the S/N is reduced because the PSF has been smeared by telescope drift. Because the bright PSF stars used for the other galaxies do not match in this case, we resorted to extracting a low-S/N PSF from a combination of six faint stars or globular clusters from the smeared image of NGC 7014. The resulting fit is acceptable, but the PSF normalization somewhat uncertain. Although Abell 3742 is among the closest clusters in our sample, the uncertainties are relatively large.
NGC 4073: This galaxy was observed by D. Geisler et al., and it is not associated with a cluster. Its globular cluster population is extensive. Worminess in some of the images contaminates the SBF measurement, and the $P_g$ correction is large. Another difficulty with the analysis of this galaxy was accurately subtracting the smooth galaxy profile. Because of the dither pattern used in this case was chosen to maximize the number of globular clusters detected, there is a hole in the image near the galaxy center that made galaxy subtraction somewhat difficult.
NGC 4709: This galaxy is part of the complex Centaurus cluster and has a significant positive peculiar radial velocity. Although it is not useful for measuring the Hubble Constant, it does have a reliable $I$-band SBF distance from WFPC-2 observations. The first of the six exposures was excluded because of low-level worminess in the background. The resulting power spectrum fits the PSF spectrum very well, and the S/N is relatively high. A modest correction for background worminess brought the individual apertures into perfect agreement
NGC 5193: A recent velocity study by Willmer et al. (1999) indicated that NGC 5193 is not a member of the Abell 3560 cluster, as had been supposed. NGC 5193 has a dust ring extending 2.6 arcsec from the center, which we masked. NGC 5193 is one of the nearest galaxies in our survey, and the S/N ratio is quite high. Nevertheless, the relatively large range of acceptable fluctuation magnitudes reflects the 0.5 mag disagreement between annuli that was not removed with a uniform $P_g$ correction.
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Lauer, T. R. & Postman, M. 1994, , 425, 418 Lauer, T. R., Tonry, J. L., Postman, M., Ajhar, E. A., & Holtzman, J. A. 1998, , 499, 577 Lineweaver, C. H., Tenorio, L., Smoot, G. F., Keegstra, P., Banday, A. J., & Lubin, P. 1996, , 470, 38 Liu, M. C., Charlot, S., & Graham, J. R. 2000, , in press Mochejska, B. J., Macri, L. M., Sasselov, D. D., & Stanek, K. Z. 2000, , 120, 810 Mould, J. R. et al. 2000, , 529, 786 Pahre, M. A. et al. 1999, , 515, 79 Parodi, B. R., Saha, A., Sandage, G. A., & Tammann, G. A. 2000, , 540, 634 Postman, M. & Lauer, T. R. 1995, , 440, 28 Saha, A., Sandage, A., Labhardt, L., Tammann, G. A., Macchetto, F. D., & Panagia, N. 1996, , 466, 55 Sakai, S. et al. 2000, , 529, 698 Schlegel D. J., Finkbeiner, D. P., & Davis, M. 1998, , 500, 525 (SFD) Stanek, K. Z. & Udalski, A. 1999, submitted (astro-ph/9909346) Stephens, A. W., Frogel, J. A., Ortolani, S., Davies, R., Jablonka, P., Renzini, A., & Rich, R. M. 2000, , 119, 419 Thomson, B., Baum, W. A., Hammergren, M., & Worthey, G. 1997, , 483, L37 Tonry, J. L., Blakeslee, J. P., Ajhar, E. A., & Dressler, A. 1997, , 475, 399 Tonry, J. L., Blakeslee, J. P., Ajhar, E. A., & Dressler, A. 2000, , 530, 625 (SBF-II) Willick, J. A. & Batra, P. 2000, , submitted Willmer, C. N. A., Maia, M. A. G., Mendes, S. O., Alonso, M. V., Rios, L. A., Chaves, O. L., & de Mello, D. F. 1999, , 118, 1131 Worthey, G. 1994, , 95, 107 Zehavi, I., Riess, A. G., Kirshner, R. P., & Dekel, A. 1998, , 503, 483
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abstract: 'A polarized variety is K-stable if, for any test configuration, the Donaldson-Futaki invariant is positive. In this paper, inspired by classical geometric invariant theory, we describe the space of test configurations as a limit of a direct system of Tits buildings. We show that the Donaldson-Futaki invariant, conveniently normalized, is a continuous function on this space. We also introduce a pseudo-metric on the space of test configurations. Recall that K-stability can be enhanced by requiring that the Donaldson-Futaki invariant is positive on any admissible filtration of the co-ordinate ring. We show that admissible filtrations give rise to Cauchy sequences of test configurations with respect to the above mentioned pseudo-metric.'
address: 'EPFL, SB MATHGEOM CAG, MA B3 635 (Bâtiment MA), Station 8, CH-1015 Lausanne'
author:
- Giulio Codogni
title: 'Tits buildings and K-stability'
---
[Introduction]{}
The Yau-Tian-Donaldson conjecture predicts that the existence of a canonical metric on a polarized variety $(X,L)$ is equivalent to an appropriate algebraic notion of stability, which should generalize the classical geometric invariant theory stability.
In classical geometric invariant theory, the Hilbert-Mumford criterion asserts that a point is stable if and only if the Hilbert-Mumford weight is positive on every non-trivial one parameter subgroup.
The suggested generalization of geometric invariant theory is K-stability, which says that a polarized variety $(X,L)$ is K-stable if for every non almost trivial test configuration the Donaldson-Futaki invariant is positive. In this theory, the role of one parameter subgroups is played by test configurations, the Donaldson-Futaki weight is a Hilbert-Mumford weight, and the Hilbert-Mumford criterion is turned into a definition.
Nowadays, it is widely accepted that the notion of K-stability should be enhanced. In [@ICM], a stronger notion is proposed: test configurations are identified with finitely generated admissible filtrations, and $(X,L)$ is called $\hat{K}$-stable if the Donaldson-Futaki invariant is positive on every admissible filtration, not just on the finitely generated ones. The Donaldson-Futaki invariant of a non-finitely generated admissible filtration is defined by approximating the filtration with honest test configurations, and then taking the limit along this approximation. We will recall the relevant definitions in Section \[sec:filtrations\].
In classical geometric invariant theory, non-zero one parameter sup-groups are parametrised by the rational points ${\Delta}({\mathbb{Q}})$ of a a space $\Delta$, which is usually called the Tits building or flag complex. The Hilbert-Mumford weight, conveniently normalized, becomes a function on $\Delta$. This space can be endowed with various geometric structures, which can be used for different goals; for example, they are used to show the existence and the uniqueness of a maximally destabilizing one parameter subgroup for unstable points, see [@Kempf] and [@Rousseau].
As observed by Y. Odaka in [@Odaka], test configurations are parametrised by an appropriate direct system of Tits buildings $\{\Delta_r({\mathbb{Q}})\}_{r\in \mathbb{N}}$, where $\Delta_r$ is the Tits building parametrising one parameter subgroups of $SL(H^0(X,rL))$. We denote by $\Delta_{\infty}({\mathbb{Q}})$ this direct limit, and we investigate two different structures that one can put on this space.
Tits building can be defined as abstract simplicial complex, this point of view gives a topology on $\Delta_r$ which we call the *simplicial topology*. The Hilbert-Mumford weight is continuous with respect to this topology. In Theorem \[thm:cont0\], we will show that the morphisms appearing in the direct system $\{\Delta_r({\mathbb{Q}})\}_{r\in \mathbb{N}}$ are continuous with respect to the simplicial topology on $\Delta_r({\mathbb{Q}})$. We call simplicial topology the direct limit topology induced on $\Delta_{\infty}({\mathbb{Q}})$. The following result, proved in Section \[sec:DF-simplicial\], is a corollary of the above mentioned continuity result.
The normalized Donaldson-Futaki weight is continuous with respect to the simplicial topology on the sub-set $\mathcal{T}$ of $\Delta_{\infty}(\mathbb{Q})$ of non almost trivial test configurations.
Let us stress that the maps appearing in the direct system $\{\Delta_r({\mathbb{Q}})\}_{r\in \mathbb{N}}$ do not preserve the simplicial structures, hence $\Delta_{\infty}({\mathbb{Q}})$ does not have a natural simplicial structure.
The second structure that we want to discuss is a metric structure. Each Tits building $\Delta_r$ can be endowed with a metric $d_r$; we call this metric the *Tits metric*, and the induced topology the *Tits topology*. The Tits topology is coarser than the simplicial topology. Using the direct system $\{\Delta_r({\mathbb{Q}})\}_{r\in \mathbb{N}}$, we are able to induce in Definition \[def:space\_conf\] a limit pseudo-metric $d_{\infty}$ on $\Delta({\mathbb{Q}})$. This metric is defined as a limsup, and in Proposition \[prop:limsup\] we show that this limsup is actually a limit. Our next result shows that this metric gives a convenient set-up to study $\hat{K}$-stability.
\[thm:intro\] Let $F$ be non-finitely generated admissible filtration with non-zero $L^2$ norm; then the sequence of points in $\Delta_{\infty}$ associated to the sequence of test configurations approximating $F$ is a Cauchy sequence for the pseudo-metric $d_{\infty}$.
The notions of admissible filtrations and $L^2$ norm will be recalled later on.
In Section \[sec:analogy\], we explain the relation between classical Tits building and symmetric spaces. We suggest a relation between the Tits building $\Delta_{\infty}$ and the space of Kähler metrics. Taking this point of view, it is natural to ask about maximal flat subspaces of the space of Kähler metrics.
The interplay between the simplicial and the Tits topology, as well as the behaviour of the Donaldson-Futaki invariant with respect to the Tits metric, are topics which deserve further investigations. Mimicking the arguments used in geometric invariant theory by [@Kempf] and [@Rousseau], a convenient convexity result about the Donaldson-Futaki invariant would imply the existence and unicity of a maximally destabilizing test configurations.
\*[Relations with other works]{}
It is possible to define a map from the space $\Delta_{\infty}$ to an appropriate quotient of the space of non-archimedean metrics on the analytification of $(X,L)$ introduced in [@BHJ2] and [@prep]. This map should be continuous for the simplicial topology. We do not investigate this topic in this note.
K-stability can also be defined in the non-projective setting, see [@DervanRoss], [@Der2] and [@Zak]. In this set-up, a test configuration is a space ${\mathcal{X}}$ endowed with a Kähler form rather than a line bundle. This configurations do not come naturally from the action of a one parameter sub-group, so Tits building are not available in this setting. It would be interesting to find an alternative way to describe the space $\Delta_{\infty}$.
The automorphism group $\operatorname{Aut}(X,L)$ acts naturally on $\Delta_{\infty}$ preserving the pseudo-metric. When $\operatorname{Aut}(X,L)$ is not reductive, the pair $(X,L)$ is expected to be not K-stable, or at least not $\hat{K}$-stable. In [@me], it is introduced a canonical admissible filtration, called Loewy filtration, which should be destabilising exactly when $\operatorname{Aut}(X,L)$ is not reductive. An interpretation of the Loewy filtration as a Cauchy sequence in $\Delta_{\infty}$ could be a useful step towards the proof of this conjecture.
\*[Notations]{} We work over an algebraically close field $\Bbbk$ of characteristic zero. We fix a normal projective variety $X$ of dimension $n$ and a very ample and projectively normal line bundle $L$ over $X$. We use the additive notation for line bundles, so $mL=L^{\otimes L}$.
\*[Acknowledgements]{} We had the pleasure and the benefit of conversations about the topics of this paper with S. Boucksom, R. Dervan, M. Jonsson, J. Ross, J. Stoppa and F. Viviani. The author was also supported by the FIRB 2012 -Moduli spaces and their applications, and the ERC StG 307119 - Stability in Algebraic and Differential Geometry.
[Tits buildings]{}\[sec:prel\] In this section, following [@Serre] and [@GIT Section 2.2], we recall the definition of the *Tits building* ${\Delta}$ associated to a finite dimensional complex vector space $V$, and some of its properties. In the literature, Tits buildings are sometime called *spherical buildings* or *flag complexes*.
Let $m$ be the dimension of $V$, and assume that $m\geq 3$. The first definition of ${\Delta}$ is as an abstract simplicial complex. Simplexes correspond to parabolic sub-group of $SL(V)$; a simplex corresponding to a parabolic group $P_1$ lies in the boundary of a simplex corresponding to a parabolic group $P_2$ if and only $P_2\subset P_1$. Vertexes are given by maximal parabolic subgroups; maximal simplexes are $m-2$ dimensional. Recall that parabolic subgroups correspond to flags of $V$: to a flag we associated its stabiliser. We thus have the following equivalent description of ${\Delta}$: each vertex corresponds to a proper vector subspace of $V$; a group of vertexes form a simplex if and only if the associated subspaces form a flag in $V$.
We can now start enhancing the structure of ${\Delta}$. We identify each simplex with the standard one, in particular we have co-ordinates $x_i$; let ${\Delta}({\mathbb{Q}})$ be the set of point with rational co-ordinates. We introduce the following definition
\[def:weighted\_flag\] A weighted flag is the data of a flag $$\{0\} \subset F_1V \subset F_2V \cdots F_{k-1} V \subset F_kV=V$$ and weights $w=(w_1, \dots , w_k)$ such that $w_i< w_{i+1}$ and $\sum w_i=0$. The weights can be either rational or real numbers. Two flags $(F,w)$ and $(G,w')$ are equivalent if there exists a constant $c$, called the scaling constant, such that $F_iV=G_iV$ and and $w_i=cw_i'$ for every $i$.
The weight of a vector $v$ is the maximum $w_i$ such that $v\in F_i V$.
We say that a basis $\{v_1, \dots, v_n\}$ of $V$ is *adapted* to a flag $F$ if, for every $i$, there exists a subset of $\{v_1, \dots, v_n\}$ which forms a basis of $F_iV$.
The Tits building ${\Delta}$ parametrise weighted flag up to equivalence: the flag corresponds to the simplex, and the weight to the co-ordinates of the point.
We now associate to each one parameter subgroup ${\lambda}$ of $SL(V)$ a weighted flag, hence a point $[{\lambda}]$ of ${\Delta}({\mathbb{Q}})$. Let $w_1,\dots, w_k$ be the weights of $\lambda$, ordered in an increasing way; let $$F_i(V)=\bigoplus_{j\leq i} V_{w_j}$$ where $V_{w_j}$ is the eigenspace of weight $w_j$ of $\lambda$. Assigning weight $w_i$ to $F_iV$, we obtain the flag associated to ${\lambda}$. As shown in [@GIT Proposition 2.6], the parabolic sub-group $P(\lambda)$ stabilising this flag consists of all $g$ in $SL(V)$ such that the limit $\lim_{t\to 0} \lambda(t)g\lambda(t)^{-1}$ does exist. This limit, when it exists, centralizes $\lambda$, so it preserves the eigenspaces of $\lambda$; see the proof of [@GIT Proposition 2.6] for a more precise description of the limit.
Two 1PS’s gives the same point in ${\Delta}$ if and only if the associated flags are equivalent. In other words, ${\Delta}({\mathbb{Q}})$ is equal to set of one parameter subgroups of $SL(V)$ modulo the equivalence relations: $$\begin{aligned}
\lambda \sim \gamma \quad \textrm{if} \quad \lambda=p\gamma p^{-1} \quad p\in P(\lambda) \\
\lambda \sim \gamma \quad \textrm{if} \quad \lambda^a=\gamma^b \quad a,b \in \mathbb{Z}\end{aligned}$$
The next piece of structure is given by the apartment. Apartments correspond to maximal tori of $SL(V)$: given a maximal torus $T$, the corresponding apparent $A_T$ is the closure in ${\Delta}$ of the one parameter sub-groups of $T$. A flag $F$ is in $A_T$ if and only if the eigenvectors $v_1,\dots , v_n$ of $T$ form a basis adapted to $F$. The key remark is that an apartment is a finite simplicial complex homeomorphic to a sphere, or a simplicial sphere for short. The following standard lemma will be very important
([@GIT Lemma II.2.9]) Given two points $p$ and $q$ of ${\Delta}$, there exists at least an apartment containing both of them.
The previous lemma can also be interpreted in the following way: given two points $p$ and $q$ in $\Delta({\mathbb{Q}})$, there exists two commuting 1PS’s $\lambda$ and $\gamma$ of $SL(V)$ such that $p=[\lambda]$ and $q=[\gamma]$. The building, so far, is an abstract simplicial complex. Looking at its geometric realisation, we can endow it with a topology, which we call the *simplicial topology*. The simplicial complex $\Delta$ is not locally of finite type, so we need some care in the description of this structure. Apartments are finite simplicial complex homeomorphic to a sphere. On the entire space $\Delta$, the topology is defined as the direct limit of the topology of finite sub-complexes. Since any finite sub-complex is contained in a finite number of apartments, a subset $U$ of $\Delta$ is open if and only if its intersection with any apartment is open. Similarly, a function on $\Delta$ is continuous with respect the simplicial topology if and only if its restriction to each apartment is continuous.
We are now in position to introduce the Tits metric on $\Delta({\mathbb{Q}})$.
\[def:TitsMetric\] Let $p$ and $q$ be two points of $\Delta({\mathbb{Q}})$; pick two commuting 1PS’s $\lambda$ and $\gamma$ such that $p=[\lambda]$ and $q=[\gamma]$, and write $\lambda=\exp tA$ and $\gamma=\exp tB$; then we let $$d(p,q)=\arccos\left(\frac{\operatorname{Tr}(AB)}{\sqrt{\operatorname{Tr}(A^2)\operatorname{Tr}(B^2)}}\right)$$
One can show that this definition is independent of the chosen one parameter subgroups, see for instance [@GIT Section 2.2]. Moreover, this metric can be extended by continuity to $\Delta$.
Let us describe an interpretation of the Tits metric as angular distance. Take a maximal torus $T$ containing both $\lambda$ and $\gamma$, this gives an apartment $A_T$ containing both $p$ and $q$. Let $\Gamma(T)$ be the lattice of one parameter subgroups of $T$. The Killing metric on $\Gamma(T)$ is a quadratic form which is equivariant for the action by conjugation of the normalizer of $T$ in $G$, it is unique up to a scalar. Denote by $E$ the space $\Gamma(T)\otimes \mathbb{R}$ equipped with the Killing metric. Then, $A_T$ can be identified with the unit sphere in $E$, and the Tits metric is nothing but the angular distance. Since the Killing metric unique up to a scalar, the angular distance on $A_T$ is uniquely defined.
Since any two points are contained in an apartment, and the apartment is isometric to a sphere endowed with the angular distance, we have that any two points can be connected by a geodesic and $\operatorname{diam}(\Delta)=\pi$. The geodesic is not unique because, for instance, two points can be contained in many different apartments, and the geodesic constructed above depends on the apartment.
The topology induced by the Tits metric on each apartment is equal to the simplicial topology. However, on $\Delta$, the topology induced by the Tits metric is coarser than the simplicial topology.
[Tits building and test configurations]{}\[sec:Tits\_and\_test\] Let $X$ be a projective variety over an algebraically closed field of characteristic zero, and $L$ a very ample and projectively normal line bundle on $X$. We also fix a generator $t$ of the space of one parameter subgroups of ${\mathbb{G}_m}$, and faithful action of ${\mathbb{G}_m}$ on ${\mathbb{A}}^1$; let $0$ be the fixed point of the action and $1$ another point of ${\mathbb{A}}^1$. We recall the definition of test configuration, which is due to S. Donaldson [@Don2 Definition 2.1.1].
\[def:tc\] Let $r$ be a positive integer. An exponent $r$ test configuration $({\mathcal{X}},{\mathcal{L}})$ for $(X,L)$ consist of the following data
1. a scheme ${\mathcal{X}}$ together with a flat map $\pi \colon {\mathcal{X}}\to {\mathbb{A}}^1$;
2. a ${\mathbb{G}_m}$ action on ${\mathcal{X}}$ such that the morphism $\pi$ is equivariant;
3. a relatively ample line bundle ${\mathcal{L}}$ on ${\mathcal{X}}$ together with a linearisation of the ${\mathbb{G}_m}$ action.
Moreover, we require that the fibre over $1$ is isomorphic to $(X,rL)$.
A test configuration is very (respectively semi-) ample if ${\mathcal{L}}$ is very (semi-) ample. A test configuration is trivial if $({\mathcal{X}},{\mathcal{L}})$ is isomorphic to $(X\times {\mathbb{A}}^1, rL\boxtimes \mathcal{O}_{{\mathbb{A}}^1})$, and the ${\mathbb{G}_m}$ action is trivial on $X$. A test configuration is normal if ${\mathcal{X}}$ is normal.
Let $\nu\colon \hat{{\mathcal{X}}}\to{\mathcal{X}}$ be the normalization, then $(\hat{{\mathcal{X}}},\nu^*{\mathcal{L}})$ has a natural structure of test configuration, we call it the normalization of $({\mathcal{X}},{\mathcal{L}})$. A test configuration is almost trivial if its normalization is trivial.
A non-polarized test configuration is the datum of an ${\mathcal{X}}$ with a ${\mathbb{G}_m}$ action as above, without the choice of a line bundle ${\mathcal{L}}$.
Basic properties of test configurations are described in [@BHJ Section 2]. There are three main operation one can perform on test configurations.
\[def:action\]
Base change
: Let $b_p\colon {\mathbb{A}}^1 \to {\mathbb{A}}^1$ be the map defined by $z\mapsto z^p$. We can make a base change $({\mathcal{X}},{\mathcal{L}})$ via $b_p$ obtaining a new test configuration.
Scaling
: Consider the trivial action of ${\mathbb{G}_m}$ on $X$, and fix a faithful lifting of this action to ${\mathcal{L}}$, so that the induced action on $H^0({\mathcal{X}}_0,{\mathcal{L}}_0)$ is a homothety. We can scale the action of ${\mathbb{G}_m}$ on ${\mathcal{L}}$ by adding $c$ times this action, where $c$ is in $\mathbb{Z}$.
Rising the line bundle
: We can replace ${\mathcal{L}}$ with $m{\mathcal{L}}$, for any positive integer $m$.
We now recall the definition of the $L^2$ norm of a test configuration. For every $k$, the test configuration gives rise to a ${\mathbb{G}_m}$ action on $H^0({\mathcal{X}}_0,k{\mathcal{L}}_0)$; let $T_k$ be an infinitesimal generator of this action. We denote by $\underline{T}_k$ the traceless part of $T_k$, in symbols $$\underline{T}_k=T_k-\frac{\operatorname{Tr}(T_k)}{h^0({\mathcal{X}}_0,k{\mathcal{L}}_0)}Id$$ Then $\operatorname{Tr}(\underline{T}_k^2)$ is, for $k$ big enough, a degree $n+2$ polynomial in $k$, where $n$ is the dimension of $X$, see for instance [@Gabor Equation 4] or [@BHJ Theorem3.1]. We let $$||({\mathcal{X}},{\mathcal{L}})||_{L^2}^2=\lim_{k\to \infty}(kr)^{-n-2}\operatorname{Tr}(\underline{T}_k^2)$$ Remark that $||({\mathcal{X}},{\mathcal{L}})||_{L^2}=||({\mathcal{X}},m{\mathcal{L}})||_{L^2}$ for every $m$.
Let now $V_r=H^0(X,rL)^{\vee}$. Given a 1PS $\lambda$ of $SL(V_r)$, we can construct a test configuration by taking the flat closure of the $\lambda$-orbit of $X$ in $\mathbb{P} V_r$. Any very ample exponent $r$ test configuration arise as orbit of a 1PS of $GL(V_r)$, see [@RossThomas Proposition 3.7]. By performing a base change and scaling the linearisation, we can always assume that this 1PS lies in $SL(V_r)$. We have now the following key observation of Odaka [@Odaka].
Let $\Delta_r$ be the Tits building of $V_r$. Then, points of $\Delta_r({\mathbb{Q}})$ are in bijective correspondence with very ample exponent $r$ test configurations, modulo base change and scaling.
The only thing we have to check is that if the weighted flags associated to two 1PS’s are equivalent, then also the corresponding test configuration are equivalent. This is done in [@Odaka Theorem 2.3].
Almost triviality of a test configuration can be characterized in term of the associated filtration of $V_r$, see [@BHJ Proposition 2.12]. The following lemma, which is contained in the proof of [@RossThomas Proposition 3.7] is also very important
There exists a ${\mathbb{G}_m}$-equivariant trivialization of $\pi_*{\mathcal{L}}$; this gives a ${\mathbb{G}_m}$-equivariant isomorphism between $H^0(X,rL)^{\vee}$ and $H^0({\mathcal{X}}_0,{\mathcal{L}}_0)$, where the ${\mathbb{G}_m}$ action on the first vector space is given by the one parameter subgroup inducing the test configuration.
From the point of view of K-stability, it is quite natural to identify the test configuration $({\mathcal{X}},{\mathcal{L}})$ and $({\mathcal{X}},m{\mathcal{L}})$. Because of this, we look at the direct systems formed by the buildings $\Delta_r$ and the morphisms $$\begin{array}{ccccc}
\iota_{r,k}\colon & \Delta_r({\mathbb{Q}}) &\to & \Delta_{rk}({\mathbb{Q}}) \\
&({\mathcal{X}},{\mathcal{L}}) & \mapsto & ({\mathcal{X}},k{\mathcal{L}})
\end{array}$$
\[thm:cont0\] For every $i$ and $k$, the map $\iota_{r,k}$ is continuous for the simplicial topology.
We postpone it to Section \[sec:map\]; let us point out that in the proof we also describe explicitly the morphisms $\iota_{r,k}$, and these morphisms do not preserve the simplicial structure of $\Delta_r$. In other words, if one sees $\Delta_r$ as a direct limit of simplicial spheres, the maps $\iota_{r,k}$ are well defined on the resulting topological space $\Delta_r$, but do not preserve the direct system structure of $\Delta_r$, and it does not make sense to ask if the two limit commute. We are now going to define the central object of study of this paper.
\[def:space\_conf\] The space of test configurations is the space $\Delta_{\infty}({\mathbb{Q}})$ defined as the direct limit $$\Delta_{\infty}({\mathbb{Q}}):=\lim_r\Delta_r({\mathbb{Q}})\,.$$ The simplicial topology on $\Delta_{\infty}({\mathbb{Q}})$ is the direct limit of the simplicial topology on $\Delta_r({\mathbb{Q}})$.
The pseudo-metric on $d_{\infty}$ is the pseudo-metric given by $$d_{\infty}(p,q)=\limsup_r d_r(p,q)$$ where $d_r$ is the Tits metric on $\Delta_r({\mathbb{Q}})$. The Tits topology on $\Delta_{\infty}({\mathbb{Q}})$ is the topology induced by $d_{\infty}$.
Remark that $p$ and $q$ can be seen as points of $\Delta_r$ for every $r$ divisible enough, so the previous expression for $d_{\infty}$ makes sense. Moreover, since $\operatorname{diam}(\Delta_r)=\pi$ for every $r$, $d_{\infty}$ is finite and $\operatorname{diam}(\Delta_{\infty})\leq \pi$.
The space $\Delta_{\infty}({\mathbb{Q}})$ parametrizes all test configurations, modulo the three operations defined introduced in \[def:action\], namely modulo scaling, base change and rising the line bundle.
\[prop:limsup\] The limsup appearing in Definition \[def:space\_conf\] is actually a limit; in other words, $$d_{\infty}(p,q)=\lim_r d_r(p,q)$$
Let $({\mathcal{X}}_1,{\mathcal{L}}_1)$ and $({\mathcal{X}}_2,{\mathcal{L}}_2)$ be very ample test configurations associated to $p$ and $q$. By raising ${\mathcal{L}}_1$ and ${\mathcal{L}}_2$ to suitable powers, we can assume that they have the same exponent. When $r$ is divisible by the exponent, we have the Tits metric $$d_r(p,q)=\arccos\left(\frac{\operatorname{Tr}(A_rB_r)}{\sqrt{\operatorname{Tr}(A_r^2)\operatorname{Tr}(B_r^2)}}\right)\,,$$ where $A_r$ and $B_r$ are generators of two commuting one parameter subgroups of $SL(H^0(X,rL)^{\vee})$ inducing respectively $({\mathcal{X}}_1,r{\mathcal{L}}_1)$ and $({\mathcal{X}}_2,r{\mathcal{L}}_2)$. The denominator of $d_r(p,q)$ is well known to be, for $r$ divisible enough, a polynomial of degree $n+2$, see for instance [@Gabor Equation 4] or [@BHJ Theorem 3.1]. We are going to show that also the numerator is a polynomial of degree $n+2$.
Choose a non-polarised test configuration ${\mathcal{X}}$ dominating equivariantly both ${\mathcal{X}}_1$ and ${\mathcal{X}}_2$; this can be constructed by resolving simultaneously the indeterminacy of the maps $X\times \mathbb{P}^1\dashrightarrow {\mathcal{X}}_i$, cf [@BHJ Section 6.6]. Let $\alpha$ be the $\mathbb{G}_m$ action on ${\mathcal{X}}$. Denote by ${\mathcal{M}}_i$ be the pull-back of ${\mathcal{L}}_i$ to ${\mathcal{X}}$. The restriction of ${\mathcal{M}}_1+ {\mathcal{M}}_2$ on ${\mathcal{X}}_t$, for $t\neq 0$, is isomorphic to $2rL$, hence $H^0({\mathcal{X}}_0,{\mathcal{M}}_1+ {\mathcal{M}}_2|_{{\mathcal{X}}_0})^{\vee}$ can be identified in a ${\mathbb{G}_m}$ equivariant way with $H^0(X,2rL)^{\vee}$ and the infinitesimal generator of the action of $\alpha$ is exactly $A_r+B_r$. By applying [@BHJ Theorem 3.1] we show that $\operatorname{Tr}(A_r+B_r)^2$ is a polynomial of degree $n+2$. Since also $\operatorname{Tr}A_r^2$ and $\operatorname{Tr}B_r^2$ are polynomial of degree $n+2$, we conclude that the same is true for $\operatorname{Tr}A_kB_k$.
It is also natural to consider the space $$\Delta_{\infty}:= \lim_r\Delta_r$$ endowed with its simplicial topology. We have a natural inclusion $\Delta_{\infty}({\mathbb{Q}})\subset \Delta_{\infty}$, and we can extend $d_{\infty}$ to a metric on $\Delta_{\infty}$. We do not know about relation between $\Delta_{\infty}$ and the completion of $\Delta_{\infty}({\mathbb{Q}})$ with respect to $d_{\infty}$.
[Donaldson-Futaki invariant and the symlicial topology]{}\[sec:DF-simplicial\] We first briefly recall some facts about the Hilbert-Mumford weight, following [@GIT Chapter 2]. Let the group $SL(V)$ act on a projective variety $Z$, and linearise the action to a line bundle $H$. Pick a closed point $z$ in $Z$. For any 1PS $\lambda$ of $SL(V)$ we can consider the Hilbert-Mumford weight $\mu(\lambda)$ with respect to $z$ and $H$.
Fix now an $SL(V)$ invariant norm $|| \, - \,||$ on the 1PS’s of $SL(V)$. The ratio $\nu(\lambda)=\mu(\lambda)/||\lambda||$ is a well defined function on the Tits building $\Delta(V)$; moreover, $\nu$ is continuous for the simplicial topology.
Following [@RossThomas Section 3] and [@Gabor], we introduce the Chow weights and the Donaldson-Futaki weight. Fix an exponent $r$, and let $V_r=H^0(X,rL)$. We choose as $SL(V_r)$-invariant norm on the space of 1PS’s of $SL(V_r)$ the norm $||\exp(tA)||=r^{-n-2}\operatorname{Tr}_{V_r}A^2$. Remark that $\lambda$ is already taken in the special linear group, so $A$ is traceless. In particular, $||\exp(tA)||$ is equal to $r^{-n-2}\operatorname{Tr}(\underline{T}_1^2)$, where $\underline{T}_1$ is the operator introduced in Section \[sec:Tits\_and\_test\].
The group $SL(V_r)$ acts on the appropriate Hilbert scheme $Z_r$, and the variety $X$ gives a point $[X]$ in $Z_r$. Choosing the correct line bundle on $Z_r$, the associated normalized Hilbert-Mumford weight is the normalised Chow weight: $$\operatorname{chow}_r\colon \Delta_r \to {\mathbb{R}}$$ This line bundle is the pull-back of the Chow line bundle from the Chow scheme, via the cycle-class map from the Hilbert scheme to the Chow scheme. The normalization of he Chow line bundle is such that the $r$-th normalized Chow weight of an exponent $r$ test configuration is $$\operatorname{chow}_r({\mathcal{X}},{\mathcal{L}})=||\lambda||^{-1}\frac{ra_0}{b_0}$$ where $h(k)=a_0k^n+O(k^{n-1})$ is the Hilbert polynomial of $(X,rL)$, and $w(k)=b_0k^{n+1}+O(k^n)$ is the trace of the operator $T_k$ introduced in Section \[sec:Tits\_and\_test\]. Remark that $w(r)=0$, because we started off with a $\lambda$ in the special linear group, however $w(k)$ is a non-trivial polynomial of degree $n+1$ for $k$ big enough.
Pulling-back via the maps $\iota_{r,k}$, we have the higher Chow weights $$\operatorname{chow}_{rk}\colon \Delta_r \to {\mathbb{R}}\, ,$$ so that $$\operatorname{chow}_{kr}({\mathcal{X}},{\mathcal{L}})=||\lambda||^{-1}\left(\frac{kra_0}{b_0}-\frac{w(k)}{h(kr)}\right)$$
Let $\mathcal{T}\subset \Delta_{\infty}({\mathbb{Q}})$ be the subset of test-configurations with non-zero $L^2$ norm, and $\mathcal{T}_r$ its intersection with $\Delta_r$. Fix a point in $\mathcal{T}_r$, the value of the Chow weight at that point is, for $k$ big enough, equal to a Laurent polynomial $$\operatorname{chow}_{kr}=\operatorname{df}+\ell(k)$$ where $\operatorname{df}$ is the constant term and $\ell(k)$ is the principal part of the Laurent polynomial, in particular $\ell(k)$ converges to zero when $k$ goes to infinity. This gives an invariant $$\operatorname{df}\colon \mathcal{T} \to {\mathbb{R}}$$ defined as $\operatorname{df}(p)=\lim_k \operatorname{chow}_{kr}(p)$, where $r$ is such that $p$ lies in $\Delta_r$.
The invariant $\operatorname{df}$ is by definition the Donaldson-Futaki invariant of a test configuration divided by its $L^2$ norm.
\[lem:uni\] Fix $r$, then there exists a positive integer $K$ such that $\operatorname{chow}_{kr}$ is a Laurent polynomial for all exponent $r$ test configuration and all $k$ divisible by $K$.
Fixed a test configuration $({\mathcal{X}},{\mathcal{L}})$, the Chow invariant $\operatorname{chow}_{kr}$ is a polynomial as soon as $H^i({\mathcal{X}}_0,k{\mathcal{L}}_0)$ vanishes for all $i>0$, see [@BHJ Theorem 3.1 and Corollary 3.2]. Fixed the exponent, central fibres are parametrized by a Hilbert scheme, so the result follows from a general statement of the form: if $T$ is a noetherian scheme, and $Y\to T$ is a projective morphism with a relatively ample line bundle $L$, then there exists a $K$ such that $H^i(Y_t,kL_t)=0$ for all $t$ in $T$, all $i>0$, and all $k$ divisible by $K$; this is well-known, see for instance[@Laz Theorem 1.2.13 and its proof].
We have now the following proposition.
The normalised Donaldson-Futaki invariant $\operatorname{df}$ is continuous with respect to the simplicial topology on $\mathcal{T}\subset \Delta_{\infty}({\mathbb{Q}})$.
Since the topology on $\Delta_{\infty}({\mathbb{Q}})$ is the direct limit topology, it is enough to show that $\operatorname{df}$ is continuous when restricted to $\mathcal{T}_r$, for every $r$. We know that $\operatorname{chow}_{kr}$ is continuous on $\mathcal{T}_{kr}$ for every $k$ and $r$; since, by Theorem \[thm:cont0\], the maps $\iota_{k,r}$ are continuous, $\operatorname{chow}_{kr}$ is continuous on $\mathcal{T}_r$. By Lemma \[lem:uni\], for $k$ divisible enough $\operatorname{chow}_{kr}$ is a Laurent polynomial in $k$, so all its coefficients have to be continuous as functions on $\mathcal{T}_r$. This proves the claim.
At least for smooth varieties over the complex numbers, because of Donaldson work [@Don], we know that $\operatorname{df}$ is bounded below on $\mathcal{T}$. The lower bound can be described in term of the curvature of Kähler metrics in the class $c_1(L)$.
When $X$ is a normal variety, a test configuration has zero $L^2$ norm if and only if it is almost trivial, [@Ruadhai_Uniform Theorem 1.3] and [@BHJ Corollary B]. Let us now give the definition of K-stability
A normal polarised variety $(X,L)$ is K-semistable if $\operatorname{df}({\mathcal{X}},{\mathcal{L}})\geq 0$ for every test configuration $({\mathcal{X}},{\mathcal{L}})$. It is K-stable if it is K-semistable and $\operatorname{df}({\mathcal{X}},{\mathcal{L}})=0$ if and only if $({\mathcal{X}},{\mathcal{L}})$ is almost trivial.
[Filtrations and the completion with respect to the Tits metric]{}\[sec:filtrations\] In this section we study filtrations of the co-ordinate ring $R$ of $(X,L)$. Recall that the K-stability of $(X,L)$ is equivalent to the K-stability of $(X,kL)$ for every $k$, so we can assume without loss of generality that $L$ is projectively normal.
\[filtration\] A filtration $F$ of $R$ is the datum of an increasing flag on each graded piece $H^0(X,kL)=V_k$, indexed by $\mathbb{Z}$. We say that the filtration is
Multiplicative
: if $$F_iV_a\otimes F_jV_b\to F_{i+j}V_{a+b}\,,$$ for every $a$, $b$, $j$ and $k$;
Point wise right bounded
: If for every fixed $k$ we have that $$F_iV_k=V_k$$ for $i>>0$; this is also said *exhaustive*;
Linearly left bounded
: If there exists a negative constant $C$ such that $$F_{Ck}V_k=\{0\}$$ for every $k$.
A filtration is *admissible* if it satisfies the three properties listed above. We let $F_iR=\oplus_k F_iH^0(X,kL)$.
There are two operations that we can perform on filtrations. We can scale them, which means replacing $F_i$ with $F_{ci}$ for some fixed constant $c$, and we can shift them, which means replacing $F_iH^0(X,kL)$ with $F_{i+ck}F_iH^0(X,kL)$, for a fixed constant $c$.
Given a multiplicative filtration $F$, we can construct its Rees algebra $$\operatorname{Rees}(F)=\bigoplus_i F_i R t^i$$ we say that a filtration is *finitely generated* if its Rees algebra is finitely generated.
As explained in [@witt] and [@Gabor], taking the Proj of the Rees algebra of an admissible finitely generated filtration one obtains a test configuration. More generally, there is a correspondence between finitely generated admissible filtrations of the rings $R$ and test configurations, see [@BHJ Proposition 2.15]. Under this correspondence, scale the filtration corresponds to a base change, shift the filtration corresponds to scale the linearisation, see Definition \[def:action\].
Following [@Gabor], we can approximate a non-finitely generated admissible filtration with finitely generated ones. Let $F$ be an admissible filtration; denote by $\chi^{(m)}$ the $\Bbbk[t]$-sub-algebra of $\operatorname{Rees}(F)$ generated by the finite dimensional vector space $\oplus_i F_i H^0(X,rmL) t^i\oplus R_it^N$, for $N$ big enough. We now let $$F^{(m)}_iH^0(X,mkL)=\{s\in H^0(X,mkL) \textrm{ s.t. } s t^i \in \chi^{(m)}\}$$ this defines a finitely generated admissible filtration of $R$. Let $({\mathcal{X}}^{(m)},{\mathcal{L}}^{(m)})$ be the corresponding test configuration. Then one defines $$||F||_{L^2}=\liminf_{m\to \infty} ||({\mathcal{X}}^{(m)},{\mathcal{L}}^{(m)})||_{L^2}$$ In [@Gabor] it is shown that this liminf is actually a limit.
Given a flag $F$, for every $m$ we can construct a weighted flag of $H^0(X,mL)$ up to scaling, in the sense of Definition \[def:weighted\_flag\]. This is done by first giving weight $i$ to the piece $F_i$, and then subtracting a common rational constant to all weights to normalise the trace. The filtration obtained in this way has rational weights; we denote by $p_m$ the corresponding point in $\Delta_m({\mathbb{Q}})$. As explained in [@Gabor Section 3.2], the test configuration associated to this weighted filtration is equivalent to the Proj of the Rees algebra of $F^{(m)}$.
If the filtration is finitely generated, then $F^{(m)}=F$ for $m$ big enough, and the sequence $\{p_m\}$ is eventually constant as a sequence in $\Delta_{\infty}({\mathbb{Q}})$. On the other hand, when the filtration is not finitely generated, the test configurations associated to the points $p_m$ are different, so a non finitely generated filtration defines a non-constant sequence in $\Delta_{\infty}$.
\[thm:cauchy\] Let $p_m$ be the sequence of points in $\Delta_{\infty}({\mathbb{Q}})$ associated to an admissible filtration $F$ such that $||F||_{L^2}\neq 0$; then, this is a Cauchy sequence for the pseudo-metric $d_{\infty}$.
We need to show that, for every $j$, the distance $d_{\infty}(p_m,p_{jm})$ converges to zero when $m$ goes to infinity. More explicitly, we have to show that $$\lim_m \limsup_k \frac{\operatorname{Tr}(A_k^{(m)}A_k^{(jm)})}{\sqrt{\operatorname{Tr}((A_k^{(m)})^2)\operatorname{Tr}((A_k^{(jm)})^2)}}=1$$ where, for each $m$, the limit is taken on all $k$ divisible by both $m$ and $jm$; the $A_k^{(m)}$ and $A_k^{(jm)}$ are infinitesimal generator of commuting 1PS representing $p_m$ and $p_{jm}$ in $\Delta_k$. Because of the hypothesis on the norm, the limit of the denominator normalised by $k^{-n-2}$ is not zero, so we can compute the limit of the numerator and the denominator separately.
To start with, let us recall that $\limsup_k \sqrt{k^{-n-2}\operatorname{Tr}((A_k^{(m)})^2)}$ converges to the $L^2$ norm of the test configuration associated to $p_m$, and the limit $\lim_m ||p_m||_{L^2}$ is equal to the $L^2$ norm $||F||_{L^2}$ of the filtration $F$, as explained in [@Gabor], see in particular Lemma 8. The same is true for $p_{jm}$.
We now have to deal with the numerator. Fix $m$ and $k$. The multiplicativity of $F$ implies, for every $j$, the following inclusion relation $$\chi^{(m)}\cap R^{(jm)}[t]\subseteq \chi^{(jm)}\,,$$ where $R^{(jm)}$ is the Veronese ring $\oplus_{\ell}H^0(X,jm\ell L)$. This inclusion in turn implies that, for every $i$ and $k$, we have $$F_i^{(jm)}H^0(X,mkjL)\subseteq F_i^{(m)}H^0(X,mkjL)\,.$$ Choosing a basis of $H^0(X,mkjL)$ adopted to both $F^{(m)}$ and $F^{(jm)}$, we can translate the above inclusions in the following inequalities. $$\operatorname{Tr}((A_k^{(m)})^2)\leq \operatorname{Tr}(A_k^{(m)}A_k^{(jm)})\leq \operatorname{Tr}((A_k^{(jm)})^2)$$
Taking the limit on $k$ and then $m$, arguing as before we conclude that $$\lim_m \limsup_k k^{-n-2}\operatorname{Tr}(A_k^{(m)}A_k^{(jm)})=||F||_{L^2}^2$$
[Description of the morphisms between Tits buildings]{}\[sec:map\] In this section we describes explicitly the maps $$\begin{array}{ccccc}
\iota_{r,k}\colon & \Delta_r({\mathbb{Q}}) &\to & \Delta_{rk}({\mathbb{Q}}) \\
&({\mathcal{X}},{\mathcal{L}}) & \mapsto & ({\mathcal{X}},{\mathcal{L}}^{\otimes k})
\end{array}$$ Our main result is the following.
\[thm:cont\] For every $i$ and $k$, the map $\iota_{r,k}$ is continuous for the simplicial topology.
We can assume without loss of generality that $r=1$; moreover, we fix $k$, so, to simplify the notation, we write $\iota$ for $\iota_{k,r}$, $\Delta$ for $\Delta_r$ and $\Delta_k$ for $\Delta_{rk}$. Let $\Delta_S$ be the Tits building of the vector space $\operatorname{Sym}^k H^0(X,L)$.
In Section \[sec:segre\], we will define a Segre map $S\colon \Delta({\mathbb{Q}})\to \Delta_S({\mathbb{Q}})$, and prove that it is continuous. In Section \[sec:retraction\], we will define a retraction map $\rho\colon \Delta_S({\mathbb{Q}})\to \Delta_k({\mathbb{Q}})$, and prove that it is continuous. In Section \[sec:prov\], we will show that $\iota$ is actually the composition of $S$ and $\rho$, concluding the proof of Theorem \[thm:cont\].
[Segre morphism of building]{}\[sec:segre\] For an algebraic group $G$, let $\Gamma(G)$ be the set of 1PS’s of $G$. Let $V=H^0(X,L)$ and $V_S=\operatorname{Sym}^k H^0(X,L)$. We have a Segre map $$\begin{array}{ccccc}
S\colon &\Gamma(SL(V)) &\to & \Gamma(SL(V_S)) \\
&\gamma & \mapsto & \gamma^{\otimes k}
\end{array}$$
The Segre map defined on 1PS’s induces a morphism of building; we describe directly this morphisms on weighted flags. Denote by $M$ be the collection of multi-indexes $I=(i_1,\dots , i_m)$ with $\sum i_j=k$. Let $\underline{v}$ be a basis of $V$ adapted to a weighted flag $(F,w)$ associated to $\gamma$. We denote by $S(\underline{v})$ the basis of $V_S$ formed by monomials in the element of $\underline{v}$; in particular, for $I\in M$ and $v\in \underline{v}$, we denote by $v^I$ the corresponding monomial in $S(\underline{v})$. Let $T(w)=\sum_{I\in M} w^I$. To each monomial $v^I$ we assign weight $w^I-T(w)$: this defines a weighted flag $(S(F),S(w))$. The weighed flag associated to $S(\gamma)$ is exactly $(S(F),S(w))$, so this gives a description of the map $$S\colon \Delta \to \Delta_S$$ The Segre map preserves apartments in the following sense. Let $A$ be an apartment of ${\Delta}$ associated to a basis $\underline{v}$. Then, $S(A)$ is contained in the apartment $A_S$ associated to the basis $S(\underline{v})$.
Let us show that $S\colon A \to A_S$ is continuous for every apartment $A$. Co-ordinates of points in an apartment are given just by the weights. In particular, for $I\in M$, the $I$-th co-ordinate of $(S(F),S(w))$ is $w^I-T(w)$; benign the new co-ordinate a polynomial in the old one, the map $S$ is continuous. Since the simplicial topology on $\Delta$ is the direct limit of the topology on the apartments, we conclude that $S$ is continuous on $\Delta$.
[Retraction of buildings]{}\[sec:retraction\] Let $i\colon W \hookrightarrow V$ be an inclusion of vector spaces. We can define the corresponding retraction of building $$\rho \colon \Delta(V) \to \Delta(W)$$ as follows. Let $(F,w)$ be a weighted flag in $\Delta(V)$. Choose a basis of $V$ adapted both to $F$ and $W$. This amounts to choose a representative $\gamma$ of $F$ which preserves globally $W$; we denote by $U$ the $\gamma$-invariant complement of $W$ in $V$. We now let $\rho((F,w))$ to be the normalised weighted flag associated to the 1PS $\gamma|_W$. Remark that the action of $\gamma$ on $W$ could be trivial; in this case, $\rho$ is not defined at $(F,w)$.
By choosing a complement $U$ of $W$ in $V$, there is a natural inclusion of $\Gamma(SL(W))$ in $\Gamma(SL(V))$, which in turn gives an inclusion of building $i\colon \Delta(W)\to \Delta(V)$; the map $\rho$ is the right-inverse of this inclusion.
We can give the following alternative description of $\rho$, which does not depend the choice of the representative $\gamma$. We let $\rho(F)$ to be the flag defined by $\rho(F)_i W=F_i V \cap W$; we assign weight $w_i$ to $\rho(F)_i$. This definition is ill-posed, and we need to refine it. The first pathology is that $\rho(F)_i W$ is not, in general, a proper sub-space of $W$, if this happens we skip this step of the flag and we relabel the indexes. If all the subspaces $\rho(F)_i W$ are not proper sub-spaces of $W$, then we do not define $\rho$ at $(F,w)$. We can also have repetitions; in symbols, for some $i$, we can have that $\rho(F)_i=\rho(F)_{i+1}$. When this happens, we skip the step $i+1$ of the flag, and we relabel the indexes. To have well-defined flag, we still have to normalise the weight. With this description, we can prove the following lemma.
The retraction $\rho$ defined above is a continuous map for the simplicial topology.
Since the simplicial topology on $\Delta$ is the direct limit of the topology on the apartments, it is enough to show that $\rho$ restricted to any apartment $A$ is continuous.
Let $A$ be an apartment in $\Delta(V)$ and $\underline{v}$ the corresponding basis. Co-ordinates on $A$ are given by the weights $w$. Let $B$ be an apartment in $\Delta(W)$, and $\underline{u}$ the corresponding basis. The map $\rho \colon U_A\cap \rho^{-1}B\to B$ is given just by the projection onto some of the co-ordinates, i.e. the weigh of $u_i$ in $(\rho(F),\rho(w))$ is just $w_j$ for an appropriate index $j$. This shows that $\rho$ restricted to $A\cap \rho^{-1}B$ is continuous. Since this holds for all apartments $B$ of $\Delta(W)$, we have proven that $\rho$ restricted $A$ is continuous.
Let us stress that $\rho$ does not preserve many geometric features of $\Delta(V)$. To start with, $\rho$ is not open: indeed, already locally on an apartment $A$, we can see that $\rho$ is like a linear projection followed by a linear inclusion, and the latter is not open. This restriction does not preserve neither the simplicial structure nor the apartments. Moreover, $\rho$ does not preserve geodesics. To see this, one can take two flags $F$ and $G$ such that does not exists an apartment which contains $F$, $G$ and $W$, where we see $W$ as a one step flag, so a vertex of $\Delta(V)$.
[Proof of Theorem \[thm:cont\]]{}\[sec:prov\] Let us start off by looking at the Segre morphism $$S\colon {\mathbb{P}}H^0(X,L)^{\vee}\to {\mathbb{P}}\operatorname{Sym}^k H^0(X,L)^{\vee}$$ A test configuration ${\mathcal{X}}$ embedded ${\mathbb{P}}H^0(X,L)^{\vee}$ can be re-embedded in a ${\mathbb{G}_m}$-equivariant way in $ {\mathbb{P}}\operatorname{Sym}^k H^0(X,L)^{\vee}$ via $S$. The test configuration $(S({\mathcal{X}}),{\mathcal{O}}(1))$ is isomorphic to $({\mathcal{X}},{\mathcal{L}}^k)$; in particular, $(S({\mathcal{X}}),{\mathcal{O}}(1))$ is trivial if and only if $({\mathcal{X}},{\mathcal{L}}^k)$ is trivial. If $\lambda$ is a 1Ps of $SL(H^0(X,L)^{\vee})$ inducing ${\mathcal{X}}$, then $S(\gamma)$ induces $(S({\mathcal{X}}),{\mathcal{L}}^k)$.
Let now $[\gamma]$ be a point of $\Delta_S$, assume it acts non-trivially on $S(X)$, so that it induces a non-trivial test configuration (we mean non-trivial in the sense of Definition \[def:tc\]). This test configuration has exponent $k$, because the restriction of ${\mathcal{O}}(1)$ to $X$ is $L^k$. We define $\rho([\gamma])$ to be the point of $\Delta_k({\mathbb{Q}})$ which represents the test configuration induced by $\gamma$ (we can think at $\Delta_k({\mathbb{Q}})$ as the moduli space of exponent $k$ test configurations, and $\rho$ as a classifying map).
The composition of $\rho$ and $S$ makes sense, because $S(\lambda)$ acts non-trivially on $X$, and it is equal to $\iota$ because of the above discussion. To prove Theorem \[thm:cont\], we have to show that the map $\rho$ defined above is the retraction of buildings introduced in Section \[sec:retraction\].
Take $V=\operatorname{Sym}^k H^0(X,L)^{\vee}$ and $W=H^0(X,kL)^{\vee}$. There is a natural inclusion of $W$ in $V$ given by the co-multiplication, let $\rho$ be the associated retraction. The embedding of $X$ in ${\mathbb{P}}V$ factors trough the embedding of $X$ in ${\mathbb{P}}W$. Let $(F,w)$ be a weighted flag in $\Delta(V)$, and take a representative $\gamma$ which preserves $W$. Remark that $\rho$ is defined at $(F,w)$ if and only if the action of $\gamma$ on $W$ is not trivial; this is equivalent to ask that the action on $X$ is not trivial, hence that $\gamma$ induces a non-trivial test-configuration. The action of $\gamma$ on $X$ is equal to the action of $\gamma|_W$ on $X$, hence $\rho((F,w))$ represent the test configuration obtained by letting $\gamma$ acting on $X\subset {\mathbb{P}}V$, as requested. This concludes the proof of Theorem \[thm:cont\].
[Analogy with classical symmetric spaces]{}\[sec:analogy\]
In this section we work over the complex numbers. The Tits building $\Delta(V)$ of a vector space $V$ can be introduced as boundary of the symmetric space $H:=SL(V)/SU(V)$, and this gives also an alternative point of view on apartments, see for example [@Borel_Li]. We briefly recall this theory, and suggest an analogy for our Tits building $\Delta_{\infty}$.
The Killing metric on the Lie algebra of $SL(V)$ defines a constant scalar curvature metric with negative curvature on the homogeneous space $H$. Let $o$ be the image of the identity in $H$. A one parameter subgroup $\lambda$ of $SL(V)$ defines a map from $\mathbb{C}^*/\mathbb{S}^1\cong\mathbb{R}^+$ to $H$, the image is a geodesic starting at $o$. One can equivalently define $\Delta(V)$ as the set of all geodesics starting at $o$. One parameter subgroups are also from this point of view rational points of $\Delta(V)$. It is then possible to define a topology on ${\bar{H}}:=H \cup \Delta(V)$, which turns ${\bar{H}}$ into a compact space.
The image of a $d$ dimensional torus of $SL(V)$ in $H$ is a flat subspace, which means that it is isometric to a $d$ dimensional Euclidean space. Maximal tori give maximal flat subspaces of $H$. One can introduce the notion of rank of $H$ as the dimension of a maximal flat subspace of $H$, and this turns out to be equal to the rank of $SL(V)$. Let $T$ be a maximal torus and $\mathbb{E}_T$ its image in $H$. The boundary of $\mathbb{E}_T$, which can be defined by intersecting the closure of $\mathbb{E}_T$ in ${\bar{H}}$ with $\Delta(V)$, is the apartment $A_T$ defined in Section \[sec:prel\]. In a more colloquial language, we can say that apartments are boundaries of maximal flat subspaces.
In view of the Yau-Tian-Donaldson conjecture, it is natural to think at $\Delta_{\infty}$ as the boundary of the space $\mathcal{H}$ of Kähler metrics on $L$. This space has a natural Riemannian metric, as advocated in [@Donaldson_Symmetric]. Since $\mathcal{H}$ is infinite dimensional, standard results of Riemannian geometry does not go trough. We can look at $\mathcal{H}$ as a metric space rather than an infinite dimensional manifold, and in this set up $\mathcal{H}$, or rather its completion, is known to be a CAT(0) spaces, see for instance [@ele Theorem 4.11] and references therein. With the formalism of CAT(0) spaces one can proves many basic result such as the uniqueness of geodesics, see [@Bridson].
This point of view suggests that the notion of apartment for $\Delta_{\infty}$ should be relate to maximally flat subspace in the space of Kähler potentials. In this set up also the notion of maximally flat subspace is troublesome, we might define them as spaces which are isometric to a Hilbert space. Again, this is discussed in [@Donaldson_Symmetric Section 6].
In classical geometric invariant theory, one can define the normalized Hilbert-Mumford weight on $\Delta(V)$ as the slope of the Kempf-Ness functional on $H$, see for instance [@BHJ2 Section 5.1]. The Kempf-Ness functional is convex and Lipschitz, so one can use results from the theory of CAT(0) spaces to prove the existence of maximally destabilising one parameter subgroups, see [@Leeb1] and [@Leeb2]. One could try a similar approach to study optimally destabilizing test configurations by replacing the Kempf-Ness functional with the Mabuchi or the Ding functional. However, none of these functionals seems to be Lipschitz, so we do not know how to generalize this approach.
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abstract: 'We investigate the dynamical properties of one-dimensional dissipative Fermi-Hubbard models, which are described by the Lindblad master equations with site-dependent jump operators. The corresponding non-Hermitian effective Hamiltonians with pure loss terms possess parity-time ($\mathcal{PT}$) symmetry after compensating an overall gain term. By solving the two-site Lindblad equation with fixed dissipation exactly, we find that the dynamics of rescaled density matrix shows an instability as the interaction increases over a threshold, which can be equivalently described in the scheme of non-Hermitian effective Hamiltonians. This instability is also observed in multi-site systems and closely related to the $\mathcal{PT}$ symmetry breaking accompanied by appearance of complex eigenvalues of the effective Hamiltonian. Moreover, we unveil that the dynamical instability of the anti-ferromagnetic Mott phase comes from the $\mathcal{PT}$ symmetry breaking in highly excited bands, although the low-energy effective model of the non-Hermitian Hubbard model in the strongly interacting regime is always Hermitian. We also provide a quantitative estimation of the time for the observation of dynamical $\mathcal{PT}$ symmetry breaking which could be probed in experiments.'
author:
- Lei Pan
- Xueliang Wang
- Xiaoling Cui
- Shu Chen
title: 'Interaction induced dynamical $\mathcal{PT}$ symmetry breaking in dissipative Fermi-Hubbard models'
---
Introduction {#sec1}
============
Dissipations are ubiquitous in nature since the physical systems are always inevitable to be influenced by surroundings. Under Markov approximation, the dynamics of dissipative quantum system is commonly described by Lindblad master equation [@Markov]. Since solving the Lindblad equation accurately is quite expensive, an approximate but efficient approach to describe the time evolution of open quantum system is directly handling the stationary Schrödinger equation, in which case the dynamics is determined by an effective non-Hermitian Hamiltonian under certain condition [@Daley; @Castin]. Under this scheme, it is feasible to study the dissipative dynamics by means of the spectrum of non-Hermitian Hamiltonian. Among various kinds of non-Hermitian Hamiltonians, there exists a fascinating one, i.e., the parity-time ($\mathcal{PT}$) symmetric Hamiltonian $\left[\mathcal{PT},H\right]=0$ whose spectra can be real and bounded below [@Bender1998]. There is a notable feature that the $\mathcal{PT}$ symmetry of a system can be spontaneously broken and the system undergoes a transition from $\mathcal{PT}$ unbroken phase to a spontaneously $\mathcal{PT}$ broken phase. It is shown that the eigenvalues in the $\mathcal{PT}$ unbroken phase are always purely real, while complex conjugated eigenvalues appear in the spectrum for the $\mathcal{PT}$ broken phase and the system exhibits many novel phenomena once $\mathcal{PT}$ symmetry broken transition happens.
During the past decade, $\mathcal{PT}$-symmetric systems and the corresponding $\mathcal{PT}$ symmetry breaking have been experimentally realized and explored in various classical systems [@Guo; @Longhi1; @Feng1; @Ruter; @Lin1; @Regensburger; @Feng2; @Feng3; @Hodaei1; @Yang3; @Hodaei2; @Yang4; @Musslimani; @Fatkhulla; @Andrey; @Wang; @Wimmer; @Schindler; @ZhangX; @Cummer; @Ding2; @YangL; @ChangL] and quantum systems including quantum gas [@LuoL], single spin system [@DuJ], synthetic lattice [@Lapp] and single photon system [@XueP]. Whereas the above experiments on quantum $\mathcal{PT}$ systems mainly focus on single particle physics [@LuoL; @DuJ; @Lapp; @XueP], recent theoretical studies have revealed intriguing physical properties with the interplay of the non-Hermitian effects and interactions [@Pan1; @Ueda1; @Cui; @Ueda4; @Ueda5; @YuZ], such as enhanced sensitivity at exceptional points [@Pan1; @YuZ], non-Hermitian superfluidity [@Ueda1] and enhanced pairing superfluidity [@Cui], $\mathcal{PT}$ symmetric quantum critical phenomena [@Ueda4; @Ueda5], anomalous slow dynamics in quantum criticality [@ZhaiH], and nontrivial non-Hermitian many-body topological phases [@NH-FQH; @XuZH; @ZhuSL; @Nori]. Particularly, several recent works reported experimental studies of dissipative many-body systems with controllable dissipation in bosonic optical lattices [@Bouganne; @Tomita1; @Tomita2], which have stimulated theoretical interests in exploring novel physical phenomena induced by the interplay between interaction and dissipation [@ZCai; @Daley; @Pan1; @Ueda1; @Cui; @Ueda4; @Ueda5; @YuZ; @ZhaiH; @NH-FQH; @XuZH; @ZhuSL; @Nori].
In this work, we study the effect of interaction on the quantum dynamics in 1D dissipative Fermi-Hubbard models with effective parity-time symmetry. To begin with, we consider the two-site system with site-dependent dissipation, which is created by a laser beam acting on one of lattice sites to ensure the emergence of $\mathcal{PT}$ symmetry, as schematically shown in Fig.\[Fig1\]. By numerically solving the Lindblad master equation for this double-well model, we find that the dissipative system can exhibit a clear dynamical signature of $\mathcal{PT}$ symmetry breaking characterized by an anomalous dynamical instability, if we rescale the density operator by multiplying an overall exponential factor. We further reveal the important role of interaction and demonstrate that a strong interaction can induce dynamical instability even in the low dissipation regime, which can be alternatively understood by the effective non-Hermitian Fermi Hubbard model with site-dependent imaginary potentials. In the strong interaction limit, the low-energy physics of half-filled fermions in the non-Hermitian dissipative lattice can be effectively described by a Hermitian anti-ferromagnetic (AFM) spin exchange model. However, the fermion dynamics is still unstable due to the existence of complex eigenvalues for the highly excited states. This instability persists in multi-site systems and leads to the dynamical instability of Mott phase. We estimate the lifetime of AFM state in Mott phase by degenerate perturbation analysis. Our results demonstrate that the interplay between interaction and dissipation leads to dynamical $\mathcal{PT}$ symmetry breaking and unstable quantum dynamics in the dissipative Fermi-Hubbard model, which are very different dynamical behaviors from its Hermitian correspondence.
The rest of the paper is organized as follows: In Sec. II we present the formalism of solving the dynamics of two-site dissipative Hubbard model by using both the Lindblad master equation and effective non-Hermitian Hamiltonian. We demonstrate that the two methods play an equivalent role in dealing with the dynamical problem for the initial state with fixed particle number. In Sec. III, we study the multi-site systems in the scheme of effective non-Hermitian Hamiltonian. Finally, we conclude and discuss the potential experimental detection in Sec. IV.
![Schematic of the experimental setup for double-well Fermi-Hubbard model with a site-dependent dissipation. A resonant laser is used to create a dissipation with strength $\gamma$ on the left well.[]{data-label="Fig1"}](Fig1.pdf){width="7.5cm"}
Dissipative Fermi-Hubbard model: Two-site system {#sec2}
================================================
To investigate the quantum dynamics of an open quantum system with dissipation, we take advantage of the Lindblad master equation ($\hbar=1$) [@Lindblad; @Lindblad2] $$\begin{aligned}
\frac{d\varrho(\tau)}{d \tau }&=&-i[H_s,\varrho]+\sum_k\Big(L_k\varrho L_k^{\dag}-\frac{1}{2}\{L_k^{\dag}L_k,\varrho\}\Big) \nonumber \\
&=&-i\big(\mathcal{H}\varrho-\varrho \mathcal{H}^{\dag}\big)+\sum_kL_k\varrho L_k^{\dag}, \label{Eq2.1}\end{aligned}$$ where $\varrho(\tau)$ denotes the density matrix, $H_s$ represents Hamiltonian of the system and $L_k$ is Lindblad operator with the index $k$ (such as space, spin degrees of freedom, etc.). Here $\mathcal{H}$ is called effective non-Hermitian Hamiltonian defined by $$\mathcal{H}\equiv H_s-\frac{i}{2}\sum_kL_k^{\dag}L_k,$$ which includes a non-Hermitian part. In this work, we shall consider the Hubbard model with $L$ sites, i.e., $$H_s = -t\sum_{j=1,\sigma}^{L-1}(c_{j\sigma}^{\dag}c_{j+1,\sigma}+h.c)+U \sum_{j=1}^{L} n_{j\uparrow}n_{j\downarrow}, \label{Hubbard}$$ where the symbol $j$ ($\sigma$) denotes the site (spin) index, $t$ represents the hopping term and $U$ the interaction strength. For convenience, we shall set $t=1$ as the energy unit throughout.
In order to present the formalism clearly and get an intuitive understanding, we firstly focus on the double well system and study its dynamics in details. The Lindblad operators are chosen as $L_{k=(1,\uparrow)}=2\sqrt{\gamma} c_{1\uparrow}, L_{k=(1,\downarrow)}=2\sqrt{\gamma} c_{1\downarrow}$, which represents that a single-particle dissipation is engineered on the left well with strength $\gamma$ (see Fig.\[Fig1\]), and then the Lindblad master equation becomes $$\begin{aligned}
\frac{d\varrho(\tau)}{d \tau}=-i\big(\mathcal{H}\varrho-\varrho \mathcal{H}^{\dag}\big)
+4\gamma c_{1\uparrow}\varrho c_{1\uparrow}^{\dag}+4\gamma c_{1\downarrow}\varrho c_{1\downarrow}^{\dag}.
\label{Eq2.2}\end{aligned}$$ The effective non-Hermitian Hamiltonian is given by $$\begin{aligned}
\mathcal{H}\equiv -t\sum_{\sigma}(c_{1\sigma}^{\dag}c_{2\sigma}+c_{2\sigma}^{\dag}c_{1\sigma})
+U\sum_{j=1}^{2}n_{j\uparrow}n_{j\downarrow}-2i\gamma n_1. \label{Eq2.3}\end{aligned}$$ The full dynamics can be obtained once the Lindblad master equation (\[Eq2.2\]) is solved. Since there is only loss term, the effective Hamiltonian does not posses $\mathcal{PT}$ symmetry and its spectrum consists of a series of complex eigenvalues. Nevertheless, the Hamiltonian is associated with a Hamiltonian with $\mathcal{PT}$ symmetry apart from a total damping term, which can be effectively obtained by a dynamical rescaling of the density operator. For convenience, we define a rescaled density matrix as $$\begin{aligned}
\widetilde{\rho}(\tau)\equiv e^{2N\gamma \tau }\varrho_N( \tau ), \label{2.4}\end{aligned}$$ where $N=n_1+n_2$ is the total particle number in the initial state and $\varrho_N(\tau)=P_N\varrho(
\tau) P_N$ ($P_N$ is the projection operator on $N$-particle subspace).
![Time evolutions of the probabilities, shown in (a1) and (b1), and rescaled probabilities, in (a2) and (b2), for both of two fermions occupying the left well and right well with dissipation strengths $\gamma/t=0$, $\gamma/t=0.25$, $\gamma/t=0.5$, $\gamma/t=0.75$, $\gamma/t=1.0$, $\gamma/t=1.25$, respectively. Here $U=0$ and the initial value is chosen by $P_L(0)=1$. The solid lines are from Lindblad master equation and dashed lines are obtained from the non-Hermitian effective Hamiltonian.[]{data-label="Fig2"}](Fig2_a1a2.pdf "fig:"){width="8.5cm"} ![Time evolutions of the probabilities, shown in (a1) and (b1), and rescaled probabilities, in (a2) and (b2), for both of two fermions occupying the left well and right well with dissipation strengths $\gamma/t=0$, $\gamma/t=0.25$, $\gamma/t=0.5$, $\gamma/t=0.75$, $\gamma/t=1.0$, $\gamma/t=1.25$, respectively. Here $U=0$ and the initial value is chosen by $P_L(0)=1$. The solid lines are from Lindblad master equation and dashed lines are obtained from the non-Hermitian effective Hamiltonian.[]{data-label="Fig2"}](Fig2_b1b2.pdf "fig:"){width="8.5cm"}
To solve the dynamical problem, one needs to provide an initial state since Eq.(\[Eq2.2\]) is the first-order differential equation. Here we choose $\varrho(0)=\ket{\uparrow\downarrow}_1\ket{0}_2 \sideset{_2}{}{\mathop{\bra{0}}}\sideset{_1}{}{\mathop{\bra{\downarrow\uparrow}}}$ as the density matrix for the initial state with both the fermions occupying the left well. We firstly discuss the non-interaction case ($U=0$). In Fig.\[Fig2\] (a1) and (a2), we display the survival probability and rescaled survival probability of two fermions in the left well, given by $P_L(\tau)=\sideset{_2}{}{\mathop{\bra{0}}}\leftidx{_1}{\bra{\downarrow\uparrow}}\rho(\tau)\ket{\uparrow\downarrow}_1\ket{0}_2 $ and $\widetilde{P}_L(\tau)=\sideset{_2}{}{\mathop{\bra{0}}}\leftidx{_1}{\bra{\downarrow\uparrow}}\widetilde{\rho}(\tau)\ket{\uparrow\downarrow}_1\ket{0}_2 $, respectively. After some straightforward calculations, we can get $$P_L(\tau)= e^{-4 \gamma \tau} f(\tau), ~~~~\widetilde{P}_L(\tau)= f(\tau),$$ where $$f(\tau) =\left| \frac{-t^2 +(2\omega^2+t^2) \cosh(2 \omega \tau) -2 \gamma \omega \sinh(2 \omega \tau)}{2 \omega^2}\right|^2$$ with $\omega=\sqrt{\gamma^2-t^2}$. It is clear that $P_L(\tau)$ always decays with the growth of dissipation strength $\gamma$. However, the rescaled $\widetilde{P}_L(\tau)$ displays divergence behavior for $\gamma/t > 1$, whereas it is an oscillating function of time for $\gamma/t<1$. In Fig.\[Fig2\] (b1) and (b2), we display the time evolution of probability and rescaled probability for two fermions occupying the right well, given by $P_R(\tau)=\sideset{_2}{}{\mathop{\bra{\uparrow\downarrow}}}\sideset{_1}{}{\mathop{\bra{0}}}\rho(\tau)\ket{0}_1\ket{\uparrow\downarrow}_2 $ and $\widetilde{P}_R(\tau)=\sideset{_2}{}{\mathop{\bra{\uparrow\downarrow}}}\sideset{_1}{}{\mathop{\bra{0}}}\widetilde{\rho}(\tau)\ket{0}_1\ket{\uparrow\downarrow}_2 $, respectively. While $P_R(\tau)$ decays very quickly as shown in Fig.\[Fig2\] (b1), we can see the rescaled amplitude of $\widetilde{P}_R(\tau)$ grows with the increase of $\gamma$ and then diverges once the $\gamma$ crosses a certain threshold ($\gamma_c/t=1$) as shown in Fig.\[Fig2\] (b2).
Now we study the effect of interaction on the quantum dynamics and consider the case with both two fermions occupying the left well as the initial state. In the absence of dissipation, the increase of interaction strength shall suppress the hopping of the fermion pair to its neighboring site and lead to the formation of repulsively bound pair [@Winkler; @WangLi], which is dynamically more stable with stronger interaction. In order to reveal the interplay between the interaction and dissipation, we display $P_L(\tau)$ and the rescaled $\widetilde{P}_L(\tau)$ with a fixed $U/t=20$ and various $\gamma$ in Fig.\[Fig3\] (a1) and (a2), and a fixed dissipative strength $\gamma/t=0.05$ and different $U$ values in Fig.\[Fig3\] (b1) and (b2), respectively. As seen in the Fig.\[Fig3\] (a2), the rescaled $\widetilde{P}_L(\tau)$ oscillates periodically over time for $\gamma/t=0$, $0.02$, and $0.03$, but a divergence appears when $\gamma/t=0.05$. Similarly, the rescaled $\widetilde{P}_L(\tau)$ is divergent when the interaction strength exceeds a threshold ($U_c/t \approx 19.9$), as shown in Fig.\[Fig3\] (b2). The divergence in the rescaled $\widetilde{P}_L(\tau)$ indicates the emergence of dynamical instability as an interplay effect of interaction and dissipation. As we will show below, this interaction induced dynamical instability stems from the emergence of imaginary parts in effective non-Hermitian Hamiltonian which is closely related to spontaneously $\mathcal{PT}$ symmetry breaking.
![ Time evolutions of the probabilities (a1) and rescaled probabilities (a2) of both two fermions occupying the left-well for different dissipation strengths $\gamma$ with fixed interaction strengths $U/t=20$. Time evolutions of the probabilities (b1) and rescaled probabilities (b2) of both two fermions occupying the left-well for different interaction strengths at $\gamma/t=0.05$. The solid lines are from Lindblad master equation and dashed lines are obtained from the non-Hermitian effective Hamiltonian.[]{data-label="Fig3"}](Fig3_a1a2.pdf "fig:"){width="8.0cm"} ![ Time evolutions of the probabilities (a1) and rescaled probabilities (a2) of both two fermions occupying the left-well for different dissipation strengths $\gamma$ with fixed interaction strengths $U/t=20$. Time evolutions of the probabilities (b1) and rescaled probabilities (b2) of both two fermions occupying the left-well for different interaction strengths at $\gamma/t=0.05$. The solid lines are from Lindblad master equation and dashed lines are obtained from the non-Hermitian effective Hamiltonian.[]{data-label="Fig3"}](Fig3_b1b2.pdf "fig:"){width="8.0cm"}
Our numerical results demonstrate the equivalence between the Lindblad master equation and the formalism in terms of effective non-Hermitian Hamiltonian in dealing with our studied dynamical problem. To get a clear understanding, we unveil why the two different methods are equivalent. Given the reality in experiment that the particle number $N$ of an initial state is a certain number, the initial density matrix $\varrho(0)$ is block diagonalized in particle number space, which results in the quantum jump term $\gamma c_{j\sigma}\varrho(0)c_{j\sigma}^{\dag}$ having no effects on the dynamics in the subspace of initial particle number. More specifically, suppose the initial density operator given by $\varrho(0)=|\psi_0(N)\rangle \langle \psi_0(N)|$ with conserving particle number $N$, the quantum jump term acts on the density matrix as $c_{j\sigma}|\psi_0(N)\rangle \langle \psi_0(N)|c_{j\sigma}^{\dag}$ and one can find the matrix elements are always zero in the subspace with the initial particle number, i.e. $P_Nc_{j\sigma}\varrho c_{j\sigma}^{\dag}P_N\equiv0$. In other words, the dynamics in the $N$-particle subspace is completely determined by non-Hermitian effective Hamiltonian $$\begin{aligned}
\varrho_N(\tau)=e^{-i\mathcal{H} \tau}\varrho_N(0)e^{i\mathcal{H}^{\dagger} \tau}, \label{2.5}\end{aligned}$$ where $\varrho_N(\tau)=P_N\varrho P_N$ and $[\mathcal{H},P_N]=0$ is also considered. Thus, from viewpoint of the effective non-Hermitian Hamiltonian, the full dynamics of $\rho_N(\tau)$ in Eq.(\[2.4\]) is obtained after solving Eq.(\[2.5\]). This demonstration is also applied to other non-Hermitian systems[@Wunner1; @Wunner2; @Wunner3; @Wunner4; @Wunner5; @ZhuBG]. The time evolution of $\rho(\tau)$ obtained by diagonalizing Eq.(\[2.5\]) ($\mathcal{H}$ and $\mathcal{H}^{\dagger}$) is completely consistent with the solution of the Lindblad master equation as shown in Fig.\[Fig2\] and Fig.\[Fig3\].
In order to reveal the origin of the instability in dynamics (see Fig.\[Fig2\] and Fig.\[Fig3\]) from the Lindblad master equation, we rewrite the effective non-Hermitian Hamiltonian as $\mathcal{H}=H_\mathcal{PT}-iN\gamma$, then $$\begin{split}
H_\mathcal{PT}&=-t\sum_{\sigma}(c_{1\sigma}^{\dag}c_{2\sigma}+c_{2\sigma}^{\dag}c_{1\sigma})
+U\sum_{j=1}^{2}n_{j\uparrow}n_{j\downarrow}\\
&-i\gamma(n_1-n_2). \label{2.6}
\end{split}$$ One can see clearly that $H_\mathcal{PT}$ is $\mathcal{PT}$ symmetric, namely $\left[\mathcal{PT},H\right]=0$ since $\mathcal{P}c_{1(2)}\mathcal{P}=c_{2(1)}$ and $\mathcal{T}i\mathcal{T}=-i$ and then we find $$\begin{aligned}
\widetilde{\rho} (\tau)=e^{-iH_\mathcal{PT}\tau}\varrho_N(0)e^{iH_\mathcal{PT}^{\dagger}\tau}, \label{2.7}\end{aligned}$$ which indicates the dynamics of rescaled density matrix $\widetilde{\rho} (\tau)$ is determined by the $\mathcal{PT}$ symmetric Hamiltonian and all the information are contained in the eigenvalues and eigenvectors of $H_\mathcal{PT}$ ($H_\mathcal{PT}^\dagger$). We can deduce that the anomalous dynamical divergence is driven by the $\mathcal{PT}$ symmetry breaking since the imaginary parts of spectrum of $H_\mathcal{PT}$ in $\mathcal{PT}$-symmetry-breaking phase always appear in pairs with complex conjugate values[@Bender2]. In Fig.\[Fig4\], we display the energy spectrum of $H_\mathcal{PT}$ with $\gamma/t=0.05$ taken the same value as in Fig.\[Fig3\] (b). It is clear that the $\mathcal{PT}$ symmetry is broken when interaction strength exceeds a critical value $U_c/t \approx 19.9 $. The dynamical instability appeared in Fig.\[Fig3\] (b2) is induced by the $\mathcal{PT}$-symmetry breaking.
![Exact energy spectrum of $\mathcal{PT}$-symmetric double-well Fermi-Hubbard model with a spin-up and spin-down fermions. Real parts (a) and imaginary parts (b) as functions of interaction strength. Here $\gamma=0.05$ and $t=1$ as the energy unit.[]{data-label="Fig4"}](Fig4.pdf){width="9cm"}
In Fig.\[Fig5\], we plot the phase diagram in $\gamma-U$ plane for the system including one spin-up and one spin-down fermions. For a given $U$, increasing the dissipative strength $\gamma$ shall induce transition from $\mathcal{PT}$ unbroken phase to broken phase. Particularly, in the non-interacting limit $U=0$, the system is in $\mathcal{PT}$-symmetry-breaking phase when $\gamma\geq1$, which explains the results shown in Fig.\[Fig2\] (a2) and (b2), where the rescaled probabilities diverge once $\gamma$ is over the critical value ($\gamma_c=1$). On the other hand, for a fixed $\gamma$, the system undergoes $\mathcal{PT}$ phase transition as the interaction strength increase over a critical value.
In strong interaction regime ($U\gg t,\gamma$), we can derive the low-energy effective Hamiltonian of Eq.(\[2.6\]), which reads as $$\begin{aligned}
H_{\rm{eff}}^{\rm{low}}=J_{\rm{eff}}\Big(\mathbf{S}_1\cdot \mathbf{S}_2-\frac{1}{4}\Big),\label{2.8}\end{aligned}$$ where $J_{\rm{eff}}=\frac{4t^2U}{U^2+4\gamma^2}$. Apparently, the effective anti-ferromagnetic (AFM) exchange model (\[2.8\]) is a Hermitian Hamiltonian, and therefore has real eigenvalues. Similarly, an effective Hamiltonian in high energy subspace can be derived by means of degenerate perturbation theory and is given by $$\begin{aligned}
H_{\rm{eff}}^{\rm{high}}=U\mathbf{I}+2i\gamma\sigma_z+\frac{2t^2}{U}\sigma_x, \label{2.9}\end{aligned}$$ where $\mathbf{I}$ denotes the identity matrix and $\sigma_x,\sigma_z$ are Pauli matrices where spin-up (spin-down) represents two fermions occupying the left (right) well. Interestingly, this simple $2\times2$ Hamiltonian predicts the $\mathcal{PT}$-symmetry broken transition when interaction crosses the critical point $U_c=\frac{t^2}{\gamma}$, which agrees with the numerical result very well in the large U regime as shown in Fig.\[Fig5\].
![Phase diagram of $\mathcal{PT}$-symmetric double-well Fermi-Hubbard model in $\gamma-U$ plane. The red solid line is the critical line below (above) which the system is in $\mathcal{PT}$ unbroken (broken) phase phase. The blue dash-dot line is the prediction by degenerate perturbation theory in strong interaction limit.[]{data-label="Fig5"}](Fig5.pdf){width="8.0cm"}
From Eq.(\[2.8\]), we see the ground state is a Mott phase with AFM order, which seems stable as the ground state energy is always real even in the presence of dissipation. Nevertheless, the Mott phase is found to be dynamically unstable when the interaction strength $U$ exceeds a threshold. For the system with initial state prepared in the AFM state, i.e., the ground state of Eq.(\[2.6\]) or equivalently Eq.(\[2.8\]) in the strong interaction regime, its dynamical properties are determined by all the eigenvalues and eigenvectors of $H_\mathcal{PT}$, instead of Eq.(\[2.8\]) solely. Therefore, the dynamical instability stems from the $\mathcal{PT}$ symmetry breaking in Eq.(\[2.9\]). In the scheme of effective Hamiltonian, the rescaled probability can be calculated via $$\begin{split}
\widetilde{P}_{\rm{AFM}}(\tau)&=\Big|\langle \mathbf{AFM}|e^{-iH_\mathcal{PT} \tau}|\mathbf{AFM}\rangle\Big|^2 \\
&=\Big|\sum_{n}\langle \mathbf{AFM}|\frac{e^{-iE_{n} \tau}|\psi_n^R\rangle \langle \psi_n^L|}{\langle \psi_n^L|\psi_n^R\rangle}|\mathbf{AFM}\rangle\Big|^2,
\end{split}
\label{2.10}$$ where $|\psi_n^R\rangle$ and $|\psi_n^L\rangle$ are respectively the right and left eigenvector, which are defined via $H_{\mathcal{PT}}|\psi_{n}^{R}\rangle=E_{n}|\psi_{n}^{R}\rangle$, $H_{\mathcal{PT}}^\dagger|\psi_{n}^{L}\rangle=E_{n}^*|\psi_{n}^{L}\rangle$ and $\langle \psi_n^{{L,R}}|=\big(|\psi_{n}^{{L,R}}\rangle\big)^\dagger$. It can be found from Eq.(\[2.10\]) that unstable dynamics is determined by combination of two factors: (positive) imaginary part of highly excited states and their overlap with AFM state. It follows from Eq.(\[2.9\]) that the imaginary part is given by $2\mathbf{Im}\sqrt{\frac{t^4}{U^2}-\gamma^2}$ and the overlap between the highly excited states and AFM state is about $\frac{\sqrt{2}t}{U}$. Accordingly, we can define a timescale (lifetime) $\tau$ beyond which the rescaled probability of AFM state diverges exponentially. The lifetime is estimated at $\tau \approx \frac{\log\big(\frac{U}{\sqrt{2}t}\big)}{\mathbf{Im} \sqrt{\frac{t^4}{U^2}-\gamma^2}}$ and $\tau\approx30$ for a typical choice of parameters $\gamma=0.1$ and $U=20$, which matches well with the numerical result (see Fig.\[Fig6\] and the inset). Basically, the instability can be viewed as a signature of dynamical $\mathcal{PT}$-symmetry breaking which inevitably occurs along the time evolution even the nonzero overlap between AFM state and highly excited state with a finite imaginary part is small.
![Time evolutions of the rescaled probabilities of the AFM state in Mott phase $\widetilde{P}_{\rm AFM}(\tau)$ for different interaction strengths. In the inset, the blue line shows an accurate exponential growth which gives the lifetime $\tau\approx30$ of Mott state. The solid line is the best fitting with an exponential function.[]{data-label="Fig6"}](Fig6.pdf){width="8.0cm"}
Dissipative Fermi-Hubbard model: Multi-site system {#sec3}
==================================================
In the preceding section, we have studied the non-Hermitian dynamics for two-site Fermi-Hubbard model. In this section, we extend two-site to multi-site system and examine the non-Hermitian dynamics. We consider the 1D Fermi-Hubbard model with balanced gain and loss on boundary, which is given by $$H=H_s +i\gamma(n_L-n_1),
\label{3.1}$$ where $H_s$ is the Hubbard Hamiltonian given by Eq.(\[Hubbard\]) and the dissipative strength is denoted by $\gamma$. $n_1$ ($n_L$) denotes particle number on the first (last) site. It is obvious that the Hamiltonian (\[3.1\]) is $\mathcal{PT}$ symmetric, i.e., $\mathcal{PT}H\mathcal{TP}=H$ according to the operations of $\mathcal{P}$ and $\mathcal{T}$ $$\begin{aligned}
\mathcal{P}c_j\mathcal{P}=c_{L+1-j},~~\mathcal{T}i\mathcal{T}=-i.\label{3.2}\end{aligned}$$ Apart from an overall exponentially damping factor, we note that Eq.(\[3.1\]) can be taken as an effective Hamiltonian of the dissipative Hubbard systems with fine-tune site-dependent dissipation parameters: $L_{k=(1,\sigma)}=\sqrt{6 \gamma} c_{1 \sigma}$, $L_{k=(i,\sigma)}=2 \sqrt{\gamma} c_{i \sigma}$ ($i=2,\cdots,L-1$), and $L_{k=(L,\sigma)}=\sqrt{2 \gamma} c_{L \sigma}$. The Fermi-Hubbard model with spin-independent boundary fields can be exactly solved by Bethe ansatz [@BA1; @BA2; @BA3; @BA4]. Here we generalize the real boundary fields to imaginary ones and derive the Bethe ansatz equations (BAEs) for Eq.(\[3.1\]): $$\begin{split}
&e^{i2k_j(L+1)}\frac{1-i\gamma e^{-ik_j}}{1-i\gamma e^{ik_j}}\frac{1+i\gamma e^{-ik_j}}{1+i\gamma e^{ik_j}}\\
&=\prod_{\beta=1}^{M}\frac{\sin k_j-\lambda_\beta+iu}{\sin k_j-\lambda_\beta-iu}\frac{\sin k_j+\lambda_\beta+iu}{\sin k_j+\lambda_\beta-iu},~~j=1,\ldots,N \\
& \prod_{l=1}^N\frac{\lambda_\alpha-\sin k_j+iu}{\lambda_\alpha-\sin k_j-iu}\frac{\lambda_\alpha+\sin k_j+iu}{\lambda_\alpha+\sin k_j-iu} \\
&=\prod_{\beta\neq\alpha}^{M}\frac{\lambda_\alpha-\lambda_\beta+2iu}{\lambda_\alpha-\lambda_\beta-2iu}
\frac{\lambda_\alpha+\lambda_\beta+2iu}{\lambda_\alpha+\lambda_\beta-2iu},~~\alpha=1,\ldots,M,
\label{BAEs}
\end{split}$$ where $L$, $N$, $M$ denote the number of lattice sites, total particle number and particle number with spin down respectively, $u=\frac{U}{4t}$ and the energy eigenvalue is expressed by $E=-2t\sum_{j=1}^N\cos k_j$. The parameter sets $\{k_j\}$ denote the charge momenta and $\{\lambda_\alpha\}$ represent the spin rapidities which are introduced to describe the motion of spin waves. In strong interaction regime with $U/t \gg 1$ and $U/\gamma \gg 1$, the spin and charge degrees of freedom are separated and the quasi-momentum $k_j$ keep finite while $\lambda_\alpha$ are proportional to $u$. Expanding up to the first order with respect to $k_j/u$, the ground state energy can be expressed by $$\begin{aligned}
E=-\frac{t}{u}\frac{N}{L}\zeta,\label{3.3}\end{aligned}$$ where $\zeta=\sum_{\alpha=1}^{M} \frac{4}{\Lambda_{\alpha}^{2}+1}$ with $\Lambda_{\alpha}=\lambda_\alpha/u$ and $\{\Lambda_{\alpha}\}$ satisfy the following equations $$2N\theta\left(2\Lambda_{\alpha}\right)=2\pi J_{\alpha}+\sum_{\beta \neq \alpha}^{M}\left[\theta\left(\Lambda_{\alpha}-\Lambda_{\beta}\right)
+\theta\left(\Lambda_{\alpha}+\Lambda_{\beta}\right)\right],\label{3.4}$$ where $\theta(x)=2\arctan(x/2)$ quantum number $J_{\alpha}$ are positive integers. At the half-filling $N=L$, Eq.(\[3.4\]) is nothing but the well-known BAEs of the open boundary Heisenberg model $$H=J\sum_{i=1}^{L-1}\big(\mathbf{P}_{i, i+1}-1\big),\label{3.5}$$ where $\mathbf{P}_{i, i+1}=\mathbf{S}_i\cdot \mathbf{S}_{i+1}+\frac{3}{4}$ is the permutation operator between the $i$th and $(i+1)$th spins. Since the ground state energy of Hamiltonian (\[3.5\]) with $J>0$ can be written as $E=-J\zeta$[@Sutherland], by comparing it with Eq.(\[3.3\]) we can find easily that the coupling constant $J=\frac{4t^2}{U}$. This is consistent with the result obtained by degenerate perturbation theory in which the half-filling Fermi-Hubbard is reduced effectively to anti-ferromagnetic Heisenberg model in single-occupied subspace when $U/t\gg1$.
![Exact energy spectrum of four-site non-Hermitian Fermi-Hubbard model with $N_\uparrow=N_\downarrow=2$. (a) displays the real parts and (b) the imaginary parts versus $\gamma$ for the system with $U=10$ and $t=1$. (c), (d) and (e) are the enlarged figures of (a). The imaginary parts first appears at the highly excited states which consists mainly of two double-occupied states whose energy scale is about 20 as shown in (c) and then at the states with one double-occupied states with energy scale 10 (see (d)). The low-energy states always have real spectrum (see (e)).[]{data-label="Fig7"}](Fig7.pdf){width="8.5cm"}
In Fig.\[Fig7\], we display the energy spectrum versus $\gamma$ for half-filling four sites system with balanced spin $N_\uparrow=N_\downarrow=2$ and $U=10$. As shown in Fig.\[Fig7\] (a), the real parts of energy spectrum are divided into three regions, which are enlarged in Fig.\[Fig7\] (c)-(d), respectively. We find that the spectrum of low-energy states are always real for arbitrary $\gamma$ in the whole region shown in Fig.\[Fig7\](e), corresponding to single-occupied Mott states described by Heisenberg model. As $\gamma$ increases, the complex conjugate pairs appear first in two double-occupied states (Fig.\[Fig7\](c)) and then in one double-occupied states (Fig.\[Fig7\](d). While all eigenvalues are real for small $\gamma$, the emergence of imaginary parts of energy eigenvalues in excited bands implies the breaking of $\mathcal{PT}$ symmetry when $\gamma$ exceeds a critical value. Similarly, if we fix $\gamma \ll 1$ and increase $U$, the complex eigenvalues will appear when $U$ exceeds a critical value.
![Time evolutions of the rescaled probabilities of all four fermions occupying the left two sites $\widetilde{P}_L(\tau)$ of four-site system for different interaction strengths. Here $\gamma/t=0.05$ and the initial value is chosen by $\widetilde{P}_L(0)=1$.[]{data-label="Fig8"}](Fig8.pdf){width="8.0cm"}
In order to reveal the signature of dynamical $\mathcal{PT}$ symmetry breaking, we define a rescaled density matrix as $$\begin{aligned}
\widetilde{\rho}(\tau)\equiv e^{4 N\gamma \tau}\varrho_N( \tau).\end{aligned}$$ Suppose that the initial state is prepared as the state with all four fermions occupying the left two sites, we plot the rescaled probabilities of the state remaining in the initial state (rescaled return probability) in Fig.\[Fig8\] for the system with $\gamma/t=0.05$ and various $U$. The interaction induced instability is clearly observed when the interaction exceeds a threshold, which is in qualitative agreement with results discussed in the double-well system. Similarly, for the dynamics of the AFM state (ground-state of Hamiltonian (\[3.5\])), the interaction induced instability is also detected in the rescaled return probability as shown in Fig.\[Fig9\], which shares the same physical origin with the double-well system.
![Time evolutions of the rescaled probabilities of AFM (Mott) state $\widetilde{P}_{\rm AFM}(\tau)$ of four-site system for different interaction strengths which take the same values as in Fig.\[Fig8\] . Here $\gamma/t=0.05$ and the initial state is chosen by $\widetilde{P}_{\rm AFM}(0)=1$.[]{data-label="Fig9"}](Fig9.pdf){width="8.0cm"}
In order to estimate time for the emergence of dynamical instability in the rescaled return probability, we employ the same expression (\[2.10\]) on this system. The characteristic time for dynamical $\mathcal{PT}$ symmetry breaking is given by $$\begin{aligned}
\tau =\frac{\log\big(1/|C|\big)}{\big|\mathbf{Im} E_{MI}\big|}, \label{3.6}\end{aligned}$$ where $C$ denotes the overlap between the AFM (Mott) state and the double-occupied excited state whose energy $E_{MI}$ has maximum imaginary part.
Besides the model (\[3.1\]), we note that some other effective Hamiltonians with $\mathcal{PT}$ symmetry can be also realized by engineering site-dependent dissipations. For example, if the dissipations are engineered only on the odd or even sites, i.e, $L_{k=(i,\sigma)}=2 \sqrt{\gamma} c_{i \sigma}$ with $i=2n-1$ or $2n$, apart from an overall damping term, we can get an effective Hamiltonian with $\mathcal{PT}$ symmetry, which reads as $$H=H_s \pm \sum_{j=1}^{L}
(-1)^j i \gamma n_j ,$$ where $H_s$ is the Hubbard Hamiltonian given by Eq.(\[Hubbard\]) and $+$ ($-$) corresponds to the dissipations engineering on odd or even sites. Although the above Hamiltonians are no longer exactly solvable, they display similar physical phenomena as we studied in this work, i.e., the interplay of interaction and dissipation also induces $\mathcal{PT}$ symmetry breaking and leads to dynamical instability in the rescaled return probability.
Summary and outlook {#sec5}
===================
In summary, we have studied the quantum dynamics in dissipative Fermi-Hubbard model with hidden dynamic $\mathcal{PT}$ symmetry breaking, which can be characterized by the emergence of dynamical instability in the evolution of rescaled density matrix. By studying the dissipative double-well system in detail, we demonstrate the equivalence of the Lindblad master equation and effective non-Hermitian Hamiltonian in dealing with the dynamical evolution of an initial state with fixed particle number. For the dissipation engineered only in one of the wells, we find that its effective Hamiltonian can possesses $\mathcal{PT}$ symmetry if we rescale the density operator, and the dynamical instability occurs when the interaction exceeds a critical value. We further unveil that the effective Hamiltonian of multi-site Hubbard models with site-dependent dissipations can possess $\mathcal{PT}$ symmetry and exhibit the interaction induced dynamical instability. This is originated from $\mathcal{PT}$ symmetry breaking of the effective Hamiltonian accompanied by the emergence of complex eigenvalues. Particularly, for an effective Hubbard Hamiltonian with $\mathcal{PT}$ symmetry boundary fields, it is exactly solvable by applying the Bethe ansatz method. By analyzing the Bethe ansatz equations of multi-site systems, we find that the highly excited states are associated with complex eigenvalues, while the low-energy states described by effective AFM Heisenberg model always have real eigenvalues. However, the low-energy AFM state remains dynamically unstable due to the existence of $\mathcal{PT}$ symmetry breaking in the highly excited double-occupied bands. Motivated by this work, it is valuable to explore more fascinating physical systems including both non-Hermitian effects and many-body correlations. Finally, we discuss the experimental detection of the interaction induced dynamical $\mathcal{PT}$ symmetry breaking in dissipative optical lattices. For simplicity, we consider the double-well system in which the dissipation is engineered only at the left well. The experiment can initially prepare two fermions with one spin-up and one spin-down at the left well and at the same time apply a resonant laser to couple the lowest hyperfine state of atoms to their highly excited state $\ket{e}$, which generates the atom loss only at left well as shown in Fig.\[Fig1\]. The initial state is $\sideset{_2}{}{\mathop{\bra{0}}}\leftidx{_1}{\bra{\downarrow\uparrow}}\varrho(\tau=0)\ket{\uparrow\downarrow}_1\ket{0}_2=1 $ and all other matrix elements equal to zero. Then, one can measure the atom number in the left (right) well at the time $\tau$. The probability of finding both two fermions located at the left (right) well is given by $p_L(\tau)=N_{\uparrow\uparrow}^L(\tau)/N(\tau)$ ($p_R(\tau)=N_{\uparrow\downarrow}^R(\tau)/N(\tau)$), where $N_{\uparrow\uparrow}^L(\tau)$ ($N_{\uparrow\downarrow}^R(\tau)$) denotes the measurement times of two fermions located at the left (right) well at time $\tau$ and $N(\tau)$ is the total number of measurement times. This probability can be calculated by $\sideset{_2}{}{\mathop{\bra{0}}}\leftidx{_1}{\bra{\downarrow\uparrow}}\varrho(\tau)\ket{\uparrow\downarrow}_1\ket{0}_2 $ ($\sideset{_2}{}{\mathop{\bra{\uparrow\downarrow}}}\sideset{_1}{}{\mathop{\bra{0}}}\varrho(\tau)\ket{0}_1\ket{\uparrow\downarrow}_2 $), where $\varrho(\tau)$ can be obtained by solving the Lindblad master equation as shown in Sec.II. After the measurements, one can obtain the rescaled probability given by $\widetilde{P}_L(\tau)=e^{4 \gamma \tau}p_L(\tau)$ ($\widetilde{P}_R(\tau)=e^{4 \gamma \tau}p_R(\tau)$). For a given dissipation strength $\gamma$, the interaction induced dynamical instability can be measured as the interaction strength increases to break the $\mathcal{PT}$ symmetry. The same principle applies to the AFM Mott phase as long as the initial state is prepared in the AFM state [@Heidelberg; @AFMHubbard; @AFMHubbard2]. The anomalous dynamics revealed in this work is expected to be observable in the current Fermi-Hubbard experiments with dissipation[@DissHubbard].
We thank Z Cai for helpful conversation. The work is supported by NSFC under Grants No.11974413 and the National Key Research and Development Program of China (2016YFA0300600 and 2016YFA0302104).
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abstract: |
In this paper we prove the Pohozaev identity for the semilinear Dirichlet problem $(-\Delta)^s u =f(u)$ in $\Omega$, $u\equiv0$ in $\mathbb R^n\backslash\Omega$. Here, $s\in(0,1)$, $(-\Delta)^s$ is the fractional Laplacian in $\R^n$, and $\Omega$ is a bounded $C^{1,1}$ domain.
To establish the identity we use, among other things, that if $u$ is a bounded solution then $u/\delta^s|_{\Omega}$ is $C^{\alpha}$ up to the boundary $\partial \Omega$, where $\delta(x)={\rm dist}(x,\partial\Omega)$. In the fractional Pohozaev identity, the function $u/\delta^s|_{\partial\Omega}$ plays the role that $\partial u/\partial\nu$ plays in the classical one. Surprisingly, from a nonlocal problem we obtain an identity with a boundary term (an integral over $\partial\Omega$) which is completely local.
As an application of our identity, we deduce the nonexistence of nontrivial solutions in star-shaped domains for supercritical nonlinearities.
address:
- 'Universitat Politècnica de Catalunya, Departament de Matemàtica Aplicada I, Diagonal 647, 08028 Barcelona, Spain'
- 'Universitat Politècnica de Catalunya, Departament de Matemàtica Aplicada I, Diagonal 647, 08028 Barcelona, Spain'
author:
- 'Xavier Ros-Oton'
- Joaquim Serra
title: The Pohozaev identity for the fractional Laplacian
---
[^1]
Introduction and results
========================
Let $s\in(0,1)$ and consider the fractional elliptic problem $$\label{eq}
\left\{ \begin{array}{rcll} (-\Delta)^s u &=&f(u)&\textrm{in }\Omega \\
u&=&0&\textrm{in }\mathbb R^n\backslash\Omega\end{array}\right.$$ in a bounded domain $\Omega\subset\mathbb R^n$, where $$\label{laps}(-\Delta)^s u (x)= c_{n,s}{\rm PV}\int_{\R^n}\frac{u(x)-u(y)}{|x-y|^{n+2s}}dy$$ is the fractional Laplacian. Here, $c_{n,s}$ is a normalization constant given by .
When $s=1$, a celebrated result of S. I. Pohozaev states that any solution of satisfies an identity, which is known as the Pohozaev identity [@P]. This classical result has many consequences, the most immediate one being the nonexistence of nontrivial bounded solutions to for supercritical nonlinearities $f$.
The aim of this paper is to give the fractional version of this identity, that is, to prove the Pohozaev identity for problem with $s\in(0,1)$. This is the main result of the paper, and it reads as follows. Here, since the solution $u$ is bounded, the notions of weak and viscosity solutions agree (see Remark \[boundedsol\]).
\[thpoh\] Let $\Omega$ be a bounded and $C^{1,1}$ domain, $f$ be a locally Lipschitz function, $u$ be a bounded solution of (\[eq\]), and $$\delta(x)={\rm dist}(x,\partial\Omega).$$ Then, $$u/\delta^s|_{\Omega}\in C^{\alpha}(\overline\Omega)\qquad \textrm{for some }\ \alpha\in(0,1),$$ meaning that $u/\delta^s|_{\Omega}$ has a continuous extension to $\overline\Omega$ which is $C^{\alpha}(\overline{\Omega})$, and the following identity holds $$(2s-n)\int_{\Omega}uf(u)dx+2n\int_\Omega F(u)dx=\Gamma(1+s)^2\int_{\partial\Omega}\left(\frac{u}{\delta^s}\right)^2(x\cdot\nu)d\sigma,$$ where $F(t)=\int_0^tf$, $\nu$ is the unit outward normal to $\partial\Omega$ at $x$, and $\Gamma$ is the Gamma function.
Note that in the fractional case the function $u/\delta^s|_{\partial\Omega}$ plays the role that $\partial u/\partial\nu$ plays in the classical Pohozaev identity. Moreover, if one sets $s=1$ in the above identity one recovers the classical one, since $u/\delta|_{\partial\Omega}=\partial u/\partial\nu$ and $\Gamma(2)=1$.
It is quite surprising that from a nonlocal problem we obtain a completely local boundary term in the Pohozaev identity. That is, although the function $u$ has to be defined in all $\R^n$ in order to compute its fractional Laplacian at a given point, knowing $u$ only in a neighborhood of the boundary we can already compute $\int_{\partial\Omega}\left(\frac{u}{\delta^s}\right)^2(x\cdot\nu)d\sigma$.
Recall that problem has an equivalent formulation given by the Caffarelli-Silvestre [@CSext] associated extension problem —a local PDE in $\R^{n+1}_+$. For such extension, some Pohozaev type identities are proved in [@BCP; @CC; @CT]. However, these identities contain boundary terms on the cylinder $\partial\Omega\times \mathbb R^+$ or in a half-sphere $\partial B_R^+\cap\mathbb R^{n+1}_+$, which have no clear interpretation in terms of the original problem in $\R^n$. The proofs of these identities are similar to the one of the classical Pohozaev identity and use PDE tools (differential calculus identities and integration by parts).
Sometimes it may be useful to write the Pohozaev identity as $$2s[u]^2_{H^s(\mathbb R^n)}-2n\mathcal{E}[u]=\Gamma(1+s)^2\int_{\partial\Omega}\left(\frac{u}{\delta^s}\right)^2(x\cdot\nu)d\sigma,$$ where $\mathcal{E}$ is the energy functional $$\label{energy}
\mathcal{E}[u]=\frac12[u]^2_{H^s(\mathbb R^n)}-\int_\Omega F(u)dx,$$ $F'=f$, and $$\label{seminorm}
[u]_{H^s(\mathbb R^n)}=\||\xi|^s\mathcal F[u]\|_{L^2(\mathbb R^n)}=\frac{c_{n,s}}{2}\int_{\mathbb R^n}\int_{\mathbb R^n}\frac{\left|u(x)-u(y)\right|^2}{|x-y|^{n+2s}}dxdy.$$ We have used that if $u$ and $v$ are $H^s(\R^n)$ functions and $u\equiv v\equiv0$ in $\R^n\setminus\Omega$, then $$\label{fip}
\int_\Omega v(-\Delta)^su\, dx=\int_{\R^n}(-\Delta)^{s/2}u(-\Delta)^{s/2}v\, dx,$$ which yields $$\int_\Omega uf(u)dx=\int_{\mathbb R^n}|(-\Delta)^{s/2}u|^2dx=[u]_{H^s(\mathbb R^n)}.$$
As a consequence of our Pohozaev identity we obtain nonexistence results for problem with supercritical nonlinearities $f$ in star-shaped domains $\Omega$. In Section \[sec2\] we will give, however, a short proof of this result using our method to establish the Pohozaev identity. This shorter proof will not require the full strength of the identity.
\[cornonexistence\] Let $\Omega$ be a bounded, $C^{1,1}$, and star-shaped domain, and let $f$ be a locally Lipschitz function. If $$\label{supercritic} \frac{n-2s}{2n}uf(u)\geq\int_0^u f(t)dt\qquad\textrm{for all}\ \ u\in\mathbb R,$$ then problem admits no positive bounded solution. Moreover, if the inequality in is strict, then admits no nontrivial bounded solution.
For the pure power nonlinearity, the result reads as follows.
\[cornonexistence2\] Let $\Omega$ be a bounded, $C^{1,1}$, and star-shaped domain. If $p\geq\frac{n+2s}{n-2s}$, then problem $$\label{eqp}\left\{ \begin{array}{rcll} (-\Delta)^s u &=&|u|^{p-1}u&\textrm{in }\Omega \\
u&=&0&\textrm{in }\mathbb R^n\backslash\Omega\end{array}\right.$$ admits no positive bounded solution. Moreover, if $p>\frac{n+2s}{n-2s}$ then admits no nontrivial bounded solution.
The nonexistence of changing-sign solutions to problem for the critical power $p=\frac{n+2s}{n-2s}$ remains open.
Recently, M. M. Fall and T. Weth [@FW] have also proved a nonexistence result for problem with the method of moving spheres. In their result no regularity of the domain is required, but they need to assume the solutions to be positive. Our nonexistence result is the first one allowing changing-sign solutions. In addition, their condition on $f$ for the nonexistence — in our Remark \[remFW\]— is more restrictive than ours, i.e., and, when $f=f(x,u)$, condition .
The existence of weak solutions $u\in H^s(\R^n)$ to problem for subcritical $f$ has been recently proved by R. Servadei and E. Valdinoci [@SV].
The Pohozaev identity will be a consequence of the following two results. The first one establishes $C^s(\R^n)$ regularity for $u$, $C^{\alpha}(\overline\Omega)$ regularity for $u/\delta^s|_\Omega$, and higher order interior Hölder estimates for $u$ and $u/\delta^s$. It is proved in our paper [@RS].
Throughout the article, and when no confusion is possible, we will use the notation $C^\beta(U)$ with $\beta>0$ to refer to the space $C^{k,\beta'}(U)$, where $k$ is the is greatest integer such that $k<\beta$, and $\beta'=\beta-k$. This notation is specially appropriate when we work with $(-\Delta)^s$ in order to avoid the splitting of different cases in the statements of regularity results. According to this, $[\cdot]_{C^{\beta}(U)}$ denotes the $C^{k,\beta'}(U)$ seminorm $$[u]_{C^\beta(U)}=[u]_{C^{k,\beta'}(U)}=\sup_{x,y\in U,\ x\neq y}\frac{|D^ku(x)-D^ku(y)|}{|x-y|^{\beta'}}.$$ Here, by $f\in C^{0,1}_{\rm loc}(\overline\Omega\times\R)$ we mean that $f$ is Lipschitz in every compact subset of $\overline\Omega\times \R$.
\[krylov\] Let $\Omega$ be a bounded and $C^{1,1}$ domain, $f\in C^{0,1}_{\rm loc}(\overline\Omega\times\R)$, $u$ be a bounded solution of $$\label{eqlin}
\left\{ \begin{array}{rcll} (-\Delta)^s u &=&f(x,u)&\textrm{in }\Omega \\
u&=&0&\textrm{in }\mathbb R^n\backslash\Omega,\end{array}\right.$$ and $\delta(x)={\rm dist}(x,\partial\Omega)$. Then,
- $u\in C^s(\R^n)$ and, for every $\beta\in[s,1+2s)$, $u$ is of class $C^{\beta}(\Omega)$ and $$[u]_{C^{\beta}(\{x\in\Omega\,:\,\delta(x)\ge\rho\})}\le C \rho^{s-\beta}\qquad \textrm{for all}\ \ \rho\in(0,1).$$
- The function $u/\delta^s|_\Omega$ can be continuously extended to $\overline\Omega$. Moreover, there exists $\alpha\in(0,1)$ such that $u/\delta^s\in C^{\alpha}(\overline{\Omega})$. In addition, for all $\beta\in[\alpha,s+\alpha]$, it holds the estimate $$[u/\delta^s]_{C^{\beta}(\{x\in\Omega\,:\,\delta(x)\ge\rho\})}\le C \rho^{\alpha-\beta}\qquad \textrm{for all}\ \ \rho\in(0,1).$$
The constants $\alpha$ and $C$ depend only on $\Omega$, $s$, $f$, $\|u\|_{L^{\infty}(\R^n)}$, and $\beta$.
\[boundedsol\] For bounded solutions of , the notions of energy and viscosity solutions coincide (see more details in Remark 2.9 in [@RS]). Recall that $u$ is an energy (or weak) solution of problem if $u\in H^s(\mathbb R^n)$, $u\equiv0$ in $\mathbb R^n\backslash\Omega$, and $$\int_{\mathbb R^n}(-\Delta)^{s/2}u(-\Delta)^{s/2}v\,dx=\int_\Omega f(x,u)v\,dx$$ for all $v\in H^s(\R^n)$ such that $v\equiv0$ in $\R^n\setminus\Omega$.
By Theorem \[krylov\] (a), any bounded weak solution is continuous up to the boundary and solve equation in the classical sense, i.e., in the pointwise sense of . Therefore, it follows from the definition of viscosity solution (see [@CS]) that bounded weak solutions are also viscosity solutions.
Reciprocally, by uniqueness of viscosity solutions [@CS] and existence of weak solution for the linear problem $(-\Delta)^sv=f(x,u(x))$, any viscosity solution $u$ belongs to $H^s(\R^n)$ and it is also a weak solution. See [@RS] for more details.
The second result towards Theorem \[thpoh\] is the new Pohozaev identity for the fractional Laplacian. The hypotheses of the following proposition are satisfied for any bounded solution $u$ of whenever $f\in C^{0,1}_{\rm loc}(\overline\Omega\times\R)$, by our results in [@RS] (see Theorem \[krylov\] above).
\[intparts\] Let $\Omega$ be a bounded and $C^{1,1}$ domain. Assume that $u$ is a $H^s(\R^n)$ function which vanishes in $\R^n\setminus\Omega$, and satisfies
- $u\in C^s(\R^n)$ and, for every $\beta\in[s,1+2s)$, $u$ is of class $C^{\beta}(\Omega)$ and $$[u]_{C^{\beta}(\{x\in\Omega\,:\,\delta(x)\ge\rho\})}\le C \rho^{s-\beta}\qquad \textrm{for all}\ \ \rho\in(0,1).$$
- The function $u/\delta^s|_\Omega$ can be continuously extended to $\overline\Omega$. Moreover, there exists $\alpha\in(0,1)$ such that $u/\delta^s\in C^{\alpha}(\overline{\Omega})$. In addition, for all $\beta\in[\alpha,s+\alpha]$, it holds the estimate $$[u/\delta^s]_{C^{\beta}(\{x\in\Omega\,:\,\delta(x)\ge\rho\})}\le C \rho^{\alpha-\beta}\qquad \textrm{for all}\ \ \rho\in(0,1).$$
- $(-\Delta)^s u$ is pointwise bounded in $\Omega$.
Then, the following identity holds $$\int_\Omega(x\cdot\nabla u)(-\Delta)^su\ dx=\frac{2s-n}{2}\int_{\Omega}u(-\Delta)^su\ dx-\frac{\Gamma(1+s)^2}{2}\int_{\partial\Omega}\left(\frac{u}{\delta^s}\right)^2(x\cdot\nu)d\sigma,$$ where $\nu$ is the unit outward normal to $\partial\Omega$ at $x$, and $\Gamma$ is the Gamma function.
Note that hypothesis (a) ensures that $(-\Delta)^s u$ is defined pointwise in $\Omega$. Note also that hypotheses (a) and (c) ensure that the integrals appearing in the above identity are finite.
By Propositions 1.1 and 1.4 in [@RS], hypothesis (c) guarantees that $u\in C^s (\R^n)$ and $u/\delta^s\in C^\alpha(\overline\Omega)$, but not the interior estimates in (a) and (b). However, under the stronger assumption $(-\Delta)^s u\in C^\alpha(\overline\Omega)$ the whole hypothesis (b) is satisfied; see Theorem 1.5 in [@RS].
As a consequence of Proposition \[intparts\], we will obtain the Pohozaev identity (Theorem \[thpoh\]) and also a new integration by parts formula related to the fractional Laplacian. This integration by parts formula follows from using Proposition \[intparts\] with two different origins.
\[corintparts\] Let $\Omega$ be a bounded and $C^{1,1}$ domain, and $u$ and $v$ be functions satisfying the hypotheses in Proposition \[intparts\]. Then, the following identity holds $$\int_\Omega (-\Delta)^su\ v_{x_i}\,dx=-\int_\Omega u_{x_i}(-\Delta)^sv\,dx+\Gamma(1+s)^2\int_{\partial\Omega}\frac{u}{\delta^{s}}\frac{v}{\delta^{s}}\,\nu_i\,d\sigma$$ for $i=1,...,n$, where $\nu$ is the unit outward normal to $\partial\Omega$ at $x$, and $\Gamma$ is the Gamma function.
To prove Proposition \[intparts\] we first assume the domain $\Omega$ to be star-shaped with respect to the origin. The result for general domains will follow from the star-shaped case, as seen in Section \[sec8\]. When the domain is star-shaped, the idea of the proof is the following. First, one writes the left hand side of the identity as $$\int_\Omega(x\cdot\nabla u)(-\Delta)^su\ dx=\left.\frac{d}{d\lambda}\right|_{\lambda=1^+}\int_{\Omega} u_\lambda (-\Delta)^s u\ dx,$$ where $$u_\lambda(x)=u(\lambda x).$$ Note that $u_\lambda\equiv0$ in $\mathbb R^n\backslash \Omega$, since $\Omega$ is star-shaped and we take $\lambda>1$ in the above derivative. As a consequence, we may use with $v=u_\lambda$ and make the change of variables $y=\sqrt{\lambda}x$, to obtain $$\int_{\Omega}u_\lambda (-\Delta)^s u\ dx=\int_{\R^n} (-\Delta)^{s/2}u_\lambda (-\Delta)^{s/2} u\ dx=\lambda^{\frac{2s-n}{2}}\int_{\mathbb R^n} w_{\sqrt{\lambda}} w_{1/\sqrt{\lambda}}\ dy,$$ where $$w(x)=(-\Delta)^{s/2}u(x).$$ Thus, $$\label{1207}\begin{split}
\int_\Omega(x\cdot\nabla u)(-\Delta)^su\ dx&= \left.\frac{d}{d\lambda}\right|_{\lambda=1^+}\left\{\lambda^{\frac{2s-n}{2}}\int_{\mathbb R^n} w_{\sqrt{\lambda}} w_{1/\sqrt{\lambda}}\ dy\right\}\\
&=\frac{2s-n}{2}\int_{\R^n}w^2dx+\left.\frac{d}{d\lambda}\right|_{\lambda=1^+}I_{\sqrt\lambda}\\
&=\frac{2s-n}{2}\int_{\R^n}u(-\Delta)^s u\,dx+\frac12\left.\frac{d}{d\lambda}\right|_{\lambda=1^+}I_{\lambda}
,\end{split}$$ where $$I_\lambda=\int_{\mathbb R^n} w_{{\lambda}} w_{1/{\lambda}}dy.$$ Therefore, Proposition \[intparts\] is equivalent to the following equality $$\label{partdificil}-\left.\frac{d}{d\lambda}\right|_{\lambda=1^+}\int_{\mathbb R^n} w_{{\lambda}} w_{1/{\lambda}}\ dy=
\Gamma(1+s)^2\int_{\partial\Omega}\left(\frac{u}{\delta^s}\right)^2(x\cdot\nu)d\sigma.$$
The quantity $\frac{d}{d\lambda}|_{\lambda=1^+}\int_{\R^n}w_{\lambda}w_{1/\lambda}$ vanishes for any $C^1(\R^n)$ function $w$, as can be seen by differentiating under the integral sign. Instead, we will prove that the function $w=(-\Delta)^{s/2} u$ has a singularity along $\partial\Omega$, and that holds.
Next we give an easy argument to give a direct proof of the nonexistence result for supercritical nonlinearities without using neither equality nor the behavior of $(-\Delta)^{s/2}u$; the detailed proof is given in Section \[sec2\].
Indeed, in contrast with the delicate equality , the inequality $$\label{num}
\left.\frac{d}{d\lambda}\right|_{\lambda=1^+}I_\lambda\leq0$$ follows easily from Cauchy-Schwarz. Namely, $$I_\lambda\leq \|w_\lambda\|_{L^2(\mathbb R^n)}\|w_{1/\lambda}\|_{L^2(\mathbb R^n)}=\|w\|_{L^2(\mathbb R^n)}^2=I_1,$$ and hence follows.
With this simple argument, leads to $$-\int_\Omega (x\cdot\nabla u)(-\Delta)^s u\ dx\geq\frac{n-2s}{2}\int_\Omega u(-\Delta)^su\ dx,$$ which is exactly the inequality used to prove the nonexistence result of Corollary \[cornonexistence\] for supercritical nonlinearities. Here, one also uses that, when $u$ is a solution of , then $$\int_\Omega (x\cdot\nabla u)(-\Delta)^s u\ dx=\int_\Omega(x\cdot\nabla u)f(u)dx=\int_{\Omega}x\cdot\nabla F(u)dx=-n\int_\Omega F(u)dx.$$
This argument can be also used to obtain nonexistence results (under some decay assumptions) for weak solutions of in the whole $\R^n$; see Remark \[remR\^n\].
The identity is the difficult part of the proof of Proposition \[intparts\]. To prove it, it will be crucial to know the precise behavior of $(-\Delta)^{s/2}u$ near $\partial\Omega$ —from both inside and outside $\Omega$. This is given by the following result.
\[thlaps/2\] Let $\Omega$ be a bounded and $C^{1,1}$ domain, and $u$ be a function such that $u\equiv0$ in $\mathbb R^n\backslash\Omega$ and that $u$ satisfies [(b)]{} in Proposition \[intparts\]. Then, there exists a $C^{\alpha}(\mathbb R^n)$ extension $v$ of $u/\delta^s|_{\Omega}$ such that $$\label{**}
(-\Delta)^{s/2}u(x)=c_1\left\{\log^-\delta(x)+c_2\chi_{\Omega}(x)\right\}v(x)+h(x)\quad \textrm{in } \ \R^n,$$ where $h$ is a $C^{\alpha}(\R^n)$ function, $\log^-t=\min\{\log t,0\}$, $$\label{1208} c_1=\frac{\Gamma(1+s)\sin\left(\frac{\pi s}{2}\right)}{\pi},\qquad{\rm and}\qquad c_2=\frac{\pi}{\tan\left(\frac{\pi s}{2}\right)}.$$
Moreover, if $u$ also satisfies [(a)]{} in Proposition \[intparts\], then for all $\beta\in(0,1+s)$ $$\label{numeret}
[(-\Delta)^{s/2}u]_{C^{\beta}(\{x\in\R^n:\,\delta(x)\ge\rho\})}\leq C\rho^{-\beta}\qquad \textrm{for all}\ \ \rho\in(0,1),$$ for some constant $C$ which does not depend on $\rho$.
The values of the constants $c_1$ and $c_2$ in arise in the expression for the $s/2$ fractional Laplacian, $(-\Delta)^{s/2}$, of the 1D function $(x_n^+)^s$, and they are computed in the Appendix.
Writing the first integral in using spherical coordinates, equality reduces to a computation in dimension 1, stated in the following proposition. This result will be used with the function $\varphi$ in its statement being essentially the restriction of $(-\Delta)^{s/2} u$ to any ray through the origin. The constant $\gamma$ will be chosen to be any value in $(0,s)$.
\[propoperador\] Let $A$ and $B$ be real numbers, and $$\varphi(t)=A\log^-|t-1|+B\chi_{[0,1]}(t)+h(t),$$ where $\log^-t=\min\{\log t,0\}$ and $h$ is a function satisfying, for some constants $\alpha$ and $\gamma$ in $(0,1)$, and $C_0>0$, the following conditions:
- $\|h\|_{C^{\alpha}([0,\infty))}\leq C_0$.
- For all $\beta\in[\gamma,1+\gamma]$ $$\|h\|_{C^{\beta}((0,1-\rho)\cup(1+\rho,2))}\leq C_0 \rho^{-\beta}\qquad \textrm{for all}\ \ \rho\in(0,1).$$
- $|h'(t)|\leq C_0 t^{-2-\gamma}$ and $|h''(t)|\leq C_0 t^{-3-\gamma}$ for all $t>2$.
Then, $$-\left.\frac{d}{d\lambda}\right|_{\lambda=1^+}\int_{0}^{\infty} \varphi\left(\lambda t\right)\varphi\left(\frac{t}{\lambda}\right)dt=A^2\pi^2+B^2.$$
Moreover, the limit defining this derivative is uniform among functions $\varphi$ satisfying (i)-(ii)-(iii) with given constants $C_0$, $\alpha$, and $\gamma$.
From this proposition one obtains that the constant in the right hand side of , $\Gamma(1+s)^2$, is given by $c_1^2(\pi^2+c_2^2)$. The constant $c_2$ comes from an involved expression and it is nontrivial to compute (see Proposition \[prop:Lap-s/2-delta-s\] in Section 5 and the Appendix). It was a surprise to us that its final value is so simple and, at the same time, that the Pohozaev constant $c_1^2(\pi^2+c_2^2)$ also simplifies and becomes $\Gamma(1+s)^2$.
Instead of computing explicitly the constants $c_1$ and $c_2$, an alternative way to obtain the constant in the Pohozaev identity consists of using an explicit nonlinearity and solution to problem in a ball. The one which is known [@G; @BGR] is the solution to problem $$\left\{ \begin{array}{rcll} (-\Delta)^s u &=&1&\textrm{in }B_r(x_0) \\
u&=&0&\textrm{in }\mathbb R^n\backslash B_r(x_0).\end{array}\right.$$ It is given by $$u(x)=\frac{2^{-2s}\Gamma(n/2)}{\Gamma\left(\frac{n+2s}{2}\right)\Gamma(1+s)}\left(r^2-|x-x_0|^2\right)^s\qquad\textrm{in}\ \ B_r(x_0).$$ From this, it is straightforward to find the constant $\Gamma(1+s)^2$ in the Pohozaev identity; see Remark \[A4\] in the Appendix.
Using Theorem \[krylov\] and Proposition \[intparts\], we can also deduce a Pohozaev identity for problem , that is, allowing the nonlinearity $f$ to depend also on $x$. In this case, the Pohozaev identity reads as follows.
\[proppoh\] Let $\Omega$ be a bounded and $C^{1,1}$ domain, $f\in C^{0,1}_{\rm loc}(\overline \Omega\times \R)$, $u$ be a bounded solution of , and $\delta(x)={\rm dist}(x,\partial\Omega)$. Then $$u/\delta^s|_{\Omega}\in C^{\alpha}(\overline\Omega)\qquad \textrm{for some }\ \alpha\in(0,1),$$ and the following identity holds $$(2s-n)\int_{\Omega}uf(x,u)dx+2n\int_\Omega F(x,u)dx=\hspace{50mm}$$ $$\hspace{30mm}=\Gamma(1+s)^2\int_{\partial\Omega}\left(\frac{u}{\delta^s}\right)^2(x\cdot\nu)d\sigma-2\int_\Omega x\cdot F_x(x,u)dx,$$ where $F(x,t)=\int_0^tf(x,\tau)d\tau$, $\nu$ is the unit outward normal to $\partial\Omega$ at $x$, and $\Gamma$ is the Gamma function.
From this, we deduce nonexistence results for problem with supercritical nonlinearities $f$ depending also on $x$. This has been done also in [@FW] for positive solutions. Our result allows changing sign solutions as well as a slightly larger class of nonlinearities (see Remark \[remFW\]).
\[cornonexistence3\] Let $\Omega$ be a bounded, $C^{1,1}$, and star-shaped domain, $f\in C^{0,1}_{\rm loc}(\overline \Omega\times \R)$, and $F(x,t)=\int_0^tf(x,\tau)d\tau$. If $$\label{supercritic2} \frac{n-2s}{2}\,uf(x,t)\geq nF(x,t)+x\cdot F_x(x,t)\qquad\textrm{for all}\ \ x\in\Omega\ \ \textrm{and}\ \ t\in\mathbb R,$$ then problem admits no positive bounded solution. Moreover, if the inequality in is strict, then admits no nontrivial bounded solution.
\[remFW\] For locally Lipschitz nonlinearities $f$, condition is more general than the one required in [@FW] for their nonexistence result. Namely, [@FW] assumes that for each $x\in\Omega$ and $t\in\mathbb R$, the map $$\label{condFW}\lambda\mapsto \lambda^{-\frac{n+2s}{n-2s}}f(\lambda^{-\frac{2}{n-2s}}x,\lambda t)\qquad \textrm{is nondecreasing for }\lambda\in (0,1].$$ Such nonlinearities automatically satisfy .
However, in [@FW] they do not need to assume any regularity on $f$ with respect to $x$.
The paper is organized as follows. In Section \[sec2\], using Propositions \[thlaps/2\] and \[propoperador\] (to be established later), we prove Proposition \[intparts\] (the Pohozaev identity) for strictly star-shaped domains with respect to the origin. We also establish the nonexistence results for supercritical nonlinearities, and this does not require any result from the rest of the paper. In Section \[sec6\] we establish Proposition \[thlaps/2\], while in Section \[sec7\] we prove Proposition \[propoperador\]. Section \[sec8\] establishes Proposition \[intparts\] for non-star-shaped domains and all its consequences, which include Theorems \[thpoh\] and \[corintparts\] and the nonexistence results. Finally, in the Appendix we compute the constants $c_1$ and $c_2$ appearing in Proposition \[thlaps/2\].
Star-shaped domains: Pohozaev identity and nonexistence {#sec2}
=======================================================
In this section we prove Proposition \[intparts\] for strictly star-shaped domains. We say that $\Omega$ is strictly star-shaped if, for some $z_0\in\R^n$, $$\label{starshaped} (x-z_0)\cdot\nu>0\qquad \textrm{for all}\ \ x\in\partial\Omega.$$ The result for general $C^{1,1}$ domains will be a consequence of this strictly star-shaped case and will be proved in Section \[sec8\].
The proof in this section uses two of our results: Proposition \[thlaps/2\] on the behavior of $(-\Delta)^{s/2}u$ near $\partial\Omega$ and the one dimensional computation of Proposition \[propoperador\].
The idea of the proof for the fractional Pohozaev identity is to use the integration by parts formula with $v=u_\lambda$, where $$u_\lambda(x)=u(\lambda x), \quad \lambda>1,$$ and then differentiate the obtained identity (which depends on $\lambda$) with respect to $\lambda$ and evaluate at $\lambda=1$. However, this apparently simple formal procedure requires a quite involved analysis when it is put into practice. The hypothesis that $\Omega$ is star-shaped is crucially used in order that $u_\lambda$, $\lambda>1$, vanishes outside $\Omega$ so that holds.
Let us assume first that $\Omega$ is strictly star-shaped with respect to the origin, that is, $z_0=0$.
Let us prove that $$\label{first}\int_\Omega(x\cdot \nabla u) (-\Delta)^su\, dx=\left.\frac{d}{d\lambda}\right|_{\lambda=1^+}\int_\Omega u_\lambda (-\Delta)^su\, dx,$$ where $\left.\frac{d}{d\lambda}\right|_{\lambda=1^+}$ is the derivative from the right side at $\lambda=1$. Indeed, let $g=(-\Delta)^s u$. By assumption (a) $g$ is defined pointwise in $\Omega$, and by assumption (c) $g\in L^\infty(\Omega)$. Then, making the change of variables $y=\lambda x$ and using that $\textrm{supp}\,u_\lambda=\frac{1}{\lambda}\Omega\subset \Omega$ since $\lambda>1$, we obtain $$\begin{split}\left.\frac{d}{d\lambda}\right|_{\lambda =1^+}&\int_\Omega u_\lambda(x)g(x) dx = \lim_{\lambda\downarrow 1}\int_\Omega \frac{u(\lambda x)-u(x)}{\lambda-1}g(x) dx\\
&\qquad=\lim_{\lambda\downarrow 1} \lambda^{-n}\int_{\lambda\Omega} \frac{u(y)-u(y/\lambda)}{\lambda-1} g(y/\lambda)dy\\
&\qquad=\lim_{\lambda\downarrow 1}\int_\Omega \frac{u(y)-u(y/\lambda)}{\lambda-1} g(y/\lambda)dy+\lim_{\lambda\downarrow1}\int_{(\lambda\Omega)\backslash\Omega}\frac{-u(y/\lambda)}{\lambda-1}
g(y/\lambda)dy.
\end{split}$$ By the dominated convergence theorem, $$\lim_{\lambda\downarrow 1}\int_\Omega \frac{u(y)-u(y/\lambda)}{\lambda-1}g(y/\lambda)\,dy=\int_\Omega (y\cdot\nabla u)g(y)\,dy,$$ since $g\in L^\infty(\Omega)$, $|\nabla u(\xi)|\leq C\delta(\xi)^{s-1}\leq C\lambda^{1-s}\delta(y)^{s-1}$ for all $\xi$ in the segment joining $y$ and $y/\lambda$, and $\delta^{s-1}$ is integrable. The gradient bound $|\nabla u(\xi)|\leq C\delta(\xi)^{s-1}$ follows from assumption (a) used with $\beta=1$. Hence, to prove it remains only to show that $$\lim_{\lambda\downarrow 1}\int_{(\lambda\Omega)\backslash\Omega}\frac{-u(y/\lambda)}{\lambda-1}g(y/\lambda)dy=0.$$ Indeed, $|(\lambda\Omega)\backslash\Omega|\leq C(\lambda-1)$ and —by (a)— $u\in C^s(\R^n)$ and $u\equiv 0$ outside $\Omega$. Hence, $\|u\|_{L^\infty((\lambda\Omega)\backslash\Omega)}\rightarrow0$ as $\lambda\downarrow1$ and follows.
Now, using the integration by parts formula with $v=u_\lambda$, $$\begin{aligned}
\int_\Omega u_\lambda (-\Delta)^su\,dx
&=& \int_{\mathbb R^n}u_\lambda (-\Delta)^su\,dx \\
&=& \int_{\mathbb R^n}(-\Delta)^{s/2}u_{\lambda}(-\Delta)^{s/2}u\,dx \\
&=& \lambda^s\int_{\mathbb R^n}\left((-\Delta)^{s/2}u\right)(\lambda x)(-\Delta)^{s/2}u(x)dx \\
&=& \lambda^s\int_{\mathbb R^n}w_{\lambda}w\ dx,\end{aligned}$$ where $$w(x)=(-\Delta)^{s/2}u(x)\qquad {\rm and}\qquad w_{\lambda}(x)=w(\lambda x).$$ With the change of variables $y=\sqrt{\lambda}x$ this integral becomes $$\lambda^s\int_{\mathbb R^n}w_{\lambda}w\, dx=\lambda^{\frac{2s-n}{2}}\int_{\mathbb R^n}w_{\sqrt{\lambda}}w_{1/\sqrt{\lambda}}\,dy,$$ and thus $$\int_\Omega u_\lambda(-\Delta)^su\, dx=\lambda^{\frac{2s-n}{2}}\int_{\mathbb R^n}w_{\sqrt{\lambda}}w_{1/\sqrt{\lambda}}\,dy.$$ Furthermore, this leads to $$\begin{aligned}
\int_\Omega(\nabla u\cdot x)(-\Delta)^su\, dx&=&\left.\frac{d}{d\lambda}\right|_{\lambda=1^+}\left\{\lambda^{\frac{2s-n}{2}}\int_{\mathbb R^n}w_{\sqrt{\lambda}}w_{1/\sqrt{\lambda}}\,dy\right\}\nonumber \\
&=&\frac{2s-n}{2}\int_{\mathbb R^n}|(-\Delta)^{s/2}u|^2\,dx +\left.\frac{d}{d\lambda}\right|_{\lambda=1^+}\int_{\mathbb R^n}w_{\sqrt{\lambda}}w_{1/\sqrt{\lambda}}\,dy\nonumber \\
&=&\frac{2s-n}{2}\int_{\Omega}u(-\Delta)^s u\, dx+ \frac 12\left.\frac{d}{d\lambda}\right|_{\lambda=1^+}\int_{\mathbb R^n}w_{{\lambda}}w_{1/{\lambda}}\,dy\label{igualtatpaluego}
.\end{aligned}$$ Hence, it remains to prove that $$\label{derivadaIlambda}-\left.\frac{d}{d\lambda}\right|_{\lambda=1^+}I_\lambda
=\Gamma(1+s)^2\int_{\partial\Omega}\left(\frac{u}{\delta^s}\right)^2(x\cdot\nu)\,d\sigma,$$ where we have denoted $$\label{Ilambda} I_\lambda=\int_{\mathbb R^n}w_{{\lambda}}w_{1/{\lambda}}\,dy.$$
Now, for each $\theta\in S^{n-1}$ there exists a unique $r_\theta>0$ such that $r_\theta\theta\in\partial\Omega$. Write the integral in spherical coordinates and use the change of variables $t=r/r_\theta$: $$\begin{aligned}
\left.\frac{d}{d\lambda}\right|_{\lambda=1^+}I_\lambda&=&
\left.\frac{d}{d\lambda}\right|_{\lambda=1^+}\int_{S^{n-1}}d\theta\int_0^\infty r^{n-1}w(\lambda r\theta)w\left(\frac{r}{\lambda}\theta\right)dr\\
&=&\left.\frac{d}{d\lambda}\right|_{\lambda=1^+}\int_{S^{n-1}}r_\theta d\theta\int_0^\infty (r_\theta t)^{n-1}w(\lambda r_\theta t\theta)w\left(\frac{r_\theta t}{\lambda}\theta\right)dt\\
&=&\left.\frac{d}{d\lambda}\right|_{\lambda=1^+}\int_{\partial\Omega}(x\cdot \nu)d\sigma(x)\int_0^\infty t^{n-1}w(\lambda tx)w\left(\frac{tx}{\lambda}\right)dt,\end{aligned}$$ where we have used that $$r_\theta^{n-1}\,d\theta = \left(\frac{x}{|x|}\cdot\nu\right)\,d\sigma=\frac{1}{r_\theta}(x\cdot\nu)\,d\sigma$$ with the change of variables $S^{n-1}\rightarrow \partial\Omega$ that maps every point in $S^{n-1}$ to its radial projection on $\partial\Omega$, which is unique because of the strictly star-shapedness of $\Omega$.
Fix $x_0\in \partial\Omega$ and define $$\varphi(t)=t^{\frac{n-1}{2}}w\left(t x_0\right)=t^{\frac{n-1}{2}}(-\Delta)^{s/2}u(t x_0).$$ By Proposition \[thlaps/2\], $$\varphi(t)=c_1\{\log^-\delta(tx_0)+c_2\chi_{[0,1]}\}v(tx_0)+h_0(t)$$ in $[0,\infty)$, where $v$ is a $C^\alpha(\R^n)$ extension of $u/\delta^s|_\Omega$ and $h_0$ is a $C^{\alpha}([0,\infty))$ function. Next we will modify this expression in order to apply Proposition \[propoperador\].
Using that $\Omega$ is $C^{1,1}$ and strictly star-shaped, it is not difficult to see that $\frac{|r-r_\theta|}{\delta(r\theta)}$ is a Lipschitz function of $r$ in $[0,\infty)$ and bounded below by a positive constant (independently of $x_0$). Similarly, $\frac{|t-1|}{\delta(t x_0)}$ and $\frac{\min\{|t-1|,1\}}{\min\{\delta(tx_0),1\}}$ are positive and Lipschitz functions of $t$ in $[0,\infty)$. Therefore, $$\log^-|t-1|- \log^- \delta(tx_0)$$ is Lipschitz in $[0,\infty)$ as a function of $t$.
Hence, for $t\in[0,\infty)$, $$\varphi(t)=c_1\{\log^-|t-1|+c_2\chi_{[0,1]}\}v(tx_0)+h_1(t),$$ where $h_1$ is a $C^{\alpha}$ function in the same interval.
Moreover, note that the difference $$v(tx_0)-v(x_0)$$ is $C^{\alpha}$ and vanishes at $t=1$. Thus, $$\varphi(t)=c_1\{\log^-|t-1|+c_2\chi_{[0,1]}(t)\}v(x_0)+h(t)$$ holds in all $[0,\infty)$, where $h$ is $C^{\alpha}$ in $[0,\infty)$ if we slightly decrease $\alpha$ in order to kill the logarithmic singularity. This is condition (i) of Proposition \[propoperador\].
From the expression $$h(t)=t^{\frac{n-1}{2}}(-\Delta)^{s/2} u \left(t x_0\right)-c_1\{\log^-|t-1|+c_2\chi_{[0,1]}(t)\}v(x_0)$$ and from in Proposition \[thlaps/2\], we obtain that $h$ satisfies condition (ii) of Proposition \[propoperador\] with $\gamma=s/2$.
Moreover, condition (iii) of Proposition \[propoperador\] is also satisfied. Indeed, for $x\in \mathbb R^n\backslash(2\Omega)$ we have $$(-\Delta)^{s/2}u(x)=c_{n,\frac s2}\int_{\Omega}\frac{-u(y)}{|x-y|^{n+s}}dy$$ and hence $$|\partial_{i}(-\Delta)^{s/2}u(x)|\leq C|x|^{-n-s-1}\ \ \textrm{ and }\ \ |\partial_{ij}(-\Delta)^{s/2}u(x)|\leq C|x|^{-n-s-2}.$$ This yields $|\varphi'(t)|\leq Ct^{\frac{n-1}{2}-n-s-1}\leq Ct^{-2-\gamma}$ and $|\varphi''(t)|\leq Ct^{\frac{n-1}{2}-n-s-2}\leq Ct^{-3-\gamma}$ for $t>2$.
Therefore we can apply Proposition \[propoperador\] to obtain $$\left.\frac{d}{d\lambda}\right|_{\lambda=1^+}\int_0^\infty \varphi(\lambda t)\varphi\left(\frac{t}{\lambda}\right)dt=\left(v(x_0)\right)^2c_1^2(\pi^2+c_2^2),$$ and thus $$\left.\frac{d}{d\lambda}\right|_{\lambda=1^+}\int_0^\infty t^{n-1}w(\lambda tx_0)w\left(\frac{tx_0}{\lambda}\right)dt=\left(v(x_0)\right)^2c_1^2(\pi^2+c_2^2)$$ for each $x_0\in\partial\Omega$.
Furthermore, by uniform convergence on $x_0$ of the limit defining this derivative (see Proposition \[lema1\] in Section \[sec7\]), this leads to $$\left.\frac{d}{d\lambda}\right|_{\lambda=1^+}I_\lambda=c_1^2(\pi^2+c_2^2)\int_{\partial\Omega}(x_0\cdot\nu)
\left(\frac{u}{\delta^s}(x_0)\right)^2dx_0.$$ Here we have used that, for $x_0\in\partial\Omega$, $v(x_0)$ is uniquely defined by continuity as $$\left(\frac{u}{\delta^s}\right)(x_0)=\lim_{x\rightarrow x_0,\ x\in\Omega} \frac{u(x)}{\delta^s(x)}.$$
Hence, it only remains to prove that $$c_1^2(\pi^2+c_2^2)=\Gamma(1+s)^2.$$ But $$c_1=\frac{\Gamma(1+s)\sin\left(\frac{\pi s}{2}\right)}{\pi}\qquad{\rm and}\qquad c_2=\frac{\pi}{\tan\left(\frac{\pi s}{2}\right)},$$ and therefore $$\begin{aligned}
c_1^2(\pi^2+c_2^2)&=&\frac{\Gamma(1+s)^2\sin^2\left(\frac{\pi s}{2}\right)}{\pi^2}\left(\pi^2+\frac{\pi^2}{\tan^2\left(\frac{\pi s}{2}\right)}\right)\\
&=&\Gamma(1+s)^2\sin^2\left(\frac{\pi s}{2}\right)\left(1+\frac{\cos^2\left(\frac{\pi s}{2}\right)}{\sin^2\left(\frac{\pi s}{2}\right)}\right)\\
&=&\Gamma(1+s)^2.\end{aligned}$$
Assume now that $\Omega$ is strictly star-shaped with respect to a point $z_0\neq0$. Then, $\Omega$ is strictly star-shaped with respect to all points $z$ in a neighborhood of $z_0$. Then, making a translation and using the formula for strictly star-shaped domains with respect to the origin, we deduce $$\label{nom}\begin{split}\int_\Omega\left\{(x-z)\cdot\nabla u\right\}(-\Delta)^su\, dx=\frac{2s-n}{2}&\int_{\Omega}u(-\Delta)^su\, dx+\\
&-\frac{\Gamma(1+s)^2}{2}\int_{\partial\Omega}\left(\frac{u}{\delta^s}\right)^2(x-z)\cdot\nu\, d\sigma\end{split}$$ for each $z$ in a neighborhood of $z_0$. This yields $$\label{eqintparts}\int_\Omega u_{x_i}(-\Delta)^su\, dx=-\frac{\Gamma(1+s)^2}{2}\int_{\partial\Omega}\left(\frac{u}{\delta^s}\right)^2\nu_i\, d\sigma$$ for $i=1,...,n$. Thus, by adding to a linear combination of , we obtain $$\int_\Omega(x\cdot\nabla u)(-\Delta)^su\, dx=\frac{2s-n}{2}\int_{\Omega}u(-\Delta)^su\, dx-\frac{\Gamma(1+s)^2}{2}\int_{\partial\Omega}\left(\frac{u}{\delta^s}\right)^2x\cdot\nu\, d\sigma.$$
Next we prove the nonexistence results of Corollaries \[cornonexistence\], \[cornonexistence2\], and \[cornonexistence3\] for supercritical nonlinearities in star-shaped domains. Recall that star-shaped means $x\cdot\nu\ge 0$ for all $x\in\partial\Omega$. Although these corollaries follow immediately from Proposition \[proppoh\] —as we will see in Section \[sec8\]—, we give here a short proof of their second part, i.e., nonexistence when the inequality or is strict. That is, we establish the nonexistence of nontrivial solutions for supercritical nonlinearities (not including the critical case).
Our proof follows the method above towards the Pohozaev identity but does not require the full strength of the identity. In addition, in terms of regularity results for the equation, the proof only needs an easy gradient estimate for solutions $u$. Namely, $$|\nabla u|\leq C\delta^{s-1}\ \mbox{ in }\ \Omega,$$ which follows from part (a) of Theorem \[krylov\], proved in [@RS].
We only have to prove Corollary \[cornonexistence3\], since Corollaries \[cornonexistence\] and \[cornonexistence2\] follow immediately from it by setting $f(x,u)=f(u)$ and $f(x,u)=|u|^{p-1}u$ respectively.
Let us prove that if $\Omega$ is star-shaped and $u$ is a bounded solution of , then $$\frac{2s-n}{2}\int_{\Omega}uf(x,u)dx+n\int_\Omega F(x,u)dx-\int_\Omega x\cdot F_x(x,u)dx\geq 0.$$ For this, we follow the beginning of the proof of Proposition \[intparts\] (given above) to obtain , i.e., until the identity $$\int_\Omega(\nabla u\cdot x)(-\Delta)^su\, dx=\frac{2s-n}{2}\int_{\Omega}u(-\Delta)^s u\, dx+\frac 12\left.\frac{d}{d\lambda}\right|_{\lambda=1^+}I_\lambda,$$ where $$I_\lambda=\int_{\mathbb R^n}w_{{\lambda}}w_{1/{\lambda}}\,dx,\qquad w(x)=(-\Delta)^{s/2}u(x), \qquad\mbox{and}\qquad w_{\lambda}(x)=w(\lambda x).$$
This step of the proof only need the star-shapedness of $\Omega$ (and not the strictly star-shapedness) and the regularity result $|\nabla u|\leq C\delta^{s-1}$ in $\Omega$, which follows from Theorem \[krylov\], proved in [@RS].
Now, since $(-\Delta)^s u=f(x,u)$ in $\Omega$ and $$(\nabla u\cdot x)(-\Delta)^su=x\cdot \nabla F(x,u)-x\cdot F_x(x,u),$$ by integrating by parts we deduce $$-n\int_\Omega F(x,u)dx-\int_\Omega x\cdot F_x(x,u)dx=\frac{2s-n}{2}\int_{\Omega}uf(x,u)dx+\frac 12\left.\frac{d}{d\lambda}\right|_{\lambda=1^+}I_\lambda.$$ Therefore, we only need to show that $$\label{negatiu}\left.\frac{d}{d\lambda}\right|_{\lambda=1^+}I_\lambda\leq0.$$ But applying Hölder’s inequality, for each $\lambda>1$ we have $$I_\lambda\leq\|w_\lambda\|_{L^2(\mathbb R^n)}\|w_{1/{\lambda}}\|_{L^2(\R^n)}=\|w\|_{L^2(\R^n)}^2=I_1,$$ and follows.
For this nonexistence result the regularity of the domain $\Omega$ is only used for the estimate $|\nabla u|\leq C\delta^{s-1}$. This estimate only requires $\Omega$ to be Lipschitz and satisfy an exterior ball condition; see [@RS]. In particular, our nonexistence result for supercritical nonlinearities applies to any convex domain, such as a square for instance.
\[remR\^n\] When $\Omega=\R^n$ or when $\Omega$ is a star-shaped domain with respect to infinity, there are two recent nonexistence results for subcritical nonlinearities. They use the method of moving spheres to prove nonexistence of bounded positive solutions in these domains. The first result is due to A. de Pablo and U. Sánchez [@PS], and they obtain nonexistence of bounded positive solutions to $(-\Delta)^s u=u^p$ in all of $\R^n$, whenever $s>1/2$ and $1<p<\frac{n+2s}{n-2s}$. The second result, by M. Fall and T. Weth [@FW], gives nonexistence of bounded positive solutions of in star-shaped domains with respect to infinity for subcritical nonlinearities.
Our method in the previous proof can also be used to prove nonexistence results for problem in star-shaped domains with respect to infinity or in the whole $\R^n$. However, to ensure that the integrals appearing in the proof are well defined, one must assume some decay on $u$ and $\nabla u$. For instance, in the supercritical case $p> \frac{n+2s}{n-2s}$ we obtain that the only solution to $(-\Delta)^s u=u^p$ in all of $\R^n$ decaying as $$|u|+|x\cdot\nabla u|\leq \frac{C}{1+|x|^{\beta}},$$ with $\beta>\frac{n}{p+1}$, is $u\equiv 0$.
In the case of the whole $\R^n$, there is an alternative proof of the nonexistence of solutions which decay fast enough at infinity. It consists of using a Pohozaev identity in all of $\R^n$, that is easily deduced from the pointwise equality $$(-\Delta)^s(x\cdot \nabla u)=2s(-\Delta)^su+x\cdot \nabla (-\Delta)^su.$$
The classification of solutions in the whole $\R^n$ for the critical exponent $p=\frac{n+2s}{n-2s}$ was obtained by W. Chen, C. Li, and B. Ou in [@CLO]. They are of the form $$u(x)=c\left(\frac{\mu}{\mu^2+|x-x_0|^2}\right)^{\frac{n-2s}{2}},$$ where $\mu$ is any positive parameter and $c$ is a constant depending on $n$ and $s$.
Behavior of $(-\Delta)^{s/2}u$ near $\partial\Omega$ {#sec6}
====================================================
The aim of this section is to prove Proposition \[thlaps/2\]. We will split this proof into two propositions. The first one is the following, and compares the behavior of $(-\Delta)^{s/2}u$ near $\partial\Omega$ with the one of $(-\Delta)^{s/2}\delta_0^s$, where $\delta_0(x)=\textrm{dist}(x,\partial\Omega)\chi_\Omega(x)$.
\[proplaps2\] Let $\Omega$ be a bounded and $C^{1,1}$ domain, $u$ be a function satisfying [(b)]{} in Proposition \[intparts\]. Then, there exists a $C^{\alpha}(\R^n)$ extension $v$ of $u/\delta^s|_\Omega$ such that $$(-\Delta)^{s/2}u(x)=(-\Delta)^{s/2}\delta_0^s(x)v(x)+h(x)\ \mbox{ in }\ \R^n,$$ where $h\in C^{\alpha}(\R^n)$.
Once we know that the behavior of $(-\Delta)^{s/2}u$ is comparable to the one of $(-\Delta)^{s/2}\delta_0^s$, Proposition \[thlaps/2\] reduces to the following result, which gives the behavior of $(-\Delta)^s\delta_0^s$ near $\partial\Omega$.
\[prop:Lap-s/2-delta-s\] Let $\Omega$ be a bounded and $C^{1,1}$ domain, $\delta(x)={\rm dist}(x,\partial\Omega)$, and $\delta_0=\delta\chi_\Omega$. Then, $$(-\Delta)^{s/2}\delta_0^s(x)=c_1\left\{\log^- \delta(x)+c_2\chi_\Omega(x)\right\}+h(x)\ \mbox{ in }\ \R^n,$$ where $c_1$ and $c_2$ are constants, $h$ is a $C^{\alpha}(\R^n)$ function, and $\log^-t=\min\{\log t,0\}$. The constants $c_1$ and $c_2$ are given by $$c_1=c_{1,\frac s2}\qquad\mbox{and}\qquad c_2=\int_0^{\infty}\left\{\frac{1-z^s}{|1-z|^{1+s}}+\frac{1+z^s}{|1+z|^{1+s}}\right\}dz,$$ where $c_{n,s}$ is the constant appearing in the singular integral expression for $(-\Delta)^{s}$ in dimension $n$.
The fact that the constants $c_1$ and $c_2$ given by Proposition \[prop:Lap-s/2-delta-s\] coincide with the ones from Proposition \[thlaps/2\] is proved in the Appendix.
In the proof of Proposition \[proplaps2\] we need to compute $(-\Delta)^{s/2}$ of the product $u=\delta_0^s v$. For it, we will use the following elementary identity, which can be derived from : $$(-\Delta)^s(w_1w_2)=w_1(-\Delta)^sw_2+w_2(-\Delta)^sw_1-I_s(w_1,w_2),$$ where $$\label{Is}I_s(w_1,w_2)(x)= c_{n,s}\textrm{PV}\int_{\R^n} \frac{\bigl(w_1(x)-w_1(y)\bigr)\bigl(w_2(x)-w_2(y)\bigr)}{|x-y|^{n+2s}}\,dy.$$ Next lemma will lead to a Hölder bound for $I_s(\delta_0^s,v)$.
\[lem-boundI2\] Let $\Omega$ be a bounded domain and $\delta_0={\rm dist}(x,\R^n\setminus\Omega)$. Then, for each $\alpha\in(0,1)$ the following a priori bound holds $$\label{eq:bound-I2}
\|I_{s/2}(\delta_0^s,w)\|_{C^{\alpha/2}(\R^n)}\le C[w]_{C^\alpha(\R^n)},$$ where the constant $C$ depends only on $n$, $s$, and $\alpha$.
Let $x_1, x_2\in \R^n$. Then, $$|I_{s/2} (\delta_0^s,w)(x_1)-I_{s/2}(\delta_0^s,w)(x_2)| \le c_{n,\frac s2}(J_1 + J_2),$$ where $$J_1= \int_{\R^n}\frac{\bigl|w(x_1)-w(x_1+z)-w(x_2)+w(x_2+z)\bigr|
\bigl|\delta_0^s(x_1)-\delta_0^s(x_1+z)\bigr|}{|z|^{n+s}}\,dz$$ and $$J_2= \int_{\R^n}\frac{\bigl|w(x_2)-w(x_2+z)\bigr|
\bigl|\delta_0^s(x_1)-\delta_0^s(x_1+z)-\delta_0^s(x_2)+\delta_0^s(x_2+z)\bigr|}{|z|^{n+s}}\,dz \,.$$ Let $r=|x_1-x_2|$. Using that $\|\delta_0^s\|_{C^{s}(\R^n)}\le 1$ and ${\rm supp}\,\delta_0^s = \overline\Omega$, $$\begin{split}
J_1&\le \int_{\R^n}\frac{\bigl|w(x_1)-w(x_1+z)-w(x_2)+w(x_2+z)\bigr|\min\{|z|^s,({\rm diam}\, \Omega)^s\} }{|z|^{n+s}}\,dz \\
&\le C\int_{\R^n} \frac{[w]_{C^\alpha(\R^n)}r^{\alpha/2}|z|^{\alpha/2}\min\{|z|^s,1\} }{|z|^{n+s}}\,dz \\
&\le C r^{\alpha/2}[w]_{C^\alpha(\R^n)}\,.
\end{split}$$ Analogously, $$J_2\le C r^{\alpha/2}[w]_{C^\alpha(\R^n)}\,.$$
The bound for $\|I_{s/2}(\delta_0^s,w)\|_{L^\infty(\R^n)}$ is obtained with a similar argument, and hence follows.
Before stating the next result, we need to introduce the following weighted Hölder norms; see Definition 1.3 in [@RS].
\[definorm\] Let $\beta>0$ and $\sigma\ge -\beta$. Let $\beta=k+\beta'$, with $k$ integer and $\beta'\in (0,1]$. For $w\in C^{\beta}(\Omega)=C^{k,\beta'}(\Omega)$, define the seminorm $$[w]_{\beta;\Omega}^{(\sigma)}= \sup_{x,y\in \Omega} \biggl(\min\{\delta(x),\delta(y)\}^{\beta+\sigma} \frac{|D^{k}w(x)-D^{k}w(y)|}{|x-y|^{\beta'}}\biggr).$$ For $\sigma>-1$, we also define the norm $\|\cdot\|_{\beta;\Omega}^{(\sigma)}$ as follows: in case that $\sigma\ge0$, $$\|w\|_{\beta;\Omega}^{(\sigma)} = \sum_{l=0}^k \sup_{x\in \Omega} \biggl(\delta(x)^{l+\sigma} |D^l w(x)|\biggr) + [w]_{\beta;\Omega}^{(\sigma)}\,,$$ while for $-1<\sigma<0$, $$\|w\|_{\beta;\Omega}^{(\sigma)} = \|w\|_{C^{-\sigma}(\overline \Omega)}+\sum_{l=1}^k \sup_{x\in \Omega} \biggl(\delta(x)^{l+\sigma} |D^l w(x)|\biggr) + [w]_{\beta;\Omega}^{(-\alpha)}.$$
The following lemma, proved in [@RS], will be used in the proof of Proposition \[proplaps2\] below —with $w$ replaced by $v$— and also at the end of this section in the proof of Proposition \[thlaps/2\] —with $w$ replaced by $u$.
\[cosaiscalpha\] Let $\Omega$ be a bounded domain and $\alpha$ and $\beta$ be such that $\alpha\le s<\beta$ and $\beta-s$ is not an integer. Let $k$ be an integer such that $\beta= k+\beta'$ with $\beta'\in(0,1]$. Then, $$\label{eq:bound-lap-s/2}
[(-\Delta)^{s/2}w]_{\beta-s;\Omega}^{(s-\alpha)}\le C\bigl( \|w\|_{C^\alpha(\R^n)}+ \|w\|_{\beta;\Omega}^{(-\alpha)}\bigl)$$ for all $w$ with finite right hand side. The constant $C$ depends only on $n$, $s$, $\alpha$, and $\beta$.
Before proving Proposition \[proplaps2\], we give an extension lemma —see [@EG Theorem 1, Section 3.1] where the case $\alpha=1$ is proven in full detail.
\[ext\] Let $\alpha\in(0,1]$ and $V\subset\R^n$ a bounded domain. There exists a (nonlinear) map $E:C^{0,\alpha}(\overline V)\rightarrow C^{0,\alpha}(\R^n)$ satisfying $$E(w)\equiv w \quad \mbox{in }\overline V,\ \ \ [E(w)]_{C^{0,\alpha}(\R^n)}\le [w]_{C^{0,\alpha}(\overline V)},\ \ \ \mbox{and}\ \ \ \|E(w)\|_{L^\infty(\R^n)}\le \|w\|_{L^\infty(V)}$$ for all $w\in C^{0,\alpha}(\overline V)$.
It is immediate to check that $$E(w)(x)=\min\left\{\min_{z\in \overline V}\left\{w(z)+
[w]_{C^{\alpha}(\overline V)}|z-x|^\alpha\right\},\|w\|_{L^\infty(V)}\right\}$$ satisfies the conditions since, for all $x,y,z$ in $\R^n$, $$|z-x|^\alpha \le |z-y|^\alpha+|y-x|^\alpha\,.$$
Now we can give the
Since $u/\delta^s|_\Omega$ is $C^{\alpha}(\overline\Omega)$ —by hypothesis (b)— then by Lemma \[ext\] there exists a $C^\alpha(\R^n)$ extension $v$ of $u/\delta^s|_\Omega$.
Then, we have that $$(-\Delta)^{s/2}u(x)=v(x)(-\Delta)^{s/2}\delta_0^s(x)+\delta_0(x)^s(-\Delta)^{s/2}v(x)-I_{s/2}(v,\delta_0^s),$$ where $$I_{s/2}(v,\delta_0^s)= c_{n,\frac s2}\int_{\R^n} \frac{\bigl(v(x)-v(y)\bigr)\bigl(\delta_0^s(x)-\delta_0^s(y)\bigr)}{|x-y|^{n+s}}\,dy \,,$$ as defined in . This equality is valid in all of $\R^n$ because $\delta_0^s\equiv0$ in $\R^n\backslash \Omega$ and $v\in C^{\alpha+s}$ in $\Omega$ —by hypothesis (b). Thus, we only have to see that $\delta_0^s(-\Delta)^{s/2}v$ and $I_{s/2}(v,\delta_0^s)$ are $C^{\alpha}(\R^n)$ functions.
For the first one we combine assumption (b) with $\beta = s+\alpha<1$ and Lemma \[cosaiscalpha\]. We obtain $$\label{paloma}
\|(-\Delta)^{s/2}v\|_{\alpha;\Omega}^{(s-\alpha)} \le C,$$ and this yields $\delta_0^s(-\Delta)^{s/2}v \in C^{\alpha}(\R^n)$. Indeed, let $w=(-\Delta)^{s/2}v$. Then, for all $x,y\in \Omega$ such that $y\in B_R(x)$, with $R=\delta(x)/2$, we have $$\frac{|\delta^s(x)w(x)-\delta^s(y)w(y)|}{|x-y|^\alpha}\leq \delta(x)^s\frac{|w(x)-w(y)|}{|x-y|^\alpha}+|w(x)|\frac{|\delta^s(x)-\delta^s(y)|}{|x-y|^\alpha}.$$ Now, since $$|\delta^s(x)-\delta^s(y)|\leq C R^{s-\alpha} |x-y|^\alpha \le C \min\{\delta(x),\delta(y)\}^{s-\alpha} |x-y|^\alpha,$$ using and recalling Definition \[definorm\] we obtain $$\frac{|\delta^s(x)w(x)-\delta^s(y)w(y)|}{|x-y|^\alpha} \le C \quad \mbox{whenever }y\in B_R(x)\,,\ R=\delta(x)/2.$$ This bound can be extended to all $x,y\in \Omega$, since the domain is regular, by using a dyadic chain of balls; see for instance the proof of Proposition 1.1 in [@RS].
The second bound, that is, $$\|I_{s/2}(v,\delta_0^s)\|_{C^{\alpha}(\R^n)}\leq C,$$ follows from assumption (b) and Lemma \[lem-boundI2\] (taking a smaller $\alpha$ if necessary).
To prove Proposition \[prop:Lap-s/2-delta-s\] we need some preliminaries.
Fixed $\rho_0>0$, define $\phi\in C^{s}(\R)$ by $$\label{phi}
\phi(x)= x^s \chi_{(0,\rho_0)}(x)+\rho_0^s \chi_{(\rho_0,+\infty)}(x).$$ This function $\phi$ is a truncation of the $s$-harmonic function $x_+^s$. We need to introduce $\phi$ because the growth at infinity of $x_+^s$ prevents us from computing its $(-\Delta)^{s/2}$.
\[claim1\] Let $\rho_0>0$, and let $\phi:\R\rightarrow\R$ be given by . Then, we have $$(-\Delta)^{s/2} \phi (x) = c_1\{\log |x| + c_2\chi_{(0,\infty)}(x)\} + h(x)$$ for $x\in(-\rho_0/2,\rho_0/2)$, where $h\in C^{s}([-\rho_0/2,\rho_0/2])$. The constants $c_1$ and $c_2$ are given by $$c_1=c_{1,\frac s2}\qquad{\rm and}\qquad c_2=\int_0^{\infty}\left\{\frac{1-z^s}{|1-z|^{1+s}}+\frac{1+z^s}{|1+z|^{1+s}}\right\}dz,$$ where $c_{n,s}$ is the constant appearing in the singular integral expression for $(-\Delta)^{s}$ in dimension $n$.
If $x<\rho_0$, $$(-\Delta)^{s/2}\phi(x) =
c_{1,\frac s2}\biggl(\int_{-\infty}^{\rho_0} \frac{x_+^s-y_+^s}{|x-y|^{1+s}}\,dy
+ \int_{\rho_0}^{\infty}\frac{x_+^s-\rho_0^s}{|x-y|^{1+s}}\,dy\biggr).$$ We need to study the first integral: $$\label{integrals}
J(x)=\int_{-\infty}^{\rho_0}\frac{x_+^s-y_+^s}{|x-y|^{1+s}}\,dy =
\begin{cases}\displaystyle
\ J_1(x)= \int_{-\infty}^{\rho_0/x}\frac{1-z_+^s}{|1-z|^{1+s}}\,dz &\mbox{if }x>0\\
&\\
\displaystyle
\ J_2(x)=\int_{-\infty}^{\rho_0/|x|}\frac{-z_+^s}{|1+z|^{1+s}} \,dz& \mbox{if }x<0\,,
\end{cases}$$ since $$\label{dosestrelles}
(-\Delta)^{s/2}\phi(x)-c_1J(x)=
c_1\int_{\rho_0}^{\infty}\frac{x_+^s-\rho_0^s}{|x-y|^{1+s}}\,dy$$ belongs to $C^{s}([-\rho_0/2,\rho_0/2])$ as a function of $x$.
Using L’Hôpital’s rule we find that $$\lim_{x\downarrow0} \frac{J_1(x)} {\log |x|}=\lim_{x\uparrow0}
\frac{J_2(x)}{\log |x|}=1.$$ Moreover, $$\begin{split}
\lim_{x\downarrow0} x^{1-s}\biggl(J_1'(x)-\frac{1}{x}\biggr)&=
\lim_{x\downarrow0} x^{1-s}\biggl(-\frac{\rho_0}{x^2}\frac{1-(\rho_0/x)^s}{((\rho_0/x)-1)^{1+s}}-\frac{1}{x}\biggr)\\
&=\rho_0^{-s}\lim_{y\downarrow0} y^{1-s} \biggl(\frac{1-y^s}{y(1-y)^{1+s}}-\frac{(1-y)^{1+s}}{y(1-y)^{1+s}}\biggr)\\
&=\rho_0^{-s}\lim_{y\downarrow0} \frac{1-y^s-(1-y)^{1+s}}{y^s}= -\rho_0^{-s}
\end{split}$$ and $$\begin{split}
\lim_{x\uparrow0} (-x)^{1-s}\biggl(J_2'(x)-\frac{1}{x}\biggr)&=
\lim_{x\uparrow0} (-x)^{1-s}\biggl(\frac{\rho_0}{x^2}\frac{-(-\rho_0/x)^s}
{(1+(-\rho_0/x))^{1+s}}-\frac{1}{-x}\biggr)\\
&=\rho_0^{-s}\lim_{y\downarrow0} y^{1-s} \biggl(\frac{-1}{y(1+y)^{1+s}}+\frac{(1+y)^{1+s}}{y(1+y)^{1+s}}\biggr)\\
&=\rho_0^{-s}\lim_{y\downarrow0} \frac{(1+y)^{1+s}-1}{y^s}= 0 \,.
\end{split}$$
Therefore, $$(J_1(x)-\log|x|)' \le C |x|^{s-1}\quad \mbox{in }(0,\rho_0/2]$$ and $$(J_2(x)-\log|x|)' \le C |x|^{s-1}\quad \mbox{in }[-\rho_0/2,0),$$ and these gradient bounds yield $$(J_1-\log|\cdot|)\in C^{s}([0,\rho_0/2])\quad \mbox{and}\quad
(J_2-\log|\cdot|)\in C^{s}([-\rho_0/2,0]).$$ However, these two Hölder functions do not have the same value at $0$. Indeed, $$\begin{split}
\lim_{x\downarrow 0} \bigl\{(J_1(x)-\log|x|)-(&J_2(-x)-\log|-x|)\bigr\}
=\lim_{x\downarrow 0}\left\{J_1(x)-J_2(-x)\right\}\\
&=\int_{-\infty}^{\infty}\left\{\frac{1-z_+^s}{|1-z|^{1+s}}+\frac{z_+^s}{|1+z|^{1+s}}\right\}dz\\
&=\int_{0}^{\infty}\left\{\frac{1-z^s}{|1-z|^{1+s}}+\frac{1+z^s}{|1+z|^{1+s}}\right\}dz=c_2.
\end{split}$$ Hence, the function $J(x)-\log|x|-c_2\chi_{(0,\infty)}(x)$, where $J$ is defined by , is $C^s([-\rho_0/2,\rho_0/2])$. Recalling , we obtain the result.
Next lemma will be used to prove Proposition \[prop:Lap-s/2-delta-s\]. Before stating it, we need the following
\[remrho0\] From now on in this section, $\rho_0>0$ is a small constant depending only on $\Omega$, which we assume to be a bounded $C^{1,1}$ domain. Namely, we assume that that every point on $\partial\Omega$ can be touched from both inside and outside $\Omega$ by balls of radius $\rho_0$. In other words, given $x_0\in \partial\Omega$, there are balls of radius $\rho_0$, $B_{\rho_0}(x_1)\subset \Omega$ and $B_{\rho_0}(x_2)\subset\R^n\setminus \Omega$, such that $\overline{B_{\rho_0}(x_1)}\cap\overline{B_{\rho_0}(x_2)}=\{x_0\}$. A useful observation is that all points $y$ in the segment that joins $x_1$ and $x_2$ —through $x_0$— satisfy $\delta(y)= |y-x_0|$.
\[claim2\] Let $\Omega$ be a bounded $C^{1,1}$ domain, $\delta(x)={\rm dist}(x,\partial\Omega)$, $\delta_0=\delta\chi_\Omega$, and $\rho_0$ be given by Remark \[remrho0\]. Fix $x_0\in\partial\Omega$, and define $$\phi_{x_0}(x)=\phi\left(-\nu(x_0)\cdot(x-x_0)\right)$$ and $$\label{Sx0}
S_{x_0}=\{x_0+t\nu(x_0),\ t\in(-\rho_0/2,\rho_0/2)\},$$ where $\phi$ is given by and $\nu(x_0)$ is the unit outward normal to $\partial\Omega$ at $x_0$. Define also $w_{x_0}=\delta_0^s-\phi_{x_0}$.
Then, for all $x\in S_{x_0}$, $$|(-\Delta)^{s/2} w_{x_0}(x)-(-\Delta)^{s/2} w_{x_0}(x_0)| \le C|x-x_0|^{s/2},$$ where $C$ depends only on $\Omega$ and $\rho_0$ (and not on $x_0$).
We denote $w=w_{x_0}$. Note that, along $S_{x_0}$, the distance to $\partial\Omega$ agrees with the distance to the tangent plane to $\partial\Omega$ at $x_0$; see Remark \[remrho0\]. That is, denoting $\delta_\pm=(\chi_\Omega-\chi_{\R^n\backslash \Omega})\delta$ and $d(x)=-\nu(x_0)\cdot(x-x_0)$, we have $\delta_\pm(x)=d(x)$ for all $x\in S_{x_0}$. Moreover, the gradients of these two functions also coincide on $S_{x_0}$, i.e., $\nabla\delta_\pm(x)=-\nu(x_0)=\nabla d(x)$ for all $x\in S_{x_0}$.
Therefore, for all $x\in S_{x_0}$ and $y\in B_{\rho_0/2}(0)$, we have $$|\delta_{\pm}(x+y)-d(x+y)|\le C|y|^2$$ for some $C$ depending only on $\rho_0$. Thus, for all $x\in S_{x_0}$ and $y\in B_{\rho_0/2}(0)$, $$\label{w(x+y)}|w(x+y)|= |(\delta_{\pm}(x+y))_+^s-(d(x+y))_+^s| \le C|y|^{2s},$$ where $C$ is a constant depending on $\Omega$ and $s$.
On the other hand, since $w\in C^{s}(\R^n)$, then $$\label{w(x+y)2}|w(x+y)-w(x_0+y)|\leq C|x-x_0|^s.$$
Finally, let $r<\rho_0/2$ to be chosen later. For each $x\in S_{x_0}$ we have $$\begin{split}
|(-\Delta)^{s/2} &w(x)-(-\Delta)^{s/2} w(x_0)|
\le C\int_{\R^n} \frac{|w(x+y)-w(x_0+y)|}{|y|^{n+s}}\,dy\\
&\le C\int_{B_r}\frac{|w(x+y)-w(x_0+y)|}{|y|^{n+s}}\,dy
+C\int_{\R^n\setminus B_r}\frac{|w(x+y)-w(x_0+y)|}{|y|^{n+s}}\,dy\\
&\le C\int_{B_r}\frac{|y|^{2s}}{|y|^{n+s}}\,dy+
C\int_{\R^n\setminus B_r}\frac{|x-x_0|^s}{|y|^{n+s}}\,dy\\
&= C(r^s +|x-x_0|^sr^{-s})\,,
\end{split}$$ where we have used and . Taking $r=|x-x_0|^{1/2}$ the lemma is proved.
The following is the last ingredient needed to prove Proposition \[prop:Lap-s/2-delta-s\].
\[claimMMM\] Let $\Omega$ be a bounded $C^{1,1}$ domain, and $\rho_0$ be given by Remark \[remrho0\]. Let $w$ be a function satisfying, for some $K>0$,
- $w$ is locally Lipschitz in $\{x\in\R^n \,:\,0<\delta(x)<\rho_0\}$ and $$|\nabla w(x)| \le K\delta(x)^{-M}\ \mbox{ in }\ \{x\in\R^n \,:\,0<\delta(x)<\rho_0\}$$ for some $M >0$.
- There exists $\alpha>0$ such that $$|w(x)-w(x^*)|\le K\delta(x)^\alpha \ \mbox{ in }\ \{x\in\R^n \,:\,0<\delta(x)<\rho_0\},$$ where $x^*$ is the unique point on $\partial \Omega$ satisfying $\delta(x)= |x-x^*|$.
- For the same $\alpha$, it holds $$\|w\|_{C^{\alpha}\left( {\{\delta\geq\rho_0\}}\right)}\le K.$$
Then, there exists $\gamma>0$, depending only on $\alpha$ and $M$, such that $$\label{claimMMMeq}
\|w\|_{C^{\gamma}(\R^n)}\le C K,$$ where $C$ depends only on $\Omega$.
First note that from (ii) and (iii) we deduce that $\|w\|_{L^\infty(\R^n)}\leq CK$. Let $\rho_1\leq\rho_0$ be a small positive constant to be chosen later. Let $x,y \in \{\delta\leq\rho_0\} $, and $r=|x-y|$.
If $r\geq\rho_1$, then $$\frac{|w(x)-w(y)|}{|x-y|^\gamma}\leq \frac{2\|w\|_{L^\infty(\R^n)}}{\rho_1^\gamma}\leq CK.$$
If $r<\rho_1$, consider $$x' = x^* + \rho_0 r^\beta \nu(x^*)\ \mbox{ and }\ y' = y^* + \rho_0 r^\beta \nu(y^*),$$ where $\beta\in(0,1)$ is to be determined later. Choose $\rho_1$ small enough so that the segment joining $x'$ and $y'$ contained in the set $\{\delta>\rho_0 r^\beta/2 \}$. Then, by (i), $$\label{1209}|w(x')-w(y')|\le CK (\rho_0 r^{\beta}/2) ^{-M} |x'-y'| \le C r^{1-\beta M}.$$ Thus, using (ii) and , $$\begin{split}
|w(x)-w(y)|&\le |w(x)-w(x^*)|+ |w(x^*)-w(x')|+
\\&\qquad +|w(y)-w(y^*)|+ |w(y^*)-w(y')|+ |w(x')-w(y')|\\
&\le K \delta(x)^\alpha + K\delta(y)^\alpha +2K (\rho_0 r^\beta)^\alpha + CK r^{1-\beta M}.
\end{split}$$ Taking $\beta<1/M$ and $\gamma= \min\{\alpha\beta, 1-\beta M\}$, we find $$|w(x)-w(y)|\le CKr^\gamma = CK|x-y|^\gamma.$$ This proves $$[w]_{C^{\gamma}\left({\{\delta\leq\rho_0\}}\right)}\le CK.$$ To obtain the bound we combine the previous seminorm estimate with (iii).
Finally, we give the proof of Proposition \[prop:Lap-s/2-delta-s\].
Let $$h(x)= (-\Delta)^{s/2}\delta_0^s(x)-c_1\left\{\log^- \delta(x)+c_2\chi_\Omega(x)\right\}.$$ We want to prove that $h\in C^{\alpha}(\R^n)$ by using Claim \[claimMMM\].
On one hand, by Lemma \[claim1\], for all $x_0\in\partial\Omega$ and for all $x\in S_{x_0}$, where $S_{x_0}$ is defined by , we have $$h (x) = (-\Delta)^{s/2}\delta_0^s(x)- (-\Delta)^{s/2} \phi_{x_0}(x) + \tilde h\bigl(\nu(x_0)\cdot(x-x_0)\bigr),$$ where $\tilde h$ is the $C^{s}([-\rho_0/2,\rho_0/2])$ function from Lemma \[claim1\]. Hence, using Lemma \[claim2\], we find $$|h(x)-h(x_0)|\le C|x-x_0|^{s/2}\quad \mbox{for all }x\in S_{x_0}$$ for some constant independent of $x_0$.
Recall that for all $x\in S_{x_0}$ we have $x^*=x_0$, where $x^*$ is the unique point on $\partial \Omega$ satisfying $\delta(x)= |x-x^*|$. Hence, $$\label{eq:pf-prop-calpha-bound1}
|h(x)-h(x^*)|\le C|x-x^*|^{s/2}\quad \mbox{ for all }x\in\{\delta<\rho_0/2\}\,.$$
Moreover, $$\label{eq:pf-prop-bound-faraway}
\|h\|_{C^{\alpha}(\{\delta\ge\rho_0/2\})} \le C$$ for all $\alpha\in (0,1-s)$, where $C$ is a constant depending only on $\alpha$, $\Omega$ and $\rho_0$. This last bound is found using that $\|\delta_0^s\|_{C^{0,1}(\{\delta\ge\rho_0/2\})} \le C$, which yields $$\|(-\Delta)^{s/2}\delta_0^s\|_{C^{\alpha}(\{\delta\ge\rho_0\})} \le C$$ for $\alpha<1-s$.
On the other hand, we claim now that if $x\notin \partial\Omega$ and $\delta(x)<\rho_0/2$, then $$\label{eq:pf-prop-grad-bound}
|\nabla h(x)|\le |\nabla(-\Delta)^{s/2}\delta_0^s(x)| + c_1|\delta(x)|^{-1}\le C|\delta(x)|^{-n-s}.$$ Indeed, observe that $\delta_0^s\equiv0$ in $\R^n\backslash\Omega$, $|\nabla\delta_0^s|\leq C\delta_0^{s-1}$ in $\Omega$, and $|D^2\delta_0^s|\leq C\delta_0^{s-2}$ in $\Omega_{\rho_0}$. Then, $r= \delta(x)/2$, $$\begin{split}
|(-\Delta)^{s/2}\nabla\delta_0^s(x)| &\le
C\int_{\R^n} \frac{|\nabla\delta_0^s(x) - \nabla\delta_0^s(x+y)|}{|y|^{n+s}}\,dy\\
&\le C\int_{B_r} \frac{C r^{s-2}|y|\,dy}{|y|^{n+s}}+ C\int_{\R^n\setminus B_r} \left(\frac{|\nabla\delta_0^s(x)|}{|y|^{n+s}}+ \frac{|\nabla\delta_0^s(x+y)|}{r^{n+s}}\right)dy\\
&\leq \frac Cr+\frac Cr+\frac{C}{r^{n+s}}\int_{\R^n} \delta_0^{s-1}\leq \frac{C}{r^{n+s}},
\end{split}$$ as claimed.
To conclude the proof, we use bounds , , and and Claim \[claimMMM\].
To end this section, we give the
The first part follows from Propositions \[proplaps2\] and \[prop:Lap-s/2-delta-s\]. The second part follows from Lemma \[cosaiscalpha\] with $\alpha=s$ and $\beta\in(s,1+2s)$.
The operator $-\left.\frac{d}{d\lambda}\right|_{\lambda=1^+}\int_{\mathbb R}w_\lambda w_{1/\lambda}$ {#sec7}
====================================================================================================
The aim of this section is to prove Proposition \[propoperador\]. In other words, we want to evaluate the operator $$\label{mathfrakI}
\mathfrak{I}(w)=-\left.\frac{d}{d\lambda}\right|_{\lambda=1^+}\int_{0}^{\infty} w\left(\lambda t\right)w\left(\frac{t}{\lambda}\right)dt$$ on $$w(t)=A\log^-|t-1|+B\chi_{[0,1]}(t)+h(t),$$ where $\log^-t=\min\{\log t,0\}$, $A$ and $B$ are real numbers, and $h$ is a function satisfying, for some constants $\alpha\in(0,1)$, $\gamma\in(0,1)$, and $C_0$, the following conditions:
- $\|h\|_{C^{\alpha}((0,\infty))}\leq C_0$.
- For all $\beta\in[\gamma,1+\gamma]$, $$\|h\|_{C^{\beta}((0,1-\rho)\cup(1+\rho,2))}\leq C_0 \rho^{-\beta}\qquad \textrm{for all}\ \ \rho\in(0,1).$$
- $|h'(t)|\leq C t^{-2-\gamma}$ and $|h''(t)|\leq C t^{-3-\gamma}$ for all $t>2$.
We will split the proof of Proposition \[propoperador\] into three parts. The first part is the following, and evaluates the operator $\mathfrak I$ on the function $$\label{w0}w_0(t)=A\log^-|t-1|+B\chi_{[0,1]}(t).$$
\[c2\] Let $w_0$ and $\mathfrak I$ be given by and , respectively. Then, $$\mathfrak I(w_0)=A^2\pi^2+B^2.$$
The second result towards Proposition \[propoperador\] is the following.
\[lema1\] Let $h$ be a function satisfying [(i)]{}, [(ii)]{}, and [(iii)]{} above, and $\mathfrak I$ be given by . Then, $$\mathfrak I(h)=0.$$ Moreover, there exist constants $C$ and $\nu>1$, depending only on the constants $\alpha$, $\gamma$, and $C_0$ appearing in [(i)-(ii)-(iii)]{}, such that $$\left|\int_{0}^{\infty} \left\{h\left(\lambda t\right)h\left(\frac{t}{\lambda}\right)-h(t)^2\right\}dt\right|\leq C|\lambda-1|^\nu$$ for each $\lambda\in(1,3/2)$.
Finally, the third one states that $\mathfrak I(w_0+h)=\mathfrak I(w_0)$ whenever $\mathfrak I(h)=0$.
\[nodependeh\] Let $w_1$ and $w_2$ be $L^2(\R)$ functions. Assume that the derivative at $\lambda=1^+$ in the expression $\mathfrak I(w_1)$ exists, and that $$\mathfrak I(w_2)=0.$$ Then, $$\mathfrak I(w_1+w_2)=\mathfrak I(w_1).$$
Let us now give the proofs of Lemmas \[c2\], \[lema1\], and \[nodependeh\]. We start proving Lemma \[nodependeh\]. For it, is useful to introduce the bilinear form $$(w_1,w_2)=-\frac12\left.\frac{d}{d\lambda}\right|_{\lambda=1^+}\int_{0}^{\infty} \left\{w_1\left(\lambda t\right)w_2\left(\frac{t}{\lambda}\right)+w_1\left(\frac{t}{\lambda}\right)w_2\left(\lambda t\right)\right\}dt,$$ and more generally, the bilinear forms $$\label{pelambda}
(w_1,w_2)_\lambda=-\frac{1}{2(\lambda-1)}\int_{0}^{\infty} \left\{w_1\left(\lambda t\right)w_2\left(\frac{t}{\lambda}\right)+w_1\left(\frac{t}{\lambda}\right)w_2\left(\lambda t\right)-2w_1(t)w_2(t)\right\}dt,$$ for $\lambda>1$.
It is clear that $\lim_{\lambda\downarrow 1}(w_1,w_2)_\lambda=(w_1,w_2)$ whenever the limit exists, and that $(w,w)=\mathfrak{I}(w)$. The following lemma shows that these bilinear forms are positive definite and, thus, they satisfy the Cauchy-Schwarz inequality.
\[scalar\] The following properties hold.
- $(w_1,w_2)_\lambda$ is a bilinear map.
- $(w,w)_\lambda\geq0$ for all $w\in L^2(\mathbb R_+)$.
- $|(w_1,w_2)_\lambda|^2\leq (w_1,w_1)_\lambda (w_2,w_2)_\lambda$.
Part (a) is immediate. Part (b) follows from the Hölder inequality $$\|w_{\lambda}w_{1/\lambda}\|_{L^1}\leq \|w_\lambda\|_{L^2}\|w_{1/\lambda}\|_{L^2}=\|w\|_{L^2}^2,$$ where $w_{\lambda}(t)=w(\lambda t)$. Part (c) is a consequence of (a) and (b).
Now, Lemma \[nodependeh\] is an immediate consequence of this Cauchy-Schwarz inequality.
By Lemma \[scalar\] (iii) we have $$0\leq |(w_1,w_2)_\lambda|\leq \sqrt{(w_1,w_1)_\lambda}\sqrt{(w_2,w_2)_\lambda}\longrightarrow0.$$ Thus, $(w_1,w_2)=\lim_{\lambda\downarrow1}(w_1,w_2)_\lambda=0$ and $$\mathfrak I(w_1+w_2)=\mathfrak I(w_1)+\mathfrak I(w_2)+2(w_1,w_2)=\mathfrak I(w_1).$$
Next we prove that $\mathfrak I(h)=0$. For this, we will need a preliminary lemma.
\[h\] Let $h$ be a function satisfying [(i)]{}, [(ii)]{}, and [(iii)]{} in Propostion \[propoperador\], $\lambda\in(1,3/2)$, and $\tau\in(0,1)$ be such that $\tau/2>\lambda-1$. Let $\alpha$, $\gamma$, and $C_0$ be the constants appearing in [(i)-(ii)-(iii)]{}. Then, $$\left|h(\lambda t)h\left(\frac{t}{\lambda}\right)-h(t)^2\right|\leq \left\{
\begin{array}{ll}\displaystyle
C\max\left\{\left|t-\lambda\right|^{\alpha},\left|t-1/\lambda\right|^{\alpha}\right\} & t\in(1-\tau,1+\tau)\vspace{2mm}\\ \displaystyle
C(\lambda-1)^{1+\gamma}|t-1|^{-1-\gamma} & t\in(0,1-\tau)\cup(1+\tau,2)\vspace{2mm}\\ \displaystyle
C(\lambda-1)^2t^{-1-\gamma} & t\in(2,\infty),
\end{array}
\right.$$ where the constant $C$ depends only on $C_0$.
Let $t\in(1-\tau,1+\tau)$. Let us denote $\widetilde h=h-h(1)$. Then, $$h\left(\lambda t\right)h\left(\frac{t}{\lambda}\right)-h(t)^2=
\widetilde h\left(\lambda t\right)\widetilde h\left(\frac{t}{\lambda}\right)-\widetilde h(t)^2+h(1)\left(\widetilde h\left(\lambda t\right)+\widetilde h\left(\frac{t}{\lambda}\right)-2\widetilde h(t)\right).$$ Therefore, using that $|\widetilde h(t)|\leq C_0|t-1|^\alpha$ and $\|h\|_{L^\infty(\R)}\leq C_0$, we obtain $$\begin{aligned}
\left|h\left(\lambda t\right)h\left(\frac{t}{\lambda}\right)-h(t)^2\right|&\leq&
C\left|\lambda t-1\right|^{\alpha}\left|\frac{t}{\lambda}-1\right|^{\alpha}+
C|t-1|^{2\alpha}+C|\lambda t-1|^{\alpha}+\\
& & +C\left|\frac{t}{\lambda}-1\right|^{\alpha}+C|t-1|^\alpha \\
&\leq&C\max\left\{\left|t-\lambda\right|^{\alpha},\left|t-\frac{1}{\lambda}\right|^{\alpha}\right\}.\end{aligned}$$
Let now $t\in(0,1-\tau)\cup(1+\tau,2)$ and recall that $\lambda\in(1,1+\tau/2)$. Define, for $\mu\in[1,\lambda]$, $$\psi(\mu)=h\left(\mu t\right)h\left(\frac{t}{\mu}\right)-h(t)^2.$$ By the mean value theorem, $\psi(\lambda)=\psi(1)+\psi'(\mu)(\lambda-1)$ for some $\mu\in(1,\lambda)$. Moreover, observing that $\psi(1)=\psi'(1)=0$, we deduce $$|\psi(\lambda)|\leq (\lambda-1)|\psi'(\mu)-\psi'(1)|.$$
Next we claim that $$\label{psimu} |\psi'(\mu)-\psi'(1)|\leq C|\mu-1|^\gamma|t-1|^{-1-\gamma}.$$ This yields the desired bound for $t\in(0,1-\tau)\cup(1+\tau,2)$.
To prove this claim, note that $$\psi'(\mu)=th'\left(\mu t\right)h\left(\frac{t}{\mu}\right)-\frac{t}{\mu^2}h\left(\mu t\right)h'\left(\frac{t}{\mu}\right).$$ Thus, using the bounds from (ii) with $\beta$ replaced by $\gamma$, $1$, and $1+\gamma$, $$\begin{split}
|\psi'(\mu)-&\psi'(1)|\leq t|h'(\mu t)-h'(t)| \left|h\left(\frac{t}{\mu}\right)\right|+t\left|h\left(\frac{t}{\mu}\right)-h(t)\right| |h'(t)|+\\
&\hspace{20mm}+t\left|h'\left(\frac{t}{\mu}\right)-h'(t)\right| \frac{|h(\mu t)|}{\mu^2}+t\left|\frac{h(\mu t)}{\mu^2}-h(t)\right||h'(t)|\\
&\leq C|\mu t-t|^\gamma m^{-1-\gamma}+
C\left|\frac t\mu-t\right|^\gamma m^{-\gamma}|t-1|^{-1}+
\frac{C}{\mu^2}\left|\frac t\mu-t\right|^\gamma m^{-1-\gamma}+\\
&\hspace{10mm}+\frac{C}{\mu^2}|\mu t-t|^\gamma m^{-\gamma}|t-1|^{-1}+C(\mu-1)|t-1|^{-1}\\
&\leq C(\mu-1)^\gamma m^{-1-\gamma},
\end{split}$$ where $m=\min\left\{|\mu t-1|,|t-1|,|t/\mu-1|\right\}$.
Furthermore, since $\mu-1<|t-1|/2$, we have $m\geq \frac14|t-1|$, and hence follows.
Finally, if $t\in(2,\infty)$, with a similar argument but using the bound (iii) instead of (ii), we obtain $$|\psi(\lambda)|\leq C(\lambda-1)^2t^{-1-\gamma},$$ and we are done.
Let us now give the
Let us call $$I_\lambda=\int_{0}^{\infty}\left\{h\left(\lambda t\right)h\left(\frac{t}{\lambda}\right)-h(t)^2\right\}dx.$$ For each $\lambda\in(1,3/2)$, take $\tau\in(0,1)$ such that $\lambda-1<\tau/2$ to be chosen later. Then, by Lemma \[h\], $$\begin{aligned}
|I_\lambda| &\leq &
C(\lambda-1)^{1+\gamma}\int_0^{1-\tau}|t-1|^{-1-\gamma}dt+C\int_{1-\tau}^1\left|t-\lambda\right|^{\alpha}dt+\\
& &+C\int_1^{1+\tau}\left|t-\frac{1}{\lambda}\right|^{\alpha}dt+C(\lambda-1)^{1+\gamma}\int_{1+\tau}^2 |t-1|^{-1-\gamma}dt+\\
&& + C(\lambda-1)^2\int_{2}^\infty t^{-1-s}dt\\
&\leq& C(\lambda-1)^{1+\gamma}\tau^{-\gamma}+C\left(\tau+\lambda-1\right)^{\alpha+1}+C(\lambda-1)^{1+\gamma}\tau^{-\gamma}+\\
&&+C\left(\tau+1-\frac{1}{\lambda}\right)^{\alpha+1}+C(\lambda-1)^2.\end{aligned}$$
Choose now $$\tau=(\lambda-1)^{\theta},$$ with $\theta<1$ to be chosen later. Then, $$\tau+\lambda-1\leq 2\tau\qquad \mbox{and} \qquad \tau+1-\frac1\lambda\leq 2\tau,$$ and hence $$\left|I_\lambda\right|\leq
C(\lambda-1)^{(\alpha+1)\theta}+C(\lambda-1)^{1+\gamma-\theta\gamma}+C(\lambda-1)^2.$$
Finally, choose $\theta$ such that $(\alpha+1)\theta>1$ and $1+\gamma-\theta\gamma>1$, that is, satisfying $$\frac{1}{1+\alpha}<\theta<1.$$ Then, for $\nu=\min\{(\alpha+1)\theta, 1+\gamma-\gamma\theta\}>1$, it holds $$\left|\int_{0}^{\infty}\left\{h\left(\lambda t\right)h\left(\frac{t}{\lambda}\right)-h(t)^2\right\}dt\right|\leq C|\lambda-1|^\nu,$$ as desired.
Next we prove Lemma \[c2\].
Let $$w_1(t)=\log^-|t-1|\qquad\textrm{and}\qquad w_2(t)=\chi_{[0,1]}(t).$$ We will compute first $\mathfrak I(w_1)$.
Define $$\Psi(t)=\int_0^t\frac{\log|r-1|}{r}dr.$$ It is straightforward to check that, if $\lambda>1$, the function $$\begin{aligned}
\vartheta_\lambda(t)&=&\left(t-\frac{1}{\lambda}\right)\log|\lambda t-1|\log\left|\frac{t}{\lambda}-1\right|+
(\lambda-t)\log\left|\frac{t}{\lambda}-1\right|\\
& &-\frac{\lambda^2-1}{\lambda}\log(\lambda^2-1)\log\left|\frac{t}{\lambda}-1\right|-
\frac{\lambda^2-1}{\lambda}\Psi\left(\frac{\lambda(\lambda-t)}{\lambda^2-1}\right)\\
& &+2t-\frac{\lambda t-1}{\lambda}\log|\lambda t-1|\end{aligned}$$ is a primitive of $\log|\lambda t-1|\log\left|\frac{t}{\lambda}-1\right|$. Denoting $I_\lambda=\int_{0}^{\infty} w_1\left(\lambda t\right)w_1\left(\frac{t}{\lambda}\right)dt$, we have $$\begin{aligned}
I_\lambda-I_1&=&\int_{0}^{\frac{2}{\lambda}}\log|\lambda t-1|\log\left|\frac{t}{\lambda}-1\right|dt-\int_0^2\log^2|t-1|dt\\
&=&\vartheta_\lambda\left(\frac{2}{\lambda}\right)-\vartheta_\lambda(0)-4\\
&=&\left(\frac{\lambda^2-1}{\lambda}\right)
\left\{\Psi\left(\frac{\lambda^2}{\lambda^2-1}\right)-
\Psi\left(\frac{\lambda^2-2}{\lambda^2-1}\right)\right\}+\left(\lambda-\frac{2}{\lambda}\right)\log\left(\frac{2}{\lambda^2}-1\right)+\\
&&+\left(\lambda-\frac{1}{\lambda}\right)\log(\lambda^2-1)\log\left(\frac{2}{\lambda^2}-1\right)
-\frac{4(\lambda-1)}{\lambda},\end{aligned}$$ where we have used that $$I_1=\int_0^2\log^2|t-1|dt=2\int_0^1\log^2t'dt'=2\int_0^\infty r^2e^{-r}dr=2\Gamma(3)=4.$$
Therefore, dividing by $\lambda-1$ and letting $\lambda\downarrow1$, $$\begin{aligned}
\left.\frac{d}{d\lambda}\right|_{\lambda=1^+}I_\lambda&=&
2\lim_{\lambda\downarrow 1}\int_{\frac{\lambda^2-2}{\lambda^2-1}}^{\frac{\lambda^2}{\lambda^2-1}}\frac{\log|t-1|}{t}\,dt+\\
& &+\lim_{\lambda\downarrow 1}\left\{2\log(\lambda^2-1)\log\left(\frac{2}{\lambda^2}-1\right)
-\frac{\log\left(\frac{2}{\lambda^2}-1\right)}{\lambda-1}-\frac{4}{\lambda}\right\}.\end{aligned}$$ The first term equals to $$\lim_{M\rightarrow +\infty}\int_{-M}^{M}\frac{2\log|t-1|}{t}dt,$$ while the second, using that $\log(1+x)\sim x$ for $x\sim 0$, equals to $$\lim_{\lambda\downarrow 1}\left\{2\log(\lambda^2-1)\left(\frac{2}{\lambda^2}-2\right)-\frac{\frac{2}{\lambda^2}-2}{\lambda-1}-\frac{4}{\lambda}\right\}=0+4-4=0.$$ Hence, $$\begin{aligned}
\left.\frac{d}{d\lambda}\right|_{\lambda=1^+}I_\lambda&=&\lim_{M\rightarrow +\infty}\int_{-M}^{M}\frac{2\log|t-1|}{t}dt=\lim_{M\rightarrow +\infty}\int_{-M}^{M}\frac{2\log|t|}{t+1}dt\\
&=&\lim_{M\rightarrow +\infty}\left\{\int_{-M}^{0}\frac{2\log(-t)}{t+1}dt+\int_{0}^{M}\frac{2\log t}{t+1}dt\right\}\\
&=&\lim_{M\rightarrow +\infty}\left\{\int_{0}^{M}\frac{2\log t}{1-t}dt+\int_{0}^{M}\frac{2\log t}{t+1}dt\right\}=\int_{0}^{+\infty}\frac{4\log t}{1-t^2}dt\\
&=&\int_{0}^{1}\frac{4\log t}{1-t^2}dt+\int_{1}^{+\infty}\frac{-4\log\frac{1}{t}}{\frac{1}{t^2}-1}\frac{dt}{t^2}=2\int_{0}^{1}\frac{4\log t}{1-t^2}dt.\end{aligned}$$ Furthermore, using that $\frac{1}{1-t^2}=\sum_{n\geq0}t^{2n}$ and that $$\int_0^1t^n\log t\ dt=-\int_0^1\frac{t^{n+1}}{n+1}\frac{1}{t}dt=-\frac{1}{(n+1)^2},$$ we obtain $$\int_{0}^{1}\frac{\log t}{1-t^2}dt=-\sum_{n\geq0}\frac{1}{(2n+1)^2}=-\frac{\pi^2}{8},$$ and thus $$\mathfrak I(w_1)=-\left.\frac{d}{d\lambda}\right|_{\lambda=1^+}I_\lambda=\pi^2.$$
Let us evaluate now $\mathfrak I(w_2)=\mathfrak I(\chi_{[0,1]})$. We have $$\int_{0}^{+\infty} \chi_{[0,1]}\left(\lambda t\right)\chi_{[0,1]}\left(\frac{t}{\lambda}\right)dt=\int_{0}^{\frac{1}{\lambda}}dt=\frac{1}{\lambda}.$$ Therefore, differentiating with respect to $\lambda$ we obtain $\mathfrak I(w_2)=1$.
Let us finally prove that $(w_1,w_2)=0$, i.e., that $$\label{derAB}
\left.\frac{d}{d\lambda}\right|_{\lambda=1^+}\left\{\int_0^{\lambda}\log|1-\lambda t|dt+\int_0^{\frac{1}{\lambda}}\log\left|1-\frac{t}{\lambda}\right|dt\right\}=0.$$ We have $$\begin{aligned}
\int_0^{\lambda}\log|1-\lambda t|dt&=&\frac{1}{\lambda}\bigl[(\lambda t-1)\log|1-\lambda t|-\lambda t\bigr]_0^\lambda\\
&=&\left(\lambda-\frac{1}{\lambda}\right)\log(\lambda^2-1)-\lambda,\end{aligned}$$ and similarly, $$\int_0^{\frac{1}{\lambda}}\log\left|1-\frac{t}{\lambda}\right|dt= \left(\frac{1}{\lambda}-\lambda\right)\log\left(1-\frac{1}{\lambda^2}\right)-\frac{1}{\lambda}.$$ Thus, $$\left|\int_0^{\lambda}\log|1-\lambda t|dt+\int_0^{\frac{1}{\lambda}}\log\left|1-\frac{t}{\lambda}\right|dt-2\int_0^1\log|1-t|dt\right|=$$ $$=\left|\frac{2(\lambda^2-1)}{\lambda}\log\lambda-\frac{(\lambda-1)^2}{\lambda}\right|\leq 4(\lambda-1)^2.$$ Therefore holds, and the proposition is proved.
Finally, to end this section, we give the:
Let us write $\varphi=w_0+h$, where $w_0$ is given by . Then, for each $\lambda>1$ we have $$(\varphi,\varphi)_\lambda=(w_0,w_0)_\lambda+2(w_0,h)_\lambda+(h,h)_\lambda,$$ where $(\cdot,\cdot)_\lambda$ is defined by . Using Lemma \[scalar\] (c) and Lemma \[lema1\], we deduce $$\left|(\varphi,\varphi)_\lambda-A^2\pi^2-B^2\right|\leq \left|(w_0,w_0)_\lambda-A^2\pi^2-B^2\right|+C|\lambda-1|^\nu.$$ The constants $C$ and $\nu$ depend only on $\alpha$, $\gamma$, and $C_0$, and by Lemma \[c2\] the right hand side goes to $0$ as $\lambda\downarrow1$, since $(w_0,w_0)_\lambda\rightarrow \mathfrak I(w_0)$ as $\lambda\downarrow1$.
Proof of the Pohozaev identity in non-star-shaped domains {#sec8}
=========================================================
In this section we prove Proposition \[intparts\] for general $C^{1,1}$ domains. The key idea is that every $C^{1,1}$ domain is locally star-shaped, in the sense that its intersection with any small ball is star-shaped with respect to some point. To exploit this, we use a partition of unity to split the function $u$ into a set of functions $u_1$, ..., $u_m$, each one with support in a small ball. However, note that the Pohozaev identity is quadratic in $u$, and hence we must introduce a bilinear version of this identity, namely $$\label{bilinear}\begin{split}\int_\Omega(x\cdot\nabla &u_1)(-\Delta)^su_2\, dx+\int_\Omega(x\cdot\nabla u_2)(-\Delta)^su_1\,dx=\frac{2s-n}{2}\int_\Omega u_1(-\Delta)^su_2\, dx+\\
&+\frac{2s-n}{2}\int_{\Omega}u_2(-\Delta)^su_1\, dx-\Gamma(1+s)^2\int_{\partial\Omega}\frac{u_1}{\delta^{s}}\frac{u_2}{\delta^{s}}(x\cdot\nu)\, d\sigma.\end{split}$$
The following lemma states that this bilinear identity holds whenever the two functions $u_1$ and $u_2$ have disjoint compact supports. In this case, the last term in the previous identity equals 0, and since $(-\Delta)^s u_i$ is evaluated only outside the support of $u_i$, we only need to require $\nabla u_i\in L^1(\R^n)$.
\[duesboles\] Let $u_1$ and $u_2$ be $W^{1,1}(\R^n)$ functions with disjoint compact supports $K_1$ and $K_2$. Then, $$\begin{split}\int_{K_1}(x\cdot\nabla u_1)(-\Delta)^su_2\, dx&+\int_{K_2}(x\cdot\nabla u_2)(-\Delta)^su_1\,dx=\\
&=\frac{2s-n}{2}\int_{K_1}u_1(-\Delta)^su_2\, dx+\frac{2s-n}{2}\int_{K_2}u_2(-\Delta)^su_1\, dx.\end{split}$$
We claim that $$\label{idcabre}
(-\Delta)^s(x\cdot\nabla u_i)=x\cdot\nabla(-\Delta)^su_i+2s(-\Delta)^su_i\qquad \mbox{in}\ \ \R^n\backslash K_i.$$ Indeed, using $u_i \equiv 0$ in $\R^n\setminus K_i$ and the definition of $(-\Delta)^s$ in , for each $x\in \R^n\backslash K_i$ we have $$\begin{split}
(-\Delta)^s(x\cdot\nabla u_i)(x)&=c_{n,s}\int_{K_i}\frac{-y\cdot\nabla u_i(y)}{|x-y|^{n+2s}}dy\\
&=c_{n,s}\int_{K_i}\frac{(x-y)\cdot\nabla u_i(y)}{|x-y|^{n+2s}}dy+c_{n,s}\int_{K_i}\frac{-x\cdot\nabla u_i(y)}{|x-y|^{n+2s}}dy\\
&=c_{n,s}\int_{K_i}{\rm div}_y\left(\frac{x-y}{|x-y|^{n+2s}}\right)u_i(y)dy+x\cdot(-\Delta)^s\nabla u_i(x)\\
&=c_{n,s}\int_{K_i}\frac{-2s}{|x-y|^{n+2s}}u_i(y)dy+x\cdot\nabla(-\Delta)^su_i(x)\\
&=2s(-\Delta)^su_i(x)+x\cdot\nabla(-\Delta)^su_i(x),
\end{split}$$ as claimed.
We also note that for all functions $w_1$ and $w_2$ in $L^1(\R^n)$ with disjoint compact supports $W_1$ and $W_2$, it holds the integration by parts formula $$\label{above}
\int_{W_1}w_1(-\Delta)^sw_2=\int_{W_1}\int_{W_2}\frac{-w_1(x)w_2(y)}{|x-y|^{n+2s}}dy\,dx
=\int_{W_2}w_2(-\Delta)^sw_1.$$
Using that $(-\Delta)^s u_2$ is smooth in $K_1$ and integrating by parts, $$\int_{K_1}(x\cdot \nabla u_1)(-\Delta)^su_2= -n\int_{K_1}u_1(-\Delta)^su_2-\int_{K_1}u_1x\cdot\nabla(-\Delta)^su_2.$$ Next we apply the previous claim and also the integration by parts formula to $w_1=u_1$ and $w_2=x\cdot \nabla u_2$. We obtain $$\begin{split}
\int_{K_1}u_1x\cdot\nabla(-\Delta)^su_2&=\int_{K_1}u_1(-\Delta)^s(x\cdot \nabla u_2)-2s\int_{K_1}u_1(-\Delta)^su_2\\
&=\int_{K_2}(-\Delta)^su_1(x\cdot \nabla u_2)-2s\int_{K_1}u_1(-\Delta)^su_2.
\end{split}$$ Hence, $$\int_{K_1}(x\cdot \nabla u_1)(-\Delta)^su_2=-\int_{K_2}(-\Delta)^su_1(x\cdot \nabla u_2)+(2s-n)\int_{K_1}u_1(-\Delta)^su_2.$$
Finally, again by the integration by parts formula we find $$\int_{K_1}u_1(-\Delta)^su_2=\frac12\int_{K_1}u_1(-\Delta)^su_2+\frac12\int_{K_2}u_2(-\Delta)^su_1,$$ and the lemma follows.
The second lemma states that the bilinear identity holds whenever the two functions $u_1$ and $u_2$ have compact supports in a ball $B$ such that $\Omega\cap B$ is star-shaped with respect to some point $z_0$ in $\Omega\cap B$.
\[unabola\] Let $\Omega$ be a bounded $C^{1,1}$ domain, and let $B$ be a ball in $\R^n$. Assume that there exists $z_0\in \Omega\cap B$ such that $$(x-z_0)\cdot\nu(x)>0\qquad\mbox{for all}\ x\in\partial\Omega\cap \overline B.$$ Let $u$ be a function satisfying the hypothesis of Proposition \[intparts\], and let $u_1=u\eta_1$ and $u_2=u\eta_2$, where $\eta_i\in C^\infty_c(B)$, $i=1,2$. Then, the following identity holds $$\int_B(x\cdot\nabla u_1)(-\Delta)^su_2\, dx+\int_B(x\cdot\nabla u_2)(-\Delta)^su_1\,dx=\frac{2s-n}{2}\int_Bu_1(-\Delta)^su_2\, dx+$$ $$+\frac{2s-n}{2}\int_{B}u_2(-\Delta)^su_1\, dx
-\Gamma(1+s)^2\int_{\partial\Omega\cap B}\frac{u_1}{\delta^{s}}\frac{u_2}{\delta^{s}}(x\cdot\nu)\, d\sigma.$$
We will show that given $\eta\in C^\infty_c(B)$ and letting $\tilde u=u\eta$ it holds $$\label{y}
\int_B(x\cdot \nabla \tilde u)(-\Delta)^{s}\tilde u\,dx=\frac{2s-n}{2}\int_B\tilde u(-\Delta)^s\tilde u\,dx-\Gamma(1+s)^2\int_{\partial\Omega\cap B}\left(\frac{\tilde u}{\delta^{s}}\right)^2(x\cdot\nu) d\sigma.$$ From this, the lemma follows by applying with $\tilde u$ replaced by $(\eta_1+\eta_2)u$ and by $(\eta_1-\eta_2)u$, and subtracting both identities.
We next prove . For it, we will apply the result for strictly star-shaped domains, already proven in Section \[sec2\]. Observe that there is a $C^{1,1}$ domain $\tilde \Omega$ satisfying $$\{\tilde u>0\}\subset \tilde\Omega\subset \Omega\cap B\quad \mbox{and}\quad(x-z_0)\cdot\nu(x)>0 \quad \mbox{for all }x\in \partial \tilde \Omega.$$ This is because, by the assumptions, $\Omega\cap B$ is a Lipschitz polar graph about the point $z_0\in \Omega\cap B$ and ${\rm supp}\,\tilde u\subset B'\subset\subset B$ for some smaller ball $B'$; see Figure \[figura2\]. Hence, there is room enough to round the corner that $\Omega \cap B$ has on $\partial \Omega\cap \partial B$.
![\[figura2\] ](dibuix_omega_tilde2.pdf)
Hence, it only remains to prove that $\tilde u$ satisfies the hypotheses of Proposition \[intparts\]. Indeed, since $u$ satisfies (a) and $\eta$ is $C^\infty_c(B')$ then $\tilde u$ satisfies $$[\tilde u]_{C^\beta(\{x\in \tilde \Omega\,:\, \tilde \delta(x) >\rho\})} \le C\rho^{s-\beta}$$ for all $\beta \in [s,1+2s)$, where $\tilde\delta(x)= {\rm dist}(x,\partial\tilde\Omega)$.
On the other hand, since $u$ satisfies (b) and we have $\eta \delta^s/{\tilde\delta}^s$ is Lipschitz in ${\rm supp}\, \tilde u$ —because ${\rm dist}(x,\partial\tilde\Omega\setminus \partial\Omega)\ge c>0$ for all $x\in{\rm supp}\,\tilde u$—, then we find $$\bigl[\tilde u/{\tilde \delta}^s\bigr]_{C^\beta(\{x\in \tilde \Omega\,:\, \tilde \delta(x) >\rho\})} \le C\rho^{\alpha-\beta}$$ for all $\beta\in [\alpha,s+\alpha]$.
Let us see now that $\tilde u$ satisfies (c), i.e., that $(-\Delta)^s \tilde u$ is bounded. For it, we use $$(-\Delta)^s(u\eta) = \eta(-\Delta)^s u + u(-\Delta)^s \eta - I_s(u,\eta)$$ where $I_s$ is given by , i.e., $$I_s(u,\eta)(x) = c_{n,s}\int_{\R^n}\frac{(u(x)-u(y))(\eta(x)-\eta(y))}{|x-y|^{n+2s}}\,dy\,.$$ The first term is bounded since $(-\Delta)^s u$ so is by hypothesis. The second term is bounded since $\eta\in C^\infty_c(\R^n)$. The third term is bounded because $u\in C^s(\R^n)$ and $\eta\in {\rm Lip}(\R^n)$.
Therefore, $\tilde u$ satisfies the hypotheses of Proposition \[intparts\] with $\Omega$ replaced by $\tilde\Omega$, and follows taking into account that for all $x_0\in \partial \tilde\Omega\cap {\rm supp}\, \tilde u = \partial \Omega\cap {\rm supp}\, \tilde u$ we have $$\lim_{x\to x_0,\, x\in\tilde\Omega} \frac{\tilde u (x)}{{\tilde \delta}^s(x)}=\lim_{x\to x_0,\, x\in\Omega} \frac{\tilde u (x)}{\delta^s(x)}.$$
We now give the
Let $B_1,...,B_m$ be balls of radius $r>0$ covering $\overline\Omega$. By regularity of the domain, if $r$ is small enough, for each $i,j$ such that $\overline{B_i}\cap\overline{B_j}\neq\varnothing$ there exists a ball $B$ containing $B_i\cup B_j$ and a point $z_0\in \Omega\cap B$ such that $$(x-z_0)\cdot\nu(x)>0\qquad\mbox{for all}\ x\in\partial\Omega\cap B.$$
Let $\{\psi_k\}_{k=1,...,m}$ be a partition of the unity subordinated to $B_1,...,B_m$, that is, a set of smooth functions $\psi_1,...,\psi_m$ such that $\psi_1+\cdots+\psi_m=1$ in $\Omega$ and that $\psi_k$ has compact support in $B_k$ for each $k=1,...,m$. Define $u_k=u\psi_k$.
Now, for each $i,j\in\{1,...,m\}$, if $\overline{B_i}\cap\overline{B_j}=\varnothing$ we use Lemma \[duesboles\], while if $\overline{B_i}\cap\overline{B_j}\neq\varnothing$ we use Lemma \[unabola\]. We obtain $$\begin{split}\int_\Omega(x\cdot\nabla u_i)&(-\Delta)^su_j\, dx+\int_\Omega(x\cdot\nabla u_j)(-\Delta)^su_i\,dx=\frac{2s-n}{2}\int_{\Omega}u_i(-\Delta)^su_j\, dx+\\
&+\frac{2s-n}{2}\int_{\Omega}u_j(-\Delta)^su_i\, dx
-\Gamma(1+s)^2\int_{\partial\Omega}\frac{u_i}{\delta^{s}}\frac{u_j}{\delta^{s}}(x\cdot\nu)\, d\sigma\end{split}$$ for each $1\leq i\leq m$ and $1\leq j\leq m$. Therefore, adding these identities for $i=1,...,m$ and for $j=1,...,m$ and taking into account that $u_1+\cdots+u_m=u$, we find $$\int_\Omega(x\cdot\nabla u)(-\Delta)^su\, dx=\frac{2s-n}{2}\int_{\Omega}u(-\Delta)^su\, dx-\frac{\Gamma(1+s)^2}{2}\int_{\partial\Omega}\left(\frac{u}{\delta^s}\right)^2(x\cdot\nu)\,d\sigma,$$ and the proposition is proved.
To end this section we prove Theorem \[thpoh\], Proposition \[proppoh\], Theorem \[corintparts\], and Corollaries \[cornonexistence\], \[cornonexistence2\], and \[cornonexistence3\].
By Theorem \[krylov\], any solution $u$ to problem satisfies the hypothesis of Proposition \[intparts\]. Hence, using this proposition and that $(-\Delta)^su=f(x,u)$, we obtain $$\int_{\Omega}(\nabla u\cdot x)f(x,u)dx=\frac{2s-n}{2}\int_\Omega uf(x,u)dx+\frac{\Gamma(1+s)^2}{2}\int_{\partial\Omega}\left(\frac{u}{\delta^s}\right)^2(x\cdot\nu) d\sigma.$$
On the other hand, note that $(\nabla u\cdot x)f(x,u)=\nabla \left(F(x,u)\right)\cdot x-x\cdot F_x(x,u)$. Then, integrating by parts, $$\int_{\Omega}(\nabla u\cdot x)f(x,u)dx=-n\int_\Omega F(x,u)dx-\int_\Omega x\cdot F_x(x,u)dx.$$ If $f$ does not depend on $x$, then the last term do not appear, as in Theorem \[thpoh\].
As shown in the final part of the proof of Proposition \[intparts\] for strictly star-shaped domains given in Section \[sec2\], the freedom for choosing the origin in the identity from this proposition leads to $$\int_\Omega w_{x_i}(-\Delta)^sw\ dx=\frac{\Gamma(1+s)^2}{2}\int_{\partial\Omega}\left(\frac{w}{\delta^s}\right)^2\nu_i\ d\sigma$$ for each $i=1,...,n$. Then, the theorem follows by using this identity with $w=u+v$ and with $w=u-v$ and subtracting both identities.
We only have to prove Corollary \[cornonexistence3\], since Corollaries \[cornonexistence\] and \[cornonexistence2\] follow immediately from it by setting $f(x,u)=f(u)$ and $f(x,u)=|u|^{p-1}u$ respectively.
By hypothesis , we have $$\frac{n-2s}{2}\int_\Omega uf(x,u)dx\geq n\int_\Omega F(x,u)dx+\int_\Omega x\cdot F_x(x,u)dx.$$ This, combined with Proposition \[proppoh\] gives $$\int_{\partial\Omega}\left(\frac{u}{\delta^s}\right)^2(x\cdot \nu)d\sigma\leq 0.$$ If $\Omega$ is star-shaped and inequality in is strict, we obtain a contradiction. On the other hand, if inequality in is not strict but $u$ is a positive solution of , then by the Hopf Lemma for the fractional Laplacian (see, for instance, [@CRS] or Lemma 3.2 in [@RS]) the function $u/\delta^s$ is strictly positive in $\overline\Omega$, and we also obtain a contradiction.
Calculation of the constants $c_1$ and $c_2$
============================================
In Proposition \[prop:Lap-s/2-delta-s\] we have obtained the following expressions for the constants $c_1$ and $c_2$: $$c_1=c_{1,\frac s2},\qquad{\rm and}\qquad c_2=\int_0^{\infty}\left\{\frac{1-x^s}{|1-x|^{1+s}}+\frac{1+x^s}{|1+x|^{1+s}}\right\}dx,$$ where $c_{n,s}$ is the constant appearing in the singular integral expression for $(-\Delta)^{s}$ in dimension $n$.
Here we prove that the values of these constants coincide with the ones given in Proposition \[thlaps/2\]. We start by calculating $c_1$.
Let $c_{n,s}$ be the normalizing constant of $(-\Delta)^{s}$ in dimension $n$. Then, $$c_{1,\frac s2}=\frac{\Gamma(1+s)\sin\left(\frac{\pi s}{2}\right)}{\pi}.$$
Recall that $$\label{cns}
c_{n,s}=\frac{s2^{2s}\Gamma\left(\frac{n+2s}{2}\right)}{\pi^{n/2}\Gamma(1-s)}.$$ Thus, $$c_{1,\frac{s}{2}}=\frac{s2^{s-1}\Gamma\left(\frac{1+s}{2}\right)}{\sqrt{\pi}\Gamma\left(1-\frac{s}{2}\right)}.$$ Now, using the properties of the Gamma function (see for example [@AAR]) $$\Gamma(z)\Gamma\left(z+\frac 12\right)=2^{1-2z}\sqrt\pi\Gamma(2z)\qquad \textrm{and}\qquad\Gamma(z)\Gamma(1-z)=\frac{\pi}{\sin(\pi z)},$$ we obtain $$c_{1,\frac{s}{2}}=\frac{s2^{s-1}}{\sqrt{\pi}}\cdot\frac{\Gamma\left(\frac{1+s}{2}\right)\Gamma\left(\frac s2\right)}{\Gamma\left(1-\frac{s}{2}\right)\Gamma\left(\frac s2\right)}=\frac{s 2^{s-1}}{\sqrt{\pi}}\cdot\frac{2^{1-s}\sqrt\pi \Gamma(s)}{\pi/\sin\left(\frac{\pi s}{2}\right)}=\frac{s\Gamma(s)\sin\left(\frac{\pi s}{2}\right)}{\pi}.$$ The result follows by using that $z\Gamma(z)=\Gamma(1+z)$.
Let us now compute the constant $c_2$.
\[constc2\] Let $0<s<1$. Then, $$\int_0^\infty \left\{\frac{1-x^s}{|1-x|^{1+s}}+\frac{1+x^s}{|1+x|^{1+s}}\right\}dx=\frac{\pi}{\tan\left(\frac{\pi s}{2}\right)}.$$
For it, we will need some properties of the hypergeometric function $\,_2F_1$, which we prove in the next lemma. Recall that this function is defined as $$\,_2F_1(a,b;c;z)=\sum_{n\geq0}\frac{(a)_n(b)_n}{(c)_n}\frac{z^n}{n!}\qquad\textrm{for }\ |z|<1,$$ where $(a)_n=a(a+1)\cdots(a+n-1)$, and by analytic continuation in the whole complex plane.
\[hypergeom\] Let $\,_2F_1(a,b;c;z)$ be the ordinary hypergeometric function, and $s\in\mathbb R$. Then,
- For all $z\in\mathbb C$, $$\frac{d}{dz}\left\{\frac{z^{s+1}}{s+1}\,_2F_1(1+s,1+s;2+s;z)\right\}=\frac{z^s}{(1-z)^{1+s}}.$$
- If $s\in(0,1)$, then $$\lim_{x\rightarrow 1}\left\{\frac{1}{s+1}\,_2F_1(1+s,1+s;2+s;x)-\frac{1}{s(1-x)^s}\right\}=-\frac{\pi}{\sin(\pi s)}.$$
- If $s\in(0,1)$, then $$\lim_{x\rightarrow +\infty}\left\{\frac{(-x)^{s+1}}{s+1}
\,_2F_1(1\hspace{-0.5mm}+\hspace{-0.5mm}s,1\hspace{-0.5mm}+\hspace{-0.5mm}s;
2\hspace{-0.5mm}+\hspace{-0.5mm}s;x)
-\frac{x^{s+1}}{s+1}
\,_2F_1(1\hspace{-0.5mm}+\hspace{-0.5mm}s,1\hspace{-0.5mm}+\hspace{-0.5mm}s;2\hspace{-0.5mm}+\hspace{-0.5mm}s;
-x)\right\}=i \pi,$$ where the limit is taken on the real line.
\(i) Let us prove the equality for $|z|<1$. In this case, $$\begin{split} \frac{d}{dz}\biggl\{\frac{z^{s+1}}{s+1}\,_2F_1&(1+s,1+s;2+s;z)\biggr\} =\frac{d}{dz}\sum_{n\geq0}\frac{(1+s)^2_n}{(2+s)_n}\frac{z^{n+1+s}}{n!(s+1)}=\\
&= \sum_{n\geq0}\frac{(1+s)_n}{n!}z^{n+s}= z^s\sum_{n\geq0}{-1-s \choose n}(-z)^n= z^s(1-z)^{-1-s},
\end{split}$$ where we have used that $(2+s)_n=\frac{n+1+s}{1+s}(1+s)_n$ and that $\frac{(a)_n}{n!}=(-1)^n{-a \choose n}$. Thus, by analytic continuation the identity holds in $\mathbb C$.
\(ii) Recall the Euler transformation (see for example [@AAR]) $$\label{euler}\,_2F_1(a,b;c;x)=(1-x)^{c-a-b}\,_2F_1(c-a,c-b;c;x),$$ and the value at $x=1$ $$\label{2F1at1}\,_2F_1(a,b;c;1)=\frac{\Gamma(c)\Gamma(c-a-b)}{\Gamma(c-a)\Gamma(c-b)}\qquad \textrm{whenever}\qquad a+b<c.$$ Hence, $$\frac{1}{s+1}\,_2F_1(1+s,1+s;2+s;x)-\frac{1}{s(1-x)^s}=\frac{\frac{1}{s+1}\,_2F_1(1,1;2+s;x)-\frac1s}{(1-x)^s},$$ and we can use l’Hôpital’s rule, $$\begin{aligned}
\lim_{x\rightarrow 1}\frac{\frac{1}{s+1}\,_2F_1(1,1;2+s;x)-\frac1s}{(1-x)^s}&=&
\lim_{x\rightarrow 1}\frac{\frac{1}{s+1}\frac{d}{dx}\,_2F_1(1,1;2+s;x)}{-s(1-x)^{s-1}}\\
&=&-\lim_{x\rightarrow 1}\frac{(1-x)^{1-s}}{s(s+1)(s+2)}\,_2F_1(2,2;3+s;x)\\
&=&-\lim_{x\rightarrow 1}\frac{1}{s(s+1)(s+2)}\,_2F_1(1+s,1+s;3+s;x)\\
&=&-\frac{1}{s(s+1)(s+2)}\,_2F_1(1+s,1+s;3+s;1)\\
&=&-\frac{1}{s(s+1)(s+2)}\frac{\Gamma(3+s)\Gamma(1-s)}{\Gamma(2)\Gamma(2)}\\
&=&-\Gamma(s)\Gamma(1-s)\\
&=&-\frac{\pi}{\sin(\pi s)}.\end{aligned}$$ We have used that $$\frac{d}{dx}\,_2F_1(1,1;2+s;x)=\frac{1}{s+2}\,_2F_1(2,2;3+s;x),$$ the Euler transformation , and the properties of the $\Gamma$ function $$x\Gamma(x)=\Gamma(x+1),\qquad \Gamma(x)\Gamma(1-x)=\frac{\pi}{\sin(\pi x)}.$$
\(iii) In [@B] it is proved that $$\label{ramanujan}\frac{\Gamma(a)\Gamma(b)}{\Gamma(a+b)}\,_2F_1(a,b;a+b;x)=
\log\frac{1}{1-x}+R+o(1)\qquad \textrm{for}\ \ x\sim 1,$$ where $$R=-\psi(a)-\psi(b)-\gamma,$$ $\psi$ is the digamma function, and $\gamma$ is the Euler-Mascheroni constant. Using the Pfaff transformation [@AAR] $$\,_2F_1(a,b;c;x)=(1-x)^{-a}\,_2F_1\left(a,c-b;c;\frac{x}{x-1}\right)$$ and , we obtain $$\begin{aligned}
\frac{(1-x)^{1+s}}{1+s}\,_2F_1(1+s,1+s;2+s;x)
&=&\frac{1}{1+s}\,_2F_1\left(1+s,1;2+s;\frac{x}{x-1}\right)\\
&=&\log\frac{1}{1-x}+R+o(1)\ \ \ \textrm{for }\ x\sim \infty.\end{aligned}$$ Thus, it also holds $$\frac{(-x)^{1+s}}{1+s}\,_2F_1(1+s,1+s;2+s;x)=\log\frac{1}{1-x}+R+o(1)\ \ \ \textrm{for }\ x\sim \infty,$$ and therefore the limit to be computed is now $$\lim_{x\rightarrow +\infty}\left\{\left(\log\frac{1}{1-x}+R\right)-\left(\log\frac{1}{1+x}+R\right)\right\}=i \pi.$$
Next we give the:
Let us compute separately the integrals $$I_1=\int_0^1\left\{\frac{1-x^s}{|1-x|^{1+s}}+\frac{1+x^s}{|1+x|^{1+s}}\right\}dx$$ and $$I_2=\int_1^\infty \left\{\frac{1-x^s}{|1-x|^{1+s}}+\frac{1+x^s}{|1+x|^{1+s}}\right\}dx.$$ By Lemma \[hypergeom\] (i), we have that $$\int\left\{\frac{1-x^s}{(1-x)^{1+s}}+\frac{1+x^s}{(1+x)^{1+s}}\right\}dx=
\frac{1}{s}(1-x)^{-s}-\frac{x^{s+1}}{s+1}\,_2F_1(1+s,1+s;2+s;x)$$ $$-\frac{1}{s}(1+x)^{-s}+\frac{x^{s+1}}{s+1}\,_2F_1(1+s,1+s;2+s;-x).$$ Hence, using \[hypergeom\] (ii), $$I_1=\frac{\pi}{\sin(\pi s)}-\frac{1}{s 2^s}+\frac{1}{s+1}\,_2F_1(1+s,1+s;2+s;-1).$$ Let us evaluate now $I_2$. As before, by Lemma \[hypergeom\] (i), $$\int\left\{\frac{1-x^s}{(x-1)^{1+s}}+\frac{1+x^s}{(x+1)^{1+s}}\right\}dx=
\frac{1}{s}(x-1)^{-s}+(-1)^s\frac{x^{s+1}}{s+1}\,_2F_1(1+s,1+s;2+s;x)$$ $$-\frac{1}{s}(1+x)^{-s}+\frac{x^{s+1}}{s+1}\,_2F_1(1+s,1+s;2+s;-x).$$ Hence, using \[hypergeom\] (ii) and (iii), $$\begin{aligned}
I_2&=&-i \pi+(-1)^s\frac{\pi}{\sin(\pi s)}+\frac{1}{s 2^s}-\frac{1}{s+1}\,_2F_1(1+s,1+s;2+s;-1)\\
&=&-i\pi+\cos(\pi s)\frac{\pi}{\sin(\pi s)}+i\sin(\pi s)\frac{\pi}{\sin(\pi s)}+\\ &&\qquad\qquad\qquad+\frac{1}{s 2^s}-\frac{1}{s+1}\,_2F_1(1+s,1+s;2+s;-1)\\
&=&\frac{\pi}{\tan(\pi s)}+\frac{1}{s 2^s}-\frac{1}{s+1}\,_2F_1(1+s,1+s;2+s;-1).\end{aligned}$$ Finally, adding up the expressions for $I_1$ and $I_2$, we obtain $$\begin{aligned}
\int_0^\infty \left\{\frac{1-x^s}{|1-x|^{1+s}}+\frac{1+x^s}{|1+x|^{1+s}}\right\}dx&=&\frac{\pi}{\sin(\pi s)}+\frac{\pi}{\tan(\pi s)}=
\pi\cdot\frac{1+\cos(\pi s)}{\sin(\pi s)}\\
&=&\pi\cdot\frac{2\cos^2\left(\frac{\pi s}{2}\right)}{2\sin\left(\frac{\pi s}{2}\right)\cos\left(\frac{\pi s}{2}\right)}=\frac{\pi}{\tan\left(\frac{\pi s}{2}\right)},\end{aligned}$$ as desired.
\[A4\] It follows from Proposition \[propoperador\] that the constant appearing in (and thus in the Pohozaev identity), $\Gamma(1+s)^2$, is given by $$c_3=c_1^2(\pi^2+c_2^2).$$ We have obtained the value of $c_3$ by computing explicitly $c_1$ and $c_2$. However, an alternative way to obtain $c_3$ is to exhibit an explicit solution of for some nonlinearity $f$ and apply the Pohozaev identity to this solution. For example, when $\Omega=B_1(0)$, the solution of $$\left\{ \begin{array}{rcll} (-\Delta)^s u &=&1&\textrm{in }B_1(0) \\
u&=&0&\textrm{in }\mathbb R^n\backslash B_1(0)\end{array}\right.$$ can be computed explicitly [@G; @BGR]: $$\label{eq:explicit-sol-ball}
u(x)=\frac{2^{-2s}\Gamma(n/2)}{\Gamma\left(\frac{n+2s}{2}\right)\Gamma(1+s)}\left(1-|x|^2\right)^s.$$ Thus, from the identity $$\label{jfk}(2s-n)\int_{B_1(0)}u\ dx+2n\int_{B_1(0)}u\ dx=c_3\int_{\partial B_1(0)}\left(\frac{u}{\delta^s}\right)^2(x\cdot\nu)d\sigma$$ we can obtain the constant $c_3$, as follows.
On the one hand, $$\begin{aligned}
\int_{B_1(0)}u\ dx&=&\frac{2^{-2s}\Gamma(n/2)}{\Gamma\left(\frac{n+2s}{2}\right)\Gamma(1+s)}\int_{B_1(0)}\left(1-|x|^2\right)^sdx\\
&=&\frac{2^{-2s}\Gamma(n/2)}{\Gamma\left(\frac{n+2s}{2}\right)\Gamma(1+s)}|S^{n-1}|\int_0^1r^{n-1}(1-r^2)^sdr\\
&=&\frac{2^{-2s}\Gamma(n/2)}{\Gamma\left(\frac{n+2s}{2}\right)\Gamma(1+s)}|S^{n-1}|\frac12\int_0^1r^{n/2-1}(1-r)^sdr\\
&=&\frac{2^{-2s}\Gamma(n/2)}{\Gamma\left(\frac{n+2s}{2}\right)\Gamma(1+s)}|S^{n-1}|\frac12\frac{\Gamma(n/2)\Gamma(1+s)}{\Gamma(n/2+1+s)},\end{aligned}$$ where we have used the definition of the Beta function $$B(a,b)=\int_0^1t^{a-1}(1-t)^{b-1}dt$$ and the identity $$B(a,b)=\frac{\Gamma(a)\Gamma(b)}{\Gamma(a+b)}.$$ On the other hand, $$\begin{aligned}
\int_{\partial B_1(0)}\left(\frac{u}{\delta^s}\right)^2(x\cdot\nu)d\sigma
&=&\left(\frac{2^{-2s}\Gamma(n/2)}{\Gamma\left(\frac{n+2s}{2}\right)\Gamma(1+s)}\right)^2|S^{n-1}|2^{2s}.\end{aligned}$$
Thus, is equivalent to $$(n+2s)\frac{2^{-2s}\Gamma(n/2)}{\Gamma\left(\frac{n+2s}{2}\right)\Gamma(1+s)} \frac12\frac{\Gamma(n/2)\Gamma(1+s)}{\Gamma(n/2+1+s)}=
c_3\left(\frac{2^{-2s}\Gamma(n/2)}{\Gamma\left(\frac{n+2s}{2}\right)\Gamma(1+s)}\right)^22^{2s}.$$ Hence, after some simplifications, $$c_3=\frac{\Gamma(1+s)^2}{\Gamma(n/2+1+s)}\frac{n+2s}{2}\Gamma\left(\frac{n+2s}{2}\right),$$ and using that $$z\Gamma(z)=\Gamma(1+z)$$ one finally obtains $$c_3=\Gamma(1+s)^2,$$ as before.
Acknowledgements {#acknowledgements .unnumbered}
================
The authors thank Xavier Cabré for his guidance and useful discussions on the topic of this paper.
[00]{}
G. E. Andrews, R. Askey, R. Roy, *Special Functions*, Cambridge University Press, 2000.
B. C. Berndt, *Ramanujan’s Notebooks, Part II*, Springer-Verlag, 1989.
K. Bogdan, T. Grzywny, M. Ryznar, *Heat kernel estimates for the fractional Laplacian with Dirichlet conditions*, Ann. of Prob. 38 (2010), 1901-1923.
C. Brandle, E. Colorado, A. de Pablo, *A concave-convex elliptic problem involving the fractional laplacian*, To appear in Proc. Roy. Soc. Edinburgh Sect. A.
X. Cabré, E. Cinti, *Sharp energy estimates for nonlinear fractional diffusion equations*, preprint.
X. Cabré, J. Tan, *Positive solutions of nonlinear problems involving the square root of the Laplacian*, Adv. Math. 224 (2010), 2052-2093.
L. Caffarelli, J. M. Roquejoffre, Y. Sire, *Variational problems in free boundaries for the fractional Laplacian*, J. Eur. Math. Soc. 12 (2010), 1151-1179.
L. Cafarelli, L. Silvestre, *Regularity theory for fully nonlinear integro-differential equations*, Comm. Pure Appl. Math. 62 (2009), 597-638.
L. Caffarelli, L. Silvestre, *An extension problem related to the fractional Laplacian*, Comm. Partial Differential Equations 32 (2007), 1245-1260.
W. Chen, C. Li, B. Ou, *Classification of solutions to an integral equation*, Comm. Pure Appl. Math. 59 (2006), 330-343.
L. C. Evans, R. F. Gariepy, *Measure Theory And Fine Properties Of Functions*, Studies in Advanced Mathematics, CRC Press, 1992.
M. M. Fall, T. Weth, *Nonexistence results for a class of fractional elliptic boundary value problems*, arXiv:1201.4007v1.
R. K. Getoor, *First passage times for symmetric stable processes in space*, Trans. Amer. Math. Soc. 101 (1961), 75–90.
D. Gilbarg, N. S. Trudinger, *Elliptic Partial Differential Equations Of Second Order*, Grundlehren der Mathematischen Wissenschaften, Vol. 224. Springer-Verlag, Berlin-New York, 1977.
A. de Pablo, U. Sánchez, *Some Liouville-type results for a fractional equation*, preprint.
S. I. Pohozaev, *On the eigenfunctions of the equation $\Delta u + \lambda f(u) = 0$*, Dokl. Akad. Nauk SSSR 165 (1965), 1408-1411.
X. Ros-Oton, J. Serra, *Fractional Laplacian: Pohozaev identity and nonexistence results*, C. R. Math. Acad. Sci. Paris 350 (2012), 505-508.
X. Ros-Oton, J. Serra, *The Dirichlet problem for the fractional Laplacian: regularity up to the boundary*, preprint arXiv, 2012.
R. Servadei, E. Valdinoci, *Mountain pass solutions for non-local elliptic operators*, J. Math. Anal. Appl. 389 (2012), 887-898.
[^1]: The authors were supported by grants MTM2008-06349-C03-01, MTM2011-27739-C04-01 (Spain), and 2009SGR345 (Catalunya)
|
---
abstract: 'Quantum key distribution (QKD) has been proved to be information-theoretically secure in theory. Unfortunately, the imperfect devices in practice compromise its security. Thus, to improve the security property of practical QKD systems, a commonly used method is to patch the loopholes in the existing QKD systems. However, in this work, we first time show an adversary’s capability of exploiting the new feature of the originally flawed component to bypass a patch. Specifically, we experimentally demonstrate that the patch of photocurrent monitor against the detector blinding attack can be defeated by the pulse illumination attack proposed in this paper. We also analyze the secret key rate under the pulse illumination attack, which theoretically confirmed that Eve can conduct the attack to learn the secret key. This work indicates the importance of inspecting the security loopholes in a detection unit to further understand their impacts on a QKD system. The method of pulse illumination attack can be a general testing item in the security evaluation standard of QKD.'
author:
- Zhihao Wu
- Anqi Huang
- Huan Chen
- 'Shi-Hai Sun'
- Jiangfang Ding
- Xiaogang Qiang
- Xiang Fu
- Ping Xu
- Junjie Wu
bibliography:
- 'paper.bib'
title: 'Hacking single-photon avalanche detector in quantum key distribution via pulse illumination'
---
Introduction
============
Information security is a core of cybersecurity in the digital era. Cryptography provides a vital tool to achieve information security in cyber environment, particularly when untrusted channels are used. However, as shown in the history of cryptography, the competition between code makers and code breakers never ended. For example, the current widely-used public-key cryptographic infrastructure is threaten by a quantum computer [@shor1997]. To defeat the threat from the quantum world, quantum key distribution (QKD) [@bennett1984] based on the laws of quantum mechanics provides a long-term solution, which has been proved its information-theoretical security. The key generated via QKD can be applied to one-time-pad algorithm, guaranteeing information-theoretically secure communication.
Due to plenty of efforts, QKD has developed with a fast pace even to be globalized and commercialized, becoming one of the most mature applications in the field of quantum information [@takesue2007; @scheidl2009; @sibson17; @ding2017; @eriksson2019]. For example, QKD networks have been built in several countries [@peev2009; @sasaki2011; @backbone], and even extend to global scale via a quantum satellite as a trusted relay [@canada_sat; @wang2013; @liao2017; @bedington2017]. Some enterprises have developed various commercial products based on QKD, including mobile phone, quantum safe service-mobile engine, and quantum security gateway [@products]. Owing to such fast development, standardization of QKD is being considered in the European Telecommunications Standards Institute (ETSI) [@ISGofETSI], the International Standard Orgnization (ISO) [@StandardsNews], and the International Telecommunication Union (ITU).
However, a practical QKD system may behave differently from its theoretical model, due to the discrepancies between ideal theory and imperfect practice. These discrepancies disclose security loopholes that can be exploited by an eavesdropper, Eve, to learn the secret key, compromising the practical security of QKD systems [@brassard2000; @makarov2006; @qi2007; @lamas-linares2007; @lamas-linares2007; @nauerth2009; @lydersen2010a; @lydersen2010b; @xu2010; @wiechers2011; @lydersen2011c; @lydersen2011b; @gerhardt2011; @sun2011; @jain2011; @tang2013; @bugge2014; @sajeed2015; @sajeed2015a; @huang2016; @sajeed2016; @makarov2016; @huang2018; @huang2018a; @huang2019; @chistiakov2019; @gras2019; @chaiwongkhot2019]. To defeat quantum hacking, an effective countermeasure is to employ an innovative protocol, like measurement-device-independent QKD (MDI QKD) [@lo2012] and twin-field QKD (TF QKD) [@lucamarini2018; @curty2019], to remove the threat from loopholes. However, the high demand of technique and relative low key rate of the innovative protocols limit their application and commercialization. Therefore, to relieve the security threat on the QKD systems in use, it is essential to patch the loopholes in the existing QKD systems [@Koehler2018; @Koehler2018a], most of which employ prepare-and-measure QKD protocol.
As countermeasures, patches, instead of ending the hacking story, lead to a new era of quantum hacking – inspire quantum attackers to conduct a new round of hacking investigation on the patched system. The quantum hackers usually take the following three strategies to counter the countermeasure. First, because a patch designer underestimates the threat of the original security loophole, Eve can apply the attack that uses the original loophole to crack the weakly patched system [@huang2016]. Second, patches challenge Eve to propose a new attack on the patches to de-functionalize them or even damage them [@sajeed2015; @makarov2016]. Third, patches motivate Eve to discover the new feature of the flawed component to attack the patched system again, which reinforces Eve’s hacking capability and shows a more general attack. Our work that bypasses a countermeasure against detector blinding attack, the photocurrent monitor, first time provides such an instance.
The detector blinding attack [@lydersen2010a; @lydersen2010b; @lydersen2011c; @gerhardt2011; @huang2016] is a quite powerful attack, which can be realized by today’s technology on commercial QKD systems [@lydersen2010a] and has been demonstrated in the full field [@gerhardt2011]. Thus, this security threat has caused much attention to propose possible countermeasures [@yuan2010; @yuan2011; @yuan2011a; @silva2012; @lim2015]. A countermeasure that monitors the abnormal photocurrent going through the avalanche photodiode (APD) as the evidence of blinding attack was believed to be effective [@yuan2010; @yuan2011]. However, no independent third party has evaluated this countermeasure. In this paper, as a third-party evaluator, we show experimentally that an APD with the countermeasure of a photocurrent monitor can be hacked again. We discover that bright *pulses* can blind APD intermittently and meanwhile bypass the alarm of photocurrent monitor, although reasonable qubit error rate (QBER) is introduced. We call this method as *pulse illumination attack*.
The pulse illumination attack is a more general type of detector blinding attack than the original one that uses continuous wave (c.w.) light. In this attack, Eve does not only exploit the mode switch between the linear mode and Geiger mode via light shining, but also makes refined use of the hysteresis current after pulse shining to create a period of full blinding and detection control. This pulse illumination attack shows a novel strategy of hacking the countermeasure – exploit the new feature of the original flawed components (the APD in our case). The deep investigation on the loopholes strengthens Eve’s hacking capability, and thus passes the challenge of protecting the QKD system to countermeasure proposers again. This study emphasizes the significance of further investigating the imperfections of single-photon detector, in order to improve the security property of the standard prepare-and-measure QKD systems that are deployed the most in field [@peev2009; @sasaki2011; @liao2017]. Moreover, this study contributes a general testing item to the security certification list of QKD standard, which is being drafted in several international organizations, e.g. ETSI, ISO, and ITU [@ISGofETSI; @ETSIwhitepaper; @StandardsNews].
From c.w. illumination attack to pulse illumination attack {#sec:theory}
==========================================================
In this section, we briefly review the origin blinding attack, i.e., c.w. illumination attack, and then introduce the countermeasure of the photocurrent monitor that is believed to be effective to blinding attacks. Finally, we propose pulse illumination attack as a new form of blinding attack that can bypass the photocurrent monitor.
For a BB84 QKD system, Eve can apply the c.w. blinding attack on APDs to eavesdrop the information. In the original c.w. blinding attack, Eve injects continuous light to generate a huge photocurrent through APD, which lowers the bias voltage and pulls the APD back to the linear mode that is insensitive to a single photon. Then, she conducts a fake-state attack as follows. She first intercepts and measures each state sent by Alice, and resends a trigger pulse encoded by her measurement result to control Bob’s clicks as the same as hers. For more details about the original c.w. blinding attack, see Supplementary \[subsec:c.w.\].
To patch the loophole exploited by the blinding attack, some QKD systems including our testing object in this work adopt a photocurrent monitor as a countermeasure against the blinding attack [@lydersen2010a; @huang2016]. This countermeasure bases on an intuitive assumption that a blinding attack will certainly generate a distinguishable low-frequency photocurrent in the circuit of the detector. The monitor extracts the low-frequency photocurrent as an alarm of the blinding attack. Once the extracted photocurrent reaches an alarming threshold, the blinding attack is considered to be launched. Please note that the extracted photocurrent is named as reported photocurrent in the following text.
However, we find that this countermeasure can not completely hinder the pulse illumination attack. This is because the aforementioned assumption about the photocurrent under a blinding attack does not stand when optical pulses are sent to blind an APD. In this attack, a group of blinding pulses accumulatively introduces a high photocurrent. This photocurrent is also able to lower the bias voltage across the APD. As a result, the detector is blinded at that time. After the blinding pulses are gone, the detector is still blinded for a certain period, because the photocurrent gradually reduces due to capacitors in the detector. Thus the detector keeps being blinded until the photocurrent becomes fairly weak. Eve can exploit this blinded period to launch the fake-state attack to eavesdrop the information. Theoretically, the length of the blinded period is positively correlated with the energy of the blinding-pulse group.
Moreover, unlike the constant high photocurrent introduced by the c.w. illumination attack, here the photocurrent varies over time. The photocurrent monitor takes this current as high-frequency noise and ignores most of it. Therefore, Eve can apply the pulse illumination attack to eavesdrop the information without being noticed by the photocurrent monitor.
\#1
(-6mm,-4.5mm) rectangle ++(2mm,0.5mm); (6mm,-4.5mm) rectangle ++(-2mm,0.5mm); (-6.5mm,-4mm) rectangle ++(13mm,8mm); (-6.25mm,-3.75mm) rectangle ++(12.5mm,7.5mm);
(-6mm,-0.5mm) rectangle (3mm,3mm); (-5.5mm,0mm) rectangle (2.5mm,2.5mm); (-10pt, 3pt) to (4pt, 3pt) to (5pt, 5pt) to (6pt, 3pt) to (10pt, 3pt); (4.9mm,1.5mm) circle \[radius=0.6mm\]; (5.15mm,1.6mm) circle \[radius=0.2mm\]; (3.5mm,0mm) to (6.3mm, 0mm); in [-5mm,-3mm, -1mm, 1mm]{} [ (,-1mm) rectangle (+1mm, -1.5mm); (,-2mm) rectangle (+1mm, -2.5mm); ]{} (4.9mm,-1.5mm) ellipse \[x radius=0.6mm, y radius=0.3mm\]; (4.5mm,4.1mm) rectangle (5.5mm,4.7mm); (6.5mm,1mm) rectangle (7.2mm,2mm); (6.5mm,-1mm) rectangle (7.2mm,-2mm); (\#1\_c1) at (7.2mm, 1.5mm); (\#1\_c2) at (7.2mm, -1.5mm); (\#1\_trg) at (5mm, 4.7mm); ;
\#1
(-6.5mm,-4.5mm) rectangle ++(2mm,0.5mm); (6.5mm,-4.5mm) rectangle ++(-2mm,0.5mm); (-7mm,-4mm) rectangle ++(14mm,8mm); (-6.75mm,-3.75mm) rectangle ++(13.5mm,7.5mm); (-6mm,-0.5mm) rectangle (3mm,3mm); (-5.5mm,0mm) rectangle (2.5mm,2.5mm); ; (4.9mm,1.5mm) circle \[radius=0.6mm\]; (5.15mm,1.6mm) circle \[radius=0.2mm\]; (3.5mm,0mm) to (6.3mm, 0mm); in [-5mm,-3mm, -1mm, 1mm]{} [ (,-1mm) rectangle (+1mm, -1.5mm); (,-2mm) rectangle (+1mm, -2.5mm); ]{} (4.9mm,-1.5mm) ellipse \[x radius=0.6mm, y radius=0.3mm\]; (4.5mm,4.1mm) rectangle (5.5mm,4.7mm); (7.1mm,1mm) rectangle (7.7mm,2mm); (7.1mm,-1mm) rectangle (7.7mm,-2mm);
\#1
(\#1) [DSG]{};
\#1\#2[ (\#2) [\#1]{}; ]{}
\#1\#2[ (\#2) [\#1]{}; ]{} \#1[ (\#1) [MVA]{}; ]{} \#1[ (\#1) [DVA]{}; ]{} \#1[ (\#1) ; in [-6mm,-3mm,-0mm, 3mm]{} (,0mm) -| ([+1mm]{}, 3mm) -| ([+2mm]{}, 0mm) – ([+3mm]{}, 0mm) ; ]{} \#1
(\#1) ;
(-6mm,0mm) – (-0.5mm, 0mm) ; (0.5mm,0mm) – (6mm, 0mm) ; (-1.5mm,0mm) -| (-0.5mm, 1mm) -| (0.5mm, 0mm) – ([1.5mm+1mm]{}, 0mm) ;
\#1
(\#1) ;
(-3mm,-1.5mm) to (-3mm, 1.5mm) ; (-3mm,-1.5mm) to (-3mm, 1.5mm) ; (-3mm,-1.5mm) to (-3mm, 1.5mm) ; (-4.5mm, 0mm) to (4.5mm, 0mm) ;
\#1
(\#1) ;
(-3mm,-1.5mm) to (-3mm, 1.5mm) ; (-3mm,-1.5mm) to (-3mm, 1.5mm) ; (-3mm,-1.5mm) to (-3mm, 1.5mm) ; (-4.5mm, 0mm) to (4.5mm, 0mm) ;
\#1[ (\#1\_west) at (-2mm, 0); (\#1\_east) at (2mm, 0); at (0,-2mm) [50:50]{}; (-2mm,-0.5mm) rectangle +(4mm, 1mm); ]{} \#1[ (\#1\_west) at (-2mm, 0); (\#1\_east) at (2mm, 0); at (0,-2mm) [50:50]{}; (\#1) ; (-2mm,-0.5mm) rectangle +(4mm, 1mm); (-2mm,0mm) to \[out=180, in=-60\] (-4mm, 2mm); (-2mm,0mm) to \[out=180, in=60\] (-4mm, -2mm); (2mm,0mm) to \[out=0, in=240\] (4mm, 2mm); (2mm,0mm) to \[out=0, in=120\] (4mm, -2mm); ]{} \#1[ (\#1) [APD]{}; (-4.5mm,-1.5mm) arc \[start angle=-90, end angle=90, radius=1.5mm\] – cycle; ]{} \#1[ (-9mm,-2mm) arc \[start angle=-90, end angle=90, radius=2mm\] – cycle; at(2mm,0)[APD’s active area]{}; ]{} \#1[ (\#1\_west) at (-4.5mm, 0) ; (\#1\_east) at (-3mm, 0); (-4.5mm,-2mm) arc \[start angle=-90, end angle=90, radius=2mm\] – cycle; ]{} \#1[ (-9mm,-2mm) arc \[start angle=-90, end angle=90, radius=2mm\] – cycle; at(2mm,0)[Power meter]{}; ]{}
\(m) at (0, 0) \[ column sep=2mm, row sep=[6 mm,between origins]{} \] [ & & & & (tmp\_relay) at (0, -0.5mm); & &\
& & & & &&\
& & & & & &\
& & & & & &\
]{};
(bsignal) to (blind); (tsignal) to (trigger); (sg\_c1) to \[out=0,in=180\] (bsignal); (sg\_c2) to \[out=0,in=180\] (tsignal); (dsg) to \[out=0,in=175\] (tmp\_relay); (tmp\_relay) to \[out=-5,in=90\] (apd); (dsg) to \[out=180,in=90\] (sg\_trg);
(blind) to (sa); (trigger) to (voa); (sa) to \[out=0,in=180\] (bs\_west); (voa) to \[out=0,in=180\] (bs\_west); (bs\_east) to \[out=0,in=180\] (apd); (bs\_east) to \[out=0,in=180\] (pm\_west);
coordinates
(0,0) (1.5,1.5) (3,0) (25,0) (26.5,1.5) (28,0) (50,0) (51.5,1.5) (53,0) (99,0)
(125,0) (150,0) (151.5,1.5) (153,0) (175,0) (176.5,1.5) (178,0) (185,0) (214,0) (225,0) (226.5,1.5) (228,0) (250,0) (251.5,1.5) (253,0) (275,0) (276.5,1.5) (278,0) (300,0) (301.5,1.5) (303,0)
; coordinates
(0,0) (35,0) (36.5,1) (38,0) (60,0) (62,1) (64,0) (85,0) (87,1) (89,0) (99,0)
(125,0) (135,0) (137,1) (139,0) (160,0) (162,1) (164,0) (185,0)
(214,0) (248.5,0) (250.5,0.3) (252.5,0) (302,0)
; (axis cs:99,0.5) – (axis cs:125,0.5); at (axis cs:112,1.3) [Blinding pulses]{}; at (axis cs:250,0.8) [Trigger\
pulse]{}; (axis cs:255,0.2) – (axis cs:273,0.2); coordinates[(273.5,0) (275.5,0.3) (277.5,0)]{}; (bld illm) at (axis cs:137, 1.1); (mid bld) at (axis cs:137, 1.5); (trg pls) at (axis cs:275.5,0.4); (mid trg) at (axis cs:275.5, 1.5); (trg pls) to (mid trg) to node \[auto,below,font=,sloped,align=center,xshift=6mm\] [inside]{} +(0, 0.47cm); (axis cs: 65, -0.1) – node \[above, align=center\] [Dead time\
()]{}(axis cs: 112, -0.1);
+ \[smooth,tension=0.2\] table\[col sep=comma, x=t, y=Ea\] [data/350cycle-ahn-zoomin.csv]{}; + \[smooth,tension=0.2\] table\[col sep=comma, x=t, y=Eh\] [data/350cycle-ahn-zoomin.csv]{}; + \[smooth,tension=0.2, mark=triangle\*\] table\[col sep=comma, x=t, y=En\] [data/350cycle-ahn-zoomin.csv]{};
+ \[smooth,tension=0.2\] table\[col sep=comma, x=t, y=Ea\] [data/400cycle-ahn-zoomin.csv]{}; + \[smooth,tension=0.2\] table\[col sep=comma, x=t, y=Eh\] [data/400cycle-ahn-zoomin.csv]{}; + \[smooth,tension=0.2, mark=triangle\*\] table\[col sep=comma, x=t, y=En\] [data/400cycle-ahn-zoomin.csv]{};
+ \[smooth,tension=0.2\] table\[col sep=comma, x=t, y=Ea\] [data/450cycle-ahn-zoomin.csv]{}; + \[smooth,tension=0.2\] table\[col sep=comma, x=t, y=Eh\] [data/450cycle-ahn-zoomin.csv]{}; + \[smooth,tension=0.2, mark=triangle\*\] table\[col sep=comma, x=t, y=En\] [data/450cycle-ahn-zoomin.csv]{};
+ \[smooth,tension=0.2\] table\[col sep=comma, x=t, y=Ea\] [data/500cycle-ahn-zoomin.csv]{}; + \[smooth,tension=0.2\] table\[col sep=comma, x=t, y=Eh\] [data/500cycle-ahn-zoomin.csv]{}; + \[smooth,tension=0.2, mark=triangle\*\] table\[col sep=comma, x=t, y=En\] [data/500cycle-ahn-zoomin.csv]{};
Experimental Demonstration {#sec:experiment}
==========================
As a third-party evaluator, we conducted tests about the pulse illumination attack on a APD-based single-photon detector module, provided by an independent party. In the tests, we assume that Eve only knows the public information of the detector as prior knowledge to show a real-life hacking scenario. For the single-photon detector module we tested, the frequency of gate signal is , and the photocurrent monitor inside the module first filters the photocurrent in the circuit of the detector via a lowpass filter to avoid high-frequency noise. The alarming threshold is set as , which is an empiric value to safely detect c.w. illumination blinding attack. The threshold is far lower than the illegitimate value () when c.w. illumination blinding attack works, as well as leaves a margin to the value of normal working state () to avoid false alarms.
Our experiment setup is shown in \[fig:bldsetup\]. A digital signal generator synchronize the whole system. The channel 1 of the waveform generator excites a laser diode to launch a group of blinding pulses towards the detector to produce a blinded period. In our experiments, the width of each blinding pulse was set as , and we kept the energy of each blinding pulse being . The blinding pulses were applied outside the gate signals to avoid unwanted clicks caused by the blinding pulses. A group of blinding pulses only triggers a click at the beginning of the group, which is followed by dead time. After that, the detector is blinded due to the accumulated photocurrent and thus no other clicks occur during the pulse illumination. This experimental result is shown in \[fig:attack\]. More experimental details are given in Supplementary \[subsec:detail\].
The channel 2 of the waveform generator excites another laser diode to launch a trigger pulse to calibrate the length of the blinded period and the fully controllable range inside. The methodology of calibrating a blinded period is shown in \[fig:calibration\]. A trigger pulse contains 67 photons, which can trigger a click in Geiger mode but is not strong enough to trigger a click in the linear mode. We first apply the trigger pulse at the gate just after the group of blinding pulses. If the trigger pulse causes no click, the detector is blinded during this gate. The trigger pulse is then moved away from the group of blinding pulses gate-by-gate to repeat the calibration process until the trigger pulse causes a click. The period of no click is the blinded period. The length of the blinded period generated by a group of 250-/300-/350-/400-/450-/500-cycle blinding pulses is shown in \[tab:blinded range\]. As the trigger pulse here contained multiple photons, the length of the blinded period here is just conservative estimations.
By a similar methodology but varying the energy of the trigger pulse, we can further calibrate a fully controllable range. As shown in \[fig:attack\]b, at each gate inside the blinded period, we vary the energy of a trigger pulse to observe the click probability and record the energy that can trigger a click with the probability of 100%/50%/0% as $E_{\text{always}}$/$E_{\frac{1}{2}}$/$E_{\text{never}}$. If $E_{\text{always}} < 2 E_{\text{never}}$, the detector at this gate in the blinded period is fully controllable by Eve for a BB84 QKD system. The experimental data of the fully controllable range in the blinded period generated by a group of 350-/400-/450-/500-cycle blinding pulses are shown in \[fig:controllablearea\]. Note that in all the testing above, the reported photocurrent keeps being far lower than the built-in alarming threshold of the photocurrent monitor () as shown in \[tab:blinded range\].
[p[1.5cm]{}<p[1.5cm]{}<p[1.7cm]{}<p[1.9cm]{}<]{} & & &\
250& 2.025& No data& 1.8\
300& 20.025& No data& 1.8\
350& 45.025& 72& 1.9\
400& 100.05& 150& 1.9\
450& 135.05& 330& 2\
500& 195.05& 690& 2.1\
Security analysis for a decoy-state BB84 QKD system {#sec:simulation}
===================================================
In this section, we analyse Eve’s maximum-profit strategy of attacking via pulse illumination, and we further study the threat of this attack to a real-life decoy-state BB84 QKD system. Here the detection parameters are from the Gobby-Yuan-Shields (GYS) experiment [@gobby2004]. The interval length, the blinded period, and the fully controllable range are from our experimental results as shown in \[tab:blinded range\].
+\[cyan, dotted\] table\[col sep=comma, x=L, y=E\_normal\] [data/simulation-data/E\_norm.csv]{}; +\[magenta, dashdotdotted\] table\[col sep=comma, x=L, y=E\_attack\] [data/simulation-data/E\_norm.csv]{}; +\[\] table\[col sep=comma\] [data/simulation-data/RL\_noattack.csv]{}; +\[\] table\[col sep=comma, x=L, y=RL\_esti\] [data/simulation-data/R\_350.csv]{}; +\[dashed\] table\[col sep=comma, x=L, y=RL\_real\] [data/simulation-data/R\_350.csv]{}; +\[dashdotted\] table\[col sep=comma, x=L, y=RU\_real\] [data/simulation-data/R\_350.csv]{}; ;
+\[\] table\[col sep=comma, x=L, y=RL\_esti\] [data/simulation-data/R\_400.csv]{}; +\[dashed\] table\[col sep=comma, x=L, y=RL\_real\] [data/simulation-data/R\_400.csv]{}; +\[dashdotted\] table\[col sep=comma, x=L, y=RU\_real\] [data/simulation-data/R\_400.csv]{};
+\[\] table\[col sep=comma, x=L, y=RL\_esti\] [data/simulation-data/R\_450.csv]{}; +\[dashed\] table\[col sep=comma, x=L, y=RL\_real\] [data/simulation-data/R\_450.csv]{}; +\[dashdotted\] table\[col sep=comma, x=L, y=RU\_real\] [data/simulation-data/R\_450.csv]{};
+\[\] table\[col sep=comma, x=L, y=RL\_esti\] [data/simulation-data/R\_500.csv]{}; +\[dashed\] table\[col sep=comma, x=L, y=RL\_real\] [data/simulation-data/R\_500.csv]{}; +\[dashdotted\] table\[col sep=comma, x=L, y=RU\_real\] [data/simulation-data/R\_500.csv]{};
(my plots c1r1.north west) node \[xshift=0.1cm, yshift=0.5cm\] [a)]{}; (my plots c2r1.north west) node \[xshift=0.1cm, yshift=0.5cm\] [b)]{}; (my plots c1r2.north west) node \[xshift=0.1cm, yshift=0.5cm\] [c)]{}; (my plots c2r2.north west) node \[xshift=0.1cm, yshift=0.5cm\] [d)]{}; (my plots c1r3.north west) node \[xshift=0.1cm, yshift=0.5cm\] [e)]{}; (my plots c2r3.north west) node \[xshift=0.1cm, yshift=0.5cm\] [f)]{};
Eve’s maximum-profit strategy
-----------------------------
The strategy of Eve’s attack is as follows. Without introducing deviation to the normal value of total gain, she launches fake-state attack during the fully controllable range, while blocks or passes the state from Alice during the unblinded time. Therefore, to eavesdrop the maximum information, she needs to optimize the parameters of her attack.
The total gain and QBER when the pulse illumination attack is applied can be written as $$\label{QE}
\begin{aligned}
Q_\omega &= \frac{1}{ N_{\text{interval}}}+ \frac{1}{2} p (1 - e^{-\omega}) \alpha \\
&+ (1 - \beta) [\gamma Y_0 + (1 - \gamma) Q_{\omega}^{\text{pass}}],\\
E_\omega &= \frac{1}{Q_\omega} [\frac{e_0}{N_{\text{interval}}} +\frac{1}{2} p (1-e^{-\omega}) \alpha e_{det} \\
&+ (1-\beta)[\gamma Y_0 e_0 + (1 - \gamma) E_{\omega}^{\text{pass}} Q_{\omega}^{\text{pass}}]],\\
\end{aligned}$$ where $\alpha = N_{\text{control}}/N_{\text{interval}}$, $\beta = (N_{\text{blind}}+N_{\text{dead}})/N_{\text{interval}}$. $\eta_\text{Bob}=4.5\%$ is the transmittance of Bob’s optical device. Here we assume $\omega \in \{\mu=0.6, \nu=0.2, 0\}$ is the mean photon number of the signal state, the decoy state, and the vacuum state. $e_0 = 0.5$ is the error rate of the background noise. $e_{\text{det}} = 3.3\%$ is the misalignment error rate of the QKD optical system. $Y_0 = 1.7 \times 10^{-6}$ is the probability of dark count per gate. $N_{\text{control}}$/$N_{\text{blind}}$/$N_{\text{dead}}$/$N_{\text{interval}}$ is the gate number of the fully controllable range/the blinded period/the dead time/the interval length for a group of blinding pulses. $p \in [0, 1]$ is the proportion of $N_{\text{control}}$ that Eve launches the fake-state attack. $\gamma \in [0, 1]$ is the ratio that Eve blocks the photons from Alice during the unblinded time in each interval. During the rest of the unblinded time, the photons are allowed to pass and the corresponding gain in these gates is $Q_{\omega}^{\text{pass}} = Y_0 + 1 - e^{-\eta_\text{Bob} \omega}$, as we assume that Eve uses a lossless channel to conduct her attack. Thus $E_{\omega}^{\text{pass}} Q_{\omega}^{\text{pass}} = e_0 Y_0 + e_\text{det} (1 - e^{-\eta_{\text{Bob}} \omega})$.
According to the principle of the attack, Eve has to keep the total gain being indistinguishable with that in the normal working state ($Q_{\omega}^{\text{normal}} = Y_0 + 1 - e^{-\eta_\text{Bob} \eta_\text{c} \omega}$, where $\eta_\text{c} = 10^{\frac{-0.21 L}{10}}$ is the transmittance of the quantum channel of the QKD system as a function of the channel length $L$) to hide her existence by modulating $p$ and $\gamma$. Moreover, she will make $p$ and $\gamma$ as high as possible. Consequently, when Eve tries to make $p = 1$ and $\gamma = 1$ initially, she may confront with two cases:
- $Q_\mu > Q_\mu^{\text{normal}}$. In this case, Eve only needs to decrease $p$ to apply less fake-state attack during the fully controllable range to ensure $Q_\mu = Q_\mu^{\text{normal}}$. Thus she can obtain almost all the information as all rounds of communication are either controlled or blocked.
- $Q_\mu < Q_\mu^{\text{normal}}$. In this case, Eve has to decrease $\gamma$ to allow some photons pass from Alice to Bob without any intervention during the unblinded time, while keeps $p=1$, and then increase $Q_\mu$ to hide herself. Therefore, she can just obtain part of the total information in the communication.
Under this strategy, the QKD system cannot be aware of Eve’s attack by checking the total gain. The QBER during the attack is shown in \[fig:simulation\]a.
The key rate estimated by Alice and Bob under pulse illumination attack
-----------------------------------------------------------------------
According to the decoy-state protocol [@ma2005], Alice and Bob can estimate the contribution of a single photon to the total gain and its error rate, which are given by $$\label{YE_1}
\begin{aligned}
Y_1^L& =\frac{\mu}{\mu \nu-\nu^{2}}\left(Q_{\nu} e^{\nu}-Q_{\mu} e^{\mu} \frac{\nu^{2}}{\mu^{2}}-\frac{\mu^{2}-\nu^{2}}{\mu^{2}} Y_{0}\right)\\
e_1^U& =\frac{E_{\nu} Q_{\nu} e^{\nu}-e_{0} Y_{0}}{Y_{1}^{L} \nu}.
\end{aligned}$$ Submitting Eq. \[QE\] and Eq. \[YE\_1\] into the GLLP [@gottesman2004], Alice and Bob can estimate the lower bound of the key rate as $$\label{GLLP}
R^L_{\text{est}} = q\left\{-Q_{\mu} f\left(E_{\mu}\right) H_{2}\left(E_{\mu}\right)+ \mu e^{-\mu} Y_{1}^L\left[1-H_{2}\left(e_{1}^U\right)\right]\right\}.$$ Here $q=1/2$ for the BB84 protocol, $f(E_\mu)=1.2$ for error correction, and $H_2(x)$ is Shannon entropy. The $R^L_{\text{est}}$ under no/350-/400-/450-/500-cycle pulse illumination attack with Eve’s strategy is shown by the blue line in \[fig:simulation\]b/c/d/e/f.
The real key rate of the QKD system under pulse illumination attack
-------------------------------------------------------------------
To judge whether Alice and Bob overestimate the key rate and thus introduce insecurity, we give the *real* upper bound and the lower bound of the key rate when the pulse illumination attack with Eve’s strategy is applied.
The real contribution of a single photon to the total gain $Y_1^{\text{attack}}$ and its error rate $e_1^{\text{attack}}$ can be calculated as $$\begin{aligned}
Y_1^{\text{attack}} &= Y_0 +\eta_{\text{Bob}} -Y_0 \eta_{\text{Bob}},\\
e_1^{\text{attack}} &=\frac{1}{Y_1^{\text{attack}}} (e_{\text{det}} \eta_{\text{Bob}} + e_0Y_0).
\end{aligned}$$ Then, the real upper bound and the lower bound of the key rate can be written as $$\label{True_kr_upper}
R^U_{\text{real}} = \frac{1}{2} (1-\beta)\gamma \mu e^{-\mu} Y_1^{\text{attack}} [1-H_2(e_1^{\text{attack}})]$$ and $$\label{True_kr_lower}
\begin{aligned}
R^L_{\text{real}} &= \frac{1}{2} (1-\beta)\gamma \{\mu e^{-\mu} Y_1^{\text{attack}} [1-H_2(e_1^{\text{attack}})] \\
&- Q_\omega^{\text{pass}} f_{EC} H_2(E_\omega^{\text{pass}})\}.
\end{aligned}$$ $R^L_{\text{real}}$ and $R^U_{\text{real}}$ under 350-/400-/450-/500-cycle pulse illumination attack with Eve’s strategy are shown in \[fig:simulation\]c/d/e/f as the dashed and dash-dot lines. The results show that Eve can successfully hack the QKD system and learn the secret key under certain communication distance between Alice and Bob. Take the scenario of 500-cycle pulse illumination attack as an example, we can easily find that when the channel length is between 20 km and 43 km, the estimated key rate by Alice and Bob is higher than the real lower bound but lower than the real upper bound. Thus, Alice and Bob overestimates the key rate. When the length of the quantum channel is longer than , we can ensure that the key rate estimated by Alice and Bob is insecure, because $R^L_{\text{est}}$ is higher than $R^U_{\text{real}}$. Moreover, when less than , Alice and Bob happen to estimate the lower bound of the key rate correctly because GLLP is conservative when the total gain is high.
Conclusion
==========
We investigate the effectiveness of a photocurrent monitor as a countermeasure against the detector blinding attack in a single-photon detector module that is provided by an independent party. The testing results show that the single-photon detector with a photocurrent monitor is vulnerable to the pulse illumination attack. Via this attack, Eve can blind the single-photon detector in a certain period and fully control its detection output, keeping the reported photocurrent of the photocurrent monitor similar to that in the normal state and thus without alarming the monitor. We also perform the theoretical security analysis to show that for a real-life QKD system under pulse illumination attack, Alice and Bob may overestimate the secret key rate and leak the key to Eve in a certain distance range. This pulse illumination attack indicates that the security issues in the detection side is still serious, which should be further investigated. As this attack seriously threatens the practical security of QKD systems, pulse illumination attack should be a standard testing item for the systematic security evaluation of a QKD system.
We also provide more details on the photocurrent of a detector under the pulse illumination attack obtained from a white-box experiment on our homemade detector (see Supplementary \[subsec:countermeasure\]), which may provide some ideas of countermeasures against the pulse illumination attack. However, patching only solves the problem in a short term. A more secure method is to model the practical single-photon detector in the security proof, if the non-MDI-QKD system would like to be immune to various blinding attacks in a long term.
Funding Information {#funding-information .unnumbered}
===================
National Natural Science Foundation of China (Grants No. 11674397, No. 61601476, No. 61901483, and No. 61632021) and National Key Research and Development Program of China (Grant No. 2019QY0702).
Acknowledgments {#acknowledgments .unnumbered}
===============
We thank Vadim Makarov for very useful discussions. Supporting from Greatwall Quantum Laboratory is also acknowledged.
Disclosures {#disclosures .unnumbered}
===========
**Disclosures.** The authors declare no conflicts of interest.
Supplemental Documents {#sec:supplement .unnumbered}
======================
Recap c.w. illumination blinding attack {#subsec:c.w.}
---------------------------------------
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The BB84 protocol is typically used in QKD implementation, especially in commercial QKD systems. The polarization-encoding BB84 protocol works as follows. Alice encodes the secret key bits randomly in $H$/$V$ basis or $+$/$-$ basis of photons. Specifically, a photon polarized in $H$ or $+$ represents bit “0”, while polarized in $V$ or $-$ represents “1”. Bob likewise randomly chooses a basis from these two bases to measure the photon sent by Alice and keeps the result. After the raw key exchange, Alice and Bob compare the basis they chose for each round via a classical channel and only keep those bits that were under same basis for both sides. Then via post-processing, Alice and Bob can share the same string of the secret key. This scheme ensures the information-theoretical security of the key even with the existence of eavesdropper, Eve. Because if she naively intercepts the photon and measures it with a random basis and resends it to Bob, she will introduce additional 25% QBER and thus will be detected [@mayers1996; @lo1999; @shor2000].
Note that the above scenario only works well under the assumption that the APDs works in Geiger mode [@cova2004], where a single photon leads to huge transient avalanche photocurrent and thus causes a click. However, a real-life QKD system deviates from the ideal model: the APD can be turned into the linear mode (be blinded) and then the clicks are controlled by Eve. One approach to achieve blinding is to illuminate the APD by carefully modulated c.w. light. The principle behind the c.w. illumination blinding is as follows. Bright c.w. light applied on the APD knocks out many electron-hole pairs, and thus a huge photocurrent is generated. According to the circuit shown in \[fig:apd-inner\]a, the APD is in series with the $R_{\text{bias}}$, so the strong photocurrent also goes through the $R_{\text{bias}}$. Because the $R_{\text{bias}}$ is a huge resistor for passive quenching, the voltage across $R_{\text{bias}}$ increases dramatically and thus $V_{\text{bias}}$ goes lower than $V_{\text{br}}$ as the total voltage is conserved.
Having been blinded, the single-photon detectors of Bob can be controlled secretly by applying trigger pulses. A trigger pulse can always trigger a click when its energy is higher than $E_{\text{always}}$, and is impossible to trigger a click when the energy is lower than $E_{\text{never}}$, as shown in the \[fig:apd-inner\]b. More specifically, with the assumptions that the all of Bob’s single-photon detectors are identical and satisfy $$\label{eq:controll sufficient}
E_{\text{always}} < 2 \times E_{\text{never}},$$ Bob can be fully controlled by the fake-state attack [@makarov2005] if the energy of the trigger pulse in $[E_{\text{always}}, 2 \times E_{\text{never}})$. In a round of the communication Eve intercepts the photon and measures it in a randomly chosen basis, and then resends a trigger pulse encoded by the measurement result to Bob. Eve can ensure that when she happens to choose the same basis as both Alice and Bob, the information of this round will be shared among Alice, Bob, and Eve (see \[fig:msrsetup\]a), while Bob will get no click and lose the information when Eve unfortunately chooses a wrong basis (see \[fig:msrsetup\]b). Afterwards, with the help of basis comparison in the post-processing (which happens in a public classical channel and can be listened by Eve), all of Alice, Bob, and Eve keep the bits that are measured with same basis. As a result, Alice and Bob innocently share the final entire secret key with Eve.
Detailed analysis on the parameters of blinding pulses {#subsec:detail}
------------------------------------------------------
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threshold]{}; (axis cs: 2.5175,32.4) – (axis cs:2.5175,26.9); (axis cs: 2.5175,32.4) – (axis cs:3.5817,32.4);
Here we first demonstrate the simplest case – 1-cycle blinding pulses in each group. In the experiment, we controlled the interval between each group of blinding pulse and the energy of each single pulse. Then we observed the reported photocurrent. \[fig:EvesusI\] shows the energy of each single blinding pulses versus the reported photocurrent with interval of 500ns/600ns/700ns. Generally, the reported photocurrent increases with the rising of single pulse energy. The reported photocurrent rises slightly at the beginning and then goes up dramatically at about . Finally, the reported photocurrent ascends linearly after . In addition, in \[fig:EvesusI\], the points where the reported photocurrent is higher than means that the low-frequency component is strong enough to blind the APD in the whole time domain (*constant blinding*).
The orange vertical arrow in \[fig:EvesusI\] shows that the reported photocurrent reduces as the interval rises. This is because for the same energy of blinding pulse, the larger interval between the groups results in the less generated photocurrent from different blinding pulses that superposes with each other. Consequently, the low-frequency components of the superposed photocurrent are less, which are reported by the photocurrent monitor. Contrarily, to increase the superposed photocurrent to constantly blind the APD when the interval is extended, higher energy of each blinding pulse is needed, as shown in the blue arrow in \[fig:EvesusI\]. From the testing result, we can see that Eve can extend the interval to reduce the reported photocurrent, and thus avoiding the alarm of the photocurrent monitor.
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To further analyse the influence introduced by the cycle number in each group, we measured the total energy of each group with 1-/2-/3-cycle blinding pulses when the APD is constantly blinded in the whole time domain (which is defined as *constant-blinding energy* in the following text) versus the interval length. The measurement results are shown in \[fig:cyclesplit\]. Comparing among the three curves in \[fig:cyclesplit\], for the same interval length, the summation energies of each 1-/2-/3-cycle group to constantly blind the APD are quite similar. Thus the equivalence between a single blinding pulse and three smaller blinding pulses is apparent. Moreover, the maximum intervals for the 1-/2-/3-cycle blinding pulses are // respectively. Taking the case of 1-cycle for illustration, if the interval is longer than its maximum value, the increased energy of pulses will no longer blind the APD but cause unwanted clicks. However, its equivalent split in 2-/3-cycle can still blind. Therefore, by using this multi-cycle approach, the corresponding blinded period is adjustable in a wider range.
The waveform of a homemade detector under pulse illumination attack {#subsec:countermeasure}
-------------------------------------------------------------------
table\[col sep=comma\] [data/homade-APD-data/b11.csv]{}; table\[col sep=comma\] [data/homade-APD-data/b12.csv]{}; table\[col sep=comma\] [data/homade-APD-data/b13.csv]{}; (axis cs: 25.9,0.0075) – (axis cs:31.9,0.0075); (axis cs: 28.9,0) – (axis cs:28.9,0.012); ;
table\[col sep=comma\] [data/homade-APD-data/a15.csv]{}; table\[col sep=comma\] [data/homade-APD-data/a16.csv]{}; table\[col sep=comma\] [data/homade-APD-data/a17.csv]{}; (axis cs: 25.9,0.0075) – (axis cs:31.9,0.0075); (axis cs: 28.59,0) – (axis cs:28.59,0.0055); ;
\[ change y base = true, xlabel=[Time]{}, ylabel=[$V_{\text{o}}$]{}, x SI prefix=nano, x unit=, y SI prefix=milli, y unit=, \] + \[no markers,magenta\] table\[col sep=comma\] [data/homade-APD-data/pulseform.csv]{};
To acquire some evidences left by a pulse illumination attack, we conducted a white-box test of the pulse illumination attack on our homemade single-photon detector whose APD is produced by Princeton Lightwave. We directly observed the waveform of the voltage $V_{\text{o}}$ across the readout resistor $R_{\text{o}}$ (for our homemade single-photon detector, $R_\text{o} = \SI{50}{\ohm}$) in the circuit of the single-photon detector shown in \[fig:apd-inner\]. The waveform of $V_{\text{o}}$ without/after blinding pulses are compared in \[fig:current-waveform\]. The fluctuation ranges of the output signals are indicated by the double-head arrows in the picture. Apparently, as shown in \[fig:current-waveform\]a, when the blinding pulses are not applied, output signals rise occasionally. These rises are strong enough to reach the comparator threshold to trigger a click because of the avalanche effect in Geiger mode. As a result, dark counts are caused in this case. On the other hand, in the blinded period under the pulse illumination attack, the output signals jump frequently but just in a much narrower range whose upper bound is far lower than the comparator threshold as shown in \[fig:current-waveform\]b. Thus, dark counts are eliminated in the blinded period.
Another interesting evidence is shown in \[fig:pulseform\], where we can clearly see that a blinding pulse causes a huge instantaneous photocurrent hill. A photocurrent monitor might be capable of figuring out this evidence by some engineering modifications, which may be not easy to be realized. Researches on reliable countermeasures against pulse illumination attack are still urgently needed.
|
---
abstract: |
The purpose of this note is to exhibit clearly how the “graphical condensation” identities of Kuo, Yan, Yeh and Zhang follow from classical Pfaffian identities by the Kasteleyn–Percus method for the enumeration of s. Knuth termed the relevant identities “overlapping Pfaffian” identities and the key concept of proof “superpositions of matchings”. In our uniform presentation of the material, we also give an apparently unpublished general “overlapping Pfaffian” identity of Krattenthaler.
A previous version of this paper contained an erroneous application of the Kasteleyn–Percus method, which is now corrected.
address: 'Fakultät für Mathematik, Nordbergstraße 15, A-1090 Wien, Austria'
author:
- Markus Fulmek
bibliography:
- 'paper.bib'
title: 'Graphical condensation, overlapping Pfaffians and superpositions of matchings'
---
[^1]
intro
graphs-matchings
superpositions
condensation
pfaffians
kasteleyn-percus
tanner
generalizations
[^1]: Research supported by the National Research Network “Analytic Combinatorics and Probabilistic Number Theory”, funded by the Austrian Science Foundation.
|
---
author:
- 'Zhen-Yu Wang, Jorge Casanova, and Martin B. Plenio'
title: Delayed entanglement echo for individual control of a large number of nuclear spins
---
**Methods for achieving quantum control and detection of individual nuclear spins by single electrons of solid-state defects play a central role for quantum information processing and nano-scale nuclear magnetic resonance (NMR) [@maurer2012room; @zhao2012sensing; @kolkowitz2012sensing; @taminiau2012detection; @liu2013noise; @taminiau2014universal; @waldherr2014quantum; @london2013sense]. However, with standard techniques, no more than 8 nuclear spins have been resolved [@zhao2012sensing; @kolkowitz2012sensing; @taminiau2012detection; @schirhagl2014nitrogen; @laraoui2013high]. Here we develop a new method that improves significantly the ability to spectrally resolve nuclear spins and demonstrate its capabilities with detailed numerical simulations by using a nitrogen-vacancy (NV) centre [@doherty2013nitrogen] as model system. Based on delayed entanglement control, a technique combining microwave and radio-frequency (rf) fields, nuclei with resonances in a broad frequency band can be unambiguously [@loretz2015spurious] and individually addressed by the sensor electron. Additionally the spectral resolution can extend beyond the electron spin relaxation time by using a long-lived qubit memory. Our method greatly increases the number of useful register qubits accessible to a defect centre and improves the signals of nano-scale NMR.**
Nuclear spins are natural quantum bits with long coherence times for quantum information tasks [@zhong2015optically] and they encode information about the structure of molecules and materials in a form that is accessible to NMR techniques [@mehring2012principles]. The NV centre in diamond represents a promising nano-scale platform for detection and coherent control of such nuclear spins [@doherty2013nitrogen; @schirhagl2014nitrogen]. In type IIa diamonds, the decoherence of the NV electron spin is dominated by the presence of $^{13}\text{C}$ nuclei. However, when properly controlled, the $^{13}\text{C}$ nuclear spins in the vicinity of an NV centre become useful resources [@liu2013noise; @taminiau2014universal; @waldherr2014quantum]. Furthermore, NV centres can be implanted close to the diamond surface to detect the signal of nuclear spins above the surface [@muller2014nuclear; @lovchinsky2016nuclear], which opens opportunities for both quantum information processing [@taminiau2014universal; @waldherr2014quantum; @cai2013large] and single molecule NMR [@lovchinsky2016nuclear] when environmental noise can be controlled.
Originally developed in NMR, dynamical decoupling (DD) techniques [@mehring2012principles; @yang2011preserving] can extend significantly qubit coherence times and they can also be applied to address single nuclear spins by an NV centre [@kolkowitz2012sensing; @taminiau2012detection; @zhao2012sensing; @london2013sense; @cai2013nmr; @casanova2015robust; @wang2015positioning]. Nevertheless standard DD techniques can only be used to address a few nuclear spins because of a number of drawbacks such as low spectral resolution [@laraoui2013high], resonance ambiguities [@loretz2015spurious], and perturbations from the electron-nuclear coupling [@casanova2015robust]. In this respect correlation spectroscopy could improve the resolution by measuring the NV signal over long evolution times [@laraoui2013high], however this technique does not provide advantages on individual spin addressing and control.
Our method overcomes these difficulties by selectively addressing target nuclear spins by radio-frequency (rf) fields in a delay window while the entanglement with the electron spin sensor is preserved by a subsequent Hahn echo [@mehring2012principles] operation. In this manner highly selective entangling gates between the electron spin and different target nuclear spins can be achieved.
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Now we describe the details of our method. A magnetic field $B_{z}$ parallel to the NV symmetry axis splits the spin triplet of the orbital ground electronic state of the NV centre. We use two of the three levels $m_{s}=0,\pm1$ to define an NV electron spin qubit [@doherty2013nitrogen]. Under strong magnetic fields such that the nuclear Zeeman energies exceed the perpendicular components, $A_{j}^{\perp}$, of the hyperfine field $\boldsymbol{A}_{j}$ at the locations of nuclear spins, see Fig. \[fig:FigSketch\]a, the interaction between the NV electron spin and its surrounding nuclear spins is described by $$H_{\text{int}}= \sigma_{z}\otimes\eta\sum_{j}A_{j}^{\parallel}I_{j}^{z}.$$ with $\sigma_{z}={|\uparrow_{e}\rangle\langle\uparrow_{e}|}-{|\downarrow_{e}\rangle\langle\downarrow_{e}|}$. Here we use ${|\uparrow_{e}\rangle}={|m_{s} = +1\rangle}$ as one of the qubit states while the second qubit state may be $|\downarrow_{e}\rangle=|m_s=0\rangle$, when $\eta=1/2$, or $|\downarrow_{e}\rangle=|-1\rangle$ with $\eta=1$ (see Supplementary Information). For each nuclear spin, $A_{j}^{\parallel}$ denotes the component of $\boldsymbol{A}_{j}$ parallel to the nuclear spin quantisation axes (see Fig. \[fig:FigSketch\]a,b). The nuclear precession frequencies are shifted by $A_{j}^{\parallel}$, which can be used to address individually nuclear spins of the same species (homonuclear spins).
An initial superposition state of the electron spin $|\psi_{e}\rangle = c_{\uparrow}|\uparrow_{e}\rangle+c_{\downarrow}|\downarrow_{e}\rangle$ loses its coherence because of the electron-nuclear coupling $H_{\text{int}}$. This effect can be removed by the Hahn echo. In our case, a microwave $\pi$ pulse exchanges the states $|\uparrow_{e}\rangle \leftrightarrow|\downarrow_{e}\rangle$ and effectively reverses $H_{\text{int}}
\rightarrow -H_{\text{int}}$. When the evolutions before and after the $\pi$ pulse have the same duration $\tau$, entanglement from all the nuclear spins is erased, preserving the electron spin coherence.
In order to preserve the coupling with a target spin, we apply rf-driving at the precession frequency of the target spin during a delay window. By flipping the target spin $j$ by an angle $\theta_{\text{rf}}=\pi$, we selectively rephase its interaction with the electron spin and realize a quantum gate $$|\uparrow_{e}\rangle\langle\uparrow_{e}|\otimes U_{+}+|\downarrow_{e}\rangle\langle\downarrow_{e}|\otimes U_{-}$$ with $U_{\pm}=\exp\left(\mp2i\eta\tau A_{j}^{\parallel}I_{j}^{z}\right)$, up to single spin rotations. This leads to the possibility of creating entanglement between the electron spin and selected nuclear spins which forms a key ingredient of our method. Note that it is not essential that $\theta_{\text{rf}}=\pi$ as almost any choice of $\theta_{\text{rf}}$ will generate entanglement with the target and lead to an observable loss of coherence in the electron spin due to entanglement with the target nucleus. Additionally, more than one nuclear spins can be addressed in the same delay window by using rf driving with different frequencies.
We will describe two strategies to achieve selective nuclear spin control in the delay window. In the first one we use DD techniques to suppress electron-nuclear interactions for protection of both the NV electron coherence and the rf nuclear spin control (Fig. \[fig:FigSketch\]e). Consider an NV centre in a diamond sample with natural $^{13}\text{C}$ abundance (1.1%) and initialise the NV-center in the equally weighted superposition state $|\psi_e\rangle$. Then the population signal $P = (1+L)/2$ is directly related to the observable of NV electron coherence [@zhao2012sensing], $L = |\langle\downarrow_{e}|\psi_e\rangle\langle
\psi_e|\uparrow_{e}\rangle|$. Fig. \[fig:FigSpectrum\]a,b shows the coherence when scanning the frequency of an rf-pulse with length $t_{\text{rf}}\approx1$ ms in the delay window protected by 100-pulse Carr-Purcell (CP) sequences [@mehring2012principles]. When the rf frequency matches one of the nuclear precession frequencies $\omega_{j}=\omega_{^{13}\text{C}} - A_{j}^{\parallel}/2$ under strong magnetic fields (with $\omega_{^{13}\text{C}}$ denoting the bare Larmor frequency of $^{13}\text{C}$), we observe a coherence $L = L(A_{j}^{\parallel},
\theta_{\text{rf}})$ where the single spin contribution $L(A_{j}^{\parallel},
\theta_{\text{rf}})=[(1-\cos\theta_{\text{rf}})\cos(2\eta A_{j}^{\parallel}\tau)
+1+\cos\theta_{\text{rf}}]/2$ (see Supplementary Information). When there are $p$ spins with the same $\omega_{j}$, the coherence signal becomes $L=[L(A_{j}^{\parallel},
\theta_{\text{rf}})]^{p}$. The one-to-one correspondence between $L$ and $\omega_{j}$ (see the dashed lines on Fig. \[fig:FigSpectrum\] where $p=1$ for C1 and C3 and $p=2$ for C2) and the coherence patterns (see Fig. \[fig:FigSpectrum\]c,d) easily identifies the number of nuclear spins in a dip, even when they are not resolved in the spectrum. Note that the signal is strong even when $A_{j}^{\perp}=0$, which contrasts previous methods [@kolkowitz2012sensing; @taminiau2012detection; @zhao2012sensing; @casanova2015robust; @wang2015positioning; @ma2016scheme] requiring relatively large $A_{j}^{\perp}$.
We can enhance the interaction and signal by transferring the electronic state $|\downarrow_{e}\rangle=|0\rangle$ to $|\downarrow_{e}\rangle=|-1\rangle$ during the interaction windows [\[]{}$(t_{1},t_{2})$ and $(t_{5},t_{6})$ in Fig. \[fig:FigSketch\][\]]{}, as shown in Fig. \[fig:FigSpectrum\]b. Fig. \[fig:FigSpectrum\]b also shows that changing the rotation angle $\theta_{\text{rf}}$ does not affect the locations of signal dips, demonstrating an intrinsic robustness of our method in nuclear spin detection.
Alternatively, during the delay window we may store the electron spin state in a long-lived nuclear spin (see Fig. \[fig:FigSketch\]f). In Fig. \[fig:FigAncilla\]a, we show the NV electron spin population signal when it addresses an isolated $^{13}\text{C}$ spin, by using an initially polarised memory qubit (see Methods). Because the electron spin is polarised to $m_{s}=+1$ during the delay window after a SWAP gate (see Supplementary Information), it is not necessary to protect electron coherence. Additionally the shifts of the nuclear spin precession frequencies (now $\omega_{j}=\omega_{^{13}\text{C}}-A_{j}^{\parallel}$ under strong magnetic fields) are larger than those achieved by the method described earlier or by DD [@kolkowitz2012sensing; @taminiau2012detection; @zhao2012sensing; @casanova2015robust; @wang2015positioning]. The electron spin relaxation creates magnetic noise on the nuclear spins and can reduce the signal contrast [@maurer2012room]. However, at low temperatures, the relaxation time $T_{1}$ of NV electron spin can reach minutes [@yang2015high]. When $T_{1}\gg
t_{\text{rf}}$ the effects of electron spin relaxation can be neglected. Additionally, we can protect the memory qubit against the electron-relaxation noise for example by strong driving [@maurer2012room]. Even for unpolarised memory qubit, the signal contrast is still large, showing that our method is insensitive to initialisation error of the qubit memory (Fig. \[fig:FigAncilla\]a).
Structural information of coupled spin clusters can also be observed by our method. The dynamics of two coupled spins under a strong magnetic field is characterised by the dipolar coupling strength $d_{j,k}$ and the difference $\delta_{j,k}=A_{j}^{\parallel}-A_{k}^{\parallel}$ between the hyperfine components [@zhao2011atomic]. A pair of $^{13}\text{C}$ spins with $d_{j,k}\approx\delta_{j,k}$ close to the NV centre can modulate the NV coherence [@zhao2011atomic; @shi2014sensing], in the absence of rf control as shown in Fig. \[fig:FigAncilla\]b. Spin pairs with $|\delta_{j,k}|\gg|d_{j,k}|$ or $|\delta_{j,k}|\ll|d_{j,k}|$, which could be useful quantum resources, were regarded as unobservable, because they form stable pseudo-spins which have negligible effects on electron coherence [@zhao2011atomic]. But using our method, we can detect, identify, and control those spin pairs close to the NV centre (see Supplementary Information for details).
To address and control nuclear spins individually in a coupled cluster the internuclear dipolar coupling needs to be suppressed. Internuclear interactions also reduce the Hahn-echo electron coherence times and, hence, the achievable interaction times $\tau$ (available values $\tau\sim0.5$ ms for natural abundance of $^{13}\text{C}$ and can be increased using lower abundance) [@cramer2015repeated; @zhao2012decoherence]. To solve the problem of spin interactions, we use the Lee-Goldburg (LG) off-resonance decoupling [@mehring2012principles] (see Methods). When the LG decoupling field is tuned such that $\sqrt{2}\Delta_{\text{LG}}\gg d_{jk}$, the dipolar coupling between nuclear spins are suppressed [@wang2015positioning; @cai2013large], giving rise to the effective interaction Hamiltonian $H_{\text{int}}\approx\eta
\sum_{j}A_{j}^{\parallel}\cos\gamma_{j}\sigma_{z}\tilde{I}_{j}^{z}/\sqrt{3}$ with $\tilde{I}_{j}^{z}$ the nuclear spin operators projected along effective rotating axes (see Supplementary Information). Fig. \[fig:FigAncilla\]b demonstrates the effect of LG decoupling with $\Delta_{\text{LG}}=2\pi\times20$ kHz. The suppression of the internuclear interactions allows us to achieve much longer interaction times $\tau$ and rf pulse lengths $t_{\text{rf}}$ for single spin addressing. The improved spectral resolution $\sim1/t_{\text{rf}}$ leads to an increase of the number of individually addressable spins (see Fig. \[fig:FigAncilla\]e).
In the case that there are other significant decoherence sources that are acting on the NV electron spin, our method can be combined with DD to further protect the NV coherence. Applying CP sequences with an inter $\pi$-pulse interval $\pi/\omega_{\text{DD}}$ during the interaction windows, noise with frequencies slower than $\omega_{\text{DD}}$ is suppressed, and at the same time the electron spin couples to nuclear spins through the interactions $2\eta/(k\pi)A_{j}^{\perp}
\sigma_{z}I_{j}^{\varphi_{j}}$ when the nuclear precession frequencies $\omega_{j}$ are resonant with $k\omega_{\text{DD}}$ ($k$ being odd integers) [@taminiau2012detection; @casanova2015robust; @wang2015positioning]. The nuclear spin operators $I_{j}^{\varphi_{j}}=I_{j}^{x}\cos\varphi_{j}+I_{j}^{y}\sin\varphi_{j}$ depends on the azimuthal angles $\varphi_{j}$ of nuclear spins relative to the magnetic field direction. Nuclear spins unresolvable by the CP sequences may nevertheless have different precession rates. To ensure the same effective Hamiltonian during the interaction windows, we apply a two-pulse CP sequence on the nuclear spins during the delay window to remove this inhomogeneity. Then adding a weak rf drive during the delay window allows us to address the target spins with high spectral resolutions. Additionally, the scheme allows us to measure the spin directions $\varphi_{j}$, because the rf driving has negligible effects when the azimuthal angle (phase) of rf control $\phi_{\text{rf}}=\varphi_{j}$ or $\varphi_{j}+\pi$ (see Fig. \[fig:FigAncilla\]c and Supplemental Information). Note that we can combine LG decoupling with DD using recently proposed protocols [@wang2015positioning; @casanova2016noise].
Our method allows to improve the spectral resolution beyond the limit set by the room-temperature electron $T_{1}$, using optical illumination that has been demonstrated to prolong the room-temperature coherence time of nuclear spin memory over one second ($\sim267$ times of $T_{1}$) [@maurer2012room]. To demonstrate the idea, we simulate the application of optical illumination and rf driving during the delay window to detect a proton spin placed 4 nm away from the NV centre with interaction times $\tau=100$ $\mu$s. A delay time $t_{\text{rf}}\approx80$ ms used in Fig. \[fig:FigAncilla\]d already provide enough frequency resolution to detect chemical shift of $\sim1$ ppm for the applied magnetic field $B_{z}\approx0.467$ T. We can apply LG decoupling when there are more target spins and internuclear interactions. In addition, we can use DD to protect the interaction windows from noise for extending the interaction time $\tau$. Electron spin coherence time of shallow NV centre has reported values of $\sim1$ ms using continuous DD (spin lock) [@lovchinsky2016nuclear].
In summary, we have proposed a method to address and control nuclear spins which were regarded as unresolvable. The method significantly increases the ability of detection and coherence control of nuclear spins and has applications in quantum information processing as well as analysis of chemical shifts and dynamics of spin clusters. The method is general and can be applied to other electron-nuclear spin systems [@widmann2015coherent; @dehollain2016bell].
**Methods**
The memory nuclear spin qubit can be initialized by swapping the initialized NV electron qubit state to the memory spin or by using dynamical nuclear polarization. Details on the swap operations are presented in Supplemental Information.
The LG decoupling field [@lee1963nuclear; @cai2013large; @wang2015positioning] can remain turned on for the entire duration of our protocol, including NV electron spin initialization and readout, because the frequency of rf decoupling field is far off-resonance to the transition frequencies of the NV electron spin. This allows for the rf decoupling field to be applied by external coils and resonators to avoid possible heating on the diamond sample. The LG decoupling with $\Delta_{\text{LG}}=2\pi \times 20$ kHz requires a rf field’s amplitude to be much smaller than the values of $\sim0.1$ T in the control fields reported in refs. .
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**Acknowledgements**\
This work was supported by the Alexander von Humboldt Foundation, the ERC Synergy grant BioQ, the EU projects DIADEMS, SIQS and EQUAM as well as the DFG via the SFB TRR/21. We thank Thomas Unden and Fedor Jelezko for discussions. Simulations were performed on the computational resource bwUniCluster funded by the Ministry of Science, Research and the Arts Baden-Württemberg and the Universities of the State of Baden-Württemberg, Germany, within the framework program bwHPC.\
\
**Author contributions**\
Z.Y.W., J.C., and M.B.P. conceived the idea. Z.Y.W. carried out the simulations and analytical work with input from J.C. and M.B.P. All authors discussed extensively on the results and contributed to the manuscript.\
\
**Additional information**\
The authors declare no competing financial interests. Correspondence and requests for materials should be addressed to Z.Y.W. (zhenyu3cn@gmail.com), J.C. (jcasanovamar@gmail.com), or M.B.P. (martin.plenio@uni-ulm.de).
**Supplementary Information for “Delayed entanglement echo for individual control of a large number of nuclear spins”**
Hamiltonian of NV centre and nuclear spins
==========================================
Under a magnetic field $\boldsymbol{B}=B_{z}\hat{z}$ along the NV symmetry axis, the Hamiltonian of NV centre electron spin and its nuclear environment reads ($\hbar=1$) $$H=H_{\text{NV}}+H_{\text{nZ}}+H_{\text{hf}}+H_{\text{nn}}.$$ Here $H_{\text{NV}}=DS_{z}^{2}-\gamma_{e}B_{z}S_{z}$ is the electron spin Hamiltonian with the spin operator $S_{z}=\sum_{m_{s}=\pm1,0}m_{s}|m_{s}\rangle\langle m_{s}|$, the ground state zero field splitting $D\approx2\pi\times2.87$ GHz, and $\gamma_{e}=-2\pi\times2.8$ MHz/G the electron spin gyromagnetic ratio [@Sdoherty2013nitrogen]. The nuclear Zeeman Hamiltonian $H_{\text{nZ}}=-\sum_{j}\gamma_{j}\boldsymbol{B}\cdot\boldsymbol{I}_{j}$, where $\gamma_{j}$ is the nuclear gyromagnetic ratio and $\boldsymbol{I}_{j}$ is the spin operator for the $j$-th nuclear spin. The dipole-dipole interactions between nuclear spins are $$H_{\text{nn}}=\sum_{j>k}\frac{\mu_{0}}{4\pi}\frac{\gamma_{j}\gamma_{k}}{|\boldsymbol{r}_{j,k}|^{3}}\left[\boldsymbol{I}_{j}\cdot\boldsymbol{I}_{k}-\frac{3(\boldsymbol{I}_{j}\cdot\boldsymbol{r}_{j,k})(\boldsymbol{r}_{j,k}\cdot\boldsymbol{I}_{k})}{|\boldsymbol{r}_{j,k}|^{2}}\right],\label{eq:Hnn}$$ with $\mu_{0}$ being the vacuum permeability, $\boldsymbol{r}_{j,k}=\boldsymbol{r}_{j}-\boldsymbol{r}_{k}$ the difference between the $k$-th and $j$-th nuclear positions. Typically the electron-nuclear flip-flop terms in the hyperfine interaction $H_{\text{hf}}$ are suppressed by the large energy mismatch between electron and nuclear spins, giving $H_{\text{hf}}=S_{z}\sum_{j}\boldsymbol{A}_{j}\cdot\boldsymbol{I}_{j}$ under the secular approximation. However, for strong hyperfine interactions the virtual flips of the electron spin could cause observable effects, an aspect that we will discuss later (see Sec. \[sub:Storage-to-N\]). For nuclear spins not too close to the NV centre, the hyperfine interaction takes the dipolar form and the hyperfine field $$\boldsymbol{A}_{j}=\frac{\mu_{0}}{4\pi}\frac{\gamma_{e}\gamma_{j}}{|\boldsymbol{r}_{j}|^{3}}\left(\hat{z}-\frac{3\hat{z}\cdot\boldsymbol{r}_{j}\boldsymbol{r}_{j}}{|\boldsymbol{r}_{j}|^{2}}\right).\label{eq:Aj}$$
Because the total Hamiltonian under secular approximation commutes with $H_{\text{NV}}$, we simply remove it by going to the rotating frame with respect to $H_{\text{NV}}$. We choose two of the three triplet states as the qubit basis states for the NV electron spin. The Hamiltonian becomes $$H_{\eta}=\eta\sigma_{z}\sum_{j}\boldsymbol{A}_{j}\cdot\boldsymbol{I}_{j}+H_{\text{n},\eta}+H_{\text{nn}}.$$ In the manifold of the electron spin levels $|\uparrow_{e}\rangle=|+1\rangle$ and $|\downarrow_{e}\rangle=|0\rangle$, we have the coupling constant $\eta=1/2$, while for the electron spin levels $|\uparrow_{e}\rangle=|+1\rangle$ and $|\downarrow_{e}\rangle=|-1\rangle$, $\eta=1$. The nuclear Hamiltonian describing nuclear precession reads $$H_{\text{n},\eta}=-\sum_{j}(\gamma_{j}\boldsymbol{B}-c_{\eta}\boldsymbol{A}_{j})\cdot\boldsymbol{I}_{j}\equiv-\omega_{j}\hat{\omega}_{j}\cdot\boldsymbol{I}_{j},$$ where the unit vectors $\hat{\omega}_{j}$ denote the directions of $\gamma_{j}\boldsymbol{B}-c_{\eta}\boldsymbol{A}_{j}$ with $c_{\eta}=1/2$ when $\eta=1/2$ and $c_{\eta}=0$ if $\eta=1$. For the case of $H_{1}$ ($\eta=1$), the electron-nuclear coupling is stronger and the nuclear precession frequencies $\omega_{j}=\gamma_{j}B_{z}$ are the bare nuclear Larmor frequencies. While for the case of $H_{\frac{1}{2}}$ ($\eta=1/2$), the electron-nuclear coupling is weaker and the precession frequencies $\omega_{j}=|\gamma_{j}\boldsymbol{B}-\frac{1}{2}\boldsymbol{A}_{j}|$ are shifted by the hyperfine field at the positions of the nuclear spins.
In the rotating frame of nuclear spin precession $H_{\text{n},\eta}$, the interaction Hamiltonian $\eta\sigma_{z}\sum_{j}\boldsymbol{A}_{j}\cdot\boldsymbol{I}_{j}$ becomes [@Scasanova2015robust] $$H_{\text{int}}=\eta\sigma_{z}\sum_{j}\left[\boldsymbol{A}_{j}^{x}\cos(\omega_{j}t)+\boldsymbol{A}_{j}^{y}\sin(\omega_{j}t)+\boldsymbol{A}_{j}^{z}\right]\cdot\boldsymbol{I}_{j},\label{eq:Hint}$$ with $$\begin{aligned}
\boldsymbol{A}_{j}^{x} & \equiv & \boldsymbol{A}_{j}-\boldsymbol{A}_{j}^{z},\\
\boldsymbol{A}_{j}^{y} & \equiv & \hat{\omega}_{j}\times\boldsymbol{A}_{j},\\
\boldsymbol{A}_{j}^{z} & \equiv & \boldsymbol{A}_{j}\cdot\hat{\omega}_{j}\hat{\omega}_{j}.\end{aligned}$$ The hyperfine components have the strengths $|\boldsymbol{A}_{j}^{x}|=|\boldsymbol{A}_{j}^{y}|=A_{j}^{\perp}$ and $|\boldsymbol{A}_{j}^{z}|=A_{j}^{\parallel}$. The time-dependent terms in Eq. (\[eq:Hint\]) do not commute with the nuclear precession $H_{\text{n},\eta}$.
Under a strong magnetic field $B_{z}\gg A_{j}^{\perp}$, $\hat{\omega}_{j}\approx\hat{z}$. The nuclear spin flips are suppressed, giving $$H_{\text{int}}\approx\eta\sigma_{z}\sum_{j}\boldsymbol{A}_{j}^{z}\cdot\boldsymbol{I}_{j}=\eta\sigma_{z}\sum_{j}A_{j}^{\parallel}I_{j}^{z}.\label{eq:HintZZ}$$ If we apply Lee-Goldburg (LG) off-resonance control [@Slee1965nuclear], we can achieve similar Hamiltonians [@Swang2015positioning; @Scai2013large] $$H_{\text{int}}^{\text{LG}}\approx\eta\sigma_{z}\sum_{j}A_{j}^{\parallel}\cos\gamma_{j}\tilde{I}_{j}^{z},\label{eq:HintZZLG}$$ where $\tilde{I}_{j}^{z}=\hat{\nu}_{j}\cdot\boldsymbol{I}_{j}$ with $\hat{\nu}_{j}$ the unit vector denoting the nuclear precession in the frame of LG control. The projection factor $\cos\gamma_{j}=\hat{\omega}_{j}\cdot\hat{\nu}_{j}\approx1/\sqrt{3}$.
The interaction Hamiltonian Eq. (\[eq:HintZZ\]) commutes with the nuclear precession. Similarly, Eq. (\[eq:HintZZLG\]) commutes with the nuclear precession around $\hat{\nu}_{j}$ in the frame of LG control. Combined with the delay entanglement control described in the main text, we can keep only terms on the target spins in $H_{\text{int}}$ or $H_{\text{int}}^{\text{LG}}$. The effective electron-nuclear interactions by delay entanglement control do not broaden the nuclear precession frequencies for addressing.
Spin addressing by dynamical decoupling
=======================================
Effective interaction Hamiltonians under dynamical decoupling\[sub:DD-Effective-interaction-Hamiltonia\]
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Nuclear spins can be addressed by dynamical decoupling (DD) [@Skolkowitz2012sensing; @Staminiau2012detection; @Szhao2012sensing; @Slondon2013detecting; @Smkhitaryan2015highly; @Scasanova2015robust; @Swang2015positioning]. The DD pulses flip the NV electron qubit. After application of $n$ $\pi$ pulses, $\sigma_{z}\rightarrow F(t)\sigma_{z}$ with the modulation function $F(t)=(-1)^{n}$. We consider periodic sequences with $F(t)=F(t+2\pi/\omega_{\text{DD}})$ in this work. The interaction Hamiltonian Eq. (\[eq:Hint\]) becomes $$H_{\text{int}}=\eta F(t)\sigma_{z} \sum_{j}\left[\boldsymbol{A}_{j}^{x}\cos(\omega_{j}t)+\boldsymbol{A}_{j}^{y}\sin(\omega_{j}t)+\boldsymbol{A}_{j}^{z}\right]\cdot\boldsymbol{I}_{j}.$$ This instantaneous-pulse control changes the electron-nuclear dynamics [@Staminiau2012detection; @Staminiau2014universal; @Sliu2013noise]. To get insight on nuclear spin sensing by DD pulse sequences, we expand the modulation function in a Fourier series, $$F(t)=\sum_{k\geq1}[f_{k}^{\text{s}}\cos(k\omega_{\text{DD}}t)+f_{k}^{\text{a}}\sin(k\omega_{\text{DD}}t)],$$ using that DD pulses have been designed to remove static noise and hence there is no static term in the Fourier series. For periodic symmetric sequences $f_{k}^{\text{a}}=0$. The frequency $\omega_{\text{DD}}$ characterises the flipping rate of the NV electron qubit. For example, for the traditional Carr-Purcell (CP) sequence [@Scarr1954effects] and its variations [@Smeiboom1958modified; @Smaudsley1986modified; @Sgullion1990new] having a time interval $\tau_{\text{CP}}$ between successive $\pi$ pulses, $\omega_{\text{DD}}=\pi/\tau_{\text{CP}}$, $f_{k}^{\text{s}}=4(k\pi)^{-1}\sin(k\pi/2)$, and $f_{k}^{\text{a}}=0$. The expansion coefficients can be tuned by adaptive XY (AXY) sequences [@Scasanova2015robust].
A nuclear spin with the precession frequency $\omega_{n}$ can be addressed by resonance to the $k_{\text{DD}}$ harmonic of the driving rate, that is, $\omega_{n}=k_{\text{DD}}\omega_{\text{DD}}$. With the additional conditions (with $j\neq n$) $|\gamma_{j}B_{z}|\gg k_{\text{DD}}|\boldsymbol{A}_{j}|$ and $$\begin{aligned}
|\omega_{n}-\omega_{j}| & \gg & |f_{k_{\text{DD}}}A_{j}^{\perp}|,\label{eq:RWAWeekFdd}\end{aligned}$$ we have single spin addressing under periodic symmetric sequences $H_{\text{int}}\approx(\eta/2) f_{k_{\text{DD}}}^{\text{s}}A_{j}^{\perp}\sigma_{z}I_{j}^{x}$ [@Scasanova2015robust]. Similar addressing Hamiltonians ($H_{\text{int}}^{\text{LG}}\propto \sigma_{z}\tilde{I}_{j}^{x}$ with $\tilde{I}_{j}^{x}$ a spin operator projected perpendicular to $\hat{\nu}_{j}$) can be achieved under LG control [@Swang2015positioning; @Scasanova2016noise].
Nuclear spins can also be addressed by continuous DD [@Slondon2013detecting]. In the rotating frame of a constant microwave driving $\Omega_{e}\sigma_{x}/2$ with the Rabi frequency $\Omega_{e}$ (the frequency of nuclear spin precession in the spin-lock frame), the Pauli operator of NV electron qubit transforms as $\sigma_{z}\rightarrow\sigma_{z}\cos(\Omega_{e}t)+\sigma_{y}\sin(\Omega_{e}t)$. The driving rate of the electron spin is $\omega_{\text{DD}}=\Omega_{e}$. When $\Omega_{e}$ is on-resonance to the nuclear spin precession frequency $\omega_{j}$, that is, $\Omega_{e}=\omega_{j}$, we have the addressing Hamiltonian $H_{\text{int}}\approx(\eta/2) A_{j}^{\perp}(\sigma_{z}I_{j}^{x}+\sigma_{y}I_{j}^{y})$ when $|\gamma_{j}B_{z}|\gg|\boldsymbol{A}_{j}|$ and $|\omega_{n}-\omega_{j}|\gg A_{j}^{\perp}$ for $j\neq n$.
Shortcomings of spin addressing by dynamical decoupling
-------------------------------------------------------
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The addressing by DD has a number of shortcomings that we are going to discuss in the following.
First, the interaction Hamiltonians achieved by DD do not commute with the nuclear precession. As a consequence, electron-nuclear interactions broaden the nuclear precession frequencies for addressing (see Figs. \[fig:smFigSpectrumCPMG\] (c) and (d)). We need the condition in Eq. (\[eq:RWAWeekFdd\]) for individual spin addressing. Reducing the effective interaction strengths between NV electron and nuclear spins improves a lot the spectral resonance by using higher harmonics $k_{\text{DD}}>1$, by alternating the phase of Rabi driving [@Smkhitaryan2015highly], or by using composite $\pi$ pulses [@Szhao2014dynamical; @Scasanova2015robust; @Sma2015resolving; @Swang2015positioning]. But the reduced coupling also makes nuclear spins that are not strongly coupled hard to detect and control (see Figs. \[fig:smFigSpectrumCPMG\] (a) and (b)).
Second, because resonances can occur at different harmonics frequencies $k\omega_{\text{DD}}$, resonance lines from different harmonic branches can have overlaps and make critical ambiguities in detection and addressing [@Staminiau2012detection; @Sloretz2015spurious; @Scasanova2015robust]. The spurious resonances caused by realistic pulse width further complicate the situation, even making false identification of different nuclear species (e.g., $^{13}\text{C}$ and $^{1}\text{H}$) [@Sloretz2015spurious].
Third, the achievable rate $\omega_{\text{DD}}$ of DD sets an upper limit on the external magnetic field for spin addressing. A strong magnetic field is the requirement in detecting the chemical shift of nuclear spins [@Smehring2012principles] and in decoupling of the nuclear dipole-dipole interactions by rf control [@Slee1965nuclear; @Scai2013large; @Swang2015positioning]. In addition, the NV electron coherence can be protected easier under strong magnetic fields [@Szhao2012decoherence]. However, nuclear spin precession frequencies $\omega_{j}$ at strong magnetic fields can be significantly larger than the achievable rate $\omega_{\text{DD}}$ of DD control. For example, the pulse number 35684 required in Fig. \[fig:smFigSpectrumCPMG\] (d) could be too many in experiments. Using resonance branches with large $k_{\text{DD}}$ can reduce the required control rate $\omega_{\text{DD}}$, but it also reduces electron-nuclear coupling and narrows spectral bandwidths (in Fig. \[fig:smFigSpectrumCPMG\] (b) the coupling is too weak to detect the nuclear spin with a weak $A_{j}^{\perp}$ and the bandwidth is about $\sim\omega_{^{13}\text{C}}/k_{\text{DD}}$ for $^{13}\text{C}$ spins).
The delayed entanglement echo technique does not suffer from the above shortcomings (compare Fig. \[fig:smFigSpectrumDEE\] with Fig. \[fig:smFigSpectrumCPMG\]), and provide some additional advantages. First, it does not require both hyperfine components $A_{j}^{\parallel}$ and $A_{j}^{\perp}$ to be strong. Second, both the electron-nuclear coupling strengths before and after the delay window are not reduced. In addition, we can use the levels $m_{s}=\pm1$ to double the interaction strength (changing $\eta=1/2$ to $\eta=1$), since the nuclear spins are addressed by the control in the delay window. In contrast, $\eta=1/2$ is necessary during the whole protocol of spin addressing by standard DD, so that homonuclear spins feeling different hyperfine fields have different precession frequencies. Third, our technique allows to simultaneously address more than one nuclear spin by applying rf driving fields at the frequencies of those spins during a delay window.
Storage of electron states to a memory qubit
=============================================
Here we present more details on storing the electron qubit states to a nuclear spin memory. During the swap operations, the NV electron qubit is working in the $m_{s}=0$, $+1$ manifold. Storage of electron spin state can be realised by SWAP gates. A SWAP gate $$\text{SWAP}=\sum_{m_{s},m_{n}=0,1}|m_{s}m_{n}\rangle\langle m_{n}m_{s}|$$ swap the electron qubit states $m_{s}$ and memory qubit states $m_{n}$. In the case that relaxations of the electron and nuclear memory qubit can be neglected during the delay window, we can also use iSWAP gate $$\text{iSWAP}=\sum_{m_{s},m_{n}=0,1}e^{i(m_{s}+m_{n})^{2}\pi/2}|m_{s}m_{n}\rangle\langle m_{n}m_{s}|,$$ which introduces a phase factor $i$ when $m_{s}\neq m_{n}$. Without relaxations of the electron and nuclear memory qubit, the whole system including the environment and the memory qubit has the evolution during the delay window $$U_{\text{delay}}=\sum_{m_{s},m_{n}=0,1}|m_{s}m_{n}\rangle\langle m_{s}m_{n}|\otimes U_{m_{s},m_{n}},$$ where $U_{m_{s},m_{n}}$ are unitary evolution operators of the environment part. The effect of the iSWAP gates, $$\text{iSWAP}^{\dagger}U_{\text{delay}}\text{iSWAP}=\sum_{m_{s},m_{n}=0,1}|m_{n}m_{s}\rangle\langle m_{n}m_{s}|\otimes U_{m_{s},m_{n}},$$ is the same as using SWAP gates.
We use protected swap gates to suppress decoherence of the NV electron spin and unwanted electron-nuclear interactions during gate implementation. Using nuclear spin addressing by DD [@Staminiau2014universal; @Scasanova2015robust; @Svan2012decoherence; @Swang2015positioning] or the delayed entanglement echo presented in the main text, we implement the elementary decoherence-protected two qubit gates $u_{z\alpha}=\exp\left(i\frac{\pi}{4}\sigma_{z}I_{\alpha}\right)$ with $\alpha=x,y,z$ as well as single qubit gates for nuclear spins. Combining the gate with electron spin rotations, we achieve the gate $u_{\alpha\alpha}=\exp\left(i\frac{\pi}{4}\sigma_{\alpha}I_{\alpha}\right)$ [\[]{}e.g., $u_{yy}=\exp\left(i\frac{\pi}{4}\sigma_{x}\right)u_{zy}\exp\left(-i\frac{\pi}{4}\sigma_{x}\right)$[\]]{}. A swap gate is constructed by $u_{zz}u_{yy}u_{xx}$, where the three gates $u_{\alpha\alpha}$ commute, while $u_{yy}u_{xx}$ gives rise to the iSWAP gate.
Storage to the intrinsic nitrogen spin\[sub:Storage-to-N\]
----------------------------------------------------------
Here we describe the details of implementation of SWAP gates between the electron qubit and the intrinsic nitrogen spin qubit. For simplicity, we consider $^{14}\text{N}$, which has $99.636\%$ natural abundance and a spin $I=1$. The Hamiltonian for the NV electron and the intrinsic nitrogen spins is $$H_{\text{NV}}=DS_{z}^{2}-\gamma_{e}B_{z}S_{z}+PI_{z}^{2}-\gamma_{N}B_{z}I_{z}+A^{\parallel}S_{z}I_{z}+A^{\perp}(S_{x}I_{x}+S_{y}I_{y}), \label{eq:ElectronIntNitrogen}$$ where $\gamma_{N}=2\pi\times0.308$ $\text{kHz}/\text{G}$. We adopt the parameters for $^{14}\text{N}$ in NV centres $A^{\perp}=-2\pi\times2.62$ MHz, $A^{\parallel}=-2\pi\times2.162$ MHz, and $P=-2\pi\times4.945$ MHz [@Schen2015measurement]. The flip-flop between electron and nuclear spins are suppressed by the large energy mismatch. We have $$H_{\text{NV}}\approx DS_{z}^{2}-\gamma_{e}B_{z}S_{z}+PI_{z}^{2}-\gamma_{N}B_{z}I_{z}+A^{\parallel}S_{z}I_{z}+\sum_{m_{s}=0,\pm1}|m_{s}\rangle\langle m_{s}|h_{m_{s}},$$ where the nitrogen operators $$h_{+1}=\frac{(A^{\perp})^{2}}{D-\gamma_{e}B_{z}}
(|0_{N}\rangle\langle0_{N}|+|-1_{N}\rangle\langle-1_{N}|),$$ $$h_{0}=\frac{(A^{\perp})^{2}}{-D+\gamma_{e}B_{z}}|+1_{N}\rangle
\langle+1_{N}|+\left[\frac{(A^{\perp})^{2}}{-D+\gamma_{e}B_{z}}
-\frac{(A^{\perp})^{2}}{D+\gamma_{e}B_{z}}\right]|0_{N}\rangle\langle0_{N}|
-\frac{(A^{\perp})^{2}}{D+\gamma_{e}B_{z}}|-1_{N}\rangle\langle-1_{N}|,$$ $$h_{-1}=\frac{(A^{\perp})^{2}}{D+\gamma_{e}B_{z}}
(|0_{N}\rangle\langle0_{N}|+|+1_{N}\rangle\langle+1_{N}|),$$ describe the energy shifts caused by virtual spin flip-flop processes.
We use the electron-nitrogen coupling to implement the entangled gate $u_{zz}=\exp(i\frac{\pi}{4}\sigma_{z}\sigma_{N,z})$ in a short duration of $0.23~\mu\text{s}$, where $\sigma_{N,z}$ is the Pauli operator of the nitrogen qubit. The iSWAP gate $\text{iSWAP}=u_{yy}u_{xx}$, where $u_{xx}=P_{y}u_{zz}P_{y}^{\dagger}$ and $u_{yy}=P_{x}u_{zz}P_{x}^{\dagger}$. Here $P_{\alpha}$ denotes decoherence-protected single-qubit $\pi/2$ gates [@Svan2012decoherence] on both nuclear and electron spins at the $\alpha$ direction. The SWAP gate can be realized by $\text{SWAP}=e^{-i\pi/4}u_{zz}\text{iSWAP}$.
We simulate the SWAP gate using the protocol and achieve a gate fidelity of $F=0.987$ (defined by $F=|\text{Tr}(GU_{g})|/\text{Tr}(GG^{\dagger})$ with $G$ the evolution of ideal SWAP gate and $U_{g}$ the actual implementation) by taking into account the energy shifts on the electron and nitrogen qubits. The microwave $\pi$ pulses for the SWAP gate have pulse duration of $12.5$ ns, and the field strength for rf $\pi/2$ pulses is $15.53$ G. The magnetic field $B_{z}=0.467$ T and the LG decoupling are the same as those used in Fig. 3b in the main text. In the simulation, we adopt the Hamiltonian Eq. (\[eq:ElectronIntNitrogen\]) and the electron and nuclear spins feel all the control fields irrespective to their frequencies. Because of the high gate fidelity, we use the nitrogen spin levels $m_{N}=0,+1_{N}$ to store the NV electron qubit by ideal swap gates in producing the figures in the main text that use the nitrogen spin as quantum memory.
After the swap operation, the NV electron spin is polarized to the state $|+1\rangle$ for the delay window. During this storage, we take into account the energy shifts on the nitrogen memory. The energy shifts can also be removed by applying DD (e.g., a Hahn echo) on the nitrogen spin.
Storage to carbon-13 memory qubits\[sub:Storage-to-carbon-13\]
--------------------------------------------------------------
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We can use AXY sequences [@Scasanova2015robust] to implement $u_{\alpha\alpha}$ gates and swap the NV electron states to a $^{13}\text{C}$ memory. Compared to traditional sequences, AXY exhibits especially good spin addressability, strong robustness against detuning and amplitude errors, and the ability to continuously tune the effective interactions between NV electron and nuclear spins [@Scasanova2015robust]. Using a symmetric version of AXY sequence (see Fig. \[fig:smFigAXY\] (a)), we have the interaction Hamiltonian $H_{\text{int}}^{x}\approx\frac{1}{4}f_{k_{\text{DD}}}^{\text{s}}A_{j}^{\perp}\sigma_{z}I_{j}^{x}$ [@Scasanova2015robust]. Similarly, for anti-symmetric sequences (see Fig. \[fig:smFigAXY\] (b)), we have $H_{\text{int}}^{y}\approx\frac{1}{4}f_{k_{\text{DD}}}^{\text{a}}A_{j}^{\perp}\sigma_{z}I_{j}^{y}$. We tune $f_{k_{\text{DD}}}^{\text{s}}=f_{k_{\text{DD}}}^{\text{a}}=f_{k_{\text{DD}}}$ and use a time $t_{g}=2\pi/(f_{k_{\text{DD}}}A_{j}^{\perp})$ to implement the operation $$\text{iSWAP=}X_{\pi/2}\exp(-iH_{\text{int}}^{y}t_{g})X_{\pi/2}^{\dagger}Y_{\pi/2}^{\dagger}\exp(-iH_{\text{int}}^{x}t_{g})Y_{\pi/2},$$ where $X_{\pi/2}$ and $Y_{\pi/2}$ are NV electron $\pi/2$ gates around the directions $x$ and $y$, respectively. The inverse gate $\text{iSWAP}^{\dagger}$ can be implemented by the interchanges $X_{\pi/2}\leftrightarrow X_{\pi/2}^{\dagger}$ and $Y_{\pi/2}\leftrightarrow Y_{\pi/2}^{\dagger}$.
Another way to implement the swap gate is by using continuous DD (i.e., using spin-locking field). For the addressed nuclear spin with a distinct precession frequency, we have the effective interaction Hamiltonian $H_{\text{int}}\approx\frac{1}{2}\eta A_{j}^{\perp}(\sigma_{z}I_{j}^{x}+\sigma_{y}I_{j}^{y})$ under continuous Rabi driving (see Sec. \[sub:DD-Effective-interaction-Hamiltonia\]). An iSWAP gate corresponds to the sequence $\exp\left(-i\frac{\pi}{4}\sigma_{y}\right)\exp\left(-iH_{\text{int}}t_{g}\right)\exp\left(i\frac{\pi}{4}\sigma_{y}\right)$.
In producing the figures in the main text with a $^{13}\text{C}$ memory, we explicitly implement the swap gate operations by ideal microwave control.
Signals of delayed entanglement echo
====================================
Single spins
------------
The evolution for a target nuclear spin is $U=e^{-i\eta A_{j}^{\parallel}\tau\sigma_{z}I_{j}^{z}}$ (without considering other spins) after an interaction window with time $\tau$. Then we apply a rotation (say, along the $x$ direction) with an angle $\theta_{\text{rf}}$ on the target nuclear spin at the delay window. Following by another interaction window with time $\tau$ between two $\pi$ pulses on the electron spin (the last $\pi$ pulse is optional, but we keep it for simplicity), we have a total evolution $U=e^{i\eta A_{j}^{\parallel}\tau\sigma_{z}I_{j}^{z}}e^{-iI_{j}^{x}\theta_{\text{rf}}}e^{-i\eta A_{j}^{\parallel}\tau\sigma_{z}I_{j}^{z}}$ for interaction between the NV electron spin and a target nuclear spins. Because $k_{B}/\hbar\approx(2\pi)\times21$ GHz/K, the thermal energies are much larger than the nuclear spin Zeeman energies at relevant temperatures, and we approximate the thermal state of nuclear spins by the identity operator up to a normalization factor. Given the initial electron spin state $|\psi_{e}\rangle=(|\uparrow_{e}\rangle+|\downarrow_{e}\rangle)/\sqrt{2}$, the population left in the original equal superposition state is $$P_{\text{NV}}=\frac{1}{2}+\frac{1}{4\mathcal{N}}\text{Tr}(U_{+}U_{-}^{\dagger}+U_{-}U_{+}^{\dagger}),$$ with $U_{\pm}=e^{\pm i\eta A_{j}^{\parallel}\tau I_{j}^{z}}e^{-i\theta_{\text{rf}}I_{j}^{x}}e^{\mp i\eta A_{j}^{\parallel}\tau I_{j}^{z}}$ and $\mathcal{N}=2I+1$ the dimension of nuclear spin ($I=1/2$ for $^{13}\text{C}$). For spin-$\frac{1}{2}$, we obtain $P_{\text{NV}}=(1+L)/2$ and the coherence $L=L(A_{j}^{\parallel},\theta_{\text{rf}})$, where the single spin contribution $$L(A_{j}^{\parallel},\theta_{\text{rf}})=\frac{1}{2}[(1-\cos\theta_{\text{rf}})\cos(2\eta A_{j}^{\parallel}\tau)+1+\cos\theta_{\text{rf}}].\label{eq:L1spin}$$ The range of single spin contribution $\cos\theta_{\text{rf}}\leq L(A_{j}^{\parallel},\theta_{\text{rf}})\leq1$. When there are a number $p$ of nuclear spins at indistinguishable Larmor frequencies, the coherence $L=L^{p}(A_{j}^{\parallel},\theta_{\text{rf}})$. For the case of a $\theta_{\text{rf}}=\pi$ rotation, $L(A_{j}^{\parallel},\theta_{\text{rf}})=\cos(2\eta A_{j}^{\parallel}\tau)$.
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Coupled spin pairs
------------------
Under strong magnetic field, there are three types of spin pairs [@Szhao2011atomic], according to their relative amplitudes between the dipolar coupling strength $$d_{j,k}=\frac{\mu_{0}}{4\pi}\frac{\gamma_{j}\gamma_{k}}{|\boldsymbol{r}_{j,k}|^{3}}\left[1-3(\hat{z}\cdot\boldsymbol{r}_{j,k}/|\boldsymbol{r}_{j,k}|)^{2}\right],$$ and the difference of the hyperfine components $\delta_{j,k}=A_{j}^{\parallel}-A_{k}^{\parallel}$. Spin pairs with $\delta_{j,k}$ and $d_{j,k}$ the same order of strengths, which we call type-s, can be detected by DD because it modulates the electron spin coherence [@Szhao2011atomic; @Sshi2014sensing]. This type of spin pairs can be detected by our method without application of rf driving at the delay window (i.e., $\theta_{\text{rf}}=0$). Other types of spin pairs, type-h ($|\delta_{j,k}|\gg|d_{j,k}|$) and type-d ($|d_{j,k}|\gg|\delta_{j,k}|$), were regarded as unobservable, because their modulation on electron coherence is negligible [@Szhao2011atomic]. Applying rf control at the delay window, we can directly detect, identify, and control type-h and -d pairs.
*1. type-h pair* ($|\delta_{j,k}|\gg|d_{j,k}|$)
For type-h pairs, homonuclear spin-flip processes are suppressed and therefore the precession frequency of each nuclear spin is split by $d_{jk}$ by the internuclear interaction $H_{\text{dip}}=d_{jk}I_{j}^{z}I_{k}^{z}$. The nuclear spin flip by the rf pulse at the delay window is conditional to the state of the other spin in the nuclear spin pairs. Similar to the calculation of single spin, when we apply a rf pulse to rotate the spin $j$, the population left in the original equal superposition state is $$P_{\text{NV}}=\frac{1}{2}+\frac{1}{4\mathcal{N}}\text{Tr}(U_{+}U_{-}^{\dagger}+U_{-}U_{+}^{\dagger}),$$ with $\mathcal{N}=4$ for the dimension of the spin pair. Here $U_{\pm}=e^{-i\theta I_{j}^{x}\otimes|\uparrow\rangle\langle\uparrow|}e^{\mp i2\eta A_{j}^{\parallel}\tau I_{j}^{z}\otimes|\uparrow\rangle\langle\uparrow|}$ for applying a $\theta_{\text{rf}}=\pi$ spin flip on nuclear spin $j$ conditioned on the state $|\uparrow\rangle$ of spin $k$. The corresponding population signal $P_{\text{NV}}=1/2+[1+\cos(2\eta A_{j}^{\parallel}\tau)]/4 \geq 1/2$ for applying $\theta_{\text{rf}}=\pi$ on resonant to the spin $j$. Note that the signal is different from the case of single spin, as shown in Figs. \[fig:smFigPair\] (a) and (c).
*2. type-d pair* ($|\delta_{j,k}|\ll|d_{j,k}|$)
For type-d pair of homonuclear spins, the interaction takes the form $H_{\text{dip}}=d_{jk}(3I_{j}^{z}I_{k}^{z}-\boldsymbol{I}_{j}\cdot\boldsymbol{I}_{k})/2$ under a strong magnetic field [@Svandersypen2005nmr]. For the nuclear spins of $I=1/2$, the composited spin cluster has a singlet state with a composited spin $J=0$, $|s_{\text{n}}\rangle=(|\uparrow\downarrow\rangle-|\downarrow\uparrow\rangle)\sqrt{2}$. The triplet states with $J=1$ are $|1_{\text{n}}\rangle=|\uparrow\uparrow\rangle$, $|0_{\text{n}}\rangle=(|\uparrow\downarrow\rangle+|\downarrow\uparrow\rangle)/\sqrt{2}$, and $|-1_{\text{n}}\rangle=|\downarrow\downarrow\rangle$. A radio frequency control $H_{rf}=\gamma_{\text{n}}B_{x}\cos(\omega_{rf}t)(I_{j}^{x}+I_{k}^{x})$ can be written as $$H_{rf}=\sqrt{2}\gamma_{\text{n}}B_{x}\cos(\omega_{rf}t)(|1_{\text{n}}\rangle+|-1_{\text{n}}\rangle)\langle0_{\text{n}}|+\text{h.c.}.$$ Tuning the rf frequency $\omega_{rf}$ around the splitting between $|0_{\text{n}}\rangle$ and $|\pm1_{\text{n}}\rangle$, the nuclear spins are rotated. The nuclear state $|0_{\text{n}}\rangle$ has an energy of $-d_{jk}/2$, while the energies for $|\pm1_{\text{n}}\rangle$ are $\pm\omega_{\text{n}}+d_{jk}/4$, where $\omega_{\text{n}}$ is the nuclear precession frequency shifted by the hyperfine field. Therefore, the transition frequencies between $|0_{\text{n}}\rangle$ and $|\pm1_{\text{n}}\rangle$ are $\omega_{n}\pm 3d_{jk}/4$, shifted by the dipolar coupling. Different from the case of single spins that a spin flip requires a rf driving time $2\pi/(\gamma_{\text{n}}B_{x})$, here the effective rf control field is increased and transitions between $|0_{\text{n}}\rangle$ and $|\pm1_{\text{n}}\rangle$ can be finished in a time of $\sqrt{2}\pi/(\gamma_{\text{n}}B_{x})$. This time difference is a signature to distinguish the signals from that of single spins. Before a spin flip of the nuclear spin pair, the interaction Hamiltonian with the NV electronic spin reads $H=\eta\sigma_{z}(A_{j}^{\parallel}I_{j}^{z}+A_{k}^{\parallel}I_{k}^{z})$, i.e., $H=\eta\sigma_{z}A^{\parallel}(|+1_{\text{n}}\rangle\langle+1_{\text{n}}|-|-1_{\text{n}}\rangle\langle-1_{\text{n}}|)$. After a time $\tau$, we flip the electron spin and the nuclear triplet state, e.g., with a transition $|0_{\text{n}}\rangle\leftrightarrow|-1_{\text{n}}\rangle$ in a delay window. We can achieve an effective interaction $H=\eta\sigma_{z}A^{\parallel}(-|0_{\text{n}}\rangle\langle0_{\text{n}}|+|-1\rangle\langle-1|)$ after the delay window. After another delay time $\tau$, the join evolution up to single qubit operations reads $U=\exp\left[-i\eta\sigma_{z}A_{z}\tau(|+1_{\text{n}}\rangle\langle+1_{\text{n}}|-|0_{\text{n}}\rangle\langle0_{\text{n}}|)\right]$. The electron spin population modulated by the spin pair signal is $$P_{\text{NV}}=\frac{1}{2}+\frac{1}{4}[\cos(2\eta A_{z}\tau_{z})+1].$$ by using $P_{\text{NV}}=1/2+\text{Tr}(U_{+}U_{-}^{\dagger}+U_{-}U_{+}^{\dagger})/(4\mathcal{N})$ with $\mathcal{N}=4$.
In summary for type-d pairs, the rf pulse drive the transitions between $|0_{\text{n}}\rangle$ and $|\pm1_{\text{n}}\rangle$ of the nuclear spin triplet, with a frequency difference of $3d_{jk}/2$ and the Rabi frequency $\sqrt{2}$ times of the one for single spins, as shown in Figs. \[fig:smFigPair\] (b) and (d).
Therefore the delayed entanglement echo enables the detection, identification, and control of differnt types of spin pairs, and it is a useful tool to extract information of more complicated spin clusters.
Combining the interaction windows with dynamical decoupling
===========================================================
We can further protect the NV centre during the interaction windows with DD. Application of a DD pulse sequence (with the sequence duration $\tau$) in an interaction window, gives the interaction of the form $H_{\text{int}}\approx(\eta/2) f_{k_{\text{DD}}}^{\text{s}}\sum_{j}^{\prime}A_{j}^{\perp}\sigma_{z}I_{j}^{x}$ for symmetric sequences (see Sec. \[sub:DD-Effective-interaction-Hamiltonia\]). Here the summation $\sum_{j}^{\prime}$ is over the spins that are not resolved by the DD sequence (with a frequency uncertainty of $\sim1/\tau$). The interaction Hamiltonian takes the same form as the effective Hamiltonian under a strong magnetic field, but with the spin operators $I_{j}^{x}$ instead of $I_{j}^{z}$ (as well as the coupling constants).
In this manner, the effective interaction $H_{\text{int}}$ during the interaction windows does not commute with the spin precession during the delay window. After the evolution during the delay window driven by $\omega_{j}I_{j}^{z}$, the nuclear spins could have suffered different evolutions on $I_{j}^{x}$ because of possible differences of $\sim1/\tau$ in their precession frequencies. This static inhomogeneous of spin precession can be removed by DD on those nuclear spins. A rf $\pi$ pulse on the nuclear spins effectively reverses the evolution driven by the precession Hamiltonian $\omega_{j}I_{j}^{z}$. Therefore, the interaction Hamiltonians in the two interaction windows before and after the delay window are the same, when we apply a two-pulse CP sequence during the delay window. With a microwave $\pi$ pulse applied on the NV electron spin before the second interaction window, the coherence of NV electron spin is preserved by the delayed entanglement echo at the end of the second interaction window.
To address desired nuclear spins, in the delay window we apply rf driving with the frequencies on the target spins to rotate the target nuclear spins. The azimuthal direction $\phi_{\text{rf}}$ of rf driving field is controlled by the rf phase and we choose $\phi_{\text{rf}}$ the same as the pulse direction of the CP sequence in the rotating frame of nuclear spin precession. When the rf driven rotation does not commute with the interaction Hamiltonian, it breaks the erasing process of delayed entanglement echo on the target spins, and thereby, we address the nuclear spins in a highly selective way. On the other hand, when the rf direction is parallel to the azimuthal angle of a target nuclear spin, the rf driving commutes with the interaction window and the electron-nuclear entanglement is removed after the echo. By measuring the rf phases $\phi_{\text{rf}}$ which cause vanishing signal dips, we obtain the relative directions of nuclear spins.
Simulation details
==================
The $^{13}\text{C}$ spins of the diamond samples are randomly distributed around the NV centre. In simulations for NV dynamics, we randomly distribute $^{13}\text{C}$ spins around the NV centre and select samples that do not contain $^{13}\text{C}$ nuclei within a distance of $0.714$ nm from the NV centre (corresponding to 274 atomic sites), so that the hyperfine interactions between the $^{13}\text{C}$ nuclei and NV electron spin are simply described by the dipolar coupling Eq. (\[eq:Aj\]). The probability of getting this kind of samples is $\sim5\%$ for natural abundance of $1.1\%$ and is higher for lower abundances. Because of low spin concentration, simulations are accurate enough by grouping nuclear spins into interacting clusters and neglecting the intercluster interactions [@Smaze2008electron]. Because of the application of control fields [\[]{}magnetic fields of the form $B_{\text{c}}\cos(\omega_{\text{c}}t+\phi_{\text{c}})$[\]]{}, the total Hamiltonians for the simulations become time-dependent. To simulate the control fields, we sample the control fields in a time step of the minimum values of $0.01\times2\pi/\omega_{\text{c}}$.
We adopt the coordinate system $\hat{z}=[111]/\sqrt{3}$ along the symmetry axis of NV centre and the orthogonal unit vectors $\hat{x}=[1\bar{1}0]/\sqrt{2}$ and $\hat{y}=[11\bar{2}]/\sqrt{6}$ to record the positions of $^{13}\text{C}$ spins $\boldsymbol{r}_{j}=[\boldsymbol{r}_{j}\cdot\hat{x},\boldsymbol{r}_{j}\cdot\hat{y},\boldsymbol{r}_{j}\cdot\hat{z}]$, which are measured relative to the location of the NV electron spin at the origin $[0,0,0]$.
The sample used for Fig. 2 of the main text contains the host nitrogen and 736 $^{13}\text{C}$ spins. The simulations are converged for clusters with up to 7 spins and intercluster interactions $\lesssim2\pi\times70$ Hz. For this sample, a spin echo $\pi$ pulse extends the electron coherence times from $T_{2}^{*}\approx4$ $\mu$s to $\sim1$ ms under magnetic fields much larger than the hyperfine fields at the nuclear spins (see Fig. \[fig:smFigHahn\]), consistent with experiments [@Scramer2015repeated] and theories [@Szhao2012decoherence]. Fig. \[fig:smFigHahn\] (d) also shows that the NV coherence time can be much longer if the nuclear-nuclear interactions are suppressed.
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In simulations for the results of Fig. 3a of the main text, the electron spin relaxation is solved by Lindblad master equations. The signals comes from the addressing to an isolated nuclear spin located at $\boldsymbol{r}_{j}=[0.0,-1.9635,-0.8925]$ nm with the hyperfine field components $A_{j}^{\parallel}=2\pi\times1.49$ kHz and $A_{j}^{\perp}=2\pi\times2.93$ kHz. The $^{13}\text{C}$ memory qubit located at $[-0.714,0.0,0.357]$ nm has $A_{m}^{\parallel}=-2\pi\times31.26$ kHz and $A_{m}^{\perp}=2\pi\times29.24$ kHz. We use AXY sequences to realize the iSWAP gate (see Sec. \[sub:Storage-to-carbon-13\]) on the $^{13}\text{C}$ qubit with a gate time $2t_{g}\approx318$ $\mu$s, using a total number of $\sim152$ composite $\pi$ pulses (explicitly, 760 elementary $\pi$ pulses since one composite pulse in AXY sequences has 5 elementary $\pi$ pulses).
In Fig. 3b of the main text, the two spins in a C-C bond are located at $\boldsymbol{r}_{j}=[-1.2495,0.714,-0.1785]$ nm and $\boldsymbol{r}_{k}=[-1.33875,0.80325,-0.26775]$ nm, which imply a dipolar coupling of $d_{j,k}=2\pi\times1.37$ kHz. The hyperfine components $A_{j}^{\parallel}=-2\pi\times4.94$ kHz and $A_{j}^{\perp}=2\pi\times5.33$ kHz for spin $j$, while $A_{k}^{\parallel}=-2\pi\times3.72$ kHz and $A_{k}^{\perp}=2\pi\times4.2$ kHz for spin $k$.
In Fig. 3c of the main text, the two separated spins located at $\boldsymbol{r}_{j}=[0.0,-1.9635,-0.8925]$ nm (i.e., the target spin in Fig. 3a) and $\boldsymbol{r}_{k}=[0.0,1.2495,1.9635]$ nm have similar hyperfine components. The values for the second spin $A_{k}^{\parallel}=2\pi\times1.43$ kHz and $A_{k}^{\perp}=2\pi\times2.28$ kHz. We protect the interaction window with $\tau=0.5$ ms, by using CP sequences with 1000 microwave $\pi$ pulses (corresponding to 200 composite $\pi$ pulses if we use AXY sequences). The two $\pi$ pulses on the $^{13}\text{C}$ spins are implemented by rf fields with a Rabi frequency of $2\pi\times20$ kHz.
In the simulation with optical illumination used in Fig. 3d of the main text, we adopt the Lindblad model and parameters of the experimental paper [@Smaurer2012room]. The memory $^{13}\text{C}$ spin is similar to the one used in ref. , with the location $\boldsymbol{r}_{m}=[-0.108,-0.295,-1.74]$ nm and hyperfine components $A_{m}^{\parallel}=-2\pi\times1.69$ kHz and $A_{m}^{\perp}=2\pi\times5.4$ kHz. We use spin lock technique on the NV electron qubit with the spin lock frequency on resonant to the precession frequency of the $^{13}\text{C}$ memory to perform a iSWAP gate (see Sec. \[sub:Storage-to-carbon-13\]). To implement a complete SWAP gate, we use delayed entanglement echo on the $^{13}\text{C}$ memory after the iSWAP operation. The delay window for the $^{13}\text{C}$ memory uses a 20-pulse CP sequence with duration $\approx100$ $\mu$s for a protected rf $\pi$ gate. The total SWAP gate time for the simulation is $\approx584$ $\mu$s, which can be reduced by a factor of two if we use the electron levels $m_{s}=\pm1$. The gate can be further protected by storing the electron state to the nitrogen spin when applying the delay window for the $^{13}\text{C}$ memory. Using a $^{13}\text{C}$ memory more strongly coupled to the NV electron can also significantly reduce the required SWAP gate times. The proton spin for detection is located $4$ nm away from the NV centre, with $\boldsymbol{r}_{j}=[2.31,2.31,2.31]$ nm (hence $A_j^{\perp}=0$ and the spin is hard to detect by traditional DD).
The procedure to detect the chemical shifts of proton spins with optical illumination is the following. We first pump the NV electron spin by optical field to initialize the NV electron spin to the state $|0\rangle$ with a fidelity $82\%$ (the fidelity is obtained for the parameters in ref. and it is higher for better samples), which is followed by using a swap operation to polarize the $^{13}\text{C}$ memory spin. Then we use optical pumping again and a microwave pulse to initialize the NV electron to a superposition state $(|0\rangle+|1\rangle)/\sqrt{2}$. After the initialization of NV electron spin and memory qubit, we let the whole system freely evolve for a time of $\tau$ to generate electron-nuclear entanglement (which can be protected by DD as shown in the main text). Then we store the NV electron spin to the memory spin by a swap gate. Subsequently we use optical illumination to decouple electron-nuclear coupling for applying a rf pulse with the length $t_{\text{rf}}\approx80$ ms. The carry frequency of rf driving is set to the target proton spins. After optical illumination, we wait for 2 $\mu$s to relax the NV electron spin back to $|0\rangle$ state. Subsequently, we use a swap gate to retrieve the quantum state of NV electron spin and to re-popularize the ancillary $^{13}\text{C}$ spin. Finally, we apply a microwave $\pi$ pulse on the electron spin and wait for another interaction window of time $\tau$ before readout of the electron spin state. We can increase the interaction to target spins by using the NV levels $m_{s}=\pm1$, as described in the main text.
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abstract: 'We study the (super-)symmetries of classical solutions in the higher spin (super-)gravity in AdS$_3$. We show that the symmetries of the solutions are encoded in the holonomy around the spatial circle. When the spatial holonomies of the solutions are trivial, they preserve maximal symmetries of the theory, and are actually the smooth conical defects. We find all the smooth conical defects in the $sl(N), so(2N+1),sp(2N), so(2N), g_2$, as well as in $sl(N|N-1)$ and $osp(2N+1|2N)$ Chern-Simons gravity theories. In the bosonic higher spin cases, there are one-to-one correspondences between the smooth conical defects and the highest weight representations of Lie group. Furthermore we investigate the higher spin black holes in $osp(3|2)$ and $sl(3|2)$ higher spin (super-)gravity and find that they are only partially symmetric. In general, the black holes break all the supersymmetries, but in some cases they preserve part of the supersymmetries.'
author:
- 'Bin Chen$^{1,2,3}$[^1], Jiang Long$^{1,2}$[^2] and Yi-Nan Wang$^{1}$[^3]'
title: 'Conical Defects, Black Holes and Higher Spin (Super-)Symmetry'
---
[*$^{1}$Department of Physics, Peking University, Beijing 100871, P.R. China\
$^{2}$State Key Laboratory of Nuclear Physics and Technology, Peking University, Beijing 100871, P.R. China\
$^{3}$Center for High Energy Physics, Peking University, Beijing 100871, P.R. China\
*]{}
Introduction
============
Symmetries of spacetime play an essential role in Einstein’s general relativity. For example, the maximally symmetric spacetime can often be taken as the vacuum of a theory. And the isometries of a black hole allow us to define the conserved quantities of the background and a test particle. Moreover, in a gravity theory with supersymmetry, supersymmetric black holes have better ultraviolet behaviors and have been one of central subjects in supergravity and string theory. Furthermore, in the context of AdS/CFT correspondence, the supersymmetries and isometries are the guide lines to find the bubbling geometry.
The notion of symmetry becomes tricky in a theory of higher spin fields. Different from usual Einstein gravity, the gauge transformation of the metric field involves the higher spin fields such that the usual notions of geometry, such as diffeomorphism and isometries, do not make much sense in the higher spin gravity. One has to find gauge invariant way to define the symmetries of a classical configuration. In the well-known AdS$_4$ Vasiliev’s higher spin theory, the fact that the higher spin gauge transformations are quite involved and there is short of classical solutions hinders us from investigating this issue. Fortunately in AdS$_3$ the higher spin gravity is much better under control in many aspects.
The higher spin gravity in AdS$_3$ has been developing quickly in the past few years. One nice feature of the AdS$_3$ higher spin gravity is that the original Vasiliev theory[^4] could be cast into a Chern-Simons gravity on a high spin algebra[@Blen1; @Blen2], and could even be truncated to a theory on a finite rank Lie algebra, if not considering the matter scalar field. Therefore the classical solutions without scalar hair can be constructed explicitly. More interestingly it was proposed[@Gaberdial] that the Vasiliev theory in AdS$_3$ could be holographically dual to a 2D ${\cal W}_{N,k}$ minimal model at the boundary. Up till now, there are two kinds of limit studied on this duality in the literature. The first one is the ’t Hooft limit, which is obtained by taking $N,k\to\infty$ while keeping the ‘t Hooft coupling $\lambda=\frac{N}{N+k}$ fixed. Under this limit the boundary theory is unitary, but the bulk theory has some troubles in counting the dual light states. See the review [@review] for recent developments and references therein. For a recent proposal, see [@Chang:2013izp]. The other one is the semi-classical limit[@Triality; @Perlmutter:2012], which is obtained by taking $c\to\infty$ while keeping $N$ fixed. In taking this limit, the level $k$ in the boundary CFT has to be negative and there are states with negative conformal weights. Hence, in this case, the theory is non-unitary. However, the bulk theory in the semi-classical limit is simpler than the one in the ‘t Hooft limit as the gauge group is of finite rank. Hence it allows us to investigate the HS/CFT correspondence in detail. As the first step to check the correspondence, one has to match the spectrum on two sides. On the CFT side, the minimal model has various representations characterized by $(\Lambda_+;\Lambda_-)$, where $\L_\pm$ are the integrable highest weight representations of the affine algebra $su(N)$ at level $k$ and $k+1$. Among them, the primary states in $(0,\L_-)$ are of particular importance. In the semiclassical limit, these primary states have conformal dimensions proportional to the central charge, indicating their non-perturbative nature. It was proposed in [@Perlmutter:2012] that the states $(0,\L_-)$ correspond to smooth conical defects(surplus) in the bulk AdS$_3$ higher spin gravity. The states $(\L_+,0)$ corresponds to scalar perturbation and the general states $(\L_+,\L_-)$ correspond to bound states of the scalar perturbation and the conical defects.
The smooth conical defects(surplus)[@Gopakumar:2012] are classical solutions of the AdS$_3$ higher spin gravity. They have the same topology as the global AdS$_3$, with a contractible spatial circle. As the usual geometric notions break down in the higher spin gravity, one has to use gauge invariant quantities to characterize these solutions. In the case of conical defect, a well-defined quantity is the holonomy of the gauge field along the contractible spatial circle. The smooth conical defect has a trivial spatial holonomy such that the corresponding gauge potential is not singular. As the corresponding states have maximally degenerate null vectors, the smooth conical defects are expected to have maximal higher spin symmetry, as the global AdS$_3$ vacuum. Another interesting class of classical solutions is higher spin black hole[@Per; @kraus:2011]. Different from the conical defects, the spatial circle of the black hole is not contractible but its time circle is. The smoothness of the higher spin black hole requires that the holonomy of the gauge field along the time circle is in the center of the gauge group. More interestingly, the trivial thermal holonomy leads to consistent thermodynamics for the higher spin black holes. On the other hand, the spatial holonomy of the gauge field for the higher spin black hole is not trivial. Hence, there is an interesting question: what is the information encoded in the spatial holonomy?
In fact, the spatial holonomy encodes the symmetry of the solution. Simply speaking, to determine how many symmetries are kept by the solution, we need to solve the following equation in holomorphic sector[^5] \_ŁA=dŁ+\[A,Ł\]=0,\[gaugetrans\] where $A$ is the flat connection. Locally the above equation could always be solved. To have a well-defined gauge transformation, we need to impose periodic boundary condition on the gauge parameter $\L$. In the end we obtain the following relation e\^[-2a\_]{}łe\^[2a\_]{}=ł, \[holo\] where $a_\phi$ is the $\phi$-component in $A$ and $\l$ is some constant matrix valued in the Lie algebra of the gauge group. When the spatial holonomy is in the center of the corresponding group, the solution is of maximal higher spin symmetries in the sense that $\l$ has maximal number of degrees of freedom. In other words, the smooth conical defect(surplus) is the maximally symmetric solution in the higher spin gravity. Actually, we show how to obtain the smooth conical defects by searching for the maximally symmetric solutions of the corresponding higher spin gravity. This turns out to be a quite effective method. We use this method to find out the smooth conical defects in $sl(N), so(2N+1),sp(2N),so(2N),g_2$ gravity theories, as well as the ones in $sl(N|N-1)$ and $osp(2N+1|2N)$ supergravities. Moreover we establish an one-to-one match between the conical defects and the highest weight representations of dual group in all cases. This exact match of the spectrum suggests that there may exist a correspondence between the finitely truncated higher spin gravities, possibly coupled to scalar matter, with some kinds of minimal models. When the spatial holonomy is non-trivial, as in the case of higher spin black hole, the solution is partially symmetric. In the case of generic higher spin black hole, the constant matrix have to be valued only in Cartan subalgebra of the gauge group, showing the black hole could have well-defined global charges.
In a higher spin supergravity, the spatial holonomy not only encodes the symmetries of the solution but also supersymmetries preserved by the solution. The supersymmetric configurations are of particular interest in a supersymmetric theory. They often have nice properties and are easier to deal with. The supersymmetric solutions in the higher spin (super-)gravity have been discussed recently in [@Justin:susy; @Tan:2012; @Hikida:3]. In [@Justin:susy], the higher spin generalization of Killing spinor equation has been proposed. And in [@Hikida:3], the maximally supersymmetric conical defects have been discussed in $sl(N|N-1)$ gravity. In the Chern-Simons supergravity, the gauge group is a supergroup. As a result, the constant matrix $\l$ in (\[holo\]) should be valued in the supergroup, with its fermionic sector being labeled by $\epsilon$. Taking into account of the boundary condition on the fermionic sector of $\L$, which could be either periodic or anti-periodic, one obtain the following relation on the spinor e\^[-2a\_]{}e\^[2a\_]{}=, \[holof\] From this relation, one may read out how many supersymmetries the solution preserve[@Justin:susy]. In usual supergravity, the extremal black holes are often supersymmetric. Therefore it is interesting to investigate if the extremal higher spin black holes can keep part of the supersymmetries as well. It turns out to be true, but the story is more interesting. Using brute force, one may solve the generalized Killing spinor equation, which is the fermionic part of (\[gaugetrans\]), to find the supersymmetric higher spin black holes. These supersymmetric black holes are exactly the ones obtained by solving holonomy equations (\[holof\]) imposed by spatial holonomy.
The structure of this paper is as follows. In section 2, we clarify the relationship of the spatial holonomy and the maximally symmetric solution, then we search for the maximally symmetric solutions in various higher spin (super-)gravity theories. In section 3, we discuss the partially (super-)symmetric solution. In section 4, we explore the black holes in the higher spin (super-)gravity theories with gauge group $osp(3|2)$, and $sl(3|2)$ and study their supersymmetries. We end this paper by some conclusion and discussions. The appendix collects the convention we use in this paper.
Maximally Symmetric Solutions
=============================
The motivation to study maximally symmetric solutions in a higher spin gravity is two-fold. Firstly, in the Einstein gravity, the maximally symmetric solution always plays important role. It is defined to be the spacetime with maximal number of globally defined Killing vectors. Actually it is unique for fixed signature and dimension, and is often regarded as the vacuum of a theory. In a supergravity theory, it could carry maximal number of Killing spinors and thus could be the maximally supersymmetric configurations. As the higher spin theory is a generalization of conventional gravity theory, it is quite interesting to search for the maximally symmetric solutions in the higher spin gauge theories. Secondly, from the HS$_3$/CFT$_2$ correspondence in the semiclassical limit, the maximally symmetric configurations in the bulk higher spin gravity should correspond to the non-perturbative state $(0,\L_-)$, which has maximally degenerate null vectors. This point has been carefully investigated in a recent paper[@Perlmutter:2012]. So searching for the maximally symmetric solutions in the higher spin gravity is a well posed and important problem. In this section, we first study this issue in the bosonic higher spin theory and then turn to the higher spin supergravity.
Maximally Symmetric Solution in Bosonic Higher Spin Gravity
-----------------------------------------------------------
First of all, we need to define what the maximally symmetric solution is in a bosonic higher spin gravity. In this section, we are searching for the solutions with the topology $D^2\times R$. We will use coordinates $(\rho, \phi, t)$, with $\phi\sim \phi+2\pi$ being a contractible cycle, and $z=x^+=t+\phi, \bar{z}=x^-=t-\phi$. We just focus on the holomorphic part of the solution here, and choose the gauge group $SL(N)$ to illustrate the problem.
To compare with states in the CFT, the solution should be asymptotic to global $AdS_3$, namely A-A\_[AdS\_3]{}\~(1).\[asym\] We can choose the highest weight gauge to set the gauge field to be of the form[@Theisen:2011] A=b\^[-1]{}a b +L\_0d\[gauge1\] where $b=\exp{L_0\rho}$ and $a=a_+ dz$ with a\_+=L\_1+\_[s=2]{}\^[N]{}\_s W\^s\_[-s+1]{}.\[gauge2\] The definition of $W^s_{-s+1}$ can be found in the appendix. We are interested in the solutions with constant $a$. And it has been shown by the asymptotic symmetry analysis [@Theisen:2011; @Henneaux:2011] that $\mathcal{W}_s$ can be identified to the spin $s$ charge. In the following, we do not distinguish $\mathcal{W}_2$ and $\mathcal{L}$ that was used in many other references.
The solution parameterized by (\[gauge1\],\[gauge2\]) has an asymptotic $W_N$ symmetry which is generated by the gauge transformation that preserve the asymptotic $AdS_3$ boundary condition (\[asym\]). To determine how many higher spin symmetries is kept by the solution, we need to solve the following equation \_A=d+\[A,\]=0,\[sym\] where $\L$ is the the parameter of the gauge transformation. This equation can always be solved locally. The $\rho$ component of the equation (\[sym\]) can be solved by =b()\^[-1]{}\_0 b(). The $+,-$ component equation of (\[sym\]) indicate that \_0=-(a\_+ z)(a\_+z) where $\lambda$ is a constant matrix taking value in $sl(N)$. Note that $\Lambda$ is nothing but the higher spin generalization of Killing vector in conventional gravity. To be globally defined, it should satisfy the periodic boundary condition in the spatial $\phi$ direction (z+2)=(z). This leads to the constraint -(2a\_+)(2a\_+)=. In order to have a maximally higher spin symmetric solution, $\lambda$ should be an arbitrary constant matrix valued in $sl(N)$. This leads to the requirement that the holonomy of the gauge field along the spatial $\phi$ cycle H\_(A)=\~ must be in the center of $SL(N)$. In other words, if $H_{\phi}(A)$ is trivial, then the solution is maximally symmetric. As a consequence, $a_+$ must be diagonizable and has different eigenvalues.
The above discussion is consistent with the results in pure gravity. In the AdS$_3$ Chern-Simons gravity, the gauge transformations of the gauge fields encodes the information of local Lorentz transformation and diffeomorphism. The maximally symmetric solution defined above is exactly the global AdS$_3$, and the constant $SL(2,\mathbb{R})$ actually correspond to the holographic one in the isometry group $SO(2,2)\simeq SL(2,\mathbb{R}) \times SL(2,\mathbb{R})$.
Obviously, the above discusion is valid for other gauge groups. Let us discuss them case by case.
### $SL(N,\mathbb{R})$ and $SL(N,\mathbb{C})$
The center of $SL(N,\mathbb{R})$ is different for odd or even $N$. For odd $N$, its center is $I$, so we have H\_(A)\~\~I.\[holo1\] The equation (\[holo1\]) only depends on the eigenvalues of $a_+$. Assumed the eigenvalue of $a_+$ to be a\_+\~diag(\_1,\_2,,\_N), the holonomy condition (\[holo1\]) tells us \_1=i n\_1, \_2=i n\_2,\_N=i n\_N\[re\] with $n_1,n_2,\cdots,n_N\in \mathbb{Z}$. For $a_+$ to be diagonalizable, we require $n_i\neq n_j$ for $i\neq j$. It is also convenient to assume $n_1>n_2\cdots>n_N$. Note that the traceless condition of $sl(N)$ requires \_[i=1]{}\^[N]{}n\_i=0\[re2\]. Moreover, for the $SL(N,\mathbb{R})$ case, $a_+$ must be real, which impose further conditions on $n_i$. Let us find the consequences of (\[re\])(\[re2\]) for the higher spin charges $\mathcal{W}$. We take $N=3$ to illustrate the point. In this case, (\[gauge2\]) becomes a\_+=L\_1+L\_[-1]{}+\_3W\_[-2]{}\^3. Note $\mathcal{L},\mathcal{W}_3$ are proportional to the trace of the power of $a_+$. Since we require the charge $\mathcal{W}_3$ to be real, the $n_i$ should be (n\_1,n\_2,n\_3)=(n, 0, -n). Thus we have =, \_3=0. Here $n=1$ corresponds to global $AdS_3$ embedded in $SL(3,\mathbb{R})$, and the other solutions with $n\ge2$ correspond to smooth conical surplus studied in [@Gopakumar:2012]. The vanishing of spin 3 charge originates from the reality condition on the gauge potential. More generally for all odd $N$, the condition of a real connection always leads to vanishing odd spin charges.
For even $N$, its center is $\pm I$, so we have H\_(A)\~\~I.\[holo2\] As before, the condition (\[holo2\]) is relevant to the eigenvalue of $a_+$. We assume a\_+\~diag(\_1, \_2,,\_N). If the holonomy is $I$, then \_i=i n\_i, n\_i. If the holonomy is $-I$, then \_i=i (n\_i+), n\_i. In the case of $N=2$, we find that =, n\^+. The holonomy is $-I$ for odd $n$, and $I$ for even $n$. When $n=1$, the solution is just global AdS$_3$, while when $n\ge 2$, the solutions are the smooth conical surplus.
We note that all the maximally symmetric solutions in $N=2,3$ have vanishing higher spin charge. However, this situation changes when $N\ge4$. We take $N=4$ as an example. We find the eigenvalues of $a_+$ to be (\_1,\_2,\_3,\_4)=i(n\_1,n\_2,-n\_2,-n\_1), where $n_i$ can be intergar or half intergar, depending on the holonomy, or the choice of the center. The spin 3 charge $\mathcal{W}_3$ is still zero, but the even spin 2 and spin 4 charges are nonzero &=&(n\_1\^2+n\_2\^2)C\_2(n)\
\_4&=&(C\_4(n)-C\_2(n)\^2) where the $C_2, C_4$ are the Casimirs in $SL(4,\mathbb{R})$ and $\rho$ is the corresponding Weyl vector . These are the conical defects or surplus in the spin 4 gravity studied in [@Gopakumar:2012]. They have nonvanishing higher spin charges. The same feature holds for all the maximally symmetric solutions when $N\ge4$.
Note that our maximally symmetric solutions in the higher spin gravity are just the smooth conical defects(surpluses) studied in [@Gopakumar:2012; @Perlmutter:2012]. However, the work in [@Gopakumar:2012] was motivated in matching the conical defects to the primary states in CFT with the same global charges, while here we have shown that the smooth conical defects(surpluses) should have been discovered by simply symmetry consideration.
To match the primary states in the $CFT$ side, the $SL(N,\mathbb{C})$ case is also important. One need do Euclidean continuation to match the spectrum. The center of $SL(N,\mathbb{C})$ is $e^{\frac{-2\pi i m}{N}}1_{N\times N}$, hence the eigenvalues of $a_+$ are \_i=i(m\_i-), m\_i, i=1,2,, N. The traceless condition of $SL(N,\mathbb{C})$ leads to \_[i=1]{}\^[N]{}m\_i=m. The $m_i$ can be shifted to set $m_N=0$. To be in match with the CFT states, one needs the Young diagram of $su(N)$. A Young diagram of $su(N)$ includes $N-1$ rows, each row has $r_i(r_N=0)$ boxes. This Young diagram is in one to one correspondence with the highest weight state $(0,\Lambda_-)$ with \_-=(\_1,,\_N) where \_i=r\_i-. To relate it to the gravity solutions, we can define r\_i=m\_i-(N-i) such that the eigenvalues of the holonomy could be rewritten as -i\_i=r\_i-+-i=\_i+\_i where $\rho_i=\frac{N+1}{2}-i$ is the Weyl vector of $su(N)$. Hence we find a one to one correspondence between the bulk maximally symmetric solution and the highest weight state $(0,\Lambda_-)$.
### $Sp(2N,\mathbb{R})$ and $Sp(2N,\mathbb{C})$
These cases are motivated by the proposed even spin minimal model hologaphy[@Ah; @Gopa; @Gaberdial:2012]. Note that the center of $Sp(2N,\mathbb{R})$ and $Sp(2N,\mathbb{C})$ are the same, which can be $\pm I$. Then the holonomy now is H\_(A)\~(2a\_+)\~I The eigenvalues of $a_+$ can be parameterized as a\_+\~diag(\_1,\_2,,\_N,-\_N,-\_[N-1]{},,-\_1). If the holonomy is chosen to be $I$, then \_i=i n\_i, n\_i. If the holonomy is chosen to be $-I$, then \_i=i(n\_i+), n\_i. To make sure $a_+$ is diagonalizable, one has an additional requirement that $n_i\neq n_j$ for all $i,j$. Hence it is convenient to assume $n_1>n_2>\cdots >n_N$.
On the other hand, a representation of $so(2N+1)$ can be parametrized by its highest weight, as $N$ numbers $r_1\geq r_2\geq\cdots\geq r_N\geq 0$ (see 1.65 in [@Dictionary]). There are two kinds of representations: the vector representation with all $r_i$’s being integers and the spinor representation with all $r_i$’s being half-integer. The relation between the weight $\Lambda$ and $r_i$ turns out to be: =\_[i=1]{}\^[N-1]{}(r\_i-r\_[i+1]{})\_i+2 r\_N\_N where $\lambda_i$ is the $i$-th fundamental weight: \_1&=&e\_1\
\_2&=&e\_1+e\_2\
&&\
\_[N-1]{}&=&e\_1+e\_2++e\_[N-1]{}\
\_N&=&
The Weyl vector is =\_[i=1]{}\^N \_i=\_[i=1]{}\^N (N+-i)e\_i. Hence in this case the correspondence reads -i\_i=\_i+\_i=r\_i+N+-i. Note that the vector representations of $so(2N+1)$ exactly correspond to half-integer valued $-i\th_i$’s, which are smooth conical defects whose holonomies are in the center $-I$ of $Sp(2N,\mathbb{R})$. For example, the trivial representation $r_1=r_2=\cdots=r_N=0$ corresponds to the $AdS_3$ vacuum with $a_+=i\hspace{1mm}diag(N-\frac{1}{2},N-\frac{3}{2},\cdots,\frac{1}{2},-\frac{1}{2},\cdots,-(N-\frac{1}{2}))$. On the other hand, the spinor representations of $so(2N+1)$ have half-integer valued $r_i$, hence exactly correspond to the conical defects whose holonomy is in the center $I$ of $Sp(2N,\mathbb{R})$. Therefore we see that each highest weight state of $so(2N+1)$ is in exact match with the smooth conical defects in the higher spin gravity with gauge group $Sp(2n,\mathbb{R})$ or $Sp(2n,\mathbb{C})$.
### $SO(2N+1,\mathbb{R})$ and $SO(2N+1,\mathbb{C})$
.
This is another realization of even spin gravity. Again, the centers of $SO(2N+1,\mathbb{R})$ and $SO(2N+1,\mathbb{C})$ are the same, being $I$. The eigenvalues of $a_+$ can be parameterized by a\_+\~diag(\_1,,\_N,0,-\_N,,-\_1). The holonomy requires \_i=i n\_i, n\_i. As before, the diagonalizable condition of $a_+$ requires $n_i$ are all distinct numbers: $n_1>n_2>\cdots>n_N$.
The highest weights of $sp(2N)$ representations are parametrized by $N$ integers $r_1\geq r_2\geq\cdots\geq r_N\geq 0$ (see 1.66 in [@Dictionary]). The relation between the weight $\Lambda$ and $r_i$ is: =\_[s=1]{}\^[N-1]{}(r\_s-r\_[s+1]{})\_s+r\_N\_N=\_[i=1]{}\^[N]{}r\_i e\_i,\
where the fundamental weights $\lambda_i$ are: \_1&=&e\_1,\
\_2&=&e\_1+e\_2,\
&&\
\_[N]{}&=&e\_1+e\_2++e\_N. The Weyl vector is =\_[i=1]{}\^N \_i=\_[i=1]{}\^N (N+1-i)e\_i. In this case the correspondence reads n\_i=-i\_i=\_i+\_i=r\_i+N+1-i. Therefore we find an exact match of the smooth conical defects in $SO(2N+1)$ Chern-Simons gravity and the highest weight representations of $sp(2N)$.
Note there is an interesting “duality” between $B_N$ and $C_N$ Lie algebras: the smooth conical defects in $B_N$ gravity could correspond to a $C_N$-type highest weight representation and the smooth conical defects in $C_N$ gravity could correspond to a $B_N$-type highest weight representation.
### $SO(2N,\mathbb{R})$ and $SO(2N,\mathbb{C})$
The groups $SO(2N,\mathbb{R})$ and $SO(2N,\mathbb{C})$ have the same center $\pm I$. For the smooth conical defects, $e^{2\pi a_+}$ has to be in the center. The generic diagonalized form of $a_+$ can be written as: a\_+=diag(\_1,\_2,,\_N,-\_N,,-\_1)=diag(in\_1,in\_2,,in\_N,-in\_N,,-in\_1). If the spatial holonomy being in the center $I$, $n_i$’s take value in $\mathbb{Z}$, while if the spatial holonomy being the center $-I$, $n_i$’s take value in $\mathbb{Z}+\frac{1}{2}$. The diagonalizable condition requires $n_1>n_2> \cdots >n_N\geq 0$.
The fundamental weights in $so(2N)$ are: \_1&=&e\_1\
\_2&=&e\_1+e\_2\
&&\
\_[N-2]{}&=&e\_1+e\_2++e\_[N-2]{}\
\_[N-1]{}&=&e\_1+e\_2++e\_[N-1]{}-e\_N\
\_N&=&e\_1+e\_2++e\_[N-1]{}+e\_N. And the Weyl vector is =\_[i=1]{}\^N \_i=\_[i=1]{}\^N (N-i)e\_i.
A highest weight representation of $so(2N)$ is labelled by $N$ numbers $r_1\geq r_2\geq \cdots\geq r_N\geq 0$, where $r_i$ could all be integer or half-integer[@Gopa] &&=\_[i=1]{}\^[N-2]{}(r\_i-r\_[i+1]{})\_i+(r\_[N-1]{}-r\_N)\_[N-1]{}+(r\_[N-1]{}+r\_N)\_N= \_[i=1]{}\^[N]{}r\_i e\_i\
&&(+)\_i=r\_i+N-i When all $r_i$’s are integers, they correspond to vector representations and if they are all half-integers, they correspond to spinor representations.
There is an one-to-one correspondence between $n_i$ and $r_i$ n\_i=-i\_i=(+)\_i=r\_i+N-i, both of which have same range. For example, the vacuum configuration has the eigenvalues $(n_1,n_2,n_3,n_4)=(3,2,1,0)$, which exactly corresponds to the trivial representation $(r_1,r_2,r_3,r_4)=(0,0,0,0)$. Therefore we establish the correspondence between the highest weight representations of $so(2N)$ and the smooth conical defects in $SO(2N)$ higher spin gravity.
### $G_2(\mathbb{R})$ and $G_2(\mathbb{C})$
In this case, the corresponding higher spin gravity has only spin 2 and spin 6 fields[@truncated]. The centers of $G_2(\mathbb{R})$ and $G_2(\mathbb{C})$ are both trivial, hence the diagonalized form of $a_+$ is a\_+=diag(in\_1,in\_2,in\_3,0,-in\_3,-in\_2,-in\_1). and $\theta_i=i n_i$ with $n_i\in \mathbb{Z}$. In this case the eigenvalue equation of $(L_1+\mathcal{L}L_{-1}+\mathcal{W}_6 W^6_{-5})$ is: =0, whose roots are $0,\pm i n_1,\pm i n_2, \pm i n_3$. If the spin-6 charge $\mathcal{W}_6=0$ then clearly one requires $\mathcal{L}=n^2/4$ and $n_1=3n,n_2=2n,n_3=n$. If the spin-6 charge $\mathcal{W}_6$ is non-vanishing, the solutions need more efforts. From the algebraic relations between $n_i$ and $\mathcal{L}$ &&n\_1\^2+n\_2\^2+n\_3\^2=56\
&&n\_1\^2 n\_2\^2+n\_2\^2 n\_3\^2+n\_3\^2 n\_1\^2=784\^2, we find $n_1^4+n_2^4+n_3^4=2(n_1^2 n_2^2+n_2^2 n_3^2+n_3^2 n_1^2)$, which requires that one of $n_i$’s equals to the sum of the other twos. Without losing generality, we choose $n_1=n_2+n_3$ and let $n_1>n_2>n_3>0$. The $n_i\neq n_j$ requirement also comes from the diagonalizable condition. Hence the maximally symmetric backgrounds are parametrized by two positive integers $n_2$ and $n_3$. Accordingly the values of $\mathcal{L}$ and $\mathcal{W}_6$ are respectively: &=&,\
\_6&=&(n\_2\^2 n\_3\^2(n\_2+n\_3)\^2-(n\_2\^2+n\_2 n\_3+n\_3\^2)\^3).
On the other hand, the representation of $g_2$ is characterized by the highest weight (see 1.63 of [@Dictionary]) $\Lambda=\frac{r_1-2r_2}{3}\lambda_1+r_2 \lambda_2$, where the fundamental weights are: \_1=-e\_1-e\_2+2e\_3, \_2=e\_3-e\_2. Therefore we have +=(r\_2-r\_1-1)e\_1+(-r\_2-r\_1-2)e\_2+(r\_1-r\_2+3)e\_3. From the representation theory of $g_2$ it is required &&r\_1+r\_2=3n, r\_1,r\_2,n\
&&r\_12r\_2.\[g2constraint\] We make the following identification between $n_i$’s and $r_i$’s: n\_3&=&-(r\_2-r\_1-1)\
n\_2&=&-(-r\_2-r\_1-2)\
n\_1&=&(r\_1-r\_2+3). Note that $n_1=n_2+n_3$ is ensured. And from these expressions we get $r_1=2n_2+n_3-5,r_2=n_2-n_3-1$, satisfying $3|(r_1+r_2)$ automatically. And $r_1-2r_2=3n_3-3$, which is non-negative as long as $n_3\geq 1$. Therefore we find the pair $(r_1,r_2)$ on the CFT side exactly corresponds to $(n_2,n_3)$ on the gravity side. For example,
1. The smallest values of $(n_2,n_3)$: $(2,1)$ corresponds to $(r_1,r_2)=(0,0)$, which is the trivial representation of $G_2$. This means the trivial representation corresponds to the global AdS$_3$ vacuum.
2. $(n_2, n_3)=(3,1)$ corresponds to $(r_1,r_2)=(2,1)$, which is the 7-dimensional representation of $G_2$.
3. $(n_2, n_3)=(3,2)$ corresponds to $(r_1,r_2)=(3,0)$, which is the 14-dimensional representation of $G_2$.
There is an exact match of the maximally higher spin symmetric solutions and the highest weight states of $G_2$ representation. The discussion on $g_2$ case shows that there need special care in dealing with the exceptional Lie group. In principle it is possible to deal with the $F_4, E_6, E_7$ and $E_8$ groups. We do not include them here.
Maximally Symmetric Solution in Higher Spin Supergravity
--------------------------------------------------------
This subsection is to search for maximally symmetric solutions in higher spin supergravity. The asymptotic analysis of the higher spin supergravity has been given in [@Cheng; @Ray]. The $\mathcal{N}=2$ supersymmetric extension of the HS/CFT duality has been proposed in [@Hikida:1], which relates the three-dimensional $\mathcal{N}=2$ supersymmetric higher spin theory [@Pro] to the Kazama-Suzuki minimal model [@KS:1; @KS:2]. The $\mathcal{N}=1$ version of duality was proposed in [@Hikida:2]. Aspects of conical defects and the higher spin black holes have been partly studied in [@Justin:susy; @Tan:2012; @Hikida:3]. Here we would like to search for the maximally symmetric solutions, which are asymptotic to AdS$_3$ and preserve the maximal symmetry of the theory. Without losing generality, we take $sl(N|N-1)$ to give an illustration[^6]. Similar to the bosonic case, we need to impose appropriate asymptotic condition on the solution. This condition is the same as (\[asym\]) A-A\_[AdS\_3]{}\~(1)\[asymp\] with A=b\^[-1]{}a b+L\_0d. But now $a=a_+dz$ is changed slightly as the bosonic spectrum changes, a\_+=L\_1+\_[s=2]{}\^[N]{}\^[(1)]{}\_sW\^[(1)s]{}\_[-s+1]{}+\_[s=2]{}\^[N-1]{}\^[(2)]{}\_sW\^[(2)s]{}\_[-s+1]{}+J where $\mathcal{W}^{(1)}_s,\mathcal{W}^{(2)}_s$ are the corresponding higher spin $s$ charges. $\mathcal{V}$ is the $U(1)$ charges. The matrix generators are given in Appendix. Note that we have turned off the fermionic generators as in supergravity when searching for classical solutions.
Similarly, from the requirement \_ A=d+\[A,\]=0, the gauge parameter could be locally written as =b\^[-1]{}e\^[-a\_+z]{}e\^[a\_+z]{}b with $\lambda$ is a constant supermatrix taking value in $sl(N|N-1)$. If we require $\l$ to be arbitrary supermatrix, then the corresponding solution is maximally symmetric.
The gauge parameter $\lambda$ can be decomposed into the bosonic parameter $\xi$ and the fermionic parameter $\epsilon$ =+. Since the background solution $A$ contains only the bosonic generator, the requirements on two parameters $\xi$ and $\epsilon$ are decoupled so can be studied separately. The bosonic gauge parameter should satisfy periodic boundary condition in the spatial $\phi$ direction, while the fermionic gauge parameter should satisfy anti-periodic or periodic boundary condition in the spatial $\phi$ direction, \_(z+2)=\_(z), \_(z+2)=\_(z) or equivalently, e\^[-2a\_+]{}e\^[2a\_+]{}=, e\^[-2a\_+]{}e\^[2a\_+]{}=\[hol\] The bosonic relation tells us that the holonomy H\_(A)\~e\^[2a\_+]{} should be the center of the bosonic subalgebra of the corresponding superalgebra. For $sl(N|N-1)$ we are discussing, it is just the center of $sl(N)\oplus sl(N-1)\oplus u(1)$. The fermionic part of (\[hol\]) is a bit more complex due to the anti-periodic boundary condition. It constrains the holonomy along the $\phi$ cycle as well. For $sl(N|N-1)$, the bosonic supermatrix can be constructed by the anticommutator of two fermionic supermatrix, hence the first condition in (\[hol\]) will be satisfied automatically provided that the second condition is satisfied. Namely, the maximally supersymmetric solution automatically has maximal bosonic symmetries.
Here we construct the maximally symmetric solutions in two higher spin supergravity to illustrate the previous discussion. We constrain ourselves to real connection. The Euclidean continuation could be important but we do not include here.
1. $sl(N|N-1)$. This has been constructed in [@Hikida:3]. The same fermionic relation has been obtained, but starting from generalized Killing spinor equation. We will not repeat the details here. It turns out the the maximally supersymmetric conical defects are in exact match with the chiral primaries in $N=(2,2) CP^N$ Kazama-Suzuki model.
2. $osp(2N+1|2N)$. The bosonic part is $so(2N+1)\oplus sp(2N)$. The bosonic spectrum includes two copies of spin $2,4,\cdots,2N$. The connection $a_+$ could be diagonalized to be $$\begin{aligned}
a_+\sim \left(\begin{array}{cc}
a_{(2N+1)\times(2N+1)}&0\\0&a_{2N\times 2N}
\end{array}\right)
\end{aligned}$$ where $a_{(2N+1)\times(2N+1)}$ and $a_{2N\times 2N}$ are a\_[(2N+1)(2N+1)]{}&\~& diag(\_1,\_2,,\_N,0,-\_N,,-\_2,-\_1),\
a\_[2N2N]{}&\~& diag(\_1,\_2,,\_N,-\_N,,-\_2,-\_1). From the discussion of the bosonic case, we can include the following two cases.
1. We choose the center to be $1_{so(2N+1)}\times 1_{sp(2N)}$. This can be satisfied by \_i&=&i n\_i, n\_i, i=1,,N\
\_i&=&i m\_i, m\_i, i=1,,N Then the fermionic boundary condition is periodic.
2. We can also choose the center to be $1_{so(2N+1)}\times (-1)_{sp(2N)}$. This can be satisfied by \_i&=&i n\_i, n\_i, i=1,,N,\
\_i&=&i (m\_i+), m\_i, i=1,,N. Then the fermionic boundary condition is anti-periodic.
In [@Hikida:2], it was proposed that in the ’t Hooft limit the $osp(2N+1|2N)$ high spin supergravity is dual to the $\mathcal{N}=(1,1)$ super coset .\[11coset\] The primary states in this coset are characterized by $(\Lambda,\Xi)$, where $\Lambda$ and $\Xi$ are the highest weight representations of $so(2n+1)$ and $so(2n)$ respectively[@Hikida:2]. They are not in match with the maximally supersymmetric conical defects found above. Actually the fact that the bosonic sector of $osp(2N+1|2N)$ high spin supergravity involves both $so(2N+1)$ and $sp(2N)$ group suggests that the possible CFT dual could be nontrivial.
Partially Symmetric Solution
============================
In the previous section, we have explored the maximally symmetric solutions in the higher spin gravity with or without supersymmetry, and found that the maximally symmetric solution were exactly the smooth conical defects which were investigated before. However, as in conventional gravity, not all the allowed solutions preserve maximal symmetries. For example, the black holes in supergravity always have less symmetries than the vacuum solution which is maximally symmetric and generically breaks the supersymmetry completely. In some cases, the extremal black holes may preserve part of supersymmetry. In other words, they are partially symmetric solutions. Here we extend the concept of partially symmetric solution to the higher spin gravity, both in the bosonic and supersymmetric case.
Partially Symmetric Solution in Bosonic Higher Spin Gravity
-----------------------------------------------------------
Here we still choose $SL(N,\mathbb{R})$ as the prototypic model, but the discussion can be easily generalized. Since we would like to include the higher spin black holes, we do not require the solution to be (\[gauge2\]), but we still require the solution to be of constant $a$. In the gauge (\[gauge1\]), the configurations we are interested in could be of the form a=a\_+ dz+a\_-d|[z]{}\[ansatz\] where $a_+$ is the same as (\[gauge1\]), but a nonvanishing $a_-$ term has been turned on to refer to the higher spin black holes. As the solutions satisfy the flatness condition, $a_+, a_-$ should commute with each other, =0.\[eqn\] Note that we still need to search for the solution of the equation \_A=d+\[A,\]=0. The $\rho$-component of the equation still gives us $\Lambda=b^{-1}\Lambda_0b$, while the $+,-$-components of the equation lead to \_0=e\^[-(a\_+z+a\_-|[z]{})]{}e\^[(a\_+z+a\_-|[z]{})]{},\[solution\] where $\lambda$ is a constant matrix taking value in $sl(N,\mathbb{R})$. To derive (\[solution\]) we have used the equation of motion (\[eqn\]). For $\Lambda$ being well-defined globally, we require the periodic condition (+2)=().\[perio\] After some elementary algebra, we find that e\^[-2a\_]{}e\^[2a\_]{}=. \[cond\] The exponential $e^{2\pi a_{\phi}}$ is actually the holonomy of the gauge potential along spatial circle for our ansatz (\[ansatz\]). Up till now, the treatment is the same as before, but this time we do not require $\lambda$ to be arbitrary. If (\[perio\]) is satisfied for some constant matrix $\lambda$ valued in $sl(N,\mathbb{R})$, there is no need to require the holonomy $H$ to be in the center of $SL(N,\mathbb{R})$. Obviously, this will lead to the solution that has only partial symmetry[^7].
As the holonomy of the gauge field needs not to be in the center, the matrix $a_\phi$ may not be diagonalizable. The discussion below separates into two cases. If the matrix $a_{\phi}$ can be diagonalized, we assume that its eigenvalues are $(\th_1,\cdots \th_N)$, i.e. a\_\~diag(\_1,,\_N) and the eigenvalues differ from each other, namely, $\th_i\not=\th_j$ for $i\not=j$. Using the identity (-2\_[k=1]{}\^[N]{}\_k E\_[kk]{}) E\_[ij]{} (2\_[k=1]{}\^[N]{}\_k E\_[kk]{})=e\^[-2(\_i-\_j)]{}E\_[ij]{} where $E_{ij}$ is the $N\times N$ matrix that is 1 in the i-th row and j-th colum, otherwise it is 0, then we find that if \_i-\_j=i n, n\[condition\] for some pairs $(i,j)$. We can have globally well-defined $\Lambda$, though it may not be arbitrary. For the most general solution, the condition (\[cond\]) can only be satisfied by $i=j$ with $n=0$. In other words, only diagonal matrix may satisfy (\[cond\]). Taking into account of the traceless condition, there are only $N-1$ independent solutions. Recall that the Cartan subagebra could be written as the traceless diagonal matrix, the general solution to (\[cond\]) could be the linear combination of the Cartan generators. This reflects the fact that there are $N-1$ well-defined global charges in the theory, corresponding to the spin 2, $\cdots$, spin $N$ charges.
If the matrix $a_{\phi}$ cannot be diagonalized, we need to solve the equation (\[cond\]) from scratch. Note that if =0\[comu\] then the equation (\[cond\]) can be satisfied automatically. The equation (\[comu\]) can always be solved by the traceless function $f(a_{\phi})$ with =f(a\_).\[fun\] Since $f$ can always be expanded as f(a\_)=\_[i=2]{}\^[N]{}c\_i (a\_\^i-tr a\_\^i) there are always $N-1$ independent solutions which are captured by the constants $c_i$. We emphasize that there may be symmetry enhancement for special configuration.
Partially Symmetric Solution in Higher Spin Supergravity
--------------------------------------------------------
The discussion in the previous subsection is a warm-up to the more interesting case we will consider now. We use the superalgebra $sl(N|N-1)$ as our prototypic model. The solution of $\Lambda_0$ is the same as (\[cond\]), but $\lambda$ can be decomposed into the bosonic part $\xi$ and the fermionic part $\epsilon$, =+The boundary condition is now \_(+2)=\_(), \_(+2)=\_(). The discussion of the bosonic part $\Lambda_{\xi}$ goes through parallel to the previous subsection, so we only focus on the fermionic part $\Lambda_{\epsilon}$. The periodic or anti-periodic boundary condition leads to e\^[-2a\_]{} e\^[2a\_]{}=. \[condf\] The independent number of solutions to (\[condf\]) tells us how many supersymmetries the configuration keeps[@Justin:susy].
Again, if the supermatrix $a_{\phi}$ can be diagonalized, we can assume $$\begin{aligned}
a_{\phi}\sim \left(\begin{array}{cc}
a_{N\times N}&0\\0&a_{(N-1)\times (N-1)}
\end{array}\right)\end{aligned}$$ with the matrix $a_{N\times N}$ and $a_{(N-1)\times (N-1)}$ to be a\_[NN]{}\~diag(\_1,,\_N),\
a\_[(N-1)(N-1)]{}\~diag(\_[|[1]{}]{},,\_[|[N]{}]{}) And we have $\th_1>\th_2>\cdots>\th_N,\ \varphi_{\bar{1}}>\varphi_{\bar{2}}>\cdots>\varphi_{\bar{N}}$. The supertraceless condition require \_[i=1]{}\^[N]{}\_i-\_[|[j]{}=1]{}\^[N-1]{}\_[|[j]{}]{}=0 The supermatrix has $2N-1$ indices: the first N indices will be denoted as $i,j$ and the last $N-1$ indices will be denoted as $\bar{i},\bar{j}$. Then a basis for the fermionic generators can be chosen to be $E_{i\bar{j}}$ and $E_{\bar{i}j}$. Due to the identity[^8] (-2(\_[i=1]{}\^N\_i E\_[ii]{}+\_[|[j]{}=1]{}\^[N-1]{}\_[|[j]{}]{}E\_[|[j]{}|[j]{}]{}))E\_[i|[j]{}]{}(2(\_[i=1]{}\^N\_i E\_[ii]{}+\_[|[j]{}=1]{}\^[N-1]{}\_[|[j]{}]{}E\_[|[j]{}|[j]{}]{}))=e\^[-2(\_i-\_[|[j]{}]{})]{}E\_[i|[j]{}]{}, the condition (\[condf\]) has a solution when there exists $i,\bar{j}$ such that \_i-\_[|[j]{}]{}=i n, n\[cri1\] for periodic boundary condition, and \_i-\_[|[j]{}]{}=i (n+), n\[cri2\] for anti-periodic boundary condition.
In the case that $a_{\phi}$ cannot be diagonalized, then we need to solve (\[condf\]) from scratch. In the following section we will find that some black holes do preserve part of supersymmetries discussed here.
Black Holes
===========
The higher spin black holes are the classical solutions of the flatness equation of motion. To have a smooth geometry, the holonomy of the gauge potential of the black hole around the thermal circle is required to be in the center of the corresponding group. The holonomy condition has been checked for the black holes in spin 3 [@Per; @kraus:2011], spin 4 [@Tan], spin $\tilde{4}$, and $G_2$ gravities[@truncated], and it has been extended to the case with supersymmetry[@Tan:2012]. In all the cases, it leads to consistent thermodynamics, together with the integrability condition.
It is interesting to compare the higher spin black holes with the conical defects discussed in the previous sections. For the smooth conical defects, the most important feature is that the holonomy around the spatial circle is trivial, H\_(A)= center.\[sp\] On the contrary, the higher spin black holes require the holonomy around the thermal circle is trivial, H\_(A)= center.\[th\] Note that we have shown that when the holonomy around the spatial circle is trivial, the solution is maximally symmetric, while if the spatial holonomy is not trivial, the solution is only partially symmetric. In particular, the solution may keep part of supersymmetries. For the higher spin black hole, its spatial holonomy is nontrivial so it is only to be partially supersymmetric.
In this section, we would like to discuss the black hole solutions in the higher spin supergravity and check if there are supersymmetric ones. First of all, we need to construct suitable higher spin black holes. The black holes we find have not been discussed before hence we will clarify these solutions in detail, including the explicit solutions, and most importantly, how the holonomies around the thermal circle lead to consistent thermodynamics. Next we discuss under what condition, the black hole become supersymmetric. We will use two methods to investigate the issue. The first one is to solve the generalized Killing equation by brute force. The second one is to discuss the relation (\[condf\]) on the holonomy. It turns out that two methods always lead to the same answer. To simplify the discussion, we work on the higher spin black holes whose explicit entropy forms are feasible. These include the ones in $osp(3|2)$ supergravity and $sl(3|2)$ supergravity.
Black Holes in $osp(3|2)$ Supergravity
--------------------------------------
The study of the black hole solution in a higher spin supergravity is not much difficult than the bosonic one. To find the black holes in the higher spin supergravity, we only need the bosonic algebra of the theory. However, from the structure of the higher spin algebra, the bosonic part is just the direct sum of decoupled semi-simple Lie algebras. This property is quite like D$_2$ gravity [@D2]. One lesson from the $D_2$ gravity is that the total entropy of the black hole can be the sum of the entropies of two decoupled system. The same is true for the black hole in the higher spin supergravity. In this subsection, we explore the black holes in $osp(3|2)$ supergravity. This supergravity can be taken the simplest higher spin supergravity as it contains a spin $5/2$ field.
### Solution
Let us first consider the solution in the $osp(3|2)$ supergravity. As in the $sl(N)$ gravity, we can choose the highest-weight gauge to set the gauge connection to be A=e\^[-L\_0]{}a e\^[L\_0]{}+L\_0 d,\
|[A]{}=e\^[L\_0]{}|[a]{}e\^[-L\_0]{}-L\_0d\[connection\] where a&=&(L\_1-L\_[-1]{}+’A\_[-1]{})dx\^++(L\_[-1]{}+\_[i=-1]{}\^[1]{}q\_i A\_i)dx\^-,\
|[a]{}&=&-(L\_[-1]{}-|L\_1+|’A\_1)dx\^–(|L\_1+\_[i=-1]{}\^[1]{}|[q]{}\_iA\_i)dx\^+ From the equation of motion, we find the solution a&=&(L\_1-L\_[-1]{}+’A\_[-1]{})dx\^++(’L\_[-1]{}+A\_1-A\_[-1]{})dx\^-,\
|[a]{}&=&-(L\_[-1]{}-|L\_1+|’A\_1)dx\^–|(A\_[-1]{}-|A\_1+|’L\_1)dx\^+. Here we have relabel the parameters $q$ and $\bar{q}$ to be $\mu$ and $\bar{\mu}$, which could be interpreted as the potentials conjugate to the spin 2 charges $\mathcal{L}'$ and $\bar{\mathcal{L}}'$. Note that this solution is exactly the same as the one in the $D_2$ gravity[@D2]. However, the $osp(3|2)$ gravity is different from the $D_2$ gravity in many aspects. First of all, the $osp(3|2)$ gravity has many fermionic degrees of freedom, which connect the decoupled bosonic degrees of freedom. Secondly, the supertrace of the supermatrix M is different from the trace in the $D_2$ gravity. Therefore this $osp(3|2)$ higher spin supergravity may not have a second order formulation as the $D_2$ gravity.
### Thermodynamics
The thermodynamics of the black holes found in previous section can be studied by obtaining its exact entropy. Now the holomorphic part of the partition function is Z=Tr We denote the entropy of the black hole to be $S$, which is a function of $\mathcal{L}$ and $\mathcal{L}'$. The parameters $\tau,\alpha$ can be related to $\mathcal{L},\mathcal{L}'$ by =, =\[mu\] Equivalently we may define four other variables $\mathcal{H},\mathcal{K},\gamma,\delta$ by &&-’, +’,\
&&-, +which have =, =.\[gamma\] The holonomy around the thermal circle is $H=e^{\oint}A\equiv e^{\omega}$, with $$\begin{aligned}
\omega=2\pi(a_+\tau-a_-\bar{\tau})=\left(\begin{array}{cc}
\omega_{3\times3}&0\\ 0&\omega_{2\times2}\end{array}
\right).\end{aligned}$$ In the bosonic higher spin gravity, the holonomy has to be in the center of the corresponding algebra[@Per; @kraus:2011]. In the higher spin supergravity we are considering, the holonomy is $$\begin{aligned}
H=\left(\begin{array}{cc}
1_{3\times3}&0\\ 0&-1_{2\times2}\end{array}
\right).\end{aligned}$$ In other words, we require the eigenvalues of $\omega_{3\times3}$ to be $2\pi i,0,-2\pi i$ and the ones of $\omega_{2\times2}$ to be $\pi i,-\pi i$. After a short computation, we find $$\begin{aligned}
\omega_{3\times3}=2\pi\left(\begin{array}{ccc}
0&\sqrt{2}\mathcal{H}\gamma\\ \sqrt{2}\gamma&0&\sqrt{2}\mathcal{H}\gamma\\0&\sqrt{2}\gamma&0\end{array}
\right),\ \ \omega_{2\times2}=2\pi\left(\begin{array}{cc}
0&\mathcal{K}\delta\\ \delta&0\end{array}\right).\end{aligned}$$ Hence the holonomy equation becomes tr\_[33]{}\^2=-8\^2, tr\_[22]{}\^2=-2\^2. From the equation (\[gamma\]) we find the entropy of the black hole to be S=kk There are four branches of the solutions
- [Branch 1: $S=\pi k(\sqrt{\mathcal{H}}+\sqrt{\mathcal{K}})$]{}
- [Branch 2: $S=\pi k(\sqrt{\mathcal{H}}-\sqrt{\mathcal{K}})$]{}
- [Branch 3: $S=\pi k(-\sqrt{\mathcal{H}}+\sqrt{\mathcal{K}})$]{}
- [Branch 4: $S=\pi k(-\sqrt{\mathcal{H}}-\sqrt{\mathcal{K}})$]{}
Note that for each branch, we can find the corresponding $\tau,\alpha$ and then $\mu$. One can explore the phase structure of the black holes as in [@Justin:thermo; @Phase], but we will not include it here. In the following, we will explore the supersymmetries of the solution.
### Supersymmetry I
In a supergravity, it is important to know how many supersymmetries the solution preserves. The supersymmetric condition is a generalized Killing spinor equation d+\[A,\]=0\[spinor\] where the spinor $\epsilon$ can be expanded as =\_r R\_r+\_s Z\_s with $r=-\frac{3}{2},-\frac{1}{2},\frac{1}{2},\frac{3}{2}$ and $s=-\frac{1}{2},\frac{1}{2}$ for $osp(3|2)$ case. We notice that in the paper [@Tan:2012], only the spin $1/2$ generators were included in the expansion. Nevertheless, we feel that it is more reasonable to include the higher spin fermonic generators, as did in [@Justin:susy; @Hikida:3].
From the form of the connection (\[connection\]) and the $\rho$ component of the Killing spinor equation (\[spinor\]), $\epsilon$ can be cast into the form =e\^[-L\_0]{}\_0 e\^[L\_0]{} where $\epsilon_0$ is a function independent of $\rho$. The $+$ component of the Killing spinor equation can be expanded as &&\_+\_+\_=0,\
&&\_+\_+2\_[-]{}+3\_-’\_=0,\
&&\_+\_[-]{}+3\_[-]{}+2\_-’\_+’\_=0,\[+\]\
&&\_+\_[-]{}+\_[-]{}-’\_[-]{}+’\_[-]{}=0,\
&&\_+\_+\_[-]{}-2’\_=0,\
&&\_+\_[-]{}+\_-’\_-’\_=0.The $-$ component of the Killing spinor equation can be expanded as &&\_-\_+\_+\_=0,\
&&\_-\_-3’\_+\_[-]{}+\_+\_[-]{}=0,\
&&\_-\_[-]{}-2’\_+\_[-]{}+\_-\_=0,\
&&\_-\_[-]{}-’\_[-]{}+\_[-]{}-\_=0,\[-\]\
&&\_-\_-\_[-]{}+\_[-]{}+2\_=0,\
&&\_-\_[-]{}-’\_-2\_[-]{}+\_+\_=0.
Let us first set $\mathcal{L}'$ to be zero. In this case we have $\mu=0$[^9]. From the equations (\[-\]) we find that $\eta_r,\k_s$ are independent of $x^-$. Then the equations (\[+\]) become &&(\_+\^2-)\_=0,\
&&(\_+\^2-)(\_+\^2-9)\_=0.\[killing\] The other components can be obtained easily. If $\mathcal{L}\not=0$, the general solution should be \_&=&c\_1 e\^[x\^+]{}+c\_2 e\^[-x\^+]{}\
\_&=&d\_1 e\^[3x\^+]{}+d\_2 e\^[x\^+]{}+d\_3 e\^[- x\^+]{}+d\_4 e\^[-3x\^+]{}. Then for general $\mathcal{L}>0$, the Killing spinor is neither periodic nor anti-periodic, so a general $BTZ$ black hole breaks the supersymmetries completely. However, for $\mathcal{L}<0$, the Killing spinor can satisfy the periodic or anti-periodic condition. If we restrict ourselves in the region $-\frac{1}{4}\le\mathcal{L}<0$, then when $\mathcal{L}=-\frac{1}{4}$ the spnior satisfy anti-periodic boundary condition. Since $\mathcal{L}=-\frac{1}{4},\mathcal{L}'=0$ corresponds to global $AdS_3$, we conclude that the global $AdS_3$ preserve all the supersymmetries. For the anti-holomorphic sector, we have the same conclusion. We use (6,6) to denote the total supersymmetries since there are six Killing spinors in each sector[^10] .
However, in the range $-\frac{1}{4}\le\mathcal{L}<0$, there are two other interesting cases. If $\mathcal{L}=-\frac{1}{9}$, we find that we can set c\_1=c\_2=d\_2=d\_3=0, and $d_1, d_4$ to be arbitrary constant. Then the configuration can preserve two supersymmetries, and the corresponding Killing spinors satisfy periodic boundary condition. If $\mathcal{L}=-\frac{1}{36}$, we may set c\_1=c\_2=d\_2=d\_3=0 and $d_1,d_4$ to be arbitrary constant, such that the configuration can preserve two supersymmetries with the corresponding Killing spinors satisfying anti-periodic boundary condition. These supercharges are coming from the spin-$5/2$ components hence are different from the conventional supercharges. Another interesting case is when $\mathcal{L}=\mathcal{L}'=0$. Then the general solution of Eq. (\[killing\]) is \_&=&c\_1 x\^++c\_2,\
\_&=&d\_1 (x\^+)\^3+d\_2 (x\^+)\^2+d\_3(x\^+)+d\_4. Only when $c_1=d_1=d_2=d_3=0$, the Killing spinor preserve periodic boundary condition. This configuration was called extreme BTZ black hole in the literature.
The results can be summarized as follows
1. For global $AdS_3$, $\mathcal{L}=\bar{\mathcal{L}}=-\frac{1}{4}$, it preserves (6,6) supersymmetry.
2. For $\mathcal{L}=-\frac{1}{4},\bar{\mathcal{L}}=-\frac{1}{9}$, it preserves (6,2) supersymmetry.
3. For $\mathcal{L}=-\frac{1}{9},\bar{\mathcal{L}}=-\frac{1}{4}$, it preserves (2,6) supersymmetry.
4. For $\mathcal{L}=\bar{\mathcal{L}}=-\frac{1}{9}$, it preserves (2,2) supersymmetry.
5. For $\mathcal{L}=-\frac{1}{4}, \bar{\mathcal{L}}=-\frac{1}{36}$, it preserves (6,2) supersymmetry.
6. For $\mathcal{L}=-\frac{1}{36}, \bar{\mathcal{L}}=-\frac{1}{4}$, it preserves (2,6) supersymmetry.
7. For $\mathcal{L}=\bar{\mathcal{L}}=-\frac{1}{36}$, it preserves (2,2) supersymmetry.
8. For $\mathcal{L}=-\frac{1}{9},\bar{\mathcal{L}}=-\frac{1}{36}$ or $\mathcal{L}=-\frac{1}{36},\bar{\mathcal{L}}=-\frac{1}{9}$, it preserves (2,2) supersymmetry.
9. For massless BTZ black hole, $\mathcal{L}=\mathcal{L}'=\bar{\mathcal{L}}=\bar{\mathcal{L}}'=0$, it preserves (2,2) supersymmetry.
10. For extreme BTZ black hole with nonzero mass, $\mathcal{L}=\mathcal{L}'=0, \bar{\mathcal{L}}\not=0,\bar{\mathcal{L}}'=0$ or $\mathcal{L}\not=0,\mathcal{L}'=0,\bar{\mathcal{L}}=\bar{\mathcal{L}}'=0$, it preserves (2,0) or (0,2) supersymmetry.
The solutions 2 to 8 listed above are not exactly the black holes. And actually they are not smooth conical defects we discussed before. They do preserve some supersymmetries, but are not smooth. Next we turn to the black hole with a nonvanishing spin 2 charge $\mathcal{L}'\not=0$. As the general solution to the Killing spinor equation would be very complicated, here we are satisfied with searching for the constant solutions. These solutions satisfy the periodic boundary condition in the $\phi$ direction so the problem reduce to search for non-zero constant solution of the equations (\[+\]) and (\[-\]). Let us first consider the equations (\[+\]). These equations reduce to a set of linear equations when the solutions are constants. They have non-zero solutions if and only if the determinant of the matrix $M_+$ $$\begin{aligned}
M_+=\left(\begin{array}{cccccc}
0&1&0&0&0&0\\3\mathcal{L}-\mathcal{L}'&0&2&0&0&0\\0&\frac{1}{3}(6\mathcal{L}-2\mathcal{L}')&0&3&\frac{4\mathcal{L}'}{3}&0\\
0&0&\frac{1}{3}(3\mathcal{L}-\mathcal{L}')&0&0&\frac{4\mathcal{L}'}{3}\\-2\mathcal{L}'&0&0&0&0&1\\0&-\frac{2\mathcal{L}'}{3}&0&0&\frac{1}{3}(3\mathcal{L}-5\mathcal{L}')&0\end{array}
\right)\end{aligned}$$ is zero: det(M\_+)=-(3-5’)\^2(+’)=0.\[+det\] From the Killing spinor equations (\[-\]), the characteristic matrix $M_-$ is $$\begin{aligned}
M_-=\mu \left(\begin{array}{cccccc}
0&\frac{1}{3}&0&0&\frac{4}{3}&0\\\mathcal{L}-3\mathcal{L}'&0&\frac{2}{3}&0&0&\frac{4}{3}\\0&\frac{1}{3}(2\mathcal{L}-6\mathcal{L}')&0&1&-\frac{4\mathcal{L}}{3}&0\\
0&0&\frac{1}{3}(\mathcal{L}-3\mathcal{L}')&0&0&-\frac{4\mathcal{L}}{3}\\2\mathcal{L}&0&-\frac{2}{3}&0&0&\frac{5}{3}\\0&\frac{2\mathcal{L}}{3}&0&-2&\frac{1}{3}(5\mathcal{L}-3\mathcal{L}')&0\end{array}
\right)\end{aligned}$$ and its determinant is det(M\_-)=-\^6(3-5’)\^2(+’).\[-det\] For the equation (\[+det\]) to be zero, there are two cases:
1. $\mathcal{L}=\frac{5\mathcal{L}'}{3}$. Then the equations (\[+\]) have the non-zero solutions \_1=0, \_[-1]{}=-\_[-1]{}, \_[-1]{}=2\_3’, \_[-3]{}=-, \_3=C\_1 \_1=C\_2\[solu1\] where $C_1, C_2$ are constants. Though (\[-det\]) is zero for this solution, the solution (\[solu1\]) does not satisfy the equations (\[-\]). Thus in general there is no supersymmetry for this configuration. However, there is one exception. That is to set $\mu$ to be zero. Then the equations (\[-\]) are satisfied. This can be achieved by $\bar{\tau}=\infty$, or equivalently, $\bar{\mathcal{L}}=\bar{\mathcal{L}}'$ or $\bar{\mathcal{L}}=-\bar{\mathcal{L}}'$. In the case that $\bar{\mathcal{L}}=\bar{\mathcal{L}}'$ or $\bar{\mathcal{L}}=-\bar{\mathcal{L}}'$, the anti-holomorphic part of the black hole is extreme. So the holomorphic part of the black hole solution will preserve 2 supersymmetries for these extreme black holes.
2. $\mathcal{L}=-\mathcal{L}'$. Then the equations (\[+\]) have the non-zero solution \_1=\_[-3]{}=\_[1]{}=0, \_[-1]{}=\_[-1]{}=-2\_3, \_3=C, where $C$ is a constant. For this configuration, the equations in (\[-\]) are indeed satisfied. In general, such kinds of configuration preserve 1 supersymmetry in the holomorphic part.
For the antiholomorphic sector, the analysis is similar. Combining the results in two sectors, we list the the configurations that preserve some supersymmetries for non-vanishing spin 2 charge $\mathcal{L}'$ in the following table
Configuration supersymmetries
---------------------------------------------------------------------------------------------------------------------------------------------- -----------------
$\mathcal{L}=\frac{5}{3}\mathcal{L}',\ \bar{\mathcal{L}}=\bar{\mathcal{L}}'$ (2,0)
$\mathcal{L}=\mathcal{L}',\ \bar{\mathcal{L}}=\frac{5}{3}\bar{\mathcal{L}}'$ (0,2)
$\mathcal{L}=\frac{5}{3}\mathcal{L}',\ \bar{\mathcal{L}}=-\bar{\mathcal{L}}'$ (2,1)
$\mathcal{L}=-\mathcal{L}',\ \bar{\mathcal{L}}=\frac{5}{3}\bar{\mathcal{L}}'$ (1,2)
$\mathcal{L}=-\mathcal{L}',\ -\bar{\mathcal{L}}<\bar{\mathcal{L}}'\le\bar{\mathcal{L}}, \bar{\mathcal{L}}'\not=\frac{3}{5}\bar{\mathcal{L}}$ (1,0)
$-\mathcal{L}<\mathcal{L}'\le\mathcal{L},\mathcal{L}'\not=\frac{3}{5}\mathcal{L}, \bar{\mathcal{L}}=-\bar{\mathcal{L}}'$ (0,1)
$\mathcal{L}=-\mathcal{L}',\ \bar{\mathcal{L}}=-\bar{\mathcal{L}}'$ (1,1)
We notice that all the solutions are extreme in at least one sector. Moreover, there is a special point $\mathcal{L}=\frac{5}{3}\mathcal{L}'$ where the supersymmetry get enhanced. This configuration is mysterious for us. It is better to have a deeper understanding. Another remarkable point is that we only collect the configurations that have some constant Killing spinors. To search for the configurations that have non-constant Killing spinors, we need to solve the Killing spinor equations, which would be quite involved. However, in the following, we can answer this question from the honolomy equation (\[condf\]).
### Supersymmetry II
Now we search for the supersymmetric black holes from the spatial holonomy condition (\[condf\]), without solving the Killing spinor equations. As the discussion before, we first set $\mathcal{L}'$ and $\bar{\mathcal{L}}'$ to be zero. Using the symbol $\th_i$, $\varphi_{\bar{j}}$ to denote the eigenvalues of $so(3)$ and $sp(2)$, we find \_1&=&2, \_2=0, \_3=-2,\
\_[|[1]{}]{}&=&, \_[|[2]{}]{}=-.
When $\mathcal{L}>0$, it is a general non-extreme $BTZ$ black hole. In this case, as the conditions (\[cri1\]) and (\[cri2\]) cannot be satisfied, the configuration breaks all the supersymmetry.
When $\mathcal{L}<0$, the supersymmetry enhancement condition is given by (\[cri1\]) or (\[cri2\]). We first consider periodic boundary condition (\[cri1\]). This can be achieved in two cases.
1. $\th_1-\varphi_{\bar{1}}=i n, n\in\mathbb{Z}$, which gives us that =-n\^2, n\^+.\[susyenhance1\] The negative integer $n$ give the same $\mathcal{L}$ as a positive $n$ and the case $n=0$ is excluded by the condition $\mathcal{L}<0$. If we restrict ourselves in the range $\mathcal{L}\ge-\frac{1}{4}$, then there is no solution for this case.
2. $\th_1-\varphi_{\bar{2}}=i n,n\in\mathbb{Z}$, which leads to =-n\^2, n\^+.\[susyenhance2\] In the range $-\frac{1}{4}\le\mathcal{L}<0$, there is only one solution =-, n=1. This configuration preserve periodic boundary condition. Since there are two pairs of $\th_i-\varphi_{\bar{j}}$($i=1,\bar{j}=2$ and $i=3,\bar{j}=1$) satisfying the condition (\[cri1\]), it preserve two fermionic symmetries. All these results are the same as the ones we found before.
We can also consider the anti-periodic boundary condition (\[cri2\]), which can be satisfied in two cases:
1. $\th_1-\varphi_{\bar{1}}=i (n+\frac{1}{2}), n\in\mathbb{Z}$. It leads to =-(n+)\^2, n.\[susyenhance3\] The unique solution is =-, n=0 in the range $-\frac{1}{4}\le\mathcal{L}<0$, leading to the global AdS$_3$. Since for arbitrary $i,\bar{j}$, the condition (\[cri2\]) can be satisfied, the configuration actually has maximal supersymmetries. The result is the same as we found before.
2. $\th_1-\varphi_{\bar{3}}=i (n+\frac{1}{2}), n\in\mathbb{Z}$. It leads to =-(n+)\^2, n.\[susyenhance3\] There are two configurations in the range $-\frac{1}{4}\le\mathcal{L}<0$,
1. $\mathcal{L}=-\frac{1}{36},\ n=0$, which preserves two fermionic symmetries and anti-periodic boundary condition.
2. $\mathcal{L}=-\frac{1}{4},\ n=1$, which is the global AdS$_3$.
Now we want to set $\mathcal{L}=0$. This is a special configuration as it corresponds to extreme $BTZ$ black hole. The eigenvalues of $a_+$ are \_i=0, \_[|[j]{}]{}=0, i{1,2,3}, |[j]{}{|[1]{},|[2]{}} In this case $a_+$ cannot be diagonalized. We need to solve the Killing spinor equation as we have done in the previous subsubsection.
Let us turn to the case $\mathcal{L}'\not=0,\ \bar{\mathcal{L}}'\not=0$. From the discussion of thermodynamics, we have $\mathcal{L}>0$ and $-\mathcal{L}\le\mathcal{L}'\le\mathcal{L}$, otherwise the solution is not a higher spin black hole. In this case we can find the following eigenvalues of $a_{\phi}$, \_1&=&2(1-), \_2=0, \_3=-2(1-),\
\_[|[1]{}]{}&=&(1+), \_[|[2]{}]{}=-(1+). Note that the extreme case is special, as when $\mathcal{L}'=\pm\mathcal{L}$, the eigenvalue is degenerate, we need to solve the Killing spinor equation from scratch. And this is the same as we discussed previously. Therefore we focus our attention to the case $\mathcal{L}'\not=\pm\mathcal{L}$. Since all the eigenvalues are real, the anti-periodic boundary condition (\[cri2\]) is impossible and the periodic boundary condition (\[cri1\]) can only be satisfied by setting $n=0$, for some $i,\ \bar{j}$. Then we find that 2(1-)=(1+).\[susyenhance5\] If we substitute the definition of $\mu=\frac{\alpha}{\bar{\tau}}$ into the equation, then we find that the condition (\[susyenhance5\]) can always be satisfied by suitable choice of $\mathcal{L},\mathcal{L}',\bar{\mathcal{L}},\bar{\mathcal{L}}'$. Thus the configuration can preserve two fermionic symmetries. This conclusion sounds different from the one in previous subsubsection. However, there is no contradiction. In the last subsection we have just solved the Killing spinor equation for the constant solution case. To have a constant solution, we require $\mu=0$ and this is achieved by setting $\bar{\mathcal{L}}'=\pm\bar{\mathcal{L}}$. Under this condition, the solution of (\[susyenhance5\]) leads to =’, which is in exact agreement with the result got before. However if we do not require a constant solution, then the equation (\[susyenhance5\]) tells us the whole story.
Moreover, there is another advantage to work with (\[susyenhance5\]). It is an algebraic equation rather than a differential equation, as a Killing spinor equation is. The condition (\[susyenhance5\]) is interesting because it can be satisfied without taking extreme limit. This is very different from the conventional supergravity, in which the supersymmetries enhancement always happens for extremal black holes. Now, in the higher spin supergravity, we find that even for non-extreme higher spin black hole, the nonconstant Killing spinors exist.
Black Holes in $sl(3|2)$ Supergravity
-------------------------------------
The $osp(3|2)$ black hole has no spin $s>2$ hair, so we now consider the black holes with higher spin hair in $sl(3|2)$ supergravity in this section.
### Thermodynamics
Since the $U(1)$ part is decoupled from the other bosonic generators, we consider the solution with vanishing $U(1)$ charge for simplicity. The solution can be parameterized as a\_+&=&L\_1-L\_[-1]{}+W\_[-2]{}-A\_[-1]{},\
a\_-&=&L\_[-1]{}+q W\_2+q\_0 W\_0+q\_[-2]{}W\_[-2]{}+p A\_1+p\_[-1]{}A\_[-1]{},\[sl32\] where $\mathcal{L},\mathcal{Y}$ are two spin 2 charges, $\mathcal{W}$ is the spin 3 charge and the constant $\nu,q_0,q_{-2},p_{-1}$ are &&=-p-2q, q\_0=-2q-2q,\
&&p\_[-1]{}=-p-2q, q\_[-2]{}=\[p+2q(+)\^2)\].\[sl32p\] The holonomy is =2(a\_+-a\_-|) In terms of q=-\_3/|, p=-\_2/|,\[pq\] the holonomy can be written as $$\begin{aligned}
\omega=\left(\begin{array}{cc}
\omega_{3\times3}&0\\ 0&\omega_{2\times2}\end{array}
\right)\end{aligned}$$ with $$\begin{aligned}
\omega_{3\times3}=\left(\begin{array}{ccc}
-\frac{8}{3}\alpha_3\pi(\mathcal{L}+\mathcal{Y})&2\sqrt{2}\pi(4\alpha_3\mathcal{W}+(\alpha_2+\tau)(\mathcal{L}+\mathcal{Y}))&4\pi(\mathcal{W}(\alpha_2+\tau)+2\alpha_3(\mathcal{L}+\mathcal{Y})^2)\\
2\sqrt{2}\pi(\alpha_2+\tau)&\frac{16}{3}\alpha_3\pi(\mathcal{L}+\mathcal{Y})&2\sqrt{2}\pi(4\alpha_3\mathcal{W}+(\alpha_2+\tau)(\mathcal{L}+\mathcal{Y})\\
8\alpha_3\pi&2\sqrt{2}\pi(\alpha_2+\tau)&-\frac{8}{3}\alpha_3\pi(\mathcal{L}+\mathcal{Y})\end{array}
\right),\end{aligned}$$ $$\begin{aligned}
\omega_{2\times2}=\left(\begin{array}{cc}
0&2\pi(\tau-\alpha_2)(\mathcal{L}-\mathcal{Y})\\
2\pi(\tau-\alpha_2)&0\end{array}
\right).\end{aligned}$$ Similar to the $osp(3|2)$ case, we use the holonomy condition that the eigenvalues of $\omega_{3\times3}$ are $2i\pi,0,-2i\pi$ and the eigenvalues of $\omega_{2\times2}$ are $i\pi,-i\pi$. We can also determine $\tau,\alpha_2,\alpha_3$ by =, \_2=, \_3=.\[tal\] Here $S$ is the entropy of the black hole. It is a function of the charges $\mathcal{L},\mathcal{Y}$ and $\mathcal{W}$. It is more convenient to redefine the charges to be $\mathcal{L}+\mathcal{Y}$,$\mathcal{L}-\mathcal{Y}$ and $\mathcal{W}$. It is easy to see that the holonomy $\omega_{3\times3}$ is only dependent of $\mathcal{L}+\mathcal{Y}$ and $\mathcal{W}$, $\alpha_2+\tau$ and $\alpha_3$. And as \_2+=, $\omega_{3\times 3}$ depends on $\mathcal{L}+\mathcal{Y}$ and $\mathcal{W}$ and their corresponding potentials. Similary, $\omega_{2\times2}$ depends on $\mathcal{L}-\mathcal{Y}$ and its potential. Therefore we can cast the entropy function to be S(,,)=S\_1(+,)+S\_2(-).\[entropyosp\] Then we can solve the holonomy equations by S\_i&=&k +k (+(i-1) ), 1i6\
S\_[i+6]{}&=&-k+k (+(i-1) ), 1i6 where the parameter $z$ is z=. Note that there are 12 branches of solutions in the holomorphic part. And for a general $sl(3|2)$ higher spin black hole with vanishing $U(1)$ charge there would be $(12\times12=)$144 branches. It could be interesting to study the phase structure of the $sl(3|2)$ black hole as in [@Phase].
The entropy of the black hole in $sl(3|2)$ is actually a sum of the entropies of the spin 2 BTZ black hole and the spin 3 black hole. This is due to the decoupling of two sets of the bosonic generators. As there are 2 branches for the BTZ black hole and 6 branches for the spin 3 black holes[@Justin:thermo; @Phase], there are totally 12 branches in each sector.
### Supersymmetry I
As the $osp(3|2)$ we studied above, we may solve the Killing spinor equation directly to read the conditions for supersymmetry d+\[A,\]=0. Here the spinor $\epsilon$ still has the form =e\^[-L\_0]{}\_0e\^[L\_0]{} where $\epsilon_0$ is independent of the $\rho$ coordinate. It can be expanded as \_0&=&\_[1]{} G\_+\_[2]{}G\_[-]{}+\_[3]{}H\_+\_[4]{}H\_[-]{}+\_[5]{}S\_+\_[6]{}S\_\
&&+\_7S\_[-]{}+\_8S\_[-]{}+\_[9]{}T\_+\_[10]{}T\_+\_[11]{}T\_[-]{}+\_[12]{}T\_[-]{}. We use the same symbol $\eta$ to simplify notation, but one should keep in mind that they are the coefficients in front of different types of fermonic generators. The $``+''$ component of the Killing spinor equation gives us &&\_+\_[1]{}+\_[2]{}+2\_[5]{}=0,\
&&\_+\_[2]{}+(3\_[1]{}-6\_[5]{}+5\_[1]{}+2\_[6]{})=0\
&&\_+\_[3]{}+\_[4]{}-2\_[9]{}=0\
&&\_+\_[4]{}+(3\_[3]{}-6\_[9]{}-2\_[10]{}+5\_3)=0,\
&&\_+\_[5]{}+\_[6]{}=0\
&&\_+\_[6]{}+2\_[7]{}+3\_[5]{}+\_[5]{}=0\[sl+\]\
&&\_+\_[7]{}+(9\_[8]{}+6\_[6]{}-6\_[5]{}-4\_[1]{}+2\_[6]{})=0\
&&\_+\_[8]{}+(3\_[7]{}+4\_[1]{}-2\_[6]{}-4\_[2]{}+\_[7]{})=0\
&&\_+\_[9]{}+\_[10]{}=0\
&&\_+\_[10]{}+2\_[11]{}+3\_[9]{}+\_[9]{}=0\
&&\_+\_[11]{}+(9\_[12]{}+6\_[10]{}+6\_[9]{}+2\_[10]{}+4\_[3]{})=0,\
&&\_+\_[12]{}+(3\_[11]{}+2\_[10]{}+4\_[3]{}+\_[11]{}+4\_[4]{})=0The $``-''$ component of Killing spinor gives us &&\_-\_[1]{}+(5\_[2]{}p-2\_[7]{}p+6\_[5]{}p+12\_[8]{}q-4\_[6]{}q+12\_[5]{}q-4\_[6]{}q)=0\
&&\_-\_[2]{}+(-6\_[8]{}p+5\_[1]{}p+2\_[6]{}p-6\_[5]{}p+4\_[7]{}q-12\_[5]{}\^2q+\
&& 16\_[1]{}q+4\_[6]{}q+3\_[1]{}p+4\_[7]{}q-24\_[5]{}q-12\_[5]{}\^2q)=0\
&&\_-\_[3]{}+(2\_[11]{}p+5\_[4]{}p-6\_[9]{}p+12\_[12]{}q-4\_[10]{}q-12\_[9]{}q-4\_[10]{}q)=0\
&&\_-\_[4]{}+(6\_[12]{}p-2\_[10]{}p+5\_[3]{}p-6\_[9]{}p+4\_[11]{}q-12\_[9]{}\^2q-\
&& 4\_[10]{}q+16\_[3]{}q+3\_[3]{}p+4\_[11]{}q-24\_[9]{}q-12\_[9]{}\^2q)=0\
&&\_-\_[5]{}+(4\_[1]{}p+\_[6]{}p-8\_[2]{}q-4\_[7]{}q+4\_[5]{}q+4\_[5]{}q)=0\
&&\_-\_[6]{}+(4\_[2]{}p+2\_[7]{}p+3\_[5]{}p-12\_[8]{}q-8\_[1]{}q-4\_[6]{}q+24\_[5]{}q+\
&& 9\_[5]{}p-8\_[1]{}q-4\_[6]{}q)=0\[sl-\]\
&&\_-\_[7]{}+(3\_[8]{}p-4\_[1]{}p+2\_[6]{}p-6\_[5]{}p+8\_2q-4\_[7]{}q-12\_5\^2q-\
&& 8\_1q+16\_6q+6\_6p+8\_2q-4\_2q-24\_5q-12\_5\^2q)=0\
&&\_-\_8+(-4\_2p+\_7p+4\_1p-2\_6p+4\_8q+8\_1\^2q-4\_6\^2q-8\_2q+\
&& 8\_7q+3\_7p+4\_8q+16\_1q-8\_6q+8\_1\^2q-4\_6\^2q)=0\
&&\_-\_9+(\_[10]{}p-4\_3p+4\_[11]{}q-8\_4q-4\_9q-4\_9q)=0\
&&\_-\_[10]{}+(2\_[11]{}p-4\_4p+3\_9p+12\_[12]{}q+4\_[10]{}q-8\_3q+24\_9q+9\_9p+\
&& 4\_[10]{}q-8\_3q)=0\
&&\_-\_[11]{}+(3\_[12]{}p+2\_[10]{}p+4\_3p+6\_9p+4\_[11]{}q+8\_4q+12\_9\^2q+16\_[10]{}q+\
&& 8\_3q+6\_[10]{}p+4\_[11]{}q+8\_4q+24\_9q+12\_9\^2q)=0\
&&\_-\_[12]{}+(\_[11]{}p+4\_4p+2\_[10]{}p+4\_3p-4\_[12]{}q+4\_[10]{}\^2q+8\_3\^2q+8\_[11]{}q+\
&& 8\_4q+3\_[11]{}p-4\_[12]{}q+8\_[10]{}q+16\_3q+4\_[10]{}\^2q+8\_3\^2q)=0
Let us first consider the $BTZ$ black hole. This black hole is parameterized by[^11] =0, =0, q=0, p=0. Then the situation is much like the case of $osp(3|2)$. The $``-''$ component equations lead to the conclusion that all $\eta_i$’s are independent of the $x^-$ coordinate. The $``+''$ component equations lead to (\_+\^2-)\_1=0,\
(\_+\^2-)\_3=0,\
(\_+\^2-)(\_+\^2-9)\_5=0,\
(\_+\^2-)(\_+\^2-9)\_[9]{}=0. As the analysis is similar to the $osp$ case, we just list the results as follows
Configuration SUSY Boundary Condition
------------------------------------------------------------- --------- --------------------
$\mathcal{L}=\bar{\mathcal{L}}=-\frac{1}{4}$ (12,12) (AP,AP)
$\mathcal{L}=-\frac{1}{4},\bar{\mathcal{L}}=-\frac{1}{9}$ (12,4) (AP,P)
$\mathcal{L}=-\frac{1}{4},\bar{\mathcal{L}}=-\frac{1}{36}$ (12,4) (AP,AP)
$\mathcal{L}=-\frac{1}{9},\bar{\mathcal{L}}=-\frac{1}{4}$ (4,12) (P,AP)
$\mathcal{L}=-\frac{1}{9},\bar{\mathcal{L}}=-\frac{1}{9}$ (4,4) (P,P)
$\mathcal{L}=-\frac{1}{9},\bar{\mathcal{L}}=-\frac{1}{36}$ (4,4) (P,AP)
$\mathcal{L}=-\frac{1}{36},\bar{\mathcal{L}}=-\frac{1}{4}$ (4,12) (AP,AP)
$\mathcal{L}=-\frac{1}{36},\bar{\mathcal{L}}=-\frac{1}{9}$ (4,4) (AP,P)
$\mathcal{L}=-\frac{1}{36},\bar{\mathcal{L}}=-\frac{1}{36}$ (4,4) (AP,AP)
Here in the table, the number in the bracket are the fermionic supercharges in the holomorphic and anti-holomorphic sectors respectively. The $AP$ or $P$ means the Killing spinor satisfy anti-periodic or periodic boundary condition in the $\phi$ direction. Besides the global AdS$_3$, the other solutions listed in the table are actually not smooth since their spatial holonomy is not in the center. Hence, they may be not allowed from the criteria of smoothness though they preserve some fermionic symmetry.
Moreover, one can also consider the massless BTZ and extreme BTZ black hole
1. For massless BTZ, $\mathcal{L}=\bar{\mathcal{L}}=0$, we have (4,4) supersymmetry.
2. For extreme BTZ with nonvanishing mass, $\mathcal{L}=0, \bar{\mathcal{L}}\not=0$ or $\mathcal{L}\not=0,\bar{\mathcal{L}}=0$, it preserves (4,0) or (0,4) supersymmetry.
For $\mathcal{L}>0, \bar{\mathcal{L}}>0$, there could be supersymmetric configurations with non-vanishing higher spin charge. As the Killing spinor equations become complicated, we can restrict ourselves to constant spinor solutions. The character matrix $M_+, M_-$ can be read from the equations (\[sl+\]) and (\[sl-\]). They are respectively $$\begin{aligned}
M_+=\left(\begin{array}{cccccccccccc}
0&1&0&0&2\mathcal{Y}&0&0&0&0&0&0&0\\
\frac{1}{3}a&0&0&0&-2\mathcal{W}&\frac{2\mathcal{Y}}{3}&0&0&0&0&0&0\\
0&0&0&1&0&0&0&0&-2\mathcal{Y}&0&0&0\\
0&0&\frac{1}{3}a&0&0&0&0&0&-2\mathcal{W}&-\frac{2\mathcal{Y}}{3}&0&0\\
0&0&0&0&0&1&0&0&0&0&0&0\\
0&0&0&0&b&0&2&0&0&0&0&0\\
-\frac{4\mathcal{Y}}{3}&0&0&0&-2\mathcal{W}&\frac{2}{3}b&0&3&0&0&0&0\\
\frac{4\mathcal{W}}{3}&-\frac{4\mathcal{Y}}{3}&0&0&0&-\frac{2\mathcal{W}}{3}&\frac{1}{3}b&0&0&0&0&0\\
0&0&0&0&0&0&0&0&0&1&0&0\\
0&0&0&0&0&0&0&0&b&0&2&0\\
0&0&\frac{4\mathcal{Y}}{3}&0&0&0&0&0&2\mathcal{W}&\frac{2}{3}b&0&3\\
0&0&\frac{4\mathcal{W}}{3}&\frac{4\mathcal{Y}}{3}&0&0&0&0&0&\frac{2\mathcal{W}}{3}&\frac{1}{3}b&0
\end{array}
\right),\nn\end{aligned}$$ $$\begin{aligned}
M_-=\left(\begin{array}{cccccccccccc}
0&\frac{5p}{3}&0&0&c_4&-\frac{1}{3}c_3&-\frac{2p}{3}&4q&0&0&0&0\\
\frac{1}{3}c_1&0&0&0&-c_5&\frac{1}{3}c_4&\frac{1}{3}c_3&-2p&0&0&0&0\\
0&0&0&\frac{5p}{3}&0&0&0&0&-c_4&-\frac{1}{3}c_3&\frac{2p}{3}&4q\\
0&0&\frac{1}{3}c_1&0&0&0&0&0&-c_5&-\frac{1}{3}c_4&\frac{1}{3}c_3&2p\\
\frac{4p}{3}&-\frac{8q}{3}&0&0&\frac{1}{3}c_3&\frac{p}{3}&-\frac{4q}{3}&0&0&0&0&0\\
-\frac{2}{3}c_3&\frac{4p}{3}&0&0&c_2&-\frac{1}{3}c_3&\frac{2p}{3}&-4q&0&0&0&0\\
-\frac{2}{3}c_4&\frac{2}{3}c_3&0&0&-c_5&\frac{2}{3}c_2&-\frac{1}{3}c_3&p&0&0&0&0\\
\frac{2}{3}c_5&-\frac{2}{3}c_4&0&0&0&-\frac{1}{3}c_5&\frac{1}{3}c_2&\frac{1}{3}c_3&0&0&0&0\\
0&0&-\frac{4p}{3}&-\frac{8q}{3}&0&0&0&0&-\frac{1}{3}c_3&\frac{p}{3}&\frac{4q}{3}&0\\
0&0&-\frac{2c_3}{3}&-\frac{4p}{3}&0&0&0&0&c_2&\frac{1}{3}c_3&\frac{2p}{3}&4q\\
0&0&\frac{2}{3}c_4&\frac{2}{3}c_3&0&0&0&0&c_5&\frac{2}{3}c_2&\frac{1}{3}c_3&p\\
0&0&\frac{2}{3}c_5&\frac{2}{3}c_4&0&0&0&0&0&\frac{1}{3}c_5&\frac{1}{3}c_2&-\frac{1}{3}c_3\\\end{array}
\right),\nn\end{aligned}$$ in which some constants are defined as &&a=3+5, b=3+, c\_1=5p+16q+3p, c\_2=p+8q+3p,\
&& c\_3=4(+)q, c\_4=2(p+2q), c\_5=2(p+2q(+)\^2). The determinant of $M_+$ and $M_-$ are respectively det(M\_+)=(-9\^3+16\^2-21\^2+5\^2+25\^3)\^2,\[cha1\] det(M\_-)&=&((729(-9\^3+16\^2-21\^2+5\^2+25\^3))p\^6\
&&+3888q(5\^2+46+77\^2)p\^5\
&&+5184q\^2(46\^4-9\^3+208\^3+153\^2+372\^2\^2+304\^3+94\^4)p\^4\
&&+27648q\^3(37\^3+27\^2+165\^2+219\^2+91\^3)p\^3\
&&-36864q\^4(+)\^2(41\^3-189\^2+105\^2+87\^2+23\^3)p\^2\
&&-442368q\^5(+)(8\^3-27\^2+24\^2+24\^2+8\^3)p\
&&+16384q\^6(8\^3-27\^2+24\^2+24\^2+8\^3)\^2)\^2. \[cha2\] The equations (\[sl+\]) has non-zero constant solution if and only if -9\^3+16\^2-21\^2+5\^2+25\^3=0\[con\] For general configuration, $\mathcal{L}\not=0,\mathcal{Y}\not=0,\mathcal{W}\not=0$, the equations (\[sl+\]) can be solved by &&\_1=C\_1, \_2=-, \_3=C\_2,\
&&\_4=, \_5=, \_6=0\
&&\_7=-, \_8=C\_1(+3), \_9=\
&&\_[10]{}=0, \_[11]{}=-, \_[12]{}=-C\_2(+3).\[kil\] However the Killing spinor equations (\[sl-\]) cannot be satisfied for the configuration (\[con\]) and the solution (\[kil\]). But for the extreme case $\bar{\tau}=\infty$, $q=p=0$, Eq. (\[sl-\]) can be satisfied automatically. The condition $\bar{\tau}=\infty$ can be achieved by |=|, |\^2=(|+|)\^3.\[con2\] Therefore a general configuration (\[con\]) and (\[con2\]) preserve (2,0) supersymmetries. Note that we do not include the case $\bar{\mathcal{L}}=-\bar{\mathcal{Y}}$ as it leads to vanishing spin 3 charge $\bar{\mathcal{W}}=0$.
There are some cases in which supersymmetry is enhanced. We consider the following cases.
1. $\mathcal{W}=0,\ q=0$. The spin 3 charge and chemical potential are zero, so that (\[cha1\]) and (\[cha2\]) reduce to det(M\_+)&=&(-9\^3-21\^2+5\^2+25\^3)\^2\
&=&(-)\^2(3+5)\^4,\
det(M\_-)&=&p\^[12]{}(-)\^2(3+5)\^4. The possible supersymmetric configurations are the following.
1. When $\mathcal{L}=-\frac{5}{3}\mathcal{Y}$, the solution of (\[sl+\]) is &&\_1=C\_1, \_2=-2 C\_3 , \_3=C\_2, \_4=2 C\_4, \_5=C\_3, \_6=0\
&&\_7=2 C\_3 , \_8=C\_1, \_9=C\_4, \_[10]{}=0, \_[11]{}=2 C\_4, \_[12]{}=-C\_2. There are four independent solutions, indicating four conserved supercharges. The equations (\[sl-\]) can be satisfied by the extreme condition (\[con2\]). Thus with the condition (\[con2\]), the configuration preserve (4,0) supersymmetries in general.
2. When $
\mathcal{L}=\mathcal{Y}$, the solution of (\[sl+\]) is &&\_1=0, \_2=-2 C\_1 , \_3=0, \_4=2C\_2 , \_5=C\_1, \_6=0\
&&\_7=-2 C\_1 , \_8=0, \_9=C\_2, \_[10]{}=0, \_[11]{}=-2C\_2 , \_[12]{}=0. There are two independent solutions, indicating two conserved supercharges. The equations (\[sl-\]) can be satisfied in this case. Thus the configuration in this case in general preserve (2,0) supersymmetries. Similar analysis in the anti-holomorphic sector suggests that the configuration $\mathcal{L}=\mathcal{Y},\bar{\mathcal{L}}=\bar{\mathcal{Y}}$ preserve (2,2) supersymmetries.
In fact, the results here is similar to the ones in the $osp(3|2)$ case. This is expected as for $\mathcal{W}=\bar{\mathcal{W}}=0$, we can embed the configurations in $osp(3|2)$ gravity into the ones in $sl(3|2)$ gravity.
2. $\mathcal{Y}=0, p=0$. In this case, the relations (\[cha1\]) and (\[cha2\]) reduce to det(M\_+)&=&(9\^3-16\^2)\^2,\
det(M\_-)&=&q\^[12]{}(8\^3-27\^2)\^4,\[det\] which can only be satisfied by 9\^3-16\^2=0, q=0 The condition $q=0$ can be achieved by extreme configuration (\[con2\]). As the solution of equations (\[sl+\]) is to set $\mathcal{Y}=0$ in the solution (\[kil\]), this case is just a special limit of the general solution (\[kil\]) and there is no more supersymmetry enhancement.
### Supersymmetry II
Now we analyze the condition from spatial holonomy to find the supersymmetric configurations. For the solution we found in (\[sl32\]),(\[sl32p\]), we find the form of $a_{\phi}$ to be $$\begin{aligned}
a_{\phi}=\left(\begin{array}{cc}
a_{\phi}^{3\times3}&0\\0&a_{\phi}^{2\times2}\end{array}
\right).\end{aligned}$$ It is a block diagonal matrix, with $$\begin{aligned}
a_{\phi}^{3\times3}=\left(\begin{array}{ccc}
\frac{4}{3}q(\mathcal{L}+\mathcal{Y})&\sqrt{2} (\mathcal{L}+\mathcal{Y} -(\mathcal{L}+\mathcal{Y})p- 4 \mathcal{W} q )&-2(\mathcal{W}(-1+p)+2q(\mathcal{L}+\mathcal{Y}))\\
-\sqrt{2}(-1+p)&-\frac{8}{3}q(\mathcal{L}+\mathcal{Y})&\sqrt{2}(\mathcal{L}+\mathcal{Y} -(\mathcal{L}+\mathcal{Y})p- 4 \mathcal{W} q )\\
-4q&-\sqrt{2}(-1+p)&\frac{4}{3}q(\mathcal{L}+\mathcal{Y})
\end{array}
\right)\nn\end{aligned}$$ and $$\begin{aligned}
a_{\phi}^{2\times2}=\left(\begin{array}{cc}
0&(1+p)(\mathcal{L}-\mathcal{Y})\\
1+p&0\end{array}
\right)\nn\end{aligned}$$ We face a problem that the eigenvalues of the $3\times3$ matrix $a_{\phi}^{3\times3}$ are involved. However, the $2\times2$ matrix $a_{\phi}^{2\times2}$ is easy to deal with. Its eigenvalues turn out to be \_[|[1]{}]{}=(1+p), \_[|[2]{}]{}=-(1+p). Here we only consider the supersymmetric black holes for which the eigenvalues of $a_{\phi}^{3\times3}$ have to be real. As the eigenvalues of $a_{\phi}^{2\times2}$ are real, then $a_{\phi}^{3\times3}$ should satisfy the equation det(a\_\^[33]{}-x 1\_[33]{})=0 where $x=\varphi_{\bar{i}}$, with $\bar{i}=1$ or $2$. This leads to the equation &&-128\^2q\^3-4(-1+p)(1+(-2+p)p+2q(-3x+8q(+)))\
&&+(-3x+16q(+))(64\^2q\^2-36(-1+p)\^2\
&&+(3x+8q)\^2+4(-9-9(-2+p)p+4q(3x+8q)))=0\[condition\] To check the consistency of (\[condition\]), we can set $\mathcal{W}=q=0$ then we find the supersymmetry enhancement condition (\[susyenhance5\])[^12]. Therefore the analysis for the $\mathcal{W}$ vanishing case is the same as before.
The interesting case comes from $\mathcal{W}\not=0$. Though the condition is given in (\[condition\]), the parameters $p,q$ are determined by (\[pq\]),(\[tal\]) and (\[entropyosp\]). In general, there are solutions to (\[condition\]), which could be quite involved. Here we would like to consider the constant solution and check the consistency with the discussion in the previous subsubsection. We may choose $q=p=0$ by setting the antiholomorphic part to be extreme. Then the condition (\[condition\]) simplifies to 4+4(+)x-x\^3=0. It is just 4=(3+5), which is the same as (\[con\]).
Discussion and Conclusion
=========================
In this paper, we discussed the symmetries of the classical configurations in 3D higher spin gravity, defined by the Chern-Simons action. We found that the holonomy of the gauge potential around the spatial circle encodes the symmetry of the solution. To be modest, we only focused on two classes of configurations: the smooth one with maximal higher spin symmetries, and the higher spin black holes which may preserve part of supersymmetries.
We obtained the maximally symmetric solutions in various higher spin (super-)gravity theories and identified them with the smooth conical defects(surplus). For smooth conical defects(surplus), the spatial holonomies are in the centers of the corresponding gauge groups. Such configurations are of particular importance in the HS/CFT correspondence, as they correspond to the primary states $(0,\L_-)$ in the dual CFT. They are equally important as the global AdS$_3$, in the sense that they are also saddle points of the Euclidean path integral and could be taken as the vacua for different sectors. It is remarkable that the uniqueness of maximally symmetric space in the usual geometric sense is lost in 3D higher spin gravity. It would be interesting to investigate if the same phenomenon happens in the higher dimensional case.
The study of the maximally higher spin symmetric configurations may shed light on the the HS/CFT correspondence in the semiclassical limit. In this limit, the central charge tends to infinity but the rank of gauge group $N$ is kept to be finite, the Vasiliev theory is simplified to a finite $sl(N)$ Chern-Simons gravity coupled to scalar matter. In [@Perlmutter:2012], it was shown that the smooth conical surplus were in precise match with the primary states $(0,\L_-)$ in dual CFT, and the excitations on these surplus could be identified with more general primary states $(\L_+, \L_-)$. The match of the spectrum gives strong support of this duality, even though the theory becomes non-unitary. Our study on the smooth conical surplus suggests that this duality could be true for the higher spin gravity theory for other gauge groups. We constructed the smooth conical defects for $Sp(2N), SO(2N+1), SO(2N), G_2$ and found precise match with the highest weight representations of $so(2N+1),sp(2 N),so(2N),g_2$. This correspondence is captured by a simple relation: -i\_i=Ł\_i+\_i,where $\th_i$’s are the eigenvalues of $a_+$, $\L_i$’s comprise the highest weight and $\rho_i$’s comprise the Weyl vector. It is interesting to see how this relation fits into some precise duality between AdS$_3$ higher spin gravity and coset minimal model.
In the higher spin supergravity theory, the picture is less clear. For the $sl(N+1|N)$ case, the smooth conical defects are indeed in match with the chiral primaries in the proposed dual CFT. However, for the $osp(2N+1|2N)$ case, the conical defects are not in agreement with the chiral primaries in the proposed supercoset. As the bosonic sector of $osp(2N+1|2N)$ involves both $so(2N+1)$ and $sp(2N)$ group, it is nontrivial for the dual CFT to match the spectrum of the smooth conical defects.
On the other hand, the smooth higher spin black holes are partially symmetric. They have trivial thermal holonomies in order to be smooth. But they have nontrivial spatial holonomies, which allow us to analyze their symmetry properties. In general, they keep only the symmetries generated by the Cartan subalgebra of the gauge groups, suggesting that the constant solutions have well-defined global charges. For the black holes in the higher spin supergravity, the supersymmetric configurations are interesting. For the higher spin supergravity, the Killing spinor should be generalized to account for the higher spin spinor. We focused on the supersymmetry of $osp(3|2),sl(3|2)$ higher spin black holes in this paper. We found that all the supersymmetric configurations with constant Killing spinor were extremal, but there were also non-extremal supersymmetric configurations which have non-constant Killing spinors. This feature may hold for the black holes in other higher spin super-gravity theories. We showed that it turned to be more efficient to work with the constraint (\[condf\]) imposed by the spatial holonomy of the configuration to find the Killing spinors, even though it is possible to solve the Killing spinor equations directly.
For the black hole in the high spin supergravity discussed in the present work, their thermodynamics are relatively easy, due to the decoupling of the bosonic sectors. Their entropies are just the sum of the ones of the black holes in the decoupled theory. In this case, the entropies should be able to be understood from boundary CFT, as suggested in [@Gaberdiel:2012yb]. Even though the black holes in $osp(3|2)$ and $sl(3|2)$ look simple, their phase structure could be rich. In particular, the high temperature phase of $sl(3|2)$ black hole may present different features from usual spin 3 black hole, considering the fact that there could be two UV theories in this case[@Peng:2012ae].
The supersymmetric black holes are of particular importance in string and supergravity, due to their better behavior under quantum corrections. In string theory, it has been found that for classes of BPS black holes, not only their entropy but also the quantum corrections are in exact agreement with the string prediction. Especially, the logarithmic corrections to the entropy function due to massless modes provide tests for the underlying quantum gravity[@Banerjee:2011jp; @Sen:2011ba]. For the high spin supersymmetric black holes, it would be interesting to understand the possible logarithmic corrections to the entropy, coming from the massless graviton and high spin fields, from dual CFT.
In our treatment of the black holes in $sl(3|2)$ gravity, we turned off the $U(1)$ field. It would be interesting to explore the black holes with non-vanishing $U(1)$ charge and search for the corresponding supersymmetric configurations. Besides, we showed for the first time that there were exact black hole solutions in higher spin super-gravity, whose entropy function could be written in an analytic form. It would be quite interesting to explore the thermodynamics of these higher spin black holes.
\
The work was in part supported by NSFC Grant No. 10975005, 11275010.
Appendix A: $sl(N), sp(2N),so(2N+1),so(2N),g_2$ {#appendix-a-sln-sp2nso2n1so2ng_2 .unnumbered}
===============================================
Generally, the principal embedding of $sl(2)$ into a Lie algebra $g$ is defined to be the unique embedding with the number of $sl(2)$ modules equals to the rank of $g$. The spin of the modules in different Lie algebras is collected in the following table:
$A_N$ $2,3,\cdots,N+1$
------- -------------------------
$B_N$ $2,4,\cdots,2N$
$C_N$ $2,4,\cdots,2N$
$D_N$ $2,4,\cdots,2N-2,N$
$E_6$ $2,5,6,8,9,12$
$E_7$ $2,6,8,10,12,14,18$
$E_8$ $2,8,12,14,18,20,24,30$
$F_4$ $2,6,8,12$
$G_2$ $2,6$
These numbers equal to the degrees of different Casimir operators in the corresponding algebra.
For a Lie algebra with rank $r$, suppose we have a set of Chevalley basis $e_i^\pm,h_i$, which obeys the following commutation relations: =A\_[ij]{}e\_j\^, \[e\_i\^+,e\_j\^-\]=\_[ij]{}h\_i where $A_{ij}$ are elements of the Cartan matrix. Then in the principal embedding the generators $L_1,L_{-1}$ can be represented as: L\_1=\_[i=1]{}\^r e\_i\^+,L\_[-1]{}=-\_[i=1]{}\^r \_[j=1]{}\^r A\^[ij]{}e\_i\^-. Here $A^{ij}=(A^{-1})_{ij}$. From the usual definition $L_0=[L_1,L_{-1}]/2$, one can easily check $[L_{\pm1},L_0]=\pm L_{\pm 1}$, hence $\{L_0,L_{\pm 1}\}$ indeed form a $sl(2)$ subalgebra.
Using this expression, as long as we have a matrix realization of the Chevalley basis and other lie algebra generators, we can construct the matrix realization of principal embedding in any Lie algebra.
In the case of $sl(N)$, we can use the set of $L_0,L_1,L_{-1}$ to generate the whole set of generators: $$\begin{aligned}
&L_0=\frac{1}{2}\left(\begin{array}{ccccc}
N-1&0&\dots&0&0\\0&N-3&\dots&0&0\\ \vdots&\vdots&\ddots&\vdots&\vdots\\0&0&\dots&-(N-3)&0\\0&0&\dots&0&-(N-1)
\end{array}\right),\nonumber\\
&L_1=\left(
\begin{array}{ccccc}
0&0&\dots&0&0\\-\sqrt{N-1}&0&\dots&0&0\\0&-\sqrt{2N-4}&\dots&0&0\\ \vdots&\vdots&\ddots&\vdots&\vdots\\0&0&\dots&-\sqrt{N-1}&0
\end{array}
\right),\nonumber\\
&L_{-1}=\left(\begin{array}{ccccc}
0&\sqrt{N-1}&0&\dots&0\\0&0&\sqrt{2N-4}&\dots&0\\ \vdots&\vdots&\vdots&\ddots&\vdots\\0&0&0&\dots&\sqrt{N-1}\\0&0&0&\dots&0
\end{array}\right).\end{aligned}$$ Here $L_{1(i+1,i)}=-\sqrt{Ni-i^2}$, $L_{-1(i,i+1)}=\sqrt{Ni-i^2}$, and they satisfy commutation relation $[L_i,L_j]=(i-j)L_{i+j}$.
The other generators are denoted by $W^s_m$, $3\leq s\leq N, -(s-1)\leq m\leq(s-1)$, and are given by $$W^s_{s-1}=L_1^{s-1},\hs{2ex}W^s_{m-1}=-\frac{1}{s+m-1}[L_{-1},W^s_m].$$ The generators of $sp(2N)$ can be obtained by truncating out odd spin generators from $sl(2N)$. The generators of $so(2N+1)$ can be obtained by truncating out odd spin generators from $sl(2N+1)$. The generators of $g_2$ can be obtained by truncating out spin 4 generators from $so(7)$.
For the Lie algebra $so(2N)$, we use the set of generators $T_{ab}=-i(E_{ab}-E_{ba})$, which satisfying the following commutation relation: =-i(\_[bc]{} T\_[ad]{}+\_[ad]{} T\_[bc]{}-\_[bd]{} T\_[ac]{}-\_[ac]{} T\_[bd]{}). The chavelley basis are && e\_[l<N]{}\^+=(T\_[2l,2l+1]{}-i T\_[2l-1,2l+1]{}-i T\_[2l,2l+2]{}-T\_[2l-1,2l+2]{})\
&& e\_[l<N]{}\^-=(T\_[2l,2l+1]{}+i T\_[2l-1,2l+1]{}+i T\_[2l,2l+2]{}-T\_[2l-1,2l+2]{})\
&& h\_[l<N]{}=T\_[2l-1,2l]{}-T\_[2l+1,2l+2]{}\
&& e\_N\^+=(T\_[2N-2,2N-1]{}-i T\_[2N-3,2N-1]{}+i T\_[2N-2,2N]{}+T\_[2N-3,2N]{})\
&& e\_N\^-=(T\_[2N-2,2N-1]{}+i T\_[2N-3,2N-1]{}-i T\_[2N-2,2N]{}+T\_[2N-3,2N]{})\
&& h\_N=T\_[2N-3,2N-2]{}+T\_[2N-1,2N]{}.
Appendix B: Superalgebra $sl(3|2)$ and $osp(3|2)$ {#appendix-b-superalgebra-sl32-and-osp32 .unnumbered}
=================================================
The generators of $sl(N|N-1)$ are classified into the bosonic ones W\^[(1)s]{}\_m(s=2,3,,N), W\^[(2)s]{}\_[m]{}(s=2,3,,N-1), J\[bosonic\] and the fermionic ones Q\^[s]{}\_[r]{}(s=1,,N), |[Q]{}\^[s]{}\_r(s=1,,N)\[fermio\] where $-s+1\le m\le s-1$ and $-s+\frac{1}{2}\le r\le s-\frac{1}{2}$. One can find their realizations in [@Racah; @Tan:2012]. The generators of $osp(2N+1|2N)$ can be obtained by truncating out all the odd spin generators in (\[bosonic\]) and one copy of the fermionic operators in (\[fermio\]). We illustrate such kind of truncation below explicitly for the $sl(3|2)$ case, as we need supermatrix realization in the main context.
Let us consider the $sl(3|2)$ higher spin supergravity. Its bosonic part is $sl(3)\oplus sl(2)\oplus u(1)$. The generators are $L_i,A_i,W_m,J$, where $L_i,A_i$ are spin 2 generators and $W_m$ are spin 3 generators, $J$ is the $u(1)$ generator. The fermionic part is generated by two spin $5/2$ and two spin $3/2$ generators, which are denoted as $S_r,T_r,G_s,H_s$ respectively. The commutation relations are &&\[L\_i,L\_j\]=(i-j)L\_[i+j]{}, \[A\_i,A\_j\]=(i-j)L\_[i+j]{}, \[L\_i,A\_j\]=(i-j)A\_[i+j]{},\
&&\[L\_i,W\_m\]=(2i-m)W\_[i+m]{}, \[A\_i,W\_m\]=(2i-m)W\_[i+m]{},\
&&\[W\_m,W\_n\]=(n-m)(2m\^2+2n\^2-m n-8)(L\_[m+n]{}+A\_[m+n]{}),&&\[L\_i,G\_r\]=(-r)G\_[i+r]{}, \[L\_i,H\_r\]=(-r)H\_[i+r]{},\
&&\[L\_i,S\_s\]=(-s)S\_[i+s]{}, \[L\_i,T\_s\]=(-s)T\_[i+s]{},\
&&\[A\_i,G\_r\]=S\_[i+r]{}+(-r)G\_[i+r]{},\
&&\[A\_i,H\_r\]=-T\_[i+r]{}+(-r)H\_[i+r]{},\
&&\[A\_i,S\_s\]=(-s)S\_[i+s]{}-(3i\^2-2is+s\^2-)G\_[i+s]{},\
&&\[A\_i,T\_s\]=(-s)T\_[i+s]{}+(3i\^2-2is+s\^2-)H\_[i+s]{},&&\[W\_m,G\_r\]=-(-2r)S\_[m+r]{}, \[W\_m,H\_r\]=-(-2r)T\_[m+r]{},\
&&\[W\_m,S\_s\]=-(2s\^2-2sm+m\^2-)S\_[m+s]{}-(4s\^3-3s\^2m+2sm\^2-m\^3-9s+m)G\_[m+s]{},\
&&\[W\_m,T\_s\]=(2s\^2-2sm+m\^2-)T\_[m+s]{}-(4s\^3-3s\^2m+2sm\^2-m\^3-9s+m)H\_[m+s]{},&&\[J,L\_i\]=0, \[J,A\_i\]=0, \[J,W\_m\]=0, \[J,G\_r\]=G\_r, \[J,H\_r\]=-H\_r,\
&&\[J,S\_r\]=S\_r, \[J,T\_r\]=-T\_r, {G\_r,G\_s}=0, {H\_r,H\_s}=0,\
&&{S\_r,S\_s}=0, {T\_r,T\_s}=0, {G\_r,S\_s}=0, {H\_r,T\_s}=0,\
&&{G\_r,H\_s}=2L\_[r+s]{}+(r-s)J,\
&&{S\_r,T\_s}=-(r-s)W\_[r+s]{}+(3s\^2-4rs+3r\^2-)(L\_[r+s]{}-3A\_[r+s]{})-(r-s)(r\^2+s\^2-)J,\
&&{G\_r,T\_s}=-W\_[r+s]{}+(3r-s)A\_[r+s]{}-(3r-s)L\_[r+s]{},\
&&{H\_r,S\_s}=-W\_[r+s]{}-(3r-s)A\_[r+s]{}+(3r-s)L\_[r+s]{}Note that one can truncate out the spin 3 and spin 1 generators and one copy of spin $3/2$ and $5/2$ generators. The resulting algebra is just $osp(3|2)$. More explicitly, one can define the remaining spin $5/2$ and spin $3/2$ generators as $R_s$ and $Z_r$, which are R\_s=S\_s-T\_s, Z\_r=G\_r+H\_r. One can show that the subalgebra $\{L_i,A_i,R_s,Z_r\}$ is closed. The commutation relations are &&\[L\_i,L\_j\]=(i-j)L\_[i+j]{}, \[A\_i,A\_j\]=(i-j)L\_[i+j]{}, \[L\_i,A\_j\]=(i-j)A\_[i+j]{},\
&&\[L\_i,R\_r\]=(-r)R\_[i+r]{}, \[L\_i,Z\_r\]=(-r)Z\_[i+r]{}\
&&\[A\_i,R\_r\]=(-r)R\_[i+r]{}-(3i\^2-2ir+r\^2-)Z\_[i+r]{}\
&&\[A\_i,Z\_r\]=R\_[i+r]{}+(-r)Z\_[i+r]{}\
&&{R\_r,R\_s}=-(3r\^2-4rs+3s\^2-)(L\_[r+s]{}-3A\_[r+s]{})\
&&{Z\_r,Z\_s}=4L\_[r+s]{},\
&&{R\_r,Z\_s}=-(3s-r)A\_[r+s]{}+(3s-r)L\_[r+s]{}The matrix realization of the $sl(3|2)$ generators are $$\begin{aligned}
&L_0=\left(\begin{array}{ccccc}
1&0&0&0&0\\0&0&0&0&0\\0&0&-1&0&0\\0&0&0&\frac{1}{2}&0\\0&0&0&0&-\frac{1}{2}
\end{array}\right),\ L_1=\left(
\begin{array}{ccccc}
0&0&0&0&0\\ \sqrt{2}&0&0&0&0\\0&\sqrt{2}&0&0&0\\0&0&0&0&0\\0&0&0&1&0
\end{array}
\right)\nonumber\\
&L_{-1}=\left(\begin{array}{ccccc}
0&-\sqrt{2}&0&0&0\\0&0&-\sqrt{2}&0&0\\ 0&0&0&0&0\\0&0&0&0&-1\\0&0&0&0&0
\end{array}\right),\ A_0=\left(\begin{array}{ccccc}
1&0&0&0&0\\0&0&0&0&\\0&0&-1&0&0\\0&0&0&-\frac{1}{2}&0\\0&0&0&0&\frac{1}{2}\end{array}
\right)\nonumber\\
&A_1=\left(\begin{array}{ccccc}
0&0&0&0&0\\ \sqrt{2}&0&0&0&0\\0&\sqrt{2}&0&0&0\\0&0&0&0&0\\0&0&0&-1&0\end{array}
\right)\ A_{-1}=\left(\begin{array}{ccccc}
0&-\sqrt{2}&0&0&0\\0&0&-\sqrt{2}&0&0\\0&0&0&0&0\\0&0&0&0&1\\0&0&0&0&0\end{array}
\right)\nonumber\\
&W_2=\left(\begin{array}{ccccc}
0&0&0&0&0\\ 0&0&0&0&0\\4&0&0&0&0\\0&0&0&0&0\\0&0&0&0&0\end{array}
\right)\ W_1=\left(\begin{array}{ccccc}
0&0&0&0&0\\ \sqrt{2}&0&0&0&0\\0&-\sqrt{2}&0&0&0\\0&0&0&0&0\\0&0&0&0&0\end{array}
\right)\nonumber\end{aligned}$$ $$\begin{aligned}
&W_0=\left(\begin{array}{ccccc}
\frac{2}{3}&0&0&0&0\\ 0&-\frac{4}{3}&0&0&0\\0&0&\frac{2}{3}&0&0\\0&0&0&0&0\\0&0&0&0&0\end{array}
\right), W_{-1}=\left(\begin{array}{ccccc}
0&-\sqrt{2}&0&0&0\\ 0&0&\sqrt{2}&0&0\\0&0&0&0&0\\0&0&0&0&0\\0&0&0&0&0\end{array}
\right)\nonumber\\
&W_{-2}=\left(\begin{array}{ccccc}
0&0&4&0&0\\ 0&0&0&0&0\\0&0&0&0&0\\0&0&0&0&0\\0&0&0&0&0\end{array}
\right)\ J=\left(\begin{array}{ccccc}
2&0&0&0&0\\ 0&2&0&0&0\\0&0&2&0&0\\0&0&0&3&0\\0&0&0&0&3\end{array}
\right)\nonumber\end{aligned}$$ $$\begin{aligned}
&G_{\frac{1}{2}}=\left(\begin{array}{ccccc}
0&0&0&0&0\\ 0&0&0&0&0\\0&0&0&0&0\\ 2&0&0&0&0\\0&\sqrt{2}&0&0&0\end{array}
\right)\ G_{-\frac{1}{2}}=\left(\begin{array}{ccccc}
0&0&0&0&0\\ 0&0&0&0&0\\0&0&0&0&0\\0&-\sqrt{2}&0&0&0\\0&0&-2&0&0\end{array}
\right)\nonumber\\
&H_{\frac{1}{2}}=\left(\begin{array}{ccccc}
0&0&0&0&0\\ 0&0&0&\sqrt{2}&0\\0&0&0&0&2\\0&0&0&0&0\\0&0&0&0&0\end{array}
\right)\ H_{-\frac{1}{2}}=\left(\begin{array}{ccccc}
0&0&0&2&0\\ 0&0&0&0&\sqrt{2}\\0&0&0&0&0\\0&0&0&0&0\\0&0&0&0&0\end{array}
\right)\nn\end{aligned}$$ $$\begin{aligned}
&S_{\frac{3}{2}}=\left(\begin{array}{ccccc}
0&0&0&0&0\\ 0&0&0&0&0\\0&0&0&0&0\\0&0&0&0&0\\-3&0&0&0&0\end{array}
\right)\ S_{\frac{1}{2}}=\left(\begin{array}{ccccc}
0&0&0&0&0\\ 0&0&0&0&0\\0&0&0&0&0\\-1&0&0&0&0\\0&\sqrt{2}&0&0&0\end{array}
\right)\nonumber\\
&S_{-\frac{1}{2}}=\left(\begin{array}{ccccc}
0&0&0&0&0\\ 0&0&0&0&0\\0&0&0&0&0\\0&\sqrt{2}&0&0&0\\0&0&-1&0&0\end{array}
\right)\ S_{-\frac{3}{2}}=\left(\begin{array}{ccccc}
0&0&0&0&0\\ 0&0&0&0&0\\0&0&0&0&0\\0&0&-3&0&0\\0&0&0&0&0\end{array}
\right)\nonumber\end{aligned}$$ $$\begin{aligned}
&T_{\frac{3}{2}}=\left(\begin{array}{ccccc}
0&0&0&0&0\\ 0&0&0&0&0\\0&0&0&-3&0\\0&0&0&0&0\\0&0&0&0&0\end{array}
\right)\ T_{\frac{1}{2}}=\left(\begin{array}{ccccc}
0&0&0&0&0\\ 0&0&0&-\sqrt{2}&0\\0&0&0&0&1\\0&0&0&0&0\\0&0&0&0&0\end{array}
\right)\nonumber\\
&T_{-\frac{1}{2}}=\left(\begin{array}{ccccc}
0&0&0&-1&0\\ 0&0&0&0&\sqrt{2}\\0&0&0&0&0\\0&0&0&0&0\\0&0&0&0&0\end{array}
\right)\ T_{-\frac{3}{2}}=\left(\begin{array}{ccccc}
0&0&0&0&3\\ 0&0&0&0&0\\0&0&0&0&0\\0&0&0&0&0\\0&0&0&0&0\end{array}
\right)\nn\end{aligned}$$ The fermionic generators of $osp(3|2)$ can be realized by $$\begin{aligned}
&R_{\frac{3}{2}}=\left(\begin{array}{ccccc}
0&0&0&0&0\\ 0&0&0&0&0\\0&0&0&3&0\\0&0&0&0&0\\-3&0&0&0&0\end{array}
\right)\ R_{\frac{1}{2}}=\left(\begin{array}{ccccc}
0&0&0&0&0\\ 0&0&0&\sqrt{2}&0\\0&0&0&0&-1\\-1&0&0&0&0\\0&\sqrt{2}&0&0&0\end{array}
\right)\nonumber\\
&R_{-\frac{1}{2}}=\left(\begin{array}{ccccc}
0&0&0&1&0\\ 0&0&0&0&-\sqrt{2}\\0&0&0&0&0\\0&\sqrt{2}&0&0&0\\0&0&-1&0&0\end{array}
\right)\ R_{-\frac{3}{2}}=\left(\begin{array}{ccccc}
0&0&0&0&-3\\ 0&0&0&0&0\\0&0&0&0&0\\0&0&-3&0&0\\0&0&0&0&0\end{array}
\right)\nn\\
&Z_{\frac{1}{2}}=\left(\begin{array}{ccccc}
0&0&0&0&0\\ 0&0&0&\sqrt{2}&0\\0&0&0&0&2\\2&0&0&0&0\\0&\sqrt{2}&0&0&0\end{array}
\right)\ Z_{-\frac{1}{2}}=\left(\begin{array}{ccccc}
0&0&0&2&0\\ 0&0&0&0&\sqrt{2}\\0&0&0&0&0\\0&-\sqrt{2}&0&0&0\\0&0&-2&0&0\end{array}
\right)\nn\end{aligned}$$ After some redefinition of the matrix we find that the commutation relations are the same as [@Justin:susy]. Our choice is to match the bosonic commutation relations given in [@Tan:2012].
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[^1]: bchen01@pku.edu.cn
[^2]: lj301@pku.edu.cn
[^3]: ynwang@pku.edu.cn
[^4]: See [@Vasi] for a review.
[^5]: Certainly we need to consider the similar equation in the anti-holomorphic sector as well
[^6]: The maximal supersymmetric conical defects in the higher spin $sl(N|N-1)$ gravity have been studied in [@Hikida:3], but our discussions can be extended to other supergroups. For completeness, we include this case.
[^7]: We should mention that all the solution has maximal symmetry locally, but not all of them preserve the maximal symmetry globally. The globally well-defined symmetry is capture by (\[perio\]). Hence the precise meaning of partial symmetry we discussed are globally partial symmetry.
[^8]: There is a similar identity for $E_{\bar{i}j}$.
[^9]: We are interested in Branch 1, then the zero $\mathcal{L}$’ leads to zero $\mu$. A general feature of higher spin black hole is that in some branches[@Phase] other than Branch 1 even if the higher spin charge $\mathcal{W}$ vanishes the corresponding $\mu$ can be non-zero. We will not include these cases here.
[^10]: Here we use $(p,\bar{p})$ to denote the number of supersymmetries. $p$($\bar{p}$) is the number of the independent killing spinors of the holomorphic (anti-holomorphic) part.
[^11]: Here we have choose Branch 1 which could be the stable phase at the low temperature.
[^12]: There is some signature flipping due to the convention of the ansatz (\[sl32\]).
|
---
author:
- |
Nassif Ghoussoub[^1] and Abbas Moameni[^2]\
Department of Mathematics, University of British Columbia,\
Vancouver BC Canada V6T 1Z2\
\
\
title: Selfdual variational principles for periodic solutions of Hamiltonian and other dynamical systems
---
Selfdual variational principles are introduced in order to construct solutions for Hamiltonian and other dynamical systems which satisfy a variety of linear and nonlinear boundary conditions including many of the standard ones. These principles lead to new variational proofs of the existence of parabolic flows with prescribed initial conditions, as well as periodic, anti-periodic and skew-periodic orbits of Hamiltonian systems. They are based on the theory of anti-selfdual Lagrangians introduced and developed recently in [@G2], [@G3] and [@G4].
Introduction
============
The existence of a selfdual variational principle for gradient flows of convex functionals was conjectured in [@BE1] and established in [@GT1]. Similar selfdual variational principles were later introduced in [@GT3] and [@GM1] for the resolution of certain gradient and Hamiltonian flows that connect two prescribed Lagrangian submanifolds. In this paper, we introduce new anti-selfdual Lagrangians in order to construct variationally solutions of evolution equations that satisfy certain nonlinear boundary conditions. These include the more traditional ones, such as the existence of flows with prescribed initial conditions, as well as periodic, anti-periodic and skew-periodic orbits. Our first variational principle typically deals with gradient flows of the form: $$-\dot x(t) = \partial\phi\big( t,x(t)\big)$$ where $\phi (t, \,)$ is a convex lower semi-continuous function on a Hilbert space $H$. Our second principle deals with Hamiltonian systems of the form: $$-J\dot x (t) \in\partial\phi (t,x(t))$$ where here $\phi (t, \cdot)$ is a convex lower semi-continuous functional on $H\times H$, and $J$ is the symplectic operator defined as $J(p,q)=(-q,p)$. In both cases, the prescribed conditions can be quite general but they include as particular cases the following more traditional ones:
- an initial value problem: $x(0) = x_0$.
- a periodic orbit: $x(0) = x(T)$,
- an anti-periodic orbit: $ x(0) = -x(T)$ or
- a skew-periodic orbit (in the case of a Hamiltonian system): $x(0)=Jx(T)$.
We are looking here for selfdual variational principles, and these depend closely on the scalar product of the underlying path space. The novelty here is in the introduction of appropriate boundary Lagrangians $G$ which, together with the main Lagrangian $L(t, x,p)$, yields an anti-selfdual Lagrangian on a path space equipped with an adequately defined scalar product. The following space (scalar product) seems to be well adapted to our framework.
Let $[0,T]$ be a fixed real interval, and let $L_H^2$ be the classical space of Bochner integrable functions from $[0,T]$ to $H$. We consider the Hilbert space $A_H^2:=\left\{ u:[0,T]{\rightarrow}H;\dot u\in L_H^2\right\}$ consisting of all absolutely continuous arcs $u:[0,T]{\rightarrow}H$ equipped with the norm $$\begin{aligned}
\| u\|_{A_H^2}={\left\{ {\big\|\frac{u(0)+u(T)}{2}\big\|^2}_H
+\int_0^T \|\dot u\|_H^2 \, dt\right\} }^{\frac{1}{2}}\end{aligned}$$ We now recall the concept of anti-selfduality introduced in [@G2].
Given a reflexive Banach space $X$, we say that a convex lower semi-continuous function $L:X\times X^*\to \R \cup \{+\infty\}$ is an [*anti-selfdual Lagrangian*]{} if $$L^*(p,x)=L(-x, -p) \quad \hbox{\rm for all $(x,p)\in X\times X^*$},$$ where here $L^*$ is the Legendre transform in both variables.
A [*time dependent anti-selfdual Lagrangian*]{} on $[0,T]\times X\times X^*$ is any function $L: [0,T]\times X\times X^*\to \R \cup \{+\infty\}$ that is measurable with respect to the $\sigma$-field generated by the products of Lebesgue sets in $[0,T]$ and Borel sets in $X\times X^*$ and such that $L(t,\cdot, \cdot)$ is an anti-selfdual Lagrangian for every $t\in [0,T]$.
The [*Hamiltonian*]{} $H_{L}$ of $L$ is the function defined on $[0,T]\times H\times H$ by: $$H_L(t, x,y)=\sup\{\langle y, p\rangle -L(t, x, p); p\in H\}$$
Here is our first variational principle
\[Principle.1\] Consider a time dependent anti-selfdual Lagrangian $L(t,x,p)$ on $[0,T] \times H \times H$ where $H$ is a Hilbert space, and let $G$ be an anti-selfdual Lagrangian on $H\times H$. Consider on $A_H^2$ the following functional $$I(x)=\int_0^T L\big( t, x(t),\dot{x}(t)\big)\, dt
+G\big(x(0)-x(T),
\frac{x(0)+ x(T)}{2}\big).$$
Assume the following conditions hold:
($A_1$)
: $-\infty < \int_0^T L(t,x(t),0)\, dt\leq C\big(1+\|
x\|_{L^2_H}^2\big) ,\quad x\in L_H^2.$
($A_2$)
: $\int_0^T H_L(t,0,x(t))\, dt{\rightarrow}+\infty\quad\mbox{as}\quad \| x\|_{L^2_H} {\rightarrow}+\infty.$
($A_3$)
: $G$ is bounded from below and $0 \in {\rm Dom}_1 (G)$.
Then, there exists $\hat x\in A_H^2$ such that $$\begin{aligned}
I(\hat x)&=&\inf\limits_{x\in A^2_H}I(x) =0 \label{P1}\\
\big(-\dot{\hat{x}}(t),-\hat x(t)\big) &\in&\partial
L\big(t,\hat x(t),\dot{\hat{x}}(t)\big)\quad {\rm for\, all}\, t\in [0,T] \label{P2}\\
\big(-\frac{\hat x(0)+\hat x(T)}{2}, \hat x (T)-\hat
x(0)\big)
& \in&\partial G\big(\hat x(0)-\hat x(T),
\frac{\hat x(0)+\hat x(T)}{2}\big). \label{P3}\end{aligned}$$
The most basic time-dependent anti-selfdual Lagrangians are of the form $
L(t,x,p)=\varphi (t, x) +\varphi^{*}(t, -p)
$ where for each $t$, the function $x\to \varphi (t, x)$ is convex and lower semi-continuous on $X$. Let now $\psi: H {\rightarrow}\R \cup\{+\infty\}$ be another convex lower semi-continuous function. The above principle then yields that if $-C\leq\int_0^T\phi\big( t,x(t)\big)\,dt\leq C\big(\|
x\|_{L_2^H}^2+1\big)$ and $\Phi (t,x):=\phi (t,x) +\frac{w}{2}|x|_H^2 +\langle f(t), x\rangle$, then the infimum of the functional $$I(x)=\int_0^T \Phi( t, x(t))+\Phi^*(t, -\dot{x}(t))\, dt + \psi (x(0)-x(T)) +\psi^*(- \frac{x(0)+ x(T)}{2})$$ on $A_H^2$ is zero and is attained at a solution $x(t)$ of the following equation $$\begin{aligned}
-\dot x(t) &=& \partial\phi\big( t,x(t)\big) +wx(t)+f(t) \quad {\rm for\, all}\, t\in [0,T] \\
-\frac{ x(0)+ x(T)}{2} & \in&\partial \psi (x(0)-x(T)).\end{aligned}$$ As to the various boundary conditions, we have to choose $\psi$ accordingly.
- Initial boundary condition $x(0)=x_0$ for a given $x_0\in
H$, then $\psi (x)= \frac{1}{4}\|x\|_H^2-{\langle x,x_0 \rangle}$.
- Periodic solutions $x(0)=x(T)$, then $\psi$ is chosen as: $$\begin{aligned}
\psi(x)=\left\{\begin{array}{ll}
0 \quad &x=0\\
+\infty &\mbox{elsewhere}.\end{array}\right.\end{aligned}$$
- Anti periodic solutions $x(0)=-x(T)$, then $\psi (x)=0$ for each $x \in H.$
It is worth noting that while the main Lagrangian $L$ is expected to be smooth and hence its subdifferential coincides with its gradient –and the differential inclusion is often an equation, it is crucial that the boundary Lagrangian $G$ be allowed to be degenerate so as its subdifferential can cover the boundary conditions discussed above.
For the case of Hamiltonian systems we consider for simplicity $H=\R^N$ and let $X=H\times H$. We shall establish the following principle.
\[Principle.2\] Let $\phi:[0,T]\times X{\rightarrow}\R$ be such that $(t,u){\rightarrow}\phi (t,u)$ is measurable in $t$ for each $u\in X$, and convex and lower semi-continuous in $u$ for a.e. $t\in [0,T]$. Let $\psi: X{\rightarrow}\R \cup \{\infty\}$ be convex and lower semi continuous on $X$ and assume the following conditions:
($B_1$)There exists $\beta \in (0,\frac{\pi}{2T})$ and $\gamma, {\alpha}\in L^2 (0,T;\mathbb{R_{+}})$ such that $- {\alpha}(t)\leq \phi(t,u)\leq\frac{\beta}{2} |u|^2+\gamma (t)$ for every $u\in H$ and all $t\in [0,T]$.
($B_2$)$\int_0^T \phi(t,u)\, dt{\rightarrow}+\infty\quad\mbox{as}\quad | u| {\rightarrow}+\infty.$
($B_3$)$\psi$ is bounded from below and $0 \in Dom (\psi).$
\(1) The infimum of the functional $$\begin{aligned}
J_1(u)&=& \int_0^T \left[ \phi (t,u(t))+\phi ^*(t,-J\dot u (t))+\langle J\dot u(t),u(t)\rangle \right] dt\\
&&\quad \quad \quad
+\langle u(T)-u(0),J\frac{u(0)+u(T)}{2}\rangle + \psi \big( u(T)-u(0)\big) +\psi^*\big(
-J\frac{u(0)+u(T)}{2}\big)\end{aligned}$$ on $A_X^2$ is then equal to zero and is attained at a solution of $$\begin{aligned}
\left\{ \begin{array}{lcl}
-J\dot u(t) &= &\partial \phi\big( t,u(t)\big),\\
-J\frac{u(T)+u(0)}{2} &\in &\partial \psi\big(
u(T)-u(0)\big).
\end{array}\right.\end{aligned}$$
\(2) The infimum of the functional $$\begin{aligned}
J_2(u) =\int_0^T\left[\phi\big( t,u(t)\big)+\phi^*(t,-J\dot u (t))+ \langle J\dot u (t),u(t) \rangle \right]\, dt
+\big( Ju(0),u(T)\big) +\psi\big( u(0)\big) +\psi^*\big( Ju(T)\big)\end{aligned}$$ on $A_X^2$ is also zero and is attained at a solution of $$\begin{aligned}
\left\{ \begin{array}{lcl}
-J\dot u (t) &=&\partial\phi\big( t,u(t)\big),\\
Ju(T) &\in& \partial \psi\big( u(0)\big).
\end{array}\right.\end{aligned}$$
In the applications, ${\psi}$ is to be chosen according to the required boundary conditions. For example:
- Initial boundary condition $x(0)=x_0$ for a given $x_0\in H$. Use the functional $J_1$ with $ \bar {\phi} (t,x) =\phi (t, x-x_0) $ and ${\psi}(x)=0$ at $0$ and $+\infty$ elsewhere.
- Periodic solutions $x(0)=x(T)$, or more generally $x(0)-x(T) \in K$ where $K$ is a closed convex subset of $H\times H$. Use the functional $J_1$ with ${\psi}$ chosen as: $$\begin{aligned}
{\psi}(x)=\left\{\begin{array}{ll}
0 \quad &x\in K\\
+\infty &\mbox{elsewhere}.\end{array}\right.\end{aligned}$$
- Anti-periodic solutions $x(0)=-x(T)$. Use the functional $J_1$ with ${\psi}(x)=0$ for each $x \in H.$
- Skew-periodic solutions $x(0)=Jx(T)$. Use the functional $J_2$ with ${\psi}(x)=\frac{1}{2}|x|^2$.
Section 2 deals with gradient flows and the proof of Theorem \[Principle.1\], while section 3 is concerned with Hamiltonian systems. This paper is self-contained but should be read in conjunction with [@G2], [@G3] and [@GT1] which introduce selfduality and [@GM1] which deals with Hamiltonian systems that link Lagrangian submanifolds.
Gradient flows with general boundary conditions
===============================================
Anti-selfdual Lagrangians on path space
---------------------------------------
We now show how a boundary anti-self dual Lagrangian allows us to “lift" a time-dependent anti-selfdual Lagrangian to the path space $A_H^2$. Note that we can and will identify the space $A_H^2$ with the product space $H\times L_H^2$, in such a way that its dual $(A_H^2)^*$ can also be identified with $H\times L_H^2$ via the formula $$\begin{aligned}
{{\langle u,(p_1,p_0) \rangle}}_{A_H^2,H\times L_H^2}
={\langle \frac{u(0)+u(T)}{2},p_1 \rangle} +\int_0^T
{\langle \dot u(t),p_0(t) \rangle}\, dt\end{aligned}$$ where $u\in A_H^2$ and $(p_1,p_0(t))\in H\times L_H^2$.
\[ASD\] Suppose $L$ is an anti-self dual Lagrangian on $[0,T]\times H\times H$ and that $G$ is an anti-selfdual Lagrangian on $H\times H$, then the Lagrangian defined on $A_H^2\times {(A_H^2)}^*=A_H^2\times (H\times L_H^2)$ by $$\begin{aligned}
{\cal M}(u,p)=\int_0^T L\big( t,u(t)+p_0(t),\dot u(t)\big)\, dt
+G\big( u(0)-u(T)+p_1,\frac{u(0)+u(T)}{2}\big)\end{aligned}$$ is anti-self dual Lagrangian on $A_H^2\times (L_H^2\times H)$.
#### Proof:
For $(q,v)\in A_H^2\times (A_H^2)^*$ with $q$ represented by $(q_0(t),q_1)$ we have $$\begin{aligned}
{\cal M}^*(q,v)&=&\sup\limits_{p_1\in H}\ \sup\limits_{p_0\in L_H^2}\ \sup\limits_{u\in A_H^2}
\Bigg\{{\langle p_1,\frac{v(0)+v(T)}{2} \rangle}+{\langle q_1,\frac{u(0)+u(T)}{2} \rangle}\\
& &\quad +\int_0^T \left[{\langle p_0(t),\dot v(t) \rangle}+{\langle q_0(t),\dot u \rangle}
-L\big( t,u(t)+p_0(t),\dot u(t)\big)\right]\, dt\\
& &\quad -G\big( u(0)-u(T)+p_1,\frac{u(0)+u(T)}{2}\big)\Bigg\},\end{aligned}$$ making a substitution $u(0)-u(T)+p_1=a\in H$ and $u(t)+p_0(t)=y(t)\in L_H^2$ we obtain $$\begin{aligned}
{\cal M}^*(q,v) &=& \sup\limits_{a\in H}\ \sup\limits_{y\in L_H^2}\ \sup\limits_{u\in A_H^2}
\Bigg\{{\langle a+u(T)-u(0),\frac{v(0)+v(T)}{2} \rangle} +{\langle q_1,\frac{u(0)+u(T)}{2} \rangle}\\
& &\quad +\int_0^T\left[{\langle y(t)-u(t),\dot v \rangle} +{\langle q_0(t),\dot u(t) \rangle}
-L\big( t,y(t),\dot u(t)\big)\right]\, dt\\
& &\quad -G\big( a,\frac{u(0)+u(T)}{2}\big)\Bigg\}.\end{aligned}$$ Since $\dot u$ and $\dot v\in L_H^2$, we have: $
\int_0^T{\langle u,\dot v \rangle}=-\int_0^T{\langle \dot u,v \rangle}+{\langle u(T),v(T) \rangle}
-{\langle v(0),u(0) \rangle}
$ which implies $$\begin{aligned}
{\cal M}^*(q,v) &=& \sup\limits_{a\in H}\ \sup\limits_{y\in L_H^2}\ \sup\limits_{u\in A_H^2}
\Bigg\{ {\langle a,\frac{v(0)+v(T)}{2} \rangle}
+{\langle u(T),\frac{v(0)+v(T)}{2}-v(T) \rangle}\\
& &\quad +{\langle u(0),v(0)-\frac{v(0)+v(T)}{2} \rangle}
+{\langle q_1,\frac{u(0)+u(T)}{2} \rangle}\\
& &\quad +\int_0^T\left[{\langle y(t),\dot v \rangle}+{\langle \dot u(t),v(t)+q_0(t) \rangle}
-L\big( t,y(t),\dot u(t)\big)\right]\, dt\\
& &\quad - G\big( a,\frac{u(0)+u(T)}{2}\big)\Bigg\} .\end{aligned}$$ Hence, $$\begin{aligned}
{\cal M}^*(q,v) &=& \sup\limits_{a\in H}\ \sup\limits_{y\in L_H^2}\ \sup\limits_{u\in A_H^2}
\Bigg\{{\langle a,\frac{v(0)+v(T)}{2} \rangle}
+{\langle q_1+v(0)-v(T),\frac{u(0)+u(T)}{2} \rangle} -G\big( a,\frac{u(0)+u(T)}{2}\big)\\
& &\quad +\int_0^T\left[{\langle y(t),\dot v(t) \rangle}
+{\langle \dot u(t),v(t)+q_0(t) \rangle}
-L\big( t,y(t),\dot u(t)\big)\right]\, dt\Bigg\}.\end{aligned}$$ Identify now $A_H^2$ with $H\times L_H^2$ via the correspondence: $$\begin{aligned}
\big( b,f(t)\big) &\in & H\times L_H^2\longmapsto b+ \frac{1}{2}\left(
\int_t^T f(s)\, ds-\int_0^t f(s)\, ds\right) \in A_H^2,\\
u &\in & A_H^2\longmapsto\big(\frac{u(0)+u(T)}{2},-\dot u(t)\big)
\in H\times L_H^2.\end{aligned}$$ We finally obtain $$\begin{aligned}
{\cal M}^*(q,v) &=& \sup\limits_{a\in H}\ \sup\limits_{b\in H}
\left\{{\langle a,\frac{v(0)+v(T)}{2} \rangle}
+{\langle q_1+v(0)-v(T),b \rangle} -G(a,b)\right\}\\
& &\quad +\sup\limits_{y\in L_H^2}\ \sup\limits_{r\in L_H^2}
\left\{\int_0^T\left[{\langle y(t),\dot v(t) \rangle}
+{\langle v(t)+q_0(t),r(t) \rangle}-L\big( t,y(t),r(t)\big)\right]\, dt\right\}\\
&=& G^*\big(\frac{v(0)+v(T)}{2},q_1+v(0)-v(T)\big)
+\int_0^T L^*\big( t,\dot v(t),v(t)+q_0(t)\big)\, dt\\
&=& G\big( -q_1-v(0)+v(T),\frac{-v(0)-v(T)}{2}\big)
+\int_0^T L\big( t,-v(t)-q_0(t),-\dot v(t)\big)\, dt\\
&=& {\cal M}(-v,-q).\end{aligned}$$
Variational principles for gradient flows with general boundary conditions
--------------------------------------------------------------------------
We now recall from [@G2] the following general result about minimizing anti-selfdual Lagrangians.
\[nassif\] Let ${\cal M}$ be a an anti-selfdual Lagrangian on a reflexive Banach space $X\times X^{*}$ such that for some $x_{0}\in X$, the function $p\to {\cal M}(x_{0},p)$ is bounded above on a neighborhood of the origin in $X^{*}$. Then there exists $\bar x\in X$, such that: $$\left\{ \begin{array}{lcl}
\label{eqn:existence}
{\cal M}(\bar x, 0)&=&\inf\limits_{x\in X} {\cal M}(x,0)=0.\\
\hfill (0, -\bar x) &\in & \partial {\cal M} (\bar x,0).
\end{array}\right.$$
We can already deduce the following version of Theorem \[Principle.1\] modulo a stronger hypothesis on the boundary Lagrangian.
\[Stringent.1\] Consider a time dependent anti-selfdual Lagrangian $L(t,x,p) $ on $[0,T] \times H \times
H$ and an anti-selfdual lagrangian $G$ on $H\times H.$ Assume the following conditions:
($A_1$) $-\infty < \int_0^T L(t,x(t),0)\, dt\leq C\big(1+\|
x\|_{L^2_H}^2\big)$ for all $x\in L_H^2$.
($A_2$) $G$ is bounded from below and $
G(a,0)\leq C\big(\| a\|_H^2+1\big)$ for all $ a\in H$.
Then the functional $
I(x)=\int_0^T L\big( t, x(t),\dot{x}(t)\big)\, dt
+G\big(x(0)-x(T),
\frac{x(0)+ x(T)}{2}\big)
$ attains its minimum at a path $\hat x\in A_H^2$ satisfying $$\begin{aligned}
I(\hat x)&=&\inf\limits_{x\in A^2_H}I(x) =0\\
\big(-\dot{\hat{x}}(t),-\hat x(t)\big) &\in&\partial
L\big(t,\hat x(t),\dot{\hat{x}}(t)\big)\quad\forall t\in [0,T]\\
\big(-\frac{\hat x(0)+\hat x(T)}{2}, \hat x (T)-\hat
x(0)\big)
& \in&\partial G\big(\hat x(0)-\hat x(T),
\frac{\hat x(0)+\hat x(T)}{2}\big) .\end{aligned}$$
#### Proof:
Apply Proposition \[nassif\] to the Lagrangian $$\begin{aligned}
{\cal M}(u,p)=\int_0^T L\big( t,u(t)+p_0(t),\dot u(t)\big)\, dt
+G\big( u(0)-u(T)+p_1,\frac{u(0)+u(T)}{2}\big)\end{aligned}$$ which is anti-selfdual on $A^2_H$ in view of Proposition \[ASD\]. Noting that $I(x)={\cal M}(x,0)$, we obtain $\hat x(t)\in A_H^2$ such that $$\begin{aligned}
\int_0^T L\big( t,\hat x(t),\dot{\hat{x}}(t)\big)\, dt+G
\left(\hat x(0)-\hat x(T),\frac{\hat x(0)+\hat x(T)}{2}\right) =0,\end{aligned}$$ which gives $$\begin{aligned}
0 &=&\int_0^T \left[L\big( t,\hat x(t),\dot{\hat{x}}(t)\big) + {\langle \hat x(t),\dot{\hat{x}}(t) \rangle}\right]\, dt- \int_0^T {\langle \hat
x(t),\dot{\hat{x}}(t) \rangle} \,dt + G\big(\hat x(0)-\hat x(T),\frac{\hat x(0)+\hat x(T)}{2}\big) \\
&=& \int_0^T \left[L\big( t,\hat x(t),\dot{\hat{x}}(t)\big) + {\langle \hat x(t),\dot{\hat{x}}(t) \rangle}\right]\, dt- \frac{1}{2} | \hat x(T) |^2+
\frac{1}{2} | \hat x(0) |^2+ G
\big(\hat x(0)-\hat x(T),\frac{\hat x(0)+\hat x(T)}{2}\big)\\
&=&\int_0^T L\big( t,\hat x(t),\dot{\hat{x}}(t)\big) + \langle \hat
x(t),\dot{\hat{x}}(t)\rangle \, dt+ {\langle \hat x(0)-\hat
x(T),\frac{\hat x(0)+\hat x(T)}{2} \rangle}+ G
\big(\hat x(0)-\hat x(T),\frac{\hat x(0)+\hat x(T)}{2}\big).\end{aligned}$$ Since $L(t,\cdot, \cdot)$ and $G$ are anti-selfdual Lagrangians we have $
L\big( t,\hat x(t),\dot{\hat{x}}(t)\big) + {\langle \hat x(t),\dot{\hat{x}}(t) \rangle}\geq 0
$ and $$\begin{aligned}
G
\big(\hat x(0)-\hat x(T),\frac{\hat x(0)+\hat x(T)}{2}\big)+ {\langle \hat x(0)-\hat
x(T),\frac{\hat x(0)+\hat x(T)}{2} \rangle} \geq 0.\end{aligned}$$ which means that $
L\big( t,\hat x(t),\dot{\hat{x}}(t)\big) + {\langle \hat x(t),\dot{\hat{x}}(t) \rangle}\,\big) = 0
$ for almost all $t\in [0,T]$, and $$\begin{aligned}
G\big(\hat x(0)-\hat x(T),\frac{\hat x(0)+\hat x(T)}{2}\big)+ {\langle \hat x(0)-\hat
x(T),\frac{\hat x(0)+\hat x(T)}{2} \rangle} = 0.\end{aligned}$$ The result follows from the above identities and the limiting case in Fenchel-Legendre duality.$\square$\
In order to complete the proof of Theorem \[Principle.1\], we need to perform an inf-convolution argument on the boundary Lagrangian $G$. We shall use the following simple estimate
Let $F:Y\mapsto
\mathbb{R}\cup\{\infty\}$ be a proper convex and lower semi continues functional on a Banach space $Y$ such that $- \beta \leq F(y)\leq \frac{{\alpha}}{p}
\|y\|^{p}_{Y}+ \gamma$ with ${\alpha}>0, p >1, \beta\geq0,$ and $\gamma
\geq 0.$ Then for every $y^* \in \partial F(y)$ we have $$\begin{aligned}
\|y^*\|_{Y^*} \leq \Big \{ p {\alpha}^{\frac{q}{p}}( \|y\|_Y +\beta
+\gamma)+1 \Big \}^{p-1}.\end{aligned}$$
We shall also make frequent use of the following lemma [@G2].
Let $G$ be an anti-selfdual Lagrangian on $X\times X^*$ and consider for each ${\lambda}>0$, its $\lambda$-regularization $$\begin{aligned}
G_{\lambda}(x,p):=\inf\left\{ G(z,p)+\frac{\| x-z\|^2}{2{\lambda}}+\frac{{\lambda}}{2}\| p\|^2; \, z\in X\right\} .\end{aligned}$$ Then,
1. $G_{\lambda}$ is also an anti-selfdual Lagrangian on $X\times X^*$ and $
G_{\lambda}(x,0)\leq G(0,0)+\frac{\| x\|^2}{2{\lambda}}.
$
2. If $(0,0)\in {\rm Dom}(G)$ and if $x_{\lambda}\rightharpoonup x$ in $X$ and $p_{\lambda}\rightharpoonup p$ weakly in $X^*$ and if $G
(x_{\lambda}, p_{\lambda})$ is bounded from above, then $
G(x,p) \leq \liminf\limits_{{\lambda}{\rightarrow}0} G_{\lambda}(x_{\lambda}, p_{\lambda}).
$
#### Proof of Theorem \[Principle.1\]:
Define for each ${\lambda}>0$, the Lagrangian $G_{\lambda}$ as in Lemma 2.2, and apply Proposition \[Stringent.1\] to obtain $x_{\lambda}\in A_H^2$ such that $$\begin{aligned}
\int_0^T L\big( t, x_{\lambda}(t),\dot x_{\lambda}(t)\big)\, dt
&+&G_{\lambda}\big( x_{\lambda}(0)-x_{\lambda}(T),\frac{x_{\lambda}(0)+x_{\lambda}(T)}{2}\big) =0\\
\big(-\dot x_{\lambda}(t) ,-x_{\lambda}(t)\big)
&\in&\partial L\big( t,x_{\lambda}(t),\dot x_{\lambda}(t)\big)\\
\big(-\frac{x_{\lambda}(0)+x_{\lambda}(T)}{2},x_{\lambda}(T)-x_{\lambda}(0)\big)
&\in&\partial G_{\lambda}\big( x_{\lambda}(0)-x_{\lambda}(T),
\frac{x_{\lambda}(0)+x_{\lambda}(T)}{2}\big).\end{aligned}$$ We shall show that $ (x_{\lambda})_{\lambda}$ is bounded in $A^2_H$.
For simplicity, we shall assume that $L$ has the form $L(t, x,p)=\phi (t, x) +\phi^*(t, -p)$. For such Lagrangians, Equation (13) yields that $-\dot x_{\lambda}(t)=\partial_1 L\big( t,x_{\lambda}(t),0\big).
$ Multiply this equation by $x_{\lambda}(t)$ and integrate over $[0,T]\times\Omega$ to get $$\begin{aligned}
\int_0^T {\langle -\dot x_{\lambda}(t), x_{\lambda}(t) \rangle} \, dt =\int_0^T
{\langle \partial_1 L (t,x_{\lambda}(t),0) ,x_{\lambda}(t) \rangle} \, dt,\end{aligned}$$ which gives $$-\frac{1}{2}| x_{\lambda}(T)|^2 +\frac{1}{2}| x_{\lambda}(0)|^2=\int_0^T
{\langle \partial_1 L (t,x_{\lambda}(t),0) ,x_{\lambda}(t) \rangle} \, dt
\geq\int_0^T H_L \big( t,0,x_{\lambda}(t)\big)\, dt.$$ Also, from (14) we have $$G_{\lambda}\big( x_{\lambda}(0)-x_{\lambda}(T),\frac{x_{\lambda}(0)+x_{\lambda}(T)}{2}\big)
=-{\langle x_{\lambda}(0)-x_{\lambda}(T),
\frac{x_{\lambda}(0)+x_{\lambda}(T)}{2} \rangle}\\
=\frac{1}{2}| x_{\lambda}(T)|^2 -\frac{1}{2}| x_{\lambda}(0)|^2.$$
Combining (15) and (16) gives that $$G_{\lambda}\big( x_{\lambda}(0)-x_{\lambda}(T),\frac{x_{\lambda}(0)+x_{\lambda}(T)}{2}\big)
+ \int_0^T H_L\big( t,0,x_{\lambda}(t)\big)\, dt
\leq 0.$$ Since $G$ is bounded from below so is $G_{\lambda}$ which together with condition $(A_2)$ imply that $\int_0^T | x_{\lambda}(t)|^2\, dt$ is bounded.
Now from condition $(A_1)$ and the boundedness of $x_{\lambda}$ in $L_H^2$, we can apply Lemma 2.1 to get that $- \dot x_{\lambda}(t)=\partial_1 L\big(
t,x_{\lambda}(t),0\big)$ is bounded in $L_H^2$. Hence, $x_{\lambda}$ is bounded in $A_H^2$, thus, up to a subsequence $ x_{\lambda}(t)\rightharpoonup \hat x(t)$ in $A_H^2$, $x_{\lambda}(0)\rightharpoonup \hat x(0)$ and $x_{\lambda}(T)\rightharpoonup \hat x(T)$ in $H$.
From (17), we have $
G_{\lambda}\big( x_{\lambda}(0)-x_{\lambda}(T),\frac{x_{\lambda}(0)+x_{\lambda}(T)}{2}\big)\leq C,
$ and we obtain from Lemma 2.2 that $$\begin{aligned}
G\big( \hat x(0)-\hat x(T),\frac{\hat x(0)+\hat x(T)}{2}\big)
\leq\liminf\limits_{{\lambda}{\rightarrow}0} G_{\lambda}\big( x_{\lambda}(0)-x_{\lambda}(T),\frac{x_{\lambda}(0)+x_{\lambda}(T)}{2}\big).\end{aligned}$$ Now, if we let ${\lambda}{\rightarrow}0$ in (12), then by considering (18) we get $$\begin{aligned}
\int_0^T L\big( t,\hat x(t),\dot {\hat x}(t)\big)\, dt+
G\big( \hat x(0)-\hat x(T),\frac{\hat x(0)+\hat x(T)}{2}\big)\leq 0.\end{aligned}$$ On the other hand, for every $x \in A^2_H$ we have $$\begin{aligned}
\int_0^T L\big( t,x(t),\dot x(t)\big)\, dt+
G\big( x(0)-x(T),\frac{x(0)+x(T)}{2}\big)\ \geq 0\end{aligned}$$ which means $I(\hat x)=0$ and as in the proof of Proposition \[Stringent.1\], $x(t)$ satisfies (\[P1\]), (\[P2\]), and (\[P3\]). $\square$\
The boundedness condition on $L$ may be too restrictive in applications, and one may want to replace the Hilbertian norm with a stronger Banach norm for which condition $(A1)$ is more likely to hold. For this situation, we have the following result.
\[Relax.1\] Let $X\subset H \subset X^*$ be an evolution pair and let $\psi:[0,T] \times X {\rightarrow}\mathbb{R} \cup \{+\infty\}$ be convex and lower semi-continuous in $x \in X$ for a.e. $t \in [0,T]$ and measurable in $t$ for every $x \in X.$ Consider the time-dependent anti-selfdual Lagrangian, $L(t,x,p)=\psi (t,x)+\psi^* (t,-p)$ on $[0,T] \times X \times X^*$ and an anti-selfdual Lagrangian $G$ on $H\times H.$ Assume the following conditions:
($A'_1)$ For some $p\geq 2$ and $C>0$, we have $-C\big(1+\| x\|_{L^p_X}^p\big) < \int_0^T L(t,x(t),0)\, dt\leq C\big(1+\| x\|_{L^p_X}^p\big)$ for every $x\in L^p_X$.
($A'_2)$ $G$ is bounded from below, $0 \in Dom (G)$ and for every $a\in H$, $G(a,b) {\rightarrow}+\infty$ as $ \| b\|_{H}{\rightarrow}+\infty. $
Then there exists $\hat x \in L^p_X$ with $\dot {\hat x} \in
L^q_{X^*}$ $(\frac{1}{p}+\frac{1}{q}=1)$, $\hat x(0), \hat x(T)
\in H$ and satisfying (\[P1\]), (\[P2\]), and (\[P3\]).
[**Proof:**]{} Here again we shall combine inf-convolution with Theorem \[Principle.1\]. For ${\lambda}>0$ consider the ${\lambda}-$regularization of $\psi$, $$\psi_{\lambda}(t,x)=\inf\limits_{y\in H}\left\{\psi (t,y)+\frac{|
x-y|_H^2}{2{\lambda}}\right\},$$ where $$\begin{aligned}
\psi (t,y)=\left\{\begin{array}{ll}
\psi (t,y)\quad &y\in X\\
+\infty &y\in H-X.\end{array}\right.\end{aligned}$$ Set $L_{\lambda}(t, x,p)=\psi_{\lambda}(t,x)+\psi^*_{\lambda}(t,-p) .$ By Theorem \[Principle.1\], there exists $x_{\lambda}(t)\in A_H^2$ such that $$\begin{aligned}
\int_0^T L\big( t, x_{\lambda}(t),\dot x_{\lambda}(t)\big)\, dt
&+&G \big( x_{\lambda}(0)-x_{\lambda}(T),\frac{x_{\lambda}(0)+x_{\lambda}(T)}{2}\big) =0\\
\big(-\dot x_{\lambda},-x_{\lambda}(t)\big)
& \in&\partial L\big( t,x_{\lambda}(t),\dot x_{\lambda}(t)\big)\\
\big(-\frac{x_{\lambda}(0)+x_{\lambda}(T)}{2},x_{\lambda}(T)-x_{\lambda}(0)\big)
&\in&\partial G\big( x_{\lambda}(0)-x_{\lambda}(T),
\frac{x_{\lambda}(0)+x_{\lambda}(T)}{2}\big)\end{aligned}$$ We now show that $(x_{\lambda})_{\lambda}$ is bounded in an appropriate function space. As in the proof of Theorem \[Principle.1\], we have $$\begin{aligned}
G\big( x_{\lambda}(0)-x_{\lambda}(T),\frac{x_{\lambda}(0)+x_{\lambda}(T)}{2}\big)
+ \int_0^T H_{L_{\lambda}} \big( t, 0, x_{\lambda}(t)\big)\, dt
\leq 0.\end{aligned}$$ Since $\psi$ is convex and lower semi-continuous, there exists $i_{\lambda}(x_{\lambda})$ such that the infimum in (21) attains at $i_{\lambda}(x_{\lambda}),$ i.e. $$\begin{aligned}
\psi_{\lambda}(t,x_{\lambda})= \psi (t,i_{\lambda}(x_{\lambda})) + \frac{\|x_{\lambda}-i_{\lambda}(x_{\lambda})\|^2}{2 {\lambda}}.\end{aligned}$$ Therefore, $$\begin{aligned}
\int_0^T H_{L_{\lambda}} \big( t, 0, x_{\lambda}(t)\big)\, dt= \int_0^T H_L \big( t, 0, i_{\lambda}(x_{\lambda}(t))\big)dt+ \frac{\|x_{\lambda}-i_{\lambda}(x_{\lambda})\|^2}{2 {\lambda}}\,
dt.\end{aligned}$$ Plug (27) in inequality (25) to get $$G\big( x_{\lambda}(0)-x_{\lambda}(T),\frac{x_{\lambda}(0)+x_{\lambda}(T)}{2}\big)
+ \int_0^T H_L \big( t, 0, i_{\lambda}(x_{\lambda}(t))\big) \, dt+ \frac{\|x_{\lambda}-i_{\lambda}(x_{\lambda})\|^2}{2 {\lambda}}\, dt
\leq 0.$$ By the coercivity assumptions in $(A'_1)$ , we obtain that $(i_{\lambda}(x_{\lambda}))_{\lambda}$ is bounded in $L^p(0,T; X)$ and $(x_{\lambda})_{\lambda}$ is bounded in $ L^2(0,T; H)$. It follows from (23) and the structure of $L$ that $
- \dot x_{\lambda}= \partial_1 L (t,i_{\lambda}(x_{\lambda}),0) ,
$ which together with the boundedness of $(i_{\lambda}(x_{\lambda}))_{\lambda}$ in $L^p(0,T;
X)$, condition $(A'_1)$, and Lemma 2.1 imply that $- (\dot x_{\lambda})_{\lambda}$ is bounded in $L^q(0,T; X^*)$. Also note that $x_{{\lambda}} (0)-x_{{\lambda}}(T)=
\int_{0}^{T}\dot {x_{\lambda}}(t)\, dt $ is therefore bounded in $X^*$. It follows from $(A'_2)$ that $x_{{\lambda}} (0)+x_{{\lambda}}(T)$ is therefore bounded in $H$ and so is in $X^*$. Hence, up to a subsequence, we have $$\begin{aligned}
i_{\lambda}(x_{\lambda}) \rightharpoonup \hat x \quad \text{ in } L^p(0,T; X),\\
\dot x_{\lambda}\rightharpoonup \dot
{ \hat x}\quad \text{ in } L^q(0,T; X^*),\\
{x_{\lambda}} \rightharpoonup \hat x \quad \text{ in } L^2(0,T; H),\\
x_{\lambda}(0)\rightharpoonup \hat x(0) \quad \text{ in } X^*,\\
x_{\lambda}(T)\rightharpoonup \hat x(T) \quad \text{ in } X^*.\end{aligned}$$\
On the other hand it follows from (22) and (26) that $$G\big( x_{\lambda}(0)-x_{\lambda}(T),\frac{x_{\lambda}(0)+x_{\lambda}(T)}{2}\big)
+ \int_0^TL \big( t,i_{\lambda}(x_{\lambda}(t)),\dot x_{\lambda}\big)+ \frac{\|x_{\lambda}-i_{\lambda}(x_{\lambda})\|^2}{2 {\lambda}} + \frac {{\lambda}}{2} \|\dot x_{\lambda}\|^2_H\, dt
= 0.$$ By letting $\lambda$ go to zero in (34), we get from (29)-(33) that $$G\big( \hat x_{\lambda}(0)-\hat x_{\lambda}(T),\frac{\hat x_{\lambda}(0)+\hat x_{\lambda}(T)}{2}\big)
+ \int_0^TL \big( t, \hat x_{\lambda}(t)),\dot {\hat x}_{\lambda}\big) ^2_H\, dt
\leq 0.$$ It follows from $(A'_1)$ and the last inequality that $\hat x \in
L^p(0,T; X)$ and $\dot { \hat x} \in L^q(0,T; X^*).$ The rest of the proof is similar to the proof of Proposition 2.3.
One can actually do without the coercivity condition on $G$ in Theorem \[Relax.1\]. Indeed, by using the ${\lambda}-$regularization $G_{{\lambda}}$ of $G$, we get the required coercivity condition on the second variable for $G_{{\lambda}}$ and we obtain from Theorem \[Relax.1\] that there exists $
x_{{\lambda}} \in L^p(0,T; X)$ with $\dot {x}_{{\lambda}} \in L^q(0,T; X^*)$ such that $$\begin{aligned}
\int_0^T L\big( t, x_{{\lambda}}(t),\dot{x}_{{\lambda}}(t)\big)\, dt
+G_{{\lambda}}\left( x_{{\lambda}}(0)- x_{{\lambda}}(T),
\frac{ x_{{\lambda}}(0)+ x_{{\lambda}}(T)}{2}\right) = 0.\end{aligned}$$ It follows from $(A_1)$ and the boundedness of $G_{{\lambda}}$ from below that $
(x_{{\lambda}})_{\lambda}$ is bounded in $ L^p(0,T; X)$, and since $(\dot {x}_{{\lambda}})_{\lambda}$ is bounded in $ L^q(0,T; X^*)$ this also means $(x_{{\lambda}}(0))_{\lambda}$ and $ (x_{{\lambda}}(T))_{\lambda}$ are bounded in $H$. Hence, up to a subsequence we have $$\begin{aligned}
x_{\lambda}\rightharpoonup \hat x \quad \text{ in } L^p(0,T; X),\\
\dot x_{\lambda}\rightharpoonup \dot
{ \hat x}\quad \text{ in } L^q(0,T; X^*),\\
x_{\lambda}(0)\rightharpoonup \hat x(0) \quad \text{ in } H,\\
x_{\lambda}(T)\rightharpoonup \hat x(T) \quad \text{ in } H.\end{aligned}$$ The rest of the proof is similar to the proof of Theorem \[Principle.1\]. $\square$
Example
-------
As mentioned in the introduction, a typical example is $$\begin{aligned}
\left\{\begin{array}{rcl}
-\dot x(t) &= &\partial\phi\big( t,x(t)\big) +wx(t)+f(t)\\
x(0) &= &x_0\quad \text{or} \quad x(0) = x(T) \quad \text{or} \quad
x(0) = -x(T),\end{array}\right.\end{aligned}$$ where $-C\leq\int_0^T\phi\big( t,x(t)\big)\,dt\leq C\big(\|
x\|_{L_2^H}^2+1\big)$ and $w >0$.
For the initial-value problem $x(0)=x_0$, we pick the boundary Lagrangian to be $G(x,p))= \frac{1}{4}|x|_H^2-{\langle x,x_0 \rangle}+|x_0-p|^2$, and so the associated functional becomes $$I(x)=\int_0^T \Phi( t, x(t))+\Phi^*(t, -\dot{x}(t))\, dt + \frac{1}{4}|x(0)-x(T)|^2-{\langle x(0)-x(T),x_0 \rangle}
+|x_0+ \frac{x(0)+ x(T)}{2}|^2$$ where $\Phi (t,x):=\phi (t,x) +\frac{w}{2}|x|_H^2 +\langle f(t), x\rangle$, The infimum of $I$ on $A_H^2$ is zero and is attained at a solution $x(t)$ of the equation. The boundary condition is then $$\begin{aligned}
-\frac{1}{2}(x(0)+x(T))=\partial_1 G\big(x(0)-
x(T), - \frac{x(0)+ x(T)}{2}\big)
=\frac{1}{2}\big(x(0)- x(T)\big) -x_0,\end{aligned}$$ which gives that $x(0)=x_0.$
We can of course relax the conditions on $\phi$ by using again inf-convolution as was done in [@GT1] in the case where $\phi$ is autonomous, or as in Theorem 2.3.
Hamiltonian systems with general boundary conditions
====================================================
For a given Hilbert space $H$, we consider the subspace $H_{T}^1$ of $A_{H}^2$ consisting of all periodic functions, equipped with the norm induced by $ A_H^2$. We also consider the space $H_{-T}^1$ consisting of all functions in $A_{H}^2$ which are anti-periodic, i.e. $u(0)=-u(T).$ The norm of $H_{-T}^1$ is given by $
\| u\|_{H_{-T}^1}=( \int_0^T |\dot u|^2 \, dt)^{\frac{1}{2}}.
$ We now establish a few useful inequalities on $H_{-T}^1$, which can be seen as the counterparts of Wirtinger’s inequality, $$\begin{aligned}
\int_0^T |u|^2 \, dt \leq \frac {T^2}{4 \pi^2} \int_0^T |\dot u|^2
\, dt \quad {\rm for}\quad u \in H_{T}^1 \text{ and }\int_0^T
u(t) \, dt=0,\end{aligned}$$ and the Sobolev inequality on $H_{T}^1$, $$\begin{aligned}
\|u\|^2_{\infty} \leq \frac {T}{12} \int_0^T |\dot u|^2
\, dt \quad {\rm for}\quad u \in H_{T}^1 \text{ and }\int_0^T
u(t) \, dt=0.\end{aligned}$$
If $u \in H_{-T}^1$ then $$\begin{aligned}
\int_0^T |u|^2 \, dt \leq \frac {T^2}{ \pi^2} \int_0^T |\dot u|^2 \,
dt,\end{aligned}$$ and $$\begin{aligned}
\|u\|^2_{\infty} \leq \frac {T}{4} \int_0^T |\dot u|^2
\, dt.\end{aligned}$$
#### Proof:
Since $u(0)=-u(T), $ $u$ has the Fourier expansion of the form $
u(t)= \sum_{k=-\infty}^{\infty}u_k \exp ((2k-1)i\pi t/T).
$ The Parseval equality implies that $$\int_0^T |\dot u|^2 \,dt =\sum_{k=-\infty}^{\infty} T \big( (2k-1)^2
\pi^2/T^2 \big) |u_k|^2 \geq
\frac{\pi^2}{T^2}\sum_{k=-\infty}^{\infty}T |u_k|^2= \frac{\pi^2}{T^2}\int_0^T |u|^2
\, dt.$$ The Cauchy-Schwarz inequality and the above imply that for $t \in [0,T],$ $$\begin{aligned}
|u(t)|^2 &\leq &\left( \sum_{k=-\infty}^{\infty}|u_k| \right )^2 \\&
\leq & \left [ \sum_{k=-\infty}^{\infty} \frac{T}{\pi^2
(2k-1)^2}\right] \left [ \sum_{k=-\infty}^{\infty} T \big( (2k-1)^2
\pi^2/T^2 \big)|u_k|^2 \right]\\ &=&\frac {T}{\pi^2}\sum_{k=-\infty}^{\infty}\frac {1}{(2k-1)^2}) \int_0^T |\dot u|^2
\, dt.\end{aligned}$$ and conclude by noting that $\sum_{k=-\infty}^{\infty}\frac {1}{(2k-1)^2} =\frac{\pi^2}{4}$.
Consider the space $A_X^2$ where $X= H \times H$ and let $J$ be the symplectic operator on $X$ defined as $J(p,q)=(-q,p)$.
1. If $H$ is any Hilbert space, then for every $u \in A_X^2$ $$\begin{aligned}
\left|\int_0^T (J\dot u,u)\, dt+\left(J\frac{u(0)+u(T)}{2},u(T)-u(0)\right)\right| \leq\frac{T}{2}\int_0^T
\big|\dot u (t)\big|^2\, dt.\end{aligned}$$
2. If $H$ is finite dimensional, then $$\begin{aligned}
\left|\int_0^T (J\dot u,u)\, dt+\left(
J\frac{u(0)+u(T)}{2},u(T)-u(0)\right)\right|
\leq\frac{T}{\pi}\int_0^T \big|\dot u (t)\big|^2\, dt.\end{aligned}$$
#### Proof:
For part (i), note that each $u\in A_X^2$ can be written as follows, $$\begin{aligned}
u(t)=\frac{1}{2}\left( \int_0^t \dot u(s)\, ds-\int_t^T\dot u (s)\,
ds\right)+\frac{u(0)+u(T)}{2}.\end{aligned}$$ where $v(t)= u(t)-\frac {u(0)+u(T)}{2}=\frac{1}{2}\left( \int_0^t \dot u(s)\, ds-\int_t^T\dot u (s)\,
ds\right)$ clearly belongs to $H_{-T}^1$. Multiplying both sides by $J\dot u$ and integrating over $[0,T]$, we get $$\begin{aligned}
\int_0^T \langle J\dot u,\, u\rangle\, dt=\frac{1}{2}\int_0^T\left\langle \int_0^t
\dot u(s)\, ds-\int_t^T \dot u(s)\, ds,\, J\dot u\right\rangle\,
dt+ \big\langle \frac{u(0)+u(T)}{2},\, \int_0^T J\dot u(t)\, dt\big\rangle\end{aligned}$$ Hence $$\begin{aligned}
\int_0^T \langle J\dot u,u\rangle\, dt-
\big\langle\frac{u(0)+u(T)}{2},\, J\big(u(T)-u(0)\big)\big\rangle=\frac{1}{2}\int_0^T\left\langle \int_0^t
\dot u(s)\, ds-\int_t^T \dot u(s)\, ds,\, J\dot u\right\rangle\,
dt\end{aligned}$$ and since $J$ is skew-symmetric, we have $$\begin{aligned}
\int_0^T \langle J\dot u,u\rangle \, dt+\big\langle
J\frac{u(0)+u(T)}{2},u(T)-u(0)\big\rangle
=\frac{1}{2}\int_0^T\left\langle \int_0^t
\dot u(s)\, ds-\int_t^T \dot u(s)\, ds,\, J\dot u\right\rangle\,
dt\end{aligned}$$ Applying Hölder’s inequality for the right hand side, we get $$\begin{aligned}
\left|\int_0^T \langle J\dot u,u\rangle \, dt+\big\langle
J\frac{u(0)+u(T)}{2},u(T)-u(0)\big\rangle\right| \leq\frac{T}{2}\int_0^T
\big|\dot u (t)\big|^2\, dt.\end{aligned}$$ For part (ii), set $v(t)= u(t)-\frac {u(0)+u(T)}{2}$ and note that $$\begin{aligned}
\int_0^T \langle J\dot u,u\rangle \, dt+\big\langle
J\frac{u(0)+u(T)}{2},u(T)-u(0)\big\rangle=\int_0^T (J\dot v,v)\, dt\end{aligned}$$ Since $v \in H_{-T}^1$, Hölder’s inequality and Proposition 3.1 imply, $$\begin{aligned}
\left |\int_0^T (J\dot v,v)\, dt \right |& \leq& \left (\int_0^T
|v|^2 \, dt \right )^{\frac{1}{2}}\left (\int_0^T |J \dot v|^2 \, dt \right )^{\frac{1}{2}}\\
&\leq & \frac {T}{\pi}\left (\int_0^T |\dot v|^2 \, dt \right
)^{\frac{1}{2}}\left (\int_0^T |J \dot v|^2 \, dt \right
)^{\frac{1}{2}}\\ &=&\frac {T}{\pi} \int_0^T |\dot v|^2 \, dt
=\frac {T}{\pi} \int_0^T |\dot u|^2 \, dt.
\end{aligned}$$ Combining this inequality with (44) yields the claimed inequality.
If $H= \mathbb{R}^N$ and $X= H \times H$, then the functional $F:A_X^2{\rightarrow}\R$ defined by $$\begin{aligned}
F(u)=\int_0^T \langle J\dot u,u\rangle \, dt+\langle u(T)-u(0),J\frac{u(T)+u(0)}{2}\rangle\end{aligned}$$ is weakly continuous.
#### Proof:
Let $u_k$ be a sequence in $A_X^2$ which converges weakly to $u$ in $A_X^2$. The injection $A_X^2$ into $C([0,T];X)$ with natural norm $\|
\ \|_\infty$ is compact, hence $u_k{\rightarrow}u$ strongly in $C([0,T];X)$ and specifically $u_k(T){\rightarrow}u(T)$ and $u_k(0){\rightarrow}u(0)$ strongly in $X$. Therefore $$\begin{aligned}
\lim\limits_{k{\rightarrow}+\infty} \left(
u_k(T)-u_k(0),J\frac{u_k(T)+u_k(0)}{2}\right)
=\left( u(T)-u(0),J\frac{u(T)+u(0)}{2}\right)\end{aligned}$$ Also, it is standard that $u{\rightarrow}\int_0^T (J\dot u,u)\, dt$ is weakly continuous (Proposition 1.2 in [@MW] ) which together with (45) imply that $F$ is weakly continuous.
A general variational principle for Hamiltonian systems
-------------------------------------------------------
In this section we establish Theorem \[Principle.2\] under the assumption that $H$ is finite dimensional ($X=\R^{2N} $). We start with the following proposition which assumes a stronger condition on the boundary Lagrangian.
\[Relax.2\] Let $\phi:[0,T]\times X{\rightarrow}\R$, such that $(t,u){\rightarrow}\phi (t,u)$ is measurable in $t$ for each $u\in X$, and is convex and lower semi-continuous in $u$ for a.e. $t\in [0,T]$. Let $\psi: X{\rightarrow}\R \cup \{\infty\}$ be convex and lower semi continuous and assume the following conditions:
($B_1$)There exists $\beta \in (0,\frac{\pi}{2T})$ and $\gamma, {\alpha}\in L^2 (0,T;\mathbb{R_{+}})$ such that $- {\alpha}(t)\leq \phi(t,u)\leq\frac{\beta}{2} |u|^2+\gamma (t)$ for every $u\in X$ and a.e. $t\in [0,T]$.
($B'_2$)There exist positive constants $ {{\alpha}}_{1}, \beta_{1}, \gamma_{1} \in \mathbb{R}$ such that, for every $u\in X$ one has $- {\alpha}_1\leq \psi(u)\leq\frac{\beta_1}{2} |u|^2+ \gamma_1$.
\(1) The infimum of the functional $$\begin{aligned}
J_1(u)&=& \int_0^T \left[ \phi (t,u(t))+\phi ^*(t,-J\dot u (t))+\langle J\dot u(t),u(t)\rangle \right] dt\\
&&\quad \quad \quad
+\langle u(T)-u(0),J\frac{u(0)+u(T)}{2}\rangle + \psi \big( u(T)-u(0)\big) +\psi^*\big(
-J\frac{u(0)+u(T)}{2}\big)\end{aligned}$$ on $A_X^2$ is then equal to zero and is attained at a solution of $$\begin{aligned}
\left\{ \begin{array}{lcl}
-J\dot u(t) &= &\partial \phi\big( t,u(t)\big)\\
-J\frac{u(T)+u(0)}{2} &= &\partial \psi\big(
u(T)-u(0)\big).
\end{array}\right.\end{aligned}$$
\(2) The infimum of the functional $$\begin{aligned}
J_2(u) =\int_0^T\left[\phi\big( t,u(t)\big)+\phi^*(t,-J\dot u (t))+\langle J\dot u (t),u(t)\rangle \right]\, dt
+\big( Ju(0),u(T)\big) +\psi\big( u(0)\big) +\psi^*\big( Ju(T)\big)\end{aligned}$$ on $A_X^2$ is also equal to zero and is attained at a solution of $$\begin{aligned}
\left\{ \begin{array}{lcl}
-J\dot u (t) &=&\partial\phi\big( t,u(t)\big)\\
Ju(T) &=& \partial \psi\big( u(0)\big).
\end{array}\right.\end{aligned}$$
The proof requires a few preliminary lemmas, but first and anticipating that the conjugate $\phi^*$ and $\psi^*$ may not be finite everywhere, we start by replacing $\phi$ and $\psi$ with the perturbations such as $\phi_{\epsilon}(t,u)= \frac
{\epsilon}{2}\|u\|^2 + \phi(t,u)$ and $ \psi_{\epsilon}(u)= \frac {\epsilon}{2}\|u\|^2 + \psi(u)$. It is then clear that $$\begin{aligned}
\frac {1}{2(\beta+\epsilon)}|u|^2-\gamma (t) \leq \phi^*_{\epsilon}(t,u)\leq \frac {1}{2 \epsilon}|u|^2+\alpha (t) ,\end{aligned}$$ and $$\begin{aligned}
\frac {1}{2(\beta_1+\epsilon)}|u|^2-\gamma_1 \leq \psi^*_{\epsilon}(u)\leq \frac {1}{2 \epsilon}|u|^2+\alpha_1 .\end{aligned}$$ We now consider the Lagrangian ${\cal L}_{\epsilon}:A_X^2\times A_X^2{\rightarrow}\R $ defined by $$\begin{aligned}
{\cal L}_{\epsilon}(v;u)&=&\int_0^T\left[ -\langle J\dot v (t) ,u(t)\rangle+\phi_{\epsilon} ^*(t,-J\dot u (t))-\phi_{\epsilon} ^*(t,-J\dot v
(t))+
\langle J\dot u(t),u(t)\rangle \right]\, dt\\
& &\quad +\left\langle u(T)-u(0),J\frac{u(T)+u(0)}{2}\right\rangle -\left\langle u(T)-u(0),J\frac{v(T)+v(0)}{2}\right\rangle
\\
& &\quad +\psi_{\epsilon}^*\big( -J\frac{u(T)+u(0)}{2}\big)
-\psi_{\epsilon}^*\big( -J\frac{v(T)+v(0)}{2}\big)\end{aligned}$$ and $$\begin{aligned}
J^{\epsilon}_1(u):&=& \int_0^T \left[ \phi_{\epsilon} (t,u(t))+\phi_{\epsilon} ^*(t,-J\dot u(t))+\big( J\dot u(t),u(t)\big)\right]\, dt\\
&& \quad +\big\langle u(T)-u(0),J\frac{u(0)+u(T)}{2}\big\rangle +\psi_{\epsilon} \big( u(T)-u(0)\big) + \psi_{\epsilon}^*\big(
-J\frac{u(0)+u(T)}{2}\big)\end{aligned}$$ To simplify the notation we use $C$ as a general positive constant.
For every $u\in A_X^2$, we have $J_1(u)\geq 0$ and $J_1^\epsilon(u) \geq 0$.
#### Proof:
By the definition of Legendre-Fenchel duality, one has $$\begin{aligned}
\phi (t,u(t))+\phi ^*(t,-J\dot u(t))+\langle J\dot
u(t),u(t)\rangle \geq 0 \quad \quad \text{for }t \in [0,T],\end{aligned}$$ and $$\begin{aligned}
\psi \big( u(T)-u(0)\big) + \psi^*\big(
-J\frac{u(0)+u(T)}{2} \big)+ \langle
u(T)-u(0),J\frac{u(0)+u(T)}{2}\rangle\geq 0,\end{aligned}$$ which means $J_1(u)\geq 0$. The same applies to $J^\epsilon_1$.
For every $u\in A_X^2$, we have $
I_{\epsilon}(u)= \sup\limits_{v\in A_X^2}{\cal L}_{\epsilon}(v,u).
$
#### Proof:
First recall that one can identify $A_X^2$ with $X\times L_X^2$ via the correspondence: $$\begin{aligned}
\big( x,f(t)\big) &\in & X\times L_X^2\longmapsto x+\frac{1}{2}\left(
\int_0^t f(s)\, ds-\int_t^T f(s)\, ds\right) \in A_X^2\\
u &\in & A_X^2\longmapsto\left(\frac{u(0)+u(T)}{2},\dot u(t)\right)
\in X\times L_X^2\end{aligned}$$ Thus, for every $u\in A_X^2$, we can write $$\begin{aligned}
\sup\limits_{v\in A_X^2}{\cal L}_{\epsilon}(v;u) &= & \sup\limits_{v\in X\times L_X^2}{\cal L}_{\epsilon}(v,u)\\
&=& \sup\limits_{f\in L^2(0,T;X)}\sup\limits_{x\in X}\Big\{\int_0^T \big[\langle -J f(t),u(t)\rangle +\phi_{\epsilon} ^*(t,-J\dot u(t))
-\phi_{\epsilon} ^*(t,-J f(t))+(J\dot u(t),u(t))\big]\, dt\Big\}\\
&&\quad \quad + \big \langle u(T)-u(0),J\frac{u(T)+u(0)}{2}\big \rangle -\big\langle u(T)-u(0),Jx\big\rangle+ \psi_{\epsilon}^*\big( -J\frac{u(T)+u(0)}{2}\big) -\psi_{\epsilon}^*(-Jx)\\
&=& \sup\limits_{f\in L^2(0,T;X)}\Big\{\int_0^T \big[\langle -J f(t),u(t)\rangle +\phi_{\epsilon} ^*(t,-J\dot u(t))
-\phi_{\epsilon} ^*(t,-J f(t))+(J\dot u(t),u(t))\big]\, dt\Big\}\\
&&\quad \quad + \sup\limits_{x\in X}\Big\{ \big \langle u(T)-u(0),J\frac{u(T)+u(0)}{2}\big \rangle -\big\langle u(T)-u(0),Jx\big\rangle\\
&&\quad \quad +\psi_{\epsilon}^*\big( -J\frac{u(T)+u(0)}{2}\big) - \psi_{\epsilon}^*(-J x)\Big\}\\
&=& \int_0^T\left[ \phi_{\epsilon} (t,u(t))+\phi_{\epsilon} ^*(t,-J\dot u(t))+(J\dot u(t),u(t))\right]\, dt\\
&&\quad \quad + \big\langle u(T)-u(0),J\frac{u(T)+u(0)}{2}\big\rangle +\psi_{\epsilon}\big( u(T)-u(0)\big)+\psi_{\epsilon}^*\big( -J\frac{u(T)+u(0)}{2}\big) \\
&=&J_1^{\epsilon}(u).\end{aligned}$$
Under the assumptions $(B_1)$ and $(B'_2)$, we have for each $0<\epsilon < \frac {1}{2}(\frac {\pi}{T}-2 \beta)$ the following coercivity condition $${\cal L}_{\epsilon}(0,u){\rightarrow}+\infty \quad {\rm when} \quad \| u\|_{A_X^2}{\rightarrow}+\infty .$$
#### Proof:
From (48) and (49) and since $\int_0^T \phi^*(t,0)\, dt$ and $\psi^*(0)$ are finite, we get $$\begin{aligned}
{\cal L}_{\epsilon}(0,u) &\geq& \frac{1}{2(\beta+\epsilon)}\int_0^T |\dot
u(t)|^2\, dt +\int_0^T \langle J\dot u(t),u(t)\rangle\, dt +\langle J\frac{u(0)+u(T)}{2}, u(T)-u(0)\rangle \\
& &\quad +\frac{1}{2(\beta_1+\epsilon)}{\left |\frac{u(0)+u(T)}{2}\right|}^2+C,\end{aligned}$$ where $C$ is a constant. From part (ii) of Proposition 3.2, we have $$\begin{aligned}
\left|\int_0^T \langle J\dot u(t) ,u (t)\rangle \, dt+ \langle u(T)-u (0),
J\frac{u(T)+u (0)}{2}\rangle \right|
\leq\frac{T}{\pi}\int_0^T |\dot u (t)|^2\, dt\end{aligned}$$ Hence, modulo a constant, we obtain $$\begin{aligned}
{\cal L}_{\epsilon}(0,u) \geq
\left(\frac{1}{2(\beta+\epsilon)}-\frac{T}{\pi}\right)\int_0^T
{|\dot u(t)|}^2\, dt+\frac{1}{2(\beta_1+\epsilon)}
{\left|\frac{u(0)+u(T)}{2}\right|}^2.\end{aligned}$$ Since $0<\epsilon < \frac {1}{2}(\frac {\pi}{T}-2 \beta)$, it follows that $\frac{1}{2(\beta+\epsilon)}-\frac{T}{\pi}>0$ and ${\cal L}_{\epsilon}(0,u){\rightarrow}+\infty$ as $\| u\|_{A_X^2}{\rightarrow}+\infty$. $\square $
Proposition 3.4 is now a consequence of the following Ky-Fan type min-max theorem which is essentially due to Brezis-Nirenberg-Stampachia (see [@BNS]).
\[KF\] Let $Y$ be a a reflexive Banach space and let ${\cal L}(x,y)$ be a real valued function on $Y\times Y $ that satisfies the following conditions:
\(1) ${\cal L}(x,x) \leq 0$ for every $x\in Y$.
\(2) For each $x\in Y$, the function $y \to {\cal L}(x,y)$ is concave.
\(3) For each $y\in Y$, the function $x\to {\cal L}(x,y)$ is weakly lower semi-continuous.
\(4) The set $Y_0=\{x\in Y; {\cal L}(x,0)\leq 0\}$ is bounded in $Y$.
Then there exists $x_{0}\in Y$ such that $\sup\limits_{y\in Y}{\cal L} (x_{0},y)\leq 0$.
#### Proof of Proposition \[Relax.2\]:
Let $0<\delta < \frac {1}{2}(\frac {\pi}{T}-2 \beta)$ and $0<
\epsilon < \delta.$ It is easy to see that the ${\cal L}_{\epsilon}:X\times
X{\rightarrow}\R $ satisfies all the hypothesis of Lemma 3.4. It follows from (48) and (49) that ${\cal L}_{\epsilon}$ is finitely valued on $ X\times X $ and that for each $u\in X\times X$, ${\cal L}_{\epsilon}(u,u)=0$. Lemma 3.3 gives that the set $Y=\{ u\in X, {\cal L}_{\epsilon}(0,u)\leq 0\}$ is bounded in $X$. Moreover, for every $u\in X$, the function $v{\rightarrow}{\cal L}_{\epsilon}(v,u)$ is concave and for every $v\in X$, $u{\rightarrow}{\cal L}_{\epsilon}(u,v)$ is weakly lower semi-continuous by Proposition 3.3. It follows that there exists $u_{\epsilon}\in X$ such that $
I_{\epsilon}(u_{\epsilon})\leq\sup\limits_{v\in A_X^2}{\cal L}_{\epsilon}(v,u_{\epsilon})\leq 0.
$\
In view of Lemma 3.1, we then have $I_{\epsilon}(u_{\epsilon})= 0$ which yields: $$\begin{aligned}
I_{\epsilon}(u_{\epsilon}) &=& \int_0^T\left[\phi_{\epsilon}
(t,u_{\epsilon}(t))
+\phi_{\epsilon}^*(t,-J\dot{u_{\epsilon}}(t))+ \langle u_{\epsilon}(t),J\dot{u_{\epsilon}}(t)\rangle \right]\, dt \nonumber \\
&& \quad +\psi_{\epsilon} \big( u_{\epsilon}(T)-u_{\epsilon}(0)\big)
+\psi_{\epsilon}^*\big(
-J\frac{u_{\epsilon}(0)+u_{\epsilon}(T)}{2}\big)\nonumber\\
&&
\quad + \langle u_{\epsilon}(T)-u_{\epsilon}(0), J\frac{u_{\epsilon}(0)+u_{\epsilon}(T)}{2} \rangle \nonumber \\
&=& 0.\end{aligned}$$ We shall show that $u_{\epsilon}$ is bounded in $X$. From Proposition 3.2, we have $$\begin{aligned}
\left|\int_0^T (J\dot u_{\epsilon} (t),u_{\epsilon}(t) )\, dt+\langle
u_\epsilon (T)-u_\epsilon (0),
J\frac{u_\epsilon (T)+u_\epsilon (0)}{2} \rangle \right|
\leq\frac{T}{\pi}\int_0^T |\dot u_\epsilon (t)|^2\, dt\end{aligned}$$ which together with (51), yield $$\begin{aligned}
\int_0^T\left[\phi_{\epsilon} (t,u_{\epsilon}(t))
+\phi_{\epsilon}^*(t,-J\dot{u_{\epsilon}}(t))\right]\, dt-\frac{T}{\pi}\int_0^T |\dot u_{\epsilon} (t)|^2\, dt
+ \psi_{\epsilon} \big( u_{\epsilon}(T)-u_{\epsilon}(0)\big)
+\psi_{\epsilon}^*\big(
-J\frac{u_{\epsilon}(0)+u_{\epsilon}(T)}{2}\big) \leq 0.\end{aligned}$$ This inequality together with the facts that $\phi_{\epsilon}$ and $\psi_{\epsilon}$ are bounded from below and $\phi^*_{\epsilon}$ and $\psi^*_{\epsilon}$ satisfy inequalities (48) and (49) respectively, guarantee the existence of a constant $C> 0$ independent of $\epsilon$ such that $$\begin{aligned}
\left(\frac{1}{2 (\beta + \delta)}-\frac{T}{\pi}\right)\int_0^T {|\dot u_{\epsilon}(t)|}^2\, dt+\frac{1}{2(\beta_1+ \delta)}
{\left|\frac{u_{\epsilon}(0)+u_{\epsilon}(T)}{2}\right|}^2 &\leq
&\\
\left(\frac{1}{2(\beta+\epsilon)}-\frac{T}{\pi}\right)\int_0^T {|\dot u_{\epsilon}(t)|}^2\, dt+\frac{1}{2(\beta_1+ \epsilon)}
{\left|\frac{u_{\epsilon}(0)+u_{\epsilon}(T)}{2}\right|}^2 &\leq & C,\end{aligned}$$ which means $(u_{\epsilon})_\epsilon$ is bounded in $A^2_X$ and so, up to a subsequence , there exists a $\bar u \in A^2_X$ such that $u_{\epsilon}\rightharpoonup \bar u$ in $A^2_X.$ It is easily seen that $$\begin{aligned}
\int_{0}^{T} \phi^*_\epsilon(t, \dot {u}_{\epsilon}(t)) \,
dt:=\inf\limits_{v \in L^2(0,T;X)}\int_{0}^{T}\left [ \phi^*(t,v(t))+\frac{| \dot {u}_{\epsilon}(t)-v(t) |^2}{2
\epsilon}\right]dt\end{aligned}$$ and since $ \phi^*$ is convex and lower semi continuous, there exists $v_{\epsilon} \in L^2(0,T;X)$ such that this infimum attains at $v_{\epsilon},$ i.e. $$\begin{aligned}
\int_{0}^{T} \phi^*_\epsilon (t,\dot {u}_{\epsilon}(t)) \,
dt=\int_{0}^{T}\left [ \phi^*(t, v_{\epsilon}(t))+\frac{| \dot
{u}_{\epsilon (t)}-v_{\epsilon}(t)|^2}{2 \epsilon}\right]dt.\end{aligned}$$ It follows from the above and the boundedness of $(u_{\epsilon})_\epsilon$ in $A^2_X,$ that there exists $C>0$ independent of $\epsilon$ such that $$\begin{aligned}
\int_{0}^{T} \phi^*_\epsilon (t, \dot {u}_{\epsilon}(t)) \,
dt=\int_{0}^{T}\left [ \phi^*(t,v_{\epsilon}(t))+\frac{| \dot
{u}_{\epsilon}(t)-v_{\epsilon}(t)|^2}{2 \epsilon}\right]dt<C.\end{aligned}$$ Since $\phi^*$ is bounded from below, we have $
\int_{0}^{T}| \dot {u}_{\epsilon}(t)-v_{\epsilon}(t)|^2dt<C \epsilon
$ which means $ v_{\epsilon} \rightharpoonup \dot {\bar{u}}$ in $L^2(0,T;X).$ Hence $$\begin{aligned}
\int_{0}^{T}\phi^*(t,\dot {{\bar u}}(t))\, dt&\leq&
\liminf\limits_{\epsilon \rightarrow 0} \int_{0}^{T}\phi^*(t,v_{\epsilon}(t))dt\nonumber\\
& \leq& \liminf\limits_{\epsilon
\rightarrow 0} \int_{0}^{T} \left [ \phi^*(t,v_{\epsilon}(t))+\frac{| \dot
{u_{\epsilon}}(t)-v_{\epsilon}(t)|^2}{2 \epsilon}\right]dt \nonumber\\
& =&\liminf\limits_{\epsilon \rightarrow 0}\int_{0}^{T}\phi^*_{\epsilon} (t, \dot
{u}_{\epsilon}(t))dt.\end{aligned}$$ Also, $$\begin{aligned}
\int_{0}^{T}\phi(t, {{\bar u}}(t))\, dt&\leq&
\liminf\limits_{\epsilon \rightarrow 0} \int_{0}^{T}\phi (t,u_{\epsilon}(t))\, dt \nonumber\\
&\leq& \liminf\limits_{\epsilon
\rightarrow
0} \int_{0}^{T} \left [ \phi(t,u_{\epsilon}(t))+\frac{ \epsilon}{2 }|u_{\epsilon}(t)|^2\right]dt \nonumber\\
& =&\liminf\limits_{\epsilon \rightarrow 0}\int_{0}^{T}\phi_{\epsilon} (t,
{u_{\epsilon}}(t))dt.\end{aligned}$$ It follows from (52) and (53) that, $$\begin{aligned}
\int_0^T \left[\phi (t, \bar u(t)) +\phi^*(t,-J \dot{
\bar u}(t)) \right] \, dt \leq \liminf\limits_{\epsilon \rightarrow
0} \int_0^T\left[\phi_{\epsilon} (t,u_{\epsilon}(t)) +\phi_{\epsilon}^*(t,-J\dot{u_{\epsilon}}(t))\right] \, dt.\end{aligned}$$ By the same argument we arrive at, $$\begin{aligned}
\psi \big( \bar u(T)- \bar u(0)\big) + \psi^*\big(
-J\frac{\bar u(0)+\bar u(T)}{2}\big) \leq \liminf\limits_{\epsilon
\rightarrow 0} \Big \{ \psi_{\epsilon} \big(
u_{\epsilon}(T)-u_{\epsilon}(0)\big) + \psi_{\epsilon}^*\big(
-J\frac{u_{\epsilon}(0)+u_{\epsilon}(T)}{2}\big)\Big \}\end{aligned}$$ Also, from Proposition 3.3, we have $$\begin{aligned}
\lim\limits_{\epsilon
\rightarrow 0}\int_0^T\langle u_{\epsilon}(t),J\dot{u_{\epsilon}}(t)\rangle\, dt &+& \langle u_{\epsilon}(T)-u_{\epsilon}(0), J \frac{u_{\epsilon}(0)+u_{\epsilon}(T)}{2}\rangle\nonumber\\
&=&\int_0^T\langle \bar u(t),J\dot{\bar u}(t)\rangle dt + \langle \bar
u(T)-\bar u(0), J \frac{\bar u(0)+\bar
u(T)}{2} \rangle.\end{aligned}$$ Combining the above yields $$\begin{aligned}
I(\bar u) &=& \int_0^T\left[\phi (t,\bar u(t))
+\phi^*(t,-J\dot{\bar{u}}(t))+(\bar{u}(t),J\dot{\bar{u}}(t))\right]\, dt\\
&+& \psi \big(\bar u(T)-\bar u(0)\big) +\psi^*\big( -J\frac{\bar u(0)+\bar
u(T)}{2}\big)
+\langle \bar u(T)-\bar u(0), J \frac{\bar u(0)+\bar u(T)}{2}\rangle\\
& \leq & \liminf\limits_{\epsilon \rightarrow 0}\Big\{
\int_0^T\left[\phi_{\epsilon} (t,u_{\epsilon}(t))
+\phi_{\epsilon}^*(t,-J\dot{u_{\epsilon}}(t))+(u_{\epsilon}(t),J\dot{u_{\epsilon}}(t))\right]\, dt\\
&+& \psi_{\epsilon} \big( u_{\epsilon}(T)-u_{\epsilon}(0)\big)
+\psi_{\epsilon}^*\big(
-J\frac{u_{\epsilon}(0)+u_{\epsilon}(T)}{2}\big)
+ \langle u_{\epsilon}(T)-u_{\epsilon}(0), J\frac{u_{\epsilon}(0)+u_{\epsilon}(T)}{2}\rangle\Big\} \\
&=& \liminf\limits_{\epsilon \rightarrow
0}I_{\epsilon}(u_{\epsilon})=0.\end{aligned}$$ On the other hand Lemma 3.1 implies that $I(\bar u)\geq 0,$ which means the latter is zero, i.e. $$\begin{aligned}
I(\bar u) &=& \int_0^T\left[\phi (t,\bar u)
+\phi^*(t,-J\dot{\bar{u}})+(\bar{u},J\dot{\bar{u}})\right]\, dt\\
&+&\psi\big(\bar u(T)-\bar u(0)\big) +\psi^*\big( -J\frac{\bar u(0)+\bar u(T)}{2}\big)
+\langle \bar u(T)-\bar u(0), J\frac{\bar u(0)+\bar u(T)}{2} \rangle=0.\end{aligned}$$ The result now follows from the following identities and from the limiting case in Legendre-Fenchel duality. $$\begin{aligned}
\phi (t,\bar u (t))+\phi^*(t,-J\dot{\bar{u}}(t))+(\bar
u(t),J\dot{\bar{u}}(t))=0\end{aligned}$$ $$\begin{aligned}
\psi\big( \bar u(T)-\bar u(0)\big) +\psi^*\big(-J\frac{\bar u(0)+\bar u(T)}{2}\big)
+ \langle \bar u(T)-\bar u(0), J\frac{\bar u(0)+\bar u(T)}{2}\rangle =0.\end{aligned}$$ $\square$\
We shall now use Proposition \[Relax.2\] to prove Theorem \[Principle.2\]. For that we shall ${\lambda}-$regularize the convex functional $\psi$, then use assumption $B_2$ of Theorem \[Principle.2\] to derive uniform bounds and ensure convergence in $A^2_X$ when ${\lambda}$ approaches to $0$. First recall that if $\psi_{\lambda}(x)=\inf\limits_{y\in X}\left\{ \psi (y)+\frac{\| x-y\|_X^2}{2{\lambda}}\right\} $ then its conjugate $\psi^*_{\lambda}$ is equal to $\psi^* (x)+\frac{{\lambda}| x|^2}{2}$, which means that if $G(x,p)$ is the anti-selfdual Lagrangian $G(x,p)=\psi (x)+\psi^*(-p)$, then its ${\lambda}$-regularization is nothing but $G_\lambda (x,p)=\psi_{\lambda}(x)+\psi^*_{\lambda}(-p)$.
#### Proof of Theorem \[Principle.2\], Part (1):
The functional $\psi_{\lambda}$ satisfies the condition $(B'_2)$ of Proposition \[Relax.2\], hence for each ${\lambda}>0$ there exists a $u_{\lambda}\in A_X^2$, such that $$\begin{aligned}
I_{\lambda}(u_{\lambda})&:=&\int_0^T\left[\phi(t,u_{\lambda}(t) )+\phi^*(t,-J\dot u_{\lambda}(t))+\langle J\dot u_{\lambda}(t) ,u_{\lambda}(t) \rangle \right]\, dt \nonumber \\
& &\quad +\langle u_{\lambda}(T)-u_{\lambda}(0),J\frac{u_{\lambda}(T)+u_{\lambda}(0)}{2}\rangle
+\psi_{\lambda}\big( u_{\lambda}(T)-u_{\lambda}(0)\big) +\psi_{\lambda}^*\big(-J\frac{u_{\lambda}(T)+u_{\lambda}(0)}{2}\big) \nonumber \\
&=& 0.\end{aligned}$$ We shall show $u_{\lambda}$ is bounded in $A_X^2$. From Proposition 3.2 we obtain $$\begin{aligned}
\left |\int_0^T \langle J\dot u_{\lambda},u_{\lambda}\rangle\, dt+\langle u_{\lambda}(T)-u_{\lambda}(0),
J\frac{u_{\lambda}(T)+u_{\lambda}(0)}{2}\rangle \right|
\leq\frac{T}{\pi}\int_0^T |\dot u_{\lambda}(t)|^2\, dt\end{aligned}$$ which together with (48) and (56) imply $$\begin{aligned}
\psi_{\lambda}\big( u_{\lambda}(T)-u_{\lambda}(0)\big) +\psi_{\lambda}^*\big(-J\frac{u_{\lambda}(T)+u_{\lambda}(0)}{2}\big)+\int_0^T \phi (t,u_{\lambda}(t))\, dt+\left(\frac{1}{2\beta}-\frac{T}{\pi}\right)
\int_0^T |\dot u_{\lambda}(t)|^2 \, dt \leq 0.\end{aligned}$$ Since $\psi$ is bounded from below so is $\psi_{\lambda}$. Also, $0\in \mbox{ Dom}(\psi)$ which means $\psi^*$ and consequently $\psi_{\lambda}^*$ is bounded from below. Therefore it follows from (57) that: $$\begin{aligned}
\psi_{\lambda}\big( u_{\lambda}(T)-u_{\lambda}(0)\big) +\psi_{\lambda}^*\big(-J\frac{u_{\lambda}(T)+u_{\lambda}(0)}{2}\big) \leq C,\end{aligned}$$ and $$\begin{aligned}
\int_0^T \phi(t,u_{\lambda})\,
dt+\left(\frac{1}{2{\alpha}}-\frac{T}{2}\right)
\int_0^T |\dot u_{\lambda}(t)|^2 \leq C,\end{aligned}$$ where $C>0$ is a positive constant. It follows from the assumption $(B_1), (B_2)$ and (59) that $| u_{\lambda}(t)|$ and $\int_0^T |\dot u_{\lambda}|^2\, dt$ are bounded. Consequently $u_{\lambda}$ is bounded in $A_X^2$ and so, up to a subsequence, $u_{\lambda}\rightharpoonup\bar{u}$ in $A_X^2$.
It follows from (58) and Lemma 2.2 that $$\begin{aligned}
\psi\big(\bar u(T)-\bar u(0)\big) +\psi^* \big( -J\frac{\bar u(T)+\bar u(0)}{2}\big)
\leq\liminf\limits_{\lambda}\psi_{\lambda}\big( u_{\lambda}(T)-u_{\lambda}(0)\big)
+\psi_{\lambda}^*\big( -J\frac{u_{\lambda}(T)+\bar u_{\lambda}(0)}{2}\big).\end{aligned}$$ Also, from Proposition 3.3, we have $$\begin{aligned}
\inf\limits_{{\lambda}\rightarrow 0}\int_0^T\langle u_{{\lambda}}(t),J\dot{u_{{\lambda}}}(t)\rangle\, dt &+& \langle u_{{\lambda}}(T)-u_{{\lambda}}(0), J \frac{u_{{\lambda}}(0)+u_{{\lambda}}(T)}{2}\rangle\nonumber\\
&=&\int_0^T\langle \bar u (t),J\dot{\bar u} (t) \rangle\, dt +\langle \bar u(T)-\bar u(0), J \frac{\bar u(0)+\bar u(T)}{2}\rangle.\end{aligned}$$ Now, taking into account (60) and (61), by letting ${\lambda}{\rightarrow}0$ in (56) we obtain, $$\begin{aligned}
I(\bar u) &=& \int_0^T \left[\phi (t,\bar u (t))
+\phi^*\big( t,-J\dot{\bar{u}} (t) \big)
+ \langle J\dot {\bar{u}}(t),\bar u (t) \rangle \right]\, dt\\
& &\quad +\langle \bar u (T)-\bar u (0),J\frac{\bar u (T)+\bar u
(0)}{2}\rangle
+\psi\big(\bar u(T)-\bar u(0)\big) +\psi^*\big( -J\frac{\bar u (T)+\bar u (0)}{2}\big)\\
&\leq &\liminf\limits_{{\lambda}\rightarrow 0} \Big \{\int_0^T\left[\phi (t,u_{\lambda}(t))+\phi^*(t,-J\dot u_{\lambda}(t))+\langle J\dot u_{\lambda}(t),u_{\lambda}(t)\rangle \right]\, dt\\
& &\quad +\langle u_{\lambda}(T)-u_{\lambda}(0),J\frac{u_{\lambda}(T)+u_{\lambda}(0)}{2}\rangle
+\psi_{\lambda}\big( u_{\lambda}(T)-u_{\lambda}(0)\big) +\psi_{\lambda}^*\big(-J\frac{u_{\lambda}(T)+u_{\lambda}(0)}{2}\big)\Big\}\\
&=& \liminf\limits_{{\lambda}\rightarrow 0}I_{\lambda}(u_{\lambda})=0.\end{aligned}$$ From Lemma 3.1, $I(\bar u)\geq 0$, which means the latter is zero. The result follows from the following identities and from the limiting case in Legendre-Fenchel duality $$\begin{aligned}
\phi (t,\bar u)+\phi^*(t,-J\dot{\bar{u}})+ \langle \bar
u,J\dot{\bar{u}}\rangle =0.\end{aligned}$$ $$\begin{aligned}
\psi\big( \bar u(T)-\bar u(0)\big) +\psi^*\big( -J\frac{\bar u(0)+\bar u(T)}{2}\big)
+ \langle \bar u(T)-\bar u(0), J \frac{\bar u(0)+\bar u(T)}{2} \rangle =0.\end{aligned}$$
#### Proof of Part (2):
Note first that $\langle J\frac{u(0)+u(T)}{2},u(T)-u(0) \rangle= \langle
Ju(0),u(T) \rangle.
$ The corresponding Lagrangian $L_{\epsilon}:X\times
X{\rightarrow}\R $ is defined as follows $$\begin{aligned}
L_{\epsilon}(v,u) &=& \int_0^T\left[ (-J\dot v (t),u (t))+\phi_{\epsilon}^* (t,-J\dot u (t))
-\phi_{\epsilon}^*(t,-J\dot v (t))+\langle J\dot u (t),u (t) \rangle \right]\, dt\\
& &\quad +\langle Ju(0),u(T)\rangle -\langle Ju(0),v(T)\rangle
+\psi_{\epsilon}^*\big( Ju(T)\big) -\psi_{\epsilon}^*\big(
Jv(T)\big),\end{aligned}$$ The rest of the proof is quite similar to Part (1) and is left to the interested reader.
Applications
------------
As mentioned in the introduction, one can choose the boundary Lagrangian $\psi$ appropriately to solve Hamiltonian systems of the form $$\begin{aligned}
\left\{\begin{array}{l}
-J\dot u (t) \in\partial\phi (t,u (t))\\
\hbox{$u(0)=u_0$, or $u(T)-u(0)\in K$, or $u(T)=-u(0)$ or $u(T)=Ju(0)$.}
\end{array}\right.\end{aligned}$$
One can also use the method to solve second order systems with convex potential and with prescribed nonlinear boundary conditions such as: $$\left\{ \begin{array}{lcl}
\hfill -\ddot{q} (t) &=& \partial \phi \big( t,q(t)\big)\\
-\frac{q(0)+q(T)}{2} &= &\partial\psi_1 \big( \dot q(T)- \dot q(0)\big),\\
\hfill \frac{\dot q(0)+ \dot q(T)}{2} &=&\partial\psi_2 \big(
q(T)-q(0)\big)
\end{array}\right.$$ and $$\left\{ \begin{array}{lcl}
\hfill \ddot{q} (t) &=& \partial \phi \big( t,q(t)\big)\\
-q(T) &= &\partial\psi_1 \big( \dot q(0)\big),\\
\hfill \dot q(T) &=&\partial\psi_2 \big( q(0)\big)
\end{array}\right.$$ where $\psi_1$ and $\psi_2$ are convex and lower semi continuous. One can deduce the following
Let $\phi: [0,T]\times H{\rightarrow}\R$ be such that $(t,q){\rightarrow}\phi (t,q)$ is measurable in $t$ for each $q\in H$, convex and lower semi-continuous in $q$ for a.e. $t\in [0,T]$, and let $\psi_i: H {\rightarrow}\R \cup \{\infty\}$, $i=1,2$ be convex and lower semi continuous on $H$. Assume that the following conditions:
$A_1$:
: There exists $\beta \in (0,\frac{\pi}{2T})$ and $\gamma, {\alpha}\in L^2 (0,T;\mathbb{R_{+}})$ such that $- {\alpha}(t)\leq \mathcal \phi (t,q)\leq\frac{\beta^2}{2} |q|^2+\gamma (t)$ for every $q\in H$ and a.e. $t\in [0,T]$.
$A_2$:
: $\int_0^T \mathcal \phi(t,q)\, dt{\rightarrow}+\infty\quad\mbox{as}\quad | q|{\rightarrow}+\infty.$
$A_3$:
: $\psi_1$ and $\psi_2$ are bounded from below and $0 \in {\rm Dom} (\psi_i)$ for $i=1,2.$
Then equations (62) and (63) have at least one solution in $A_H^2.$
#### Proof:
Define $\Psi: H \times H{\rightarrow}\R \cup \{\infty\}$ by $\Psi (p,q):=\psi_1(p)+\psi_2(q)$ and $\Phi:[0,T] \times H \times H{\rightarrow}\R$ by $\Phi (t,u):=\frac
{\beta}{2} |p|^2+ \frac {1}{\beta}\phi \big( t,q(t)\big)$ where $u=(p,q).$ It is easily seen that $\Phi$ is convex and lower semi continuous in $u$ and that $$\begin{aligned}
\hbox{$- {\alpha}(t)\leq \Phi(t,u)\leq\frac{\beta}{2} |u|^2+\frac {\gamma
(t) }{\beta}$ and $\int_0^T \Phi(t,u)\, dt{\rightarrow}+\infty\quad\mbox{as}\quad | u| {\rightarrow}+\infty.$}\end{aligned}$$ Also, from $A_3$, the function $\Psi$ is bounded from below and $0 \in {\rm Dom} (\Psi).$ By Theorem \[Principle.2\], the infimum of the functional $$\begin{aligned}
I(u):&=& \int_0^T \left[ \Phi (t,u (t))+\Phi^*(t,-J\dot u (t))+\langle J\dot u(t),u(t)\rangle \right]\, dt\\
&& \quad + \langle u(T)-u(0),J\frac{u(0)+u(T)}{2}\rangle +\Psi\big(
u(T)-u(0)\big) +\Psi^*\big( -J\frac{u(0)+u(T)}{2}\big),\end{aligned}$$ on $A_X^2$ is zero and is attained at a solution of $$\begin{aligned}
\left\{\begin{array}{l}
-J\dot u (t)\in\partial\Phi (t,u(t)),\\
-J \frac {u(T)+u(0)}{2}=\partial\Psi (u(T)-u(0)).\end{array}\right.\end{aligned}$$ Now if we rewrite this problem for $u=(p,q),$ we get $$\begin{aligned}
- \dot p(t) &=& \frac {1}{\beta}\partial \phi \big( t,q(t)\big),\\
\dot q(t)&= &\beta p(t),\\
-\frac{q(T)+q(0)}{2} &= &\partial\psi\big( p(T)- p(0)\big),\\
\frac{p(T)+ p(0)}{2} &= &\partial\psi\big( q(T)-q(0)\big),\end{aligned}$$ and hence $q \in A_H^2$ is a solution of (61).
As in the case of Hamiltonian systems, one can then solve variationally the differential equation $ -\ddot{q} (t) = \partial \phi \big( t,q(t)\big)$ with any one of the following boundary conditions:
(i)
: Periodic: $ \dot q(T)= \dot q(0)$ and $ q(T)= q(0).$
(ii)
: Antiperiodic: $ \dot q(T)= -\dot q(0)$ and $ q(T)=- q(0).$
(iii)
: Initial value condition: $q(0)=q_0$ and $\dot
q(0)=q_1$ for given $q_0, q_1 \in H.$
[99]{}
H. Brezis, I. Ekeland, [*Un principe variationnel associé à certaines equations paraboliques. Le cas independant du temps*]{}, C.R. Acad. Sci. Paris Sér. A [**282**]{} (1976), 971–974.
H. Brezis, L. Nirenberg, G. Stampachia, [*A remark on Ky Fan’s Minimax Principle*]{}, Bollettino U. M. I (1972), 293-300
N. Ghoussoub, [*Anti-selfdual Lagrangians: Variational resolutions of non self-adjoint equations and dissipative evolutions*]{}, Submitted (2005).
N. Ghoussoub, [*Anti-selfdual Hamiltonians: Variational resolution for Navier-Stokes equations and other nonlinear evolutions*]{}, Submitted (2005)
N. Ghoussoub, [*A class of selfdual partial differential equations and its variational principles*]{}, In preparation (2005)
N. Ghoussoub, A. Moameni, [*On the existence of Hamiltonian paths connecting Lagrangian submanifolds*]{}, Submitted (2005)
N. Ghoussoub, L. Tzou. [*A variational principle for gradient flows*]{}, Math. Annalen, Vol 30, 3 (2004) p. 519-549.
N. Ghoussoub, L. Tzou. [*Iterations of anti-selfdual Lagrangians and applications to Hamiltonian systems and multiparameter gradient flows*]{}, Submitted (2005).
J. Mawhin, M. Willem: [*Critical point theory and Hamiltonian systems*]{}. Applied Mathematical Sciences, [**74**]{}, Springer Verlag (1989).
[^1]: Partially supported by a grant from the Natural Sciences and Engineering Research Council of Canada.
[^2]: Research supported by a postdoctoral fellowship at the University of British Columbia.
|
---
abstract: 'We demonstrate that for a broad class of local Calabi-Yau geometries built around a string of $\IP^1$’s – those whose toric diagrams are given by triangulations of a strip – we can derive simple rules, based on the topological vertex, for obtaining expressions for the topological string partition function in which the sums over Young tableaux have been performed. By allowing non-trivial tableaux on the external legs of the corresponding web diagrams, these strips can be used as building blocks for more general geometries. As applications of our result, we study the behavior of topological string amplitudes under flops, as well as check Nekrasov’s conjecture in its most general form.'
---
SMS-0402\
SLAC-PUB-10804\
SU-ITP 4/39\
hep-th/0410174\
\
0.5cm
[$^{\spadesuit}$School of Mathematical Sciences\
GC University,\
Lahore, 54600, Pakistan.\
]{} 0.5cm
[$^{\clubsuit}$ Department of Physics and SLAC\
Stanford University,\
Stanford, CA 94305/94309, U.S.A.\
]{}0.5cm
[$^{\spadesuit}$ Department of Mathematics\
University of Washington,\
Seattle, WA, 98195, U.S.A.\
]{}
Introduction
============
In the last few years, dramatic progress has been made in techniques for calculating the partition function of the topological string on toric (hence non-compact) Calabi-Yau manifolds [@Diaconescu:2002sf; @Diaconescu:2002qf; @Aganagic:2002qg]. The culmination of this effort has been the formulation of the topological vertex [@Aganagic:2003db] (see [@Li] for a recent mathematical treatment). With it, a set of diagrammatic rules can be formulated which allow an expression for the topological string partition function to be read off from the web diagram of the toric manifold. While the expressions obtained such are algorithmically complete, they contain unwieldy sums over Young tableaux, one sum for each internal line of the web diagram. Starting with [@Iqbal:2003ix], methods were developed to perform a portion of these sums [@Iqbal:2003zz; @Eguchi:2003sj; @Zhou:2003zp; @Eguchi:2003it]. In this note, we show how to perform all sums which arise in an arbitrary smooth triangulation of a strip toric diagram, such as
with arbitrary representations on all external legs but the first and last.
The ultimate goal of this program is to provide a technique for efficiently extracting the Gopakumar-Vafa invariants from the expressions the topological vertex yields for the topological string partition function. We will outline the obstacles to this goal using the methods of this paper as we proceed.
As other applications, we offer an analysis of the behavior of the topological amplitude under flops of the target manifold. We demonstrate that the Gopakumar-Vafa invariants for all toric geometries decomposable into strips are invariant under flops. We also show that our results provide the framework to check Nekrasov’s results [@Nekrasov:2003af] in the most general case of product $U(N)$ gauge groups with any number of allowed hypermultiplets.
The organization of this paper is as follows. In section 2, we elucidate the geometries we are considering and present and interpret the rules for obtaining the topological string partition function on them. In section 3, we derive these results. We include a brief review of the topological vertex at the beginning of this section, and end it with a comparison to the natural 4-vertex obtained from Chern-Simons theory. We discuss the behavior of Gopakumar-Vafa invariants under flops on geometries decomposable into strips in section 4.1. Section 4.2 provides the basic building blocks to study Nekrasov’s conjecture. We end with conclusions. An appendix gives a brief introduction to Schur functions, and collects the identities for Schur functions used throughout the paper.
The results
===========
Geometry of the strip
---------------------
Recall that a simple way of visualizing the geometries given by toric diagrams is to think of them as $T^n$ fibrations over $n$ dimensional base manifolds with corners (see eg. section 4.1 of [@fulton93]). Locally, one can introduce complex coordinates on the toric manifold. The base manifold is then locally given by the absolute value of these coordinates, the $T^n$ by the phases. The boundary of the base is where some of these coordinates vanish, entailing a degeneration of the corresponding number of fiber directions.
In 3 complex dimensions, the 3 real dimensional base has a 2 dimensional boundary with edges and corner. Web diagrams, easily obtained from toric diagrams as sketched in figure \[torictoweb\], represent the projection of the edges and corners of the base on to the plane. There is a full $T^3$ fibered over each point above the plane, corresponding to the interior of the base manifold. On a generic point on the plane, representing a generic point on the boundary, one cycle of the fiber degenerates, two degenerate on the lines of the web diagram, which correspond to edges of the base, and the entire fiber degenerates at the vertices of the diagram, the corners of the base.
![Relation between toric and web diagram. \[torictoweb\]](stripwithweb.eps)
Returning to figure \[torictoweb\], we now see the string of $\IP^1$’s (in red and blue) emerging by following the $S^1$ fibration along the internal line running through the web diagram. It is capped off to $\IP^1$’s by the $S^1$’s degenerating at each vertex. The two non-compact directions of the geometry locally correspond to the sum over two line bundles over each $\IP^1$. The two local geometries that arise on the strip are $(\O(-2) \oplus \O) \rightarrow \IP^1$ (in red) and $(\O(-1) \oplus \O(-1))\rightarrow \IP^1$ (in blue). We refer to the respective $\IP^1$’s as $(-2,0)$ and $(-1,-1)$ curves in the following.
Rules on the strip
------------------
Each vertex has one non-trivial Young tableau associated to it, the two outer vertices in addition have one leg carrying the trivial tableau. All other indices of the vertices are summed over. We label the non-trivial tableaux by $\beta_i$, with $i$ indexing the vertex. The internal lines carry a factor $Q_i = e^{-t_i}$, where $t_i$ is the Kähler parameter of the curve the internal line represents.
- [ Each vertex contributes a factor of $\W_{\beta_i}=s_{\beta_i}(q^\rho)$ (the notation for the argument of the Schur function is explained in the next section).]{}
- [Each pair of vertices (not just adjacent ones) contributes a factor to the amplitude, which is a pairing of the non-trivial tableaux carried by the pair. The interpretation of this observation is that branes wrapping the curves consisting of touching $\IP^1$’s in the web diagram contribute to the Gopakumar-Vafa index just as those wrapping the individual $\IP^1$’s.]{}
- While the pairing itself is symmetric, for the purpose of book keeping, we will choose one of the two natural orderings of the vertices along the string of $\IP^1$’s. We will speak of the first or second slot of the pairing with reference to this ordering. To determine the pairing factor, note that two types of curves occur on the strip: $(-2,0)$ curves and $(-1,-1)$ curves. Up to $SL(2,\IZ)$ transformations, these are represented by the toric/web diagrams depicted in figure \[twogoodlinks\].
The contribution of the pairing to the amplitude depends essentially on whether an even or odd number of $(-1,-1)$ curves lie between the two vertices. The geometric interpretation of this fact is that the curves consisting of touching $\IP^1$’s in the web diagram have normal bundle $\O(-2) \oplus \O$ or $\O(-1) \oplus \O(-1)$, depending on whether the string of $\IP^1$’s contains an even or odd number of smooth $(-1,-1)$ curves. This is suggested both by the toric diagram and by the expressions for the two pairings, as we will see next.
- [We denote the two types of pairings between vertices carrying the Young tableaux $\alpha$ and $\beta$ as $\{ \alpha \beta\}$ and $[\alpha \beta ]=\{\alpha \beta\}^{-1}$ (we use the notation $\[ \alpha \beta \]$ when we make statements valid for both types of pairing). The pairing $\{\alpha \beta\}$ is given by the expression {}\_Q &=& \_k (1- Q q\^k)\^[C\_k(, )]{} . The product over $k$ is over a finite range of integers (possibly negative), $C_k(\alpha,\beta)$ are numbers which depend on the two Young tableaux that are being paired, given by \_k C\_k(, )q\^k &=& ( 1 + (q-1)\^2 \_[i=1]{}\^[d\_]{} q\^[-i]{} \_[j=0]{}\^[\_i-1]{} q\^j ) ( 1 + (q-1)\^2 \_[i=1]{}\^[d\_]{} q\^[-i]{} \_[j=0]{}\^[\_i-1]{} q\^j )\
& & - . The factor $Q$ is the product of all $Q_i$ labeling the internal lines connecting the two vertices. Note that, as advertised above, if we take the two Young tableaux to be trivial, the contribution from the pairing $\{\cdot \cdot\}$ is exactly that of a $(-1,-1)$ curve, and likewise, the contribution of $[\cdot \cdot]=\{\cdot \cdot\}^{-1}$ is that of a $(-2,0)$ curve.]{}
- [To keep track of the contribution from two paired vertices, we can divide the vertices into two relative types, $A$ and $B$, such that the type of a vertex depends on that of the preceding vertex: two vertices connected by a $(-2,0)$ curve are of same type, two connected by a $(-1,-1)$ curve of opposite type. If the vertices $i$ and $j$ are of same type (i.e. have an even number of $(-1,-1)$ curves between them), the pairing factor is $[\beta_i^{\cdot} \beta_j^{\cdot}]$, else $\{\beta_i^{\cdot} \beta_j^{\cdot} \} = [\beta_i^{\cdot} \beta_j^{\cdot}]^{-1} $.]{}
- [The upper case dot indicates that either $\beta$ or $\beta^t$ is the correct entry. Either all pairings involving $\beta_i$ are of the form $\[ \beta_i \cdot \]$ and $\[ \cdot \beta_i^t \]$, or they are of the form $\[\beta_i^t \cdot\]$ and $\[\cdot \beta_i\]$. To determine which of the two options apply to $\beta_i$ for each $i$, we anchor the relative types $A$ and $B$ as follows: we will take the first vertex of the string of $\IP^1$’s to be of type $A$ if, labeling the legs in clockwise order, it is given by $C_{\alpha_1 \bullet \beta_1}$. Otherwise, it must be given by $C_{\bullet \alpha_1 \beta_1}$, and we will classify it as type $B$. With this convention, && $$\beta_i \cdot$$ $$\cdot \beta_i^t$$ ,\
&& $$\beta_i^t \cdot$$ $$\cdot \beta_i$$ . ]{}
As an example, consider the diagram in figure \[example\].
Starting from the left, the curves are of type $(-2,0)$, $(-1,-1)$, $(-2,0)$. The first vertex is $C_{\alpha_1 \bullet \beta_1}$, hence of type $A$. This determines the sequence of vertices to be $(A,A,B,B)$. By the rules above, we now obtain the following expression for the amplitude, s\_[\_1]{} s\_[\_2]{} s\_[\_3]{} s\_[\_4]{} \[\_1 \_2\^t\]\_[Q\_1]{} {\_1 \_3 }\_[Q\_1 Q\_2]{} {\_1 \_4 }\_[Q\_1 Q\_2 Q\_3]{} {\_2 \_3}\_[Q\_2]{} {\_2 \_4}\_[Q\_2 Q\_3]{} \[\_3\^t \_4\]\_[Q\_3]{} =\
s\_[\_1]{} s\_[\_2]{} s\_[\_3]{} s\_[\_4]{} , where we have omitted the arguments $q^\rho$ of the Schur functions.
Derivation
==========
Review of the vertex
--------------------
Locally, any complex manifold is isomorphic to $\IC^n$. The topology and complex structure of the manifold are obtained by specifying how these $\IC^n$-patches are to be glued together. The insight underlying the topological vertex [@Aganagic:2003db] is that the topological string partition function on a toric CY can also be pieced together patchwise. The patching conditions are implemented by placing non-compact Lagrangian D-branes along the three legs of the web diagram of $\IC^3$, intersecting the curves extending along these legs (recall that the legs indicate where 2 of the 3 cycles of the $T^3$ fibration have degenerated) in $S^1$’s. The topological string on each such patch counts the holomorphic curves ending on the branes, weighted by the appropriate Wilson lines from the boundaries of the worldsheet, \[zpatch\] Z\_[patch]{} &=& \_[\^[(1)]{},\^[(2)]{},\^[(3)]{}]{} C\_[\^[(1)]{}\^[(2)]{}\^[(3)]{}]{} . The vectors $\vec{k}^{(i)}$ encode that $k_j^{(i)}$ holes of winding number $j$ are ending on the $i$-th brane. The Wilson loop factors are given by &=& \_[i=1]{}\^[3]{} \_[j=1]{}\^(V\^j)\^[k\_j\^[(i)]{}]{} . The trace here is taken in the fundamental representation. The definition of the vertex we will use in the following arises when rewriting (\[zpatch\]) in the representation basis [@Aganagic:2003db], \_[\_1, \_2, \_3]{} C\_[\_1 \_2 \_3]{} \_[i=1]{}\^[3]{} \_[\_i]{} V\_i &=& \_[\^[(1)]{},\^[(2)]{},\^[(3)]{}]{} C\_[\^[(1)]{}\^[(2)]{}\^[(3)]{}]{} \_[i=1]{}\^[3]{} \_[j=1]{}\^(V\^j)\^[k\_j\^[(i)]{}]{} , \[repbasis\] where the $\alpha_i$ now denote Young tableaux. This equation is to be understood in the limit when the number of D-branes on each leg is taken to infinity, such that the sum extends over Young tableaux with an arbitrary number of rows. An application of the Frobenius formula lets us solve (\[repbasis\]) for $C_{\alpha_1 \alpha_2 \alpha_3}$.
The non-compactness of the Lagrangian D-branes gives rise to an integer ambiguity [@Aganagic:2001nx; @Katz:2001vm], which necessitates specifying one integer per leg to full determine the vertex. [@Aganagic:2003db] refer to this choice as the framing of the vertex, since the integer ambiguity maps to the framing ambiguity of Chern-Simons theory under geometric transitions. The vertex in canonical framing is given by [@Okounkov:2003sp] $$\begin{aligned}
C_{\lambda \mu \nu} = q^{\frac{\kappa(\lambda)}{2}}s_{\nu}(q^\rho)\sum_\eta s_{\lambda^t/\eta}(q^{\nu+\rho})s_{\mu/\eta}(q^{\nu^t+\rho}) \,.\end{aligned}$$ The notation is $s(q^{\nu+\rho}) = s(\{q^{\nu_i -i +\frac{1}{2}}\})$. The $s_{\mu / \eta}$ are skew Schur functions, defined by \[skew\] s\_[/ ]{} = \_c\^\_ s\_, where the $c^{\mu}_{\eta \nu}$ are tensor product coefficients, and $\kappa(\lambda) = \sum \lambda_i(\lambda_i - 2i+1)$. In the following, we will use the abbreviated notation $s_{\mu/\eta}(q^{\nu+\rho}) = \frac{\mu}{\eta}(q^{\nu+\rho})=\frac{\mu}{\eta}(\nu)$ whenever convenient, and imply a sum over repeated tableaux.
A framing must be specified for each leg of the vertex. If we represent each leg by an integer vector $v$, we can encode the framing by an integer vector $f$ that satisfies $f \wedge v =1$. The notation $f \wedge v$ denotes the symplectic product $f_1 v_2 -f_2 v_1$. The condition $f \wedge v =1$ determines $f$ up to integer multiples of $v$. Having chosen a canonical framing, we can hence classify relative framing by an integer $n$. To this end, we label the legs of the vertex in counter-clockwise order by $v_1, v_2, v_3$, s.t. $v_{i} \wedge v_{i+1}=1$. The natural choice for a framing is then $(f_1,f_2,f_3)=(v_3,v_1,v_2)$. Given a framing $f_i=v_{i-1}-n v_{i}$, the integer $n$ (the framing relative to the fiducial choice) is determined via $n = f_i \wedge v_{i-1}$. Under shifts of framing, the vertex transforms as follows [@Aganagic:2003db], C\_[\_1 \_2 \_3]{}\^[f\_1 - n\_1 v\_1,f\_2 - n\_2 v\_2,f\_3 - n\_3 v\_3]{} &=& (-1)\^[\_i n\_i |\_i|]{} q\^[\_i n\_i ]{} C\_[\_1 \_2 \_3]{}\^[f\_1,f\_2,f\_3]{} . Gluing two vertices together along $v_1$ and $v_1^\prime$ requires the framings along this leg to be opposite.[^1] If we are gluing along $v_1$, and have canonical framing $f_1=v_3$ along $v_1$ for the first vertex, then the second vertex must have non-canonical framing $-v_3= v_3^\prime - n v_1^\prime$. We thus obtain the gluing rule \_[\_1]{} C\_[\_2 \_3 \_1]{} e\^[-|\_1| t]{} (-1)\^[ |\_1|]{} C\_[\_1\^t \_2\^\_3\^]{}\^[-f\_1,f\_2\^,f\_3\^]{}&=&\_[\_1]{} C\_[\_2 \_3 \_1]{} e\^[-|\_1| t]{} (-1)\^[ |\_1|]{} C\_[\_1\^t \_2\^\_3\^]{}\^[-v\_3,f\_2\^,f\_3\^]{}\
&=&\_[\_1]{} C\_[\_2 \_3 \_1]{} e\^[-|\_1| t]{} (-1)\^[|\_1|]{} C\_[\_1\^t \_2\^\_3\^]{}\^[v\_3\^-n v\_1\^,f\_2\^,f\_3\^]{}\
&=&\_[\_1]{} C\_[\_2 \_3 \_1]{} e\^[-|\_1| t]{} (-1)\^[(n+1) |\_1|]{} q\^[-n \_[\_1]{}/2]{} C\_[\_1\^t \_2\^\_3\^]{}\^[f\_1\^,f\_2\^,f\_3\^]{} , with $n= v_3^\prime \wedge (v_3^\prime + v_3) = v_3^\prime \wedge v_3$.
Performing the sums
-------------------
In performing the sums, we will make use of the following two identities for skew Schur polynomials [@Macdonald], \_s\_[/\_1]{}(x)s\_[/\_2]{}(y)&=& \_[i,j]{} (1-x\_i y\_j)\^[-1]{} \_s\_[\_2/]{}(x)s\_[\_1/]{}(y) , \[sumruleone\]\
\_s\_[\^t/\_1]{}(x)s\_[/\_2]{}(y) &=& \_[i,j]{} (1+x\_i y\_j) \_s\_[\_2\^t/\^t]{}(x)s\_[\_1\^t/]{}(y) .\[sumruletwo\] In the abbreviated notation introduced above, these sum rules become (x)(y) &=& \_[i,j]{} (1-x\_i y\_j)\^[-1]{} (x)(y) ,\
(x)(y) &=& \_[i,j]{} (1+x\_i y\_j) (x)(y) . In the following, it will be convenient to rewrite the infinite products as follows, (1-x\_i y\_j)\^[-1]{} &=&\
&=&\
&=& ,\
(1+x\_i y\_j) &=& .
Aside from the skew Schur functions, the expression for the topological vertex contains two additional elements: powers of the exponential of the Kähler parameters, $(\pm Q)^{|\alpha|}$, and powers of the exponential of the string coupling, $q^{\kappa(\alpha)}$. With recourse to the homogeneity property of the skew Schur polynomials, the former is easy to deal with. The latter poses a greater difficulty and is at the root of our calculations being confined to the strip. We hope to return to this difficulty in forthcoming work. For the two types of pairing occurring on the strip, the dependence on this factor cancels, as we demonstrate next.
#### Type $[\beta_i \beta_j]$: (-2,0) curves
Consider figure \[-2\].
According to the rules reviewed above, the corresponding partition function is \_[\_2]{} C\_[\_2 \_1 \_1]{} C\_[\_3 \_2\^t \_2]{} (-1)\^[|\_2|]{}Q\^[|\_2|]{} (-1)\^[n |\_2|]{} q\^[-]{} . The $n$-dependence arises from the framing factors, where $n$ is given by n=v\_[\_2]{} v\_[\_1]{} = (0,1)(-1,-1) = 1 . Inserting the expression for the vertex, this yields \[exprlink\] \_[\_2]{} Q\^[|\_2|]{} q\^[-]{}\
. We see that the $q^{\kappa(\alpha_2)}$ dependence of the vertex cancels against the framing factor, so that we can apply (\[sumruleone\]) to perform the sum over $\alpha_2$, once we deal with the $Q^{|\alpha_2|}$ dependence. As alluded to above, this does not pose any difficulty due to the homogeneity of the Schur functions, $s_{\lambda}(c\,x) = c^{|\lambda|} s_{\lambda}(x)$. Since the tensor product coefficients $c^{\mu}_{\eta \nu}$ vanish unless $|\mu|=|\eta|+|\nu|$, we easily deduce the homogeneity property of the skew Schur polynomials to be $s_{\mu/\eta}(c\,x) =\sum_{\nu} c^{\mu}_{\eta \nu} s_{\nu} (c\,x) = c^{|\mu|-|\eta|}s_{\mu/\eta}(x)$. We can now incorporate the Kähler parameters into our calculation. We obtain \_ (\_1) (\_2) Q\^[||]{} &=& \_ (q\^[+\_1]{}Q) (q\^[+\_2]{}) Q\^[|\_1|]{}\
&=& \_ (q\^[+\_1]{}Q) (q\^[+\_2]{}) Q\^[|\_1|]{}\
&=&\
& &\_ (q\^[+\_1]{}) (q\^[+\_2]{}) Q\^[|\_1|+|\_2|-||]{}\
&=& \[\_1 \_2\]\_[Q]{} \_ (q\^[+\_1]{}) (q\^[+\_2]{}) Q\^[|\_1|+|\_2|-||]{} , where in the last step, we have defined the pairing $[\cdot \cdot]_Q$.
#### Type $\{\beta_i \beta_j\}$: (-1,-1) curves:
The second type of pairing arises for (-1,-1) curves, as depicted in figure \[m1m1curve\].
The corresponding expression is \[typeiisum\] C\_ C\_[\_3 \_2]{} (-1)\^[||]{} Q\^[||]{} &=& (-1)\^[||]{} Q\^[||]{} . Note that here, the $\kappa(\alpha_2)$ dependence cancels between the two vertices, by $\kappa(\alpha^t)=-\kappa(\alpha)$. The framing factor $n$ vanishes. Again, this allows us to perform the sum over $\alpha_2$, utilizing (\[sumruletwo\]). We obtain \_(\_1) (\_2) (-Q)\^[||]{} &=& \[defpair2\]\
& & \_ ((-Q)q\^[+\_1]{}) (q\^[+\_2]{}) (-Q)\^[|\_1|]{}\
&=& {\_1 \_2 }\_[Q]{} \_ (q\^[+\_1]{}) (q\^[+\_2]{}) (-Q)\^[|\_1|+|\_2|-||]{} . The last step defines the pairing $\{\cdot \cdot\}_Q= \frac{1}{[\cdot \cdot]_Q}$.
Stringing the curves together
-----------------------------
The rules we have proposed in section 2 are easily proved by induction. Here, we wish to give some intuition as to how they arise. This is best done by considering an example. Let’s therefore revisit figure \[example\]. We organize the calculation in a diagram, see figure \[calc\].
The following items should help explain and interpret the diagram.
- [The dominos in the first row correspond to the vertices of the web diagram describing the geometry. The connecting lines in between dominos indicate applications of the rules (\[sumruleone\],\[sumruletwo\]) for summing over skew Schur polynomials. The dominos in the second row arise after a depth one application of the summing rules, etc.]{}
- [In each domino in the first row, either both of the top representations of the skew Schur functions (this terminology is to refer to the $\alpha$ in $\frac{\alpha}{\eta}$) are transposed or neither of them are. The former are vertices of type $C_{\alpha_{i+1} \alpha_i^t \beta_i}$ (type $A$), the latter of type $C_{\alpha_i^t \alpha_{i+1} \beta_i}$ (type $B$). In all the following rows, whether a skew Schur function carrying the argument $\beta_i^\cdot$ has transposed top representation or not depends on what is the case for the Schur function in the first row with argument $\beta_i^\cdot$. Hence, whether the pairing of $\beta_i^\cdot$ with $\beta_j^\cdot$ is of type $\{ \cdot \cdot \}$ or $[\cdot \cdot]$ is determined by the relative type of vertex the $\beta$’s descended from in the first row.]{}
- [All $\[\cdot \beta_i^\cdot\]$ pairings descend from the Schur function $\frac{\alpha_i^\cdot}{\eta_i}(\beta_i^\cdot)$ in the $i$-th vertex in the first row of the diagram, all $\[\beta_i^\cdot \cdot\]$ pairings from the Schur function $\frac{\alpha_{i+1}^\cdot}{\eta_i}(\beta_i^\cdot)$ in the same vertex. Hence, whether the correct entry is $\beta_i$ or $\beta_i^t$ again depends on whether the $i$-th vertex is of type $C_{\alpha_{i+1} \alpha_i^t \beta_i}$ (type $A$) or $C_{\alpha_i^t \alpha_{i+1} \beta_i}$ (type $B$).]{}
- [The calculation terminates, because the first and last domino in each row contain a trivial skew Schur function $\frac{\bullet}{\eta}$, s.t. the sum over $\eta$ collapses.]{}
- [The factors of $Q$ can be thought of as flowing along the connecting lines. Consider the second domino in the second line of the diagram. After applying (\[sumruletwo\]), we obtain the $Q$ factor $Q_2^{|\eta_2| + |\eta_3| - |\kappa_2|}$. The diagram shows into which pairing these factors are incorporated. Note that the factors $Q_2^{|\eta_2|}$ and $Q_2^{|\eta_3|}$ enter into the next level of evaluation (sums over $\eta_2$ and $\eta_3$), whereas $Q_2^{-|\kappa_2|}$, as indicated by the arrows, enters in the level after next (the sum over $\kappa_2$).]{}
- [Finally, a word on the factors $(-1)^{|\alpha|}$ that appear in sums that lead to the pairing $\{\cdot \cdot\}$, see (\[typeiisum\]). One could combine these with their kin factors $Q^{|\alpha|}$, such that all $Q$’s associated to $(-1,-1)$ curves would come with a minus sign. The sign of the product of all $Q$ factors contributing to a pairing would then depend on whether an odd or even number of $(-1,-1)$ curves lie between the two vertices being paired. This of course is the criterion that distinguishes between the two pairings $\{\cdot \cdot\}$ and $[\cdot \cdot]$. Hence, this sign is taken into account correctly by incorporating it into the definition of $\{\cdot \cdot\}$ in (\[defpair2\]).]{}
Simplifying the two pairings
----------------------------
The two pairings $\{\alpha \beta\}$ and $[\alpha \beta]$ are exponentials of the argument \[infsum\] . In this section, we perform the representation dependent part of this infinite sum. The calculation already appeared in [@Iqbal:2003ix; @Iqbal:2003zz]. Our goal will be to write the product of Schur functions (up to a correction term) as a sum $\sum_{\rm finite} C_k q^{kn}$, which will allow us to subsume the infinite sum over $n$ in a logarithm. First, let us take a closer look at the Schur polynomials. $\sf(x)=\sum_i x_i$, and hence, (q\^[+ ]{}) = \_[i=1]{}\^ q\^[\_i - i + ]{} . Apart from the finite number of terms involving the Young tableau $\alpha$, this is a geometric series, (q\^[+ ]{}) &=& ( \_[i=1]{}\^ q\^[-i]{} + \_[i=1]{}\^[d\_]{} (q\^[\_i-i]{} - q\^[-i]{}) )\
&=& ( + (q-1) \_[i=1]{}\^[d\_]{} q\^[-i]{} )\
&=& ( 1 + (q-1)\^2 \_[i=1]{}\^[d\_]{} q\^[-i]{} \_[j=0]{}\^[\_i-1]{} q\^j ) .Here, $d_\alpha$ denotes the number of rows of $\alpha$. We now see the desired structure in the product $\sf(q^{n(\rho + \alpha)}) \sf(q^{n(\rho + \beta)})$ emerging, (q\^[+ ]{}) (q\^[+ ]{})&=& \_k C\_k(, ) q\^k + , where \[defck\] \_k C\_k(, )q\^k &=& ( 1 + (q-1)\^2 \_[i=1]{}\^[d\_]{} q\^[-i]{} \_[j=0]{}\^[\_i-1]{} q\^j ) ( 1 + (q-1)\^2 \_[i=1]{}\^[d\_]{} q\^[-i]{} \_[j=0]{}\^[\_i-1]{} q\^j )\
& & - . The infinite sum in (\[infsum\]) can now be expressed in a more compelling form. For the minus sign which corresponds to the pairing $\{\cdot \cdot\}$, we obtain -\_[n=1]{}\^ (q\^[n(+ )]{}) (q\^[n(+ )]{}) &=& - \_[n=1]{}\^\
&=& \_k C\_k(, ) (1- Q q\^k) - \_[n=1]{}\^ . Recalling that $q=e^{i g_s}$, the remaining infinite sum takes a familiar form, \_[n=1]{}\^ &=& -\_[n=1]{}\^ . We hence obtain the tidy expression, {}\_Q &=& \_k (1- Q q\^k)\^[ C\_k(, )]{} . When considering flops further below, we will need the relation between $\{\alpha \beta \}$ and $\{\alpha^t \beta^t\}$. The sum (\[defck\]) satisfies the property \_k C\_k(, )q\^k &=& \_k C\_k(\^t, \^t)q\^[-k]{} . By the symmetry of the correction term $\frac{q}{(1-q)^2}$ under $q \rightarrow \frac{1}{q}$, it follows that { \^t \^t }(q) &=& { }() \[transposed\].
CS calculations in the light of the vertex {#CS}
------------------------------------------
All genus results for the topological string on non-compact Calabi-Yau were originally obtained using Chern-Simons theory as the target space description of the open topological string, combined with open/closed duality via geometric transitions [@Diaconescu:2002sf; @Diaconescu:2002qf; @Aganagic:2002qg]. In the context of Chern-Simons theory, the natural object is a 4-vertex, the normalized expectation value of a Hopf link with gauge fields in representations $\alpha$ and $\beta$ on the two unknots. It is given by \[defW\] W\_(q,)=W\_(q,)(q)\^[||/2]{}S\_(E\_(t)), where \[defE\] E\_(t)=(1+\_[n=1]{}\^(\_[i=1]{}\^[n]{})t\^[n]{})(\_[j=1]{}\^[d]{}), and W\_ = (q)\^[||/2]{} S\_(E\_(t)) . We explain the relation of $S_\alpha(E(t))$ to the Schur functions $s_\alpha(x)$ of the previous sections in the appendix. The relevant fact for our purposes is that for $E(t) = \prod(1+x_i t)$, $S_\alpha(E(t)) = s_\alpha(x)$. To bring (\[defE\]) into this form, we note that the first factor in that expression can be expressed as (see page 27, example 5 in [@Macdonald]), 1+\_[n=1]{}\^(\_[i=1]{}\^[n]{})t\^[n]{}&=& \_[i=0]{}\^\
&=& \_[i=0]{}\^ (1+\^[-1]{}q\^[i]{}t) \_[i=1]{}\^ (1 + q\^[-i]{}t) . Hence, E\_(t)= \_[i=0]{}\^ (1+\^[-1]{}q\^[i]{}t) \_[j=1]{}\^(1+q\^[\_[j]{}-j]{}t) . In this equation, we have set $\alpha_j=0$ for $j>d$. By absorbing the factor of $q^{|\alpha|/2}$ in (\[defW\]) into the definition of $E$, \_(t)=\_[i=1]{}\^ (1+\^[-1]{}q\^[i-]{}t) \_[j=1]{}\^(1+q\^[\_[j]{}-j+]{}t) , we can now express $W_{\alpha \beta}$ in terms of ordinary Schur functions, W\_&=& \^ S\_(\_(t)) S\_(\_(t))\
&=& \^ s\_(\^[-1]{}q\^[-]{},q\^) s\_(\^[-1]{}q\^[-]{}, q\^[+]{}) . Through a series of transformations, we can bring this expression into a form in which it can readily be related to the topological vertex, W\_&=& \^ (q\^) \^t ((-)\^[-1]{}q\^) (q\^[+]{}) \^t ((-)\^[-1]{}q\^) \[cstovertex\]\
&=& \^ (q\^) (q\^[+]{})\
&=& \^ (q\^) \^t(q\^[+ \^t]{}) (q\^[+]{}) (q\^[+ \^t]{}) \^t(q\^) (-1)\^[||]{}Q\^[||]{}\
&=& \^ , where $Q= \lambda^{-1}$. In the course of these manipulations, we have used virtually all of the identities listed in the appendix. The relation (\[cstovertex\]) between the CS 4-vertex and the topological vertex is depicted in figure \[dcstovertex\].
The mismatch of the factor $\lambda^{\frac{|\alpha|+|\beta|}{2}}$ was first noticed in [@Diaconescu:2002sf].
Applications
============
Flops
-----
A natural question to study is the behavior of the string partition function under flops of the target geometry. We can analyze this question for all geometries whose toric diagrams decompose into strips.
First, let us recall the behavior of the conifold under a flop. The instanton piece of the partition function is given by Z\_[conifold]{}\^[inst]{}&=&\
&=& \_[k=1]{}\^ (1- Q q\^[-k]{})\^[k]{} . In addition, the genus 0 and 1 free energy contain further polynomial dependence on $t$. The behavior of the polynomial contributions to $F_0$ under flops can be considered separately from that of the instanton contributions [@Witten:1993yc; @Gopakumar:1998ki]. It turns out that the polynomial contribution to $F_1$, $\frac{1}{24}t$, is naturally considered together with the instanton contribution to the partition function. By analytic continuation, we obtain the partition function of the topological string on the flopped geometry from the partition function on the pre-flop geometry, Q\^ Z\_[conifold]{}\^[inst]{}(Q) Q\^[-]{} Z\_[conifold]{}\^[inst]{}() . To obtain the Gopakumar-Vafa invariants of the flopped geometry, we must now expand the RHS in the correct variable, $Q$, Q\^[-]{} Z\_[conifold]{}\^[inst]{}(Q\^[-1]{}) &=& Q\^[-]{} \_[k=1]{}\^ (1- )\^[k]{}\
&=&Q\^[-]{} Q\^[-(-1)]{} q\^[-(-2)]{}\_[k=1]{}\^ (Q q\^[k]{} -1)\^[k]{}\
&=&(-1)\^[(-1)]{} Q\^ \_[k=1]{}\^ (1-Q q\^[-k]{})\^[k]{} . In the last line, we have used that $Z_{conifold}^{instanton}$ is invariant under $q \rightarrow \frac{1}{q}$, and $\zeta(-1)=-\frac{1}{12}$. We see that up to a phase, the partition function is invariant when analytically continued from $Q$ to $Q^{-1}$, and then re-expanded in powers of $Q$. It follows that the Gopakumar-Vafa invariants are in fact invariant under this flop.
Now let’s turn to flops on the strip. The normal bundles of the curves neighboring the flopped curve are affected by the flop. On the strip, two geometries are to be distinguished: the $(-1,-1)$ curve to be flopped is connected along the strip to a $(-1,-1)$ curve on both sides (figure \[flopcase1\]), or to a $(-1,-1)$ curve on one side, and a $(-2,0)$ curve on the other (figure \[flopcase2\]). After the flop, the $(-1,-1)$ curves become $(-2,0)$ curves in the former case, and the $(-1,-1)$ and $(-2,0)$ curves are swapped in the latter.
We will consider the first case in detail. The second works out in exact analogy. Before the flop, we have three $(-1,-1)$ curves. The first vertex is $C_{\bullet \alpha_1 \beta_1}$, hence of type $B$, from which we can determine the type of all vertices to be $BABA$, yielding . \[preflop\] After the flop, note the important fact that the ordering of the vertices does not coincide with the ordering of the indices of the external Young tableaux the vertices carry. Assembling the data required to apply our rules in short hand: {$(-2,0)$, $(-1,-1)$, $(-2,0)$ curves, first vertex of type $B$} $\rightarrow BBAA$, we obtain . \[postflop\] To compare these two expressions, we can express the post-flop expression in terms of the pre-flop Kähler parameters, and then reexpand as in the case of the conifold considered above. The identification of the Kähler parameters that we propose is obtained by matching corresponding curves before and after the flop. Considering e.g. the $(-1,-1)$ curves before and after the flop, we obtain the relations Q\_1 &=& \_1 \_2 ,\
Q\_2 &=& ,\
Q\_3 &=& \_2 \_3 , which are consistent with the identification we obtain by considering $(-2,0)$ curves, Q\_1 Q\_2 &=& \_1 ,\
Q\_2 Q\_3 &=& \_3 . This identification of Kähler parameters is to be contrasted to the naive substitution $Q_2 \mapsto Q_2^{-1}$ for each curve whose Kähler parameter has $Q_2$ dependence. Upon making these substitutions, the only factor in (\[postflop\]) which must be reexpanded is $\{\beta_2^t \beta_3^t\}_{Q_2^{-1}}$. Using relation (\[transposed\]) and the definition of the pairing, we obtain (dropping indices and renaming tableaux for ease of notation), {\^t \^t}\_[Q\^[-1]{}]{}(q) &=& {}\_[Q\^[-1]{}]{}() \[floppin\]\
&=& \_k (1- Q\^[-1]{} q\^[-k]{})\^[ C\_k(, )]{}\
&=& \_k (Q\^[-1]{}q\^[-k]{})\^[C\_k(,)]{} (Q q\^[k]{}-1)\^[C\_k(, )]{} (-Q)\^\
&=& Q\^[-||-||]{}q\^[-]{}\_k (Q q\^[k]{}-1)\^[C\_k(, )]{} (-Q)\^\
&=& (-Q)\^[-||-||]{}q\^[-]{} (-Q)\^ {}\_Q (q), where we have used [@Iqbal:2003ix] \_k C\_k(,) &=& ||+|| ,\
\_k k C\_k(,) &=& . This almost coincides with the partition function (\[preflop\]) for the pre-flop geometry. To interpret the coefficients, let’s include the two curves that are connected to the flopped curve via a sum over the representations $\beta_2$ and $\beta_3$ into our considerations. Figure \[floptopbottom\] shows an example.
The $\beta_2$ and $\beta_3$ dependent factors from the upper and lower part of the diagram before the flop are Q\_t\^[|\_2|]{} (-1)\^[n\_t |\_2|]{} q\^[-]{} Q\_b\^[|\_3|]{} (-1)\^[n\_b |\_3|]{} q\^[-]{} = Q\_t\^[|\_2|]{} (-1)\^[|\_2|]{} q\^[-]{} Q\_b\^[|\_3|]{} \[precoef\] , where $n_t = (-1, 0) \wedge (1, -1) = 1$ and $n_b = (1, -1) \wedge (-1, 1) = 0$. After the flop, we have \_t\^[|\_2|]{} (-1)\^[\_t |\_2|]{} q\^[-]{} \_b\^[|\_3|]{} (-1)\^[\_b |\_3|]{} q\^[-]{}=\_t\^[|\_2|]{} \_b\^[|\_3|]{} (-1)\^[- |\_3|]{} q\^ \[postcoef\] , with $\tilde{n}_t = (-1 ,0) \wedge (1, 0) = 0$ and $\tilde{n}_b = (1, -1) \wedge (-1, 0) = -1$. Combining (\[postcoef\]) with the coefficients from (\[floppin\]), we obtain (\[precoef\]), (-Q\_2)\^[-|\_2|-|\_3|]{}q\^[-]{} \_t\^[|\_2|]{} \_b\^[|\_3|]{} (-1)\^[- |\_3|]{} q\^ &=& Q\_t\^[|\_2|]{} Q\_b\^[|\_3|]{} (-1)\^[ |\_2|]{} q\^[-]{} . We see that the coefficients in (\[floppin\]) are exactly those needed to maintain invariance of the Gopakumar-Vafa invariants under flops.
This result continues to hold for the situation depicted in figure \[flopcase2\], as well as the other possible completions of the lines carrying the Young tableaux $\beta_2$ and $\beta_3$ by curves of type $(-2,0)$ or $(-1,-1)$.
Can we conclude that Gopakumar-Vafa invariants for toric Calabi-Yau are invariant under flops in general? The above arguments were valid for geometries which contained only $(-1,-1)$ and $(-2,0)$ curves. For the general case, the topological string partition function Z=can be put in the product form [@Hollowood:2003cv; @Katz:2004js] Z=\_ ( \_[k=1]{}\^ (1- q\^k Q\_)\^[k n\^0\_]{} \_[g=1]{}\^ \_[k=0]{}\^[2g-2]{} (1 - q\^[g-1-k]{}Q\^ )\^[(-1)\^[k+g]{} n\^[g]{}\_ [2g-2 k]{}]{}) , where $\vec{n}$ encodes the classes of the various holomorphic curves relative to the basis specified by the $Q_i$. By the same argument as above, the only factor which needs to be re-expanded after expressing the post-flop partition function in pre-flop variables is the one counting contributions from just the flopped curve. As above, this factor is invariant, up to a coefficient, under this operation. However, it remains to argue that for the curves we identify before and after the flop, say $\vec{n}$ and $\vec{\tilde{n}}$, the relation $n^g_{\vec{n}} = n^g_{\vec{\tilde{n}}}$ holds. For the case of geometries with only $(-1,-1)$ and $(-2,0)$ curves, this equality followed from the comparison of (\[preflop\]) and (\[postflop\]), and the interpetation of the coefficients in (\[floppin\]).
Note that after an appropriate number of blowups and flops, any toric CY can be decomposed into the strips we have been considering in this paper. Hence, if Gopakumar-Vafa invariants indeed prove to be invariant under flops in general, the vertex calculations on a strip performed in this paper become relevant for any toric CY.
Geometric Engineering
---------------------
A natural physical playground for the formalism developed in this paper is in the context of geometric engineering of $\N=2$ gauge theories by compactification on local CY. The toric geometries that give rise to linear chains of $U(N)$ gauge groups (i.e. theories with product gauge group with $U(N_i)$ factors and bifundamental matter between adjacent gauge groups) can be decomposed into the strips we consider here. Thanks to Nekrasov’s construction, the full string partition function (vs. only its field theory limit) can be extracted from such gauge theories. The basic building block of the geometry which engineers such gauge theories is the triangulation of the strip given by . The corresponding web diagram is shown in figure \[bblock\].
All of the curves in figure (\[bblock\]) are (-1,-1) curves, and the first vertex is of type $A$, hence we have an alternating succession of vertices $ABABAB\ldots$. By the rules derived above, we obtain K\^[\_[1]{}\_[N]{}]{}\_[\_[1]{}\_[N]{}]{} &=& \[basic\]\
&=&[W]{}\_[\_[1]{}]{}[W]{}\_[\_[1]{}]{}\_[\_[N]{}]{}[W]{}\_[\_[N]{}]{}\
&&\_[k]{} The Kähler parameters $Q_{\alpha,\beta}$ are given by Q\_[\_[i]{}\_[j]{}]{}&=&Q\_[ij]{}\
Q\_[\_[i]{}\_[j]{}]{}&=&Q\_[ij]{}Q\_[m,j]{}\
Q\_[\_[i]{}\_[j]{}]{}&=&Q\_[ij]{}Q\_[m,i]{}\^[-1]{},\
Q\_[\_[i]{}\_[j]{}]{}&=&Q\_[ij]{}Q\_[m,i]{}\^[-1]{}Q\_[m,j]{}, where $Q_{ij}=\prod_{k=i}^{j-1}Q_{m,k}Q_{f,k}$.
${\cal K}^{\alpha_{1} \cdots \alpha_{N}}_{\beta_{1}\cdots\beta_{N}}$ is the building block for the partition function $Z_{Nekrasov}$ of ${\cal N}=2$ gauge theories with product gauge groups and bi-fundamental matter. The partition function is given by products of ${\cal
K}^{\alpha_{1},\cdots, \alpha_{N}}_{\beta_{1},\cdots,\beta_{N}}$, in the field theory limit, summed over $\alpha_{i},\beta_{i}$, where the field theory limit of ${\cal K}^{\alpha_{1},\cdots,
\alpha_{N}}_{\beta_{1},\cdots,\beta_{N}}$ is given by K\^[\_[1]{},,\_[N]{}]{}\_[\_[1]{},,\_[N]{}]{} \_[\_[1]{}]{}[L]{}\_[\_[1]{}]{}\_[\_[N]{}]{}[L]{}\_[\_[N]{}]{}\_[k]{} with \_=\_[q1]{}(q-1)\^[||]{}[W]{}\_.
As an example, consider the CY in figure (\[sunwithmatter\]) which can be used to engineer $U(N)$ with $N_{f}=2N$.
The partition function is obtained by gluing two ${\cal K}$ type expressions together. Note however that in the upper strip, the order of vertices is $BABABA\ldots$. A moment’s thought teaches us that $\K_{upper}(\alpha,\beta) = \K_{lower}(\alpha^t,\beta^t)$. Hence, Z&=&\_[\_[1]{},,\_N]{} Q\_[b]{}\^[|\_[1]{}|++|\_[N]{}|]{}[K]{}\^[\_[1]{} \_[N]{}]{}\_(Q)[K]{}\^\_[\_[1]{} \_[N]{}]{}()\
&=&\_[\_[1]{},,\_N]{}Q\_[b]{}\^[|\_[1]{}|++|\_[N]{}|]{}[W]{}\_[\_[1]{}]{}\^[2]{}\_[\_[N]{}]{}\^[2]{}\_[k]{}\_[i=1]{}\^[N]{}(1-q\^[k]{}Q\_[m,i]{})\^[C\_[k]{}(\_[i]{},)]{} (1-q\^[k]{}\_[m,i]{})\^[C\_[k]{}(,\_[i]{})]{}\
&&\_[i<j]{} . \[PF\]Defining Q\_[ij]{}&=&e\^[-(a\_[i]{}-a\_[j]{})]{},\
Q\_[m,i]{}&=&e\^[-(a\_[i]{}+m\_[i]{})]{},\
\_[m,i]{}&=&e\^[-(a\_[i]{}+m\_[i+N]{})]{},\
q&=&e\^[-]{}, the field theory limit is given by $\beta \rightarrow 0$. In this limit, (\[PF\]) yields =\_[\_[1,,N]{}]{}Q\_[b]{}\^[|\_[1]{}|++|\_[N]{}|]{} [Z]{}\^[(0)]{}\_[\_[1]{}\_[N]{}]{}\_[k]{}\_[i,j]{}(a\_[i]{}+m\_[j]{}+k)\^[C\_[k]{}(\_[i]{})]{}(a\_[i]{}+m\_[j+N]{}+k)\^[C\_[k]{}(\_[i]{})]{} , which is Nekrasov’s partition function (equation (1.8) in [@Nekrasov:2003af]) for $N_{f}=2N$ after using the identities given in [@Iqbal:2003zz].
Conclusion
==========
How to move off the strip? We saw that an obstacle to taking a turn off the strip was performing the sums (\[sumruleone\]) and (\[sumruletwo\]) with factors of type $q^{\kappa(\alpha)/2}$ included in the sum over $\alpha$. This obstacle does not appear insurmountable, and efforts are underway to evaluate such sums. With them, all sums in the expression for the topological partition function of toric manifolds whose web diagram consists of a closed loop with external lines attached could be performed. To go further, one would need to perform sums over the Young tableaux which are the arguments of the Schur functions which appear in the topological vertex.
Acknowledgements {#acknowledgements .unnumbered}
================
We would like to thank Cumrun Vafa for many valuable discussions. AK would also like to thank Mina Aganagic, Paul Aspinwall, and Bogdan Florea for helpful conversations, as well as the Simons Workshop in Mathematics and Physics and the Harvard High Energy Physics Group for a very productive atmosphere in which part of this work was completed.
The work of AK was supported by the U.S. Department of Energy under contract number DE-AC02-76SF00515.
Getting to know Schur functions
===============================
Since Schur functions feature prominently in this text, we wish to briefly present them in their natural habitat of symmetric functions in this appendix. Readers who wish to learn more are referred to, e.g., [@Macdonald; @Fulton].
Schur polynomials $s_\lambda(x_1,\ldots,x_k)$ present a basis of the symmetric polynomials in $k$ variables. They arise in representation theory as the characters of the Schur functor. Two perhaps more intuitive choices of basis for the symmetric polynomials are the following. The complete symmetric polynomials $h_r$ are defined as the sum of all monomials in $k$ variables of degree $r$, e.g. for $k=2$, $r=2$, $h_2=x_1^2+x_2^2+x_1 x_2$, and the elementary symmetric polynomials $e_r$ as the sum of all monomials of degree $r$ in distinct variables, e.g. $e_2=x_1 x_2$. To a Young tableaux $\lambda$ with at most $k$ rows, one now introduces the polynomial $h_\lambda = h_{\lambda_1} \cdots h_{\lambda_n}$, where $\lambda_i$ denotes the number of boxes in the $i$-th row of $\lambda$, and likewise, to a Young tableaux $\mu$ such that $\mu^t$ has at most $k$ rows, $e_\mu = e_{\mu_1} \cdots e_{\mu_n}$. Both sets $\{h_\lambda\}$, $\{e_\mu\}$ comprise a basis for symmetric polynomials. The Schur polynomials can be expressed in terms of these, using the so-called determinantal formulae, s\_&=& | h\_[\_i + j-i]{}|\
&=& | e\_[\^t\_i + j -i]{}| . \[JT\] The skew Schur polynomials, which we introduced in the text via their relation to the ordinary Schur polynomials, $s_{\lambda/\mu}(x) = \sum_\nu c^\lambda_{\mu \nu} s_\nu(x)$, also satisfy determinantal identities,[^2] s\_[/]{}&=& | h\_[\_i - \_j + j-i]{}| \[detss1\]\
&=& | e\_[\^t\_i -\^t\_j + j -i]{}| . \[detss2\] The generating function for the elementary symmetric functions $e_i$ is $\prod (1+x_i t)$, i.e. the coefficient of $t^i$ in this power series is the $i$-th elementary symmetric function $e_i$. We now define the functions $S_\lambda(E(t))$, which we encountered in section (\[CS\]), in accordance with the determinantal formula (\[JT\]), where $e_i$ is replaced by the coefficient of $t^i$ in the power series $E(t)$. Clearly, for $E(t)= \prod (1+x_i t)$, $S_\lambda(E(t)) = s_\lambda(x)$.
Next, we collect the identities for the Schur functions we use in the text. \_s\_[/\_1]{}(x)s\_[/\_2]{}(y)&=& \_[i,j]{} (1-x\_i y\_j)\^[-1]{} \_s\_[\_2/]{}(x)s\_[\_1/]{}(y) ,\
\_s\_[\^t/\_1]{}(x)s\_[/\_2]{}(y) &=& \_[i,j]{} (1+x\_i y\_j) \_s\_[\_2\^t/\^t]{}(x)s\_[\_1\^t/]{}(y) ,\
\_s\_[/]{}(x) s\_(y) &=& s\_(x,y) ,\
s\_(q\^[+ ]{}) &=& (-1)\^[||]{} s\_[\^t]{}(q\^[--\^t]{}) ,\
s\_(q\^) &=& q\^ s\_[\^t]{}(q\^) . By invoking the cyclicity of the vertex, we further obtain s\_(q\^) s\_(q\^[+]{}) &=& s\_(q\^) s\_(q\^[+]{}) ,\
q\^ s\_(q\^) s\_[\^t]{}(q\^[+ \^t]{}) &=& \_s\_[/]{}(q\^) s\_[/]{}(q\^) .
In our applications, we need to work within the ring of symmetric functions with countably many independent variables. This ring can be obtained from the rings of symmetric polynomials in finitely many variables via an inverse limit construction [@Macdonald].
[99]{} D. E. Diaconescu, B. Florea and A. Grassi, “Geometric transitions and open string instantons,” Adv. Theor. Math. Phys. [**6**]{}, 619 (2003) \[arXiv:hep-th/0205234\]. D. E. Diaconescu, B. Florea and A. Grassi, “Geometric transitions, del Pezzo surfaces and open string instantons,” Adv. Theor. Math. Phys. [**6**]{}, 643 (2003) \[arXiv:hep-th/0206163\]. M. Aganagic, M. Marino and C. Vafa, “All loop topological string amplitudes from Chern-Simons theory,” Commun. Math. Phys. [**247**]{}, 467 (2004) \[arXiv:hep-th/0206164\]. M. Aganagic, A. Klemm, M. Marino and C. Vafa, “The topological vertex,” arXiv:hep-th/0305132. J. Li, C. Liu, K. Liu, J. Zhou, “A Mathematical Theory of the Topological Vertex," arXiv:math.AG/0408426.
A. Iqbal and A. K. Kashani-Poor, “Instanton counting and Chern-Simons theory,” Adv. Theor. Math. Phys. [**7**]{}, 457 (2004) \[arXiv:hep-th/0212279\]. A. Iqbal and A. K. Kashani-Poor, “SU(N) geometries and topological string amplitudes,” arXiv:hep-th/0306032. T. Eguchi and H. Kanno, “Topological strings and Nekrasov’s formulas,” JHEP [**0312**]{}, 006 (2003) \[arXiv:hep-th/0310235\]. J. Zhou, “Curve counting and instanton counting,” arXiv:math.ag/0311237. T. Eguchi and H. Kanno, “Geometric transitions, Chern-Simons gauge theory and Veneziano type amplitudes,” Phys. Lett. B [**585**]{}, 163 (2004) \[arXiv:hep-th/0312234\]. N. A. Nekrasov, “Seiberg-Witten prepotential from instanton counting,” arXiv:hep-th/0306211. M. Aganagic, A. Klemm and C. Vafa, “Disk instantons, mirror symmetry and the duality web,” Z. Naturforsch. A [**57**]{}, 1 (2002) \[arXiv:hep-th/0105045\]. S. Katz and C. C. Liu, “Enumerative Geometry of Stable Maps with Lagrangian Boundary Conditions and Adv. Theor. Math. Phys. [**5**]{}, 1 (2002) \[arXiv:math.ag/0103074\]. A. Okounkov, N. Reshetikhin and C. Vafa, “Quantum Calabi-Yau and classical crystals,” arXiv:hep-th/0309208. E. Witten, “Phases of N = 2 theories in two dimensions,” Nucl. Phys. B [**403**]{}, 159 (1993) \[arXiv:hep-th/9301042\]. R. Gopakumar and C. Vafa, “On the gauge theory/geometry correspondence,” Adv. Theor. Math. Phys. [**3**]{}, 1415 (1999) \[arXiv:hep-th/9811131\]. T. J. Hollowood, A. Iqbal and C. Vafa, “Matrix models, geometric engineering and elliptic genera,” arXiv:hep-th/0310272.
S. Katz, “Gromov-Witten, Gopakumar-Vafa, and Donaldson-Thomas invariants of Calabi-Yau arXiv:math.ag/0408266.
W. Fulton, “Introduction to Toric Varieties,” Princeton University Press, Princeton, 1993.
I. G. Macdonald, “Symmetric functions and Hall polynomials,” Second Edition, Oxford University Press, New York, 1995.
W. Fulton, J. Harris, “Representation Theory," Springer Verlag, New York, 1991.
[^1]: Thanks to Andy Neitzke for a discussion on this point that lead to the correction of a sign error in a previous version of this paper.
[^2]: The difference of two Young tableaux (performed row-wise), as it appears in the determinantal formulae (\[detss1\]) and (\[detss2\]), is also called a skew Young tableau, hence the name [*skew*]{} Schur polynomials.
|
---
author:
- 'S. A. Owerre'
title: Photoinduced Topological Phase Transitions in Topological Magnon Insulators
---
**Topological magnon insulators are the bosonic analogs of electronic topological insulators. They are manifested in magnetic materials with topologically nontrivial magnon bands as realized experimentally in a quasi-two-dimensional (quasi-2D) kagomé ferromagnet Cu(1-3, bdc), and they also possess protected magnon edge modes. These topological magnetic materials can transport heat as well as spin currents, hence they can be useful for spintronic applications. Moreover, as magnons are charge-neutral spin-${\bf 1}$ bosonic quasiparticles with a magnetic dipole moment, topological magnon materials can also interact with electromagnetic fields through the Aharonov-Casher effect. In this report, we study photoinduced topological phase transitions in intrinsic topological magnon insulators in the kagomé ferromagnets. Using magnonic Floquet-Bloch theory, we show that by varying the light intensity, periodically driven intrinsic topological magnetic materials can be manipulated into different topological phases with different sign of the Berry curvatures and the thermal Hall conductivity. We further show that, under certain conditions, periodically driven gapped topological magnon insulators can also be tuned to synthetic gapless topological magnon semimetals with Dirac-Weyl magnon cones. We envision that this work will pave the way for interesting new potential practical applications in topological magnetic materials.**
Topological insulators have captivated the attention of researchers in recent years and they currently represent one of the active research areas in condensed matter physics [@top1; @top2; @top3; @top4; @top5]. These nontrivial insulators can be realized in electronic systems with strong spin-orbit coupling and a nontrivial gap in the energy band structures. They also possess Chern number or $\mathbb{Z}_2$ protected metallic edge or surface modes that can transport information without backscattering [@top5]. In principle, however, the ubiquitous notion of topological band theory is independent of the statistical nature of the quasiparticle excitations. In other words, the concept of Berry curvature and Chern number can be defined for any topological band structure irrespective of the quasiparticle excitations. Consequently, these concepts have been extended to bosonic systems with charge-neutral quasiparticle excitations such as magnons [@alex0; @zhh; @alex4; @alex4h; @cao; @sm; @rold; @sol; @sol1; @alex5b; @alex5c; @ruck; @cherny; @swang; @kova; @alex4a; @pant], triplons [@rom; @mcc], phonons [@pho1; @pho3], and photons [@ling].
Topological magnon insulators [@alex0; @zhh; @alex4; @alex4h; @cao; @sm; @rold; @sol; @sol1; @alex5b; @alex5c; @alex4a] are the bosonic analogs of electronic topological insulators. They result from the nontrivial low-energy excitations of insulating quantum magnets with spin-orbit coupling or Dzyaloshinskii-Moriya (DM) interaction [@dm; @dm2], and exhibit topologically nontrivial magnon bands and Chern number protected magnon edge modes, with similar properties to those of electrons in topological insulators [@top1; @top2; @top3; @top4; @top5]. Theoretically, topological magnetic excitations can arise in different lattice geometries with DM interaction, however their experimental observation is elusive in real magnetic materials. Recently, intrinsic topological magnon insulator has been observed experimentally in a quasi-2D kagomé ferromagnet Cu(1-3, bdc) [@alex5b]. Moreover, recent evidence of topological triplon bands have also been reported in a dimerized quantum magnet SrCu$_2$(BO$_3$)$_2$ [@rom; @mcc]. These magnetic materials have provided an interesting transition from electronic to bosonic topological insulators.
Essentially, the intrinsic properties of a specific topological magnon insulator are material constants that cannot be tuned, thereby hindering a topological phase transition in the material. In many cases of physical interest, however, manipulating the intrinsic properties of topological magnetic materials could be a stepping stone to promising practical applications, and could also provide a platform for studying new interesting features such as photo-magnonics [@benj], magnon spintronics [@magn; @benja], and ultrafast optical control of magnetic spin currents [@ment; @tak4; @tak4a; @walo]. One way to achieve these scientific goals is definitely through light-matter interaction induced by photo-irradiation. In recent years, the formalism of photo-irradiation has been a subject of intensive investigation in electronic systems such as graphene and others [@foot3; @foot4; @foot5; @gru; @fot; @fot1; @jot; @fla; @we1; @we2; @we3; @we4; @we5; @we6; @gol; @buk; @eck1; @ste; @ple; @ew; @dik; @lin; @du; @du1; @delp; @eza; @zhai; @saha; @roy; @roy1]. Basically, photo-irradiation allows both theorists and experimentalists to engineer topological phases from trivial systems and also induce photocurrents and phase transitions in topologically nontrivial systems. In a similar manner to the notion of bosonic topological band theory, one can also extend the mechanism of photo-irradiation to bosonic systems.
In this report, we theoretically investigate photo-irradiated intrinsic topological magnon insulators in the kagomé ferromagnets and their associated topological phase transitions. One of the main objectives of this report is to induce tunable parameters in intrinsic topological magnon insulators, which subsequently drive the system into a topological phase transition. We achieve this objective by utilizing the quantum theory of magnons, which are charge-neutral spin-$1$ bosonic quasiparticles and carry a magnetic dipole moment. Therefore, magnons can couple to both time-independent[@mei; @mei1; @xr; @xr1] and periodic time-dependent (see Methods) [@ow1] electric fields through the Aharonov-Casher (AC) effect [@aha], in the same manner that electronic charged particles couple through the Aharonov-Bohm (AB) effect [@aha1]. Quite distinctively, for the periodic time-dependent electric fields (see Methods) [@ow1], this results in a periodically driven magnon system, and thus can be studied by the Floquet-Bloch theory. Using this formalism, we show that intrinsic topological magnon insulators can be tuned from one topological magnon insulator to another with different Berry curvatures, Chern numbers, and thermal Hall conductivity. Moreover, we show that, by manipulating the light intensity, periodically driven intrinsic topological magnon insulators can also transit to synthetic gapless topological magnon semimetals. Therefore, the magnon spin current in topological magnetic materials can be manipulated by photo-irradiation, which could be a crucial step towards potential practical applications.
**Results**
**Topological magnon insulators.** We consider the simple microscopic spin Hamiltonian for intrinsic topological magnon insulators in the kagomé ferromagnets [@alex5b] $$\begin{aligned}
\mathcal H&=\sum_{\la \ell\ell^\prime \ra } \big[ -J{\vec S}_{\ell}\cdot{\vec S}_{\ell^\prime}+\vec{D}_{\ell\ell^\prime}\cdot ({\vec S}_{\ell}\times{\vec S}_{\ell^\prime})\big]-\vec{B}\cdot\sum_{\ell} \vec{S}_{\ell}.
\label{kham}\end{aligned}$$ The first summation is taken over nearest-neighbour (NN) sites $\ell$ and $\ell^\prime$ on the 2D kagomé lattice, and $\vec{D}_{\ell\ell^\prime}$ is the DM vector between the NN sites due to lack of an inversion center as depicted in Fig. a. The last term is the Zeeman coupling to an external magnetic field $\vec{B}=g\mu_B\vec{H}$, where $\mu_B$ is the Bohr magneton and $g$ the spin $g$-factor. Topological magnon insulators [@alex0; @zhh; @alex4; @alex4h; @cao; @sm; @rold; @sol; @sol1; @alex5b; @alex5c] can be captured by transforming the spin Hamiltonian to a bosonic hopping model using the Holstein-Primakoff (HP) spin-boson transformation. In this formalism, only the DM vector parallel to the magnetic field contributes to the noninteracting bosonic Hamiltonian [@alex5b; @alex0], but other components of the DM vector can be crucial when considering magnon-magnon interactions [@cherny]. Here, we limit our study to noninteracting magnon system as it captures all the topological aspects of the system [@alex5b; @alex0]. We consider then an external magnetic field along the $z$ (out-of-plane) direction, $\vec{B}=B\hat{z}$, and take the DM vector as $\vec{D}_{\ell\ell^\prime}=D\hat z$. The Holstein-Primakoff (HP) spin-boson transformation is given by $S_{\ell}^{ z}= S-a_{\ell}^\dagger a_{\ell},~S_{\ell}^+\approx \sqrt{2S}a_{\ell}=(S_{\ell}^-)^\dg$, where $a_{\ell}^\dagger (a_{\ell})$ are the bosonic creation (annihilation) operators, and $S^\pm_{\ell}= S^x_{\ell} \pm i S^y_{\ell}$ denote the spin raising and lowering operators. Applying the transformation to Eq. yields the bosonic (magnon) hopping Hamiltonian $$\begin{aligned}
\mathcal H&=-t_0\sum_{\la \ell\ell^\prime \ra} \big(a_{\ell}^\dagger a_{\ell^\prime}e^{i\varphi_{\ell\ell^\prime}} + \text{H.c}\big) +t_z\sum_{\ell}n_\ell,
\label{ham1}\end{aligned}$$ where $n_\ell=a_{\ell}^\dagger a_{\ell}$ is the number operator; $t_0=JS\sqrt{1+(D/J)^2}$ and $t_z=4JS+B$ with $t_z>t_0$. The phase $\varphi_{\ell\ell^\prime}=\pm\varphi=\pm\tan^{-1}(D/J)$ is the fictitious magnetic flux in each unit triangular plaquette of the kagomé lattice [@alex0], in analogy to the Haldane model [@top1]. The Fourier transform of the magnon Hamiltonian is given by $\mathcal H=\sum_{\vec{k}} \psi_{\vec{k}}^\dg \mathcal H({\vec{k}})\psi_{\vec{k}}$, with $\psi_{\vec{k}}=(a_{\vec{k},1},a_{\vec{k},2},a_{\vec{k},3})^{\text{T}}$, where $\mathcal H(\vec{k})=t_z \text{I}_{3\times 3}- \Lambda(\vec{k})$, $$\begin{aligned}
\Lambda(\vec{k}) &=2t_0
\begin{pmatrix}
0& \cos k_2e^{-i\varphi}& \cos k_3e^{i\varphi}\\
\cos k_2e^{i\varphi}&0&\cos k_1e^{-i\varphi}\\
\cos k_3e^{-i\varphi}&\cos k_1e^{i\varphi}&0
\end{pmatrix},
\label{khama}\end{aligned}$$ with $k_i=\vec{k}\cdot\vec{a}_i$, and $\vec{a}_1=(1,0)$, $\vec{a}_2=(1/2,\sqrt{3}/2)$, $\vec{a}_3=\vec{a}_2-\vec{a}_1$ are the lattice vectors. Diagonalizing the Hamiltonian gives three magnon branches of the kagomé ferromagnet. In the following we set $B=0$ as it simply shifts the magnon bands to high energy. As shown in Fig. c, without the DM interaction, i.e. $D/J=0$ or $\varphi=0$, the two lower dispersive bands form Dirac magnon cones at $\pm{\bf K}$ (see Fig. b), whereas the flat band has the highest energy and touches one of the dispersive bands quadratically at ${\bf \Gamma}$. In Fig. d, we include a small DM interaction $D/J=0.15$ applicable to Cu(1-3, bdc) [@alex5b]. Now the flat band acquires a dispersion and all the bands are separated by a finite energy gap with well-defined Chern numbers. Thus, the system becomes a topological magnon insulator [@zhh; @alex4; @alex5b; @alex5c].
![(a) Schematic of the kagomé lattice with three sublattices A, B, C as indicated by coloured dots, and the out-of-plane DM interaction is indicated by open circles. (b) The first Brillouin zone of the kagomé lattice with two inequvilaent high symmetry points at $\pm{\bf K}$. The red and green dots denote the photoinduced Dirac-Weyl magnon nodes as will be discussed later. (c) Magnon bands of undriven insulating kagomé ferromagnets with $D/J=0$, showing Dirac magnon nodes at $\pm{\bf K}$, formed by the two lower dispersive bands. (d) Topological magnon bands of undriven insulating kagomé ferromagnets with $D/J=0.15$.[]{data-label="klat"}](kagome_lattice){width="5in"}
**Periodically driven topological magnon insulators.** In this section, we introduce the notion of periodically driven intrinsic topological magnon insulators. Essentially, this concept will be based on the charge-neutrality of magnons in combination with their magnetic dipole moment $\vec{ \mu} =g\mu_B\hat{z}$. Let us suppose that magnons in insulating quantum magnetic systems are exposed to an electromagnetic field with a dominant time-dependent electric field vector $\vec E(\tau)$. Then the effects of the field on the system can be described by a vector potential defined as $\vec E(\tau)=-\partial \vec A(\tau)/\partial \tau$, where A()=\[-A\_x(+), A\_y(),0\], \[eqn4a\] with amplitudes $A_x$ and $A_y$, frequency $\omega$, and phase difference $\phi$. The vector potential has time-periodicity: $\vec A(\tau+T)=\vec A(\tau)$, with $T=2\pi/\omega$ being the period. Here $\phi=\pi/2$ corresponds to circularly-polarized light, whereas $\phi=0$ or $\pi$ corresponds to linearly-polarized light.
Using the AC effect for charge-neutral particles [@aha], we consider magnon quasiparticles with magnetic dipole $\mu$ moving in the background of a time-dependent electric field. In this scenario they will acquire a time-dependent AC phase (see Methods) given by \_[\^]{}()=\_m\_[\_]{}\^[\_[\^]{}]{} ()d, \[eqn4\] where $\mu_m = g\mu_B/\hbar c^2$ and $\vec{r}_\ell$ is the coordinate of the lattice at site $\ell$. We have used the notation $ \vec \Xi(\tau) = \vec{E}(\tau)\times \hat z$ for brevity. From Eq. we obtain $$\begin{aligned}
\vec \Xi(\tau)=[E_y\sin(\omega \tau),E_x\sin(\omega \tau+\phi),0],
\end{aligned}$$ where $E_x$ and $E_y$ are the amplitudes of the electric field and $\phi$ is the phase factor. By virtue of the time-dependent Peierls substitution, the periodically driven magnon Hamiltonian is succinctly given by
$$\begin{aligned}
\mathcal H(\tau)&=-t_0\sum_{\la \ell\ell^\prime \ra} \big(e^{i[\varphi_{\ell\ell^\prime} +\theta_{\ell\ell^\prime}(\tau)]} a_{\ell}^\dagger a_{\ell^\prime}+ \text{H.c}\big) +t_z\sum_{\ell}n_\ell.
\label{kham1}\end{aligned}$$
Therefore, the time-dependent momentum space Hamiltonian $\mathcal H(\vec{k},\tau)$ corresponds to making the time-dependent Peierls substitution $\vec{k}\to \vec{k} +\mu_m\vec \Xi(\tau)$ in Eq. . We note that previous studies based on the AC effect in insulating magnets considered a time-independent electric field gradient, which leads to magnonic Landau levels [@mei1; @mei; @xr1; @xr]. In stark contrast to those studies, the time-dependent version can lead to Floquet topological magnon insulators in insulating quantum magnets with inversion symmetry, e.g. the honeycomb lattice [@ow1] or the Lieb lattice (see Supplementary Information). We note that Floquet topological magnon insulators can also be generated by driving a gapped trivial magnon insulator with vanishing Chern number, in a similar manner to Dirac magnons. A comprehensive study of this case is beyond the purview of this report. In the current study, however, the kagomé lattice quantum ferromagnets naturally lack inversion symmetry, and thus allows an intrinsic DM interaction as depicted in Fig. a.
![ Top panel. Topological magnon bands of periodically driven topological magnon insulator at $D/J=0.15$. (a) $\mathcal{E}_x=\mathcal{E}_y=1.7$ and $\phi=\pi/2$. (b) $\mathcal{E}_x=\mathcal{E}_y=2.5$ and $\phi=\pi/2$. Bottom panel. Tunable Berry curvatures of periodically driven intrinsic topological magnon insulator on the kagomé lattice at $k_y=0$ and $D/J=0.15$. (c) $\mathcal{E}_x=\mathcal{E}_y=1.7$ and $\phi=\pi/2$. (d) $\mathcal{E}_x=\mathcal{E}_y=2.5$ and $\phi=\pi/2$.[]{data-label="BC"}](BC){width="6in"}
![Topological phase diagram of periodically driven intrinsic topological magnon insulator on the kagomé lattice. (a) Chern number phase diagram for $D/J=0.15$ and $\phi=\pi/2$. (b) Thermal Hall conductivity phase diagram for $D/J=0.15$, $\phi=\pi/2$, and $T=0.75$.[]{data-label="Phase"}](phases){width="5.5in"}
Now, we apply the magnonic Floquet-Bloch theory in Methods. For simplicity, we consider the magnonic Floquet Hamiltonian in the off-resonant regime, when the driving frequency $\omega$ is larger than the magnon bandwidth $\Delta$ of the undriven system, i.e. $\omega\gg\Delta$. In this limit, the Floquet bands are decoupled, and it suffices to consider the zeroth order time-independent Floquet magnon Hamiltonian $\mathcal H^0(\vec{k})=t_z \text{I}_{3\times 3}- \Lambda^0(\vec{k}) $, where
$$\begin{aligned}
\Lambda^0(\vec{k}) &=
\begin{pmatrix}
0& t_0^{AB}\cos k_2e^{-i\varphi}& t_0^{CA}\cos k_3e^{i\varphi}\\
t_0^{AB}\cos k_2e^{i\varphi}&0&t_0^{BC}\cos k_1e^{-i\varphi}\\
t_0^{CA}\cos k_3e^{-i\varphi}&t_0^{BC}\cos k_1e^{i\varphi}&0
\end{pmatrix},
\label{kflo}\end{aligned}$$
and $$\begin{aligned}
t_0^{AB}&=2t_0\mathcal J_{0}\lb \frac{1}{2}\sqrt{\mathcal{E}_x^2+3\mathcal{E}_y^2+2\sqrt{3}\mathcal{E}_x\mathcal{E}_y\cos\phi}\rb,\\
t_0^{BC}&=2t_0\mathcal J_{0}\lb|\mathcal{E}_x|\rb,\\
t_0^{CA}&=2t_0\mathcal J_{0}\lb \frac{1}{2}\sqrt{\mathcal{E}_x^2+3\mathcal{E}_y^2-2\sqrt{3}\mathcal{E}_x\mathcal{E}_y\cos\phi}\rb,\end{aligned}$$ where $\mathcal J_n$ is the Bessel function of order $n$. The dimensionless quantity characterizing the intensity of light is different from that of electronic systems and it is given by $$\begin{aligned}
\mathcal{E}_i =\frac{g\mu_B E_i a}{\hbar c^2},\end{aligned}$$ where $ \quad i = x,y$ and $a$ is the lattice constant. Evidently, a direct consequence of photo-irradiation is that the magnonic Floquet Hamiltonian is equivalent to that of a distorted kagomé ferromagnet with unequal tunable interactions $t_0^{AB}\neq t_0^{BC}\neq t_0^{CA}$. In the following, we shall discuss the topological aspects of this model. The Berry curvature is one of the main important quantities in topological systems. It is the basis of many observables in topological insulators. To study the photoinduced topological phase transitions in driven topological magnon insulators, we define the Berry curvature of a given magnon band $\alpha$ as $$\begin{aligned}
\Omega_{\alpha}(\vec k)=-2\text{Im}\sum_{\alpha^\prime \neq \alpha}\frac{\big(\braket{\psi_{\vec k, \alpha}|\hat v_x|\psi_{\vec k, \alpha^\prime}}\braket{\psi_{\vec k, \alpha^\prime}|\hat v_y|\psi_{\vec k, \alpha}}\big)}{\big(\epsilon_{\vec k, \alpha}- \epsilon_{\vec k, \alpha^\prime}\big)^2},
\label{chern2}\end{aligned}$$ where $\hat v_{x,y}=\partial \mathcal{H}^0(\vec k)/\partial k_{x,y}$ are the velocity operators, $\psi_{\vec k, \alpha}$ are the magnon eigenvectors, and $\epsilon_{\vec k, \alpha}$ are the magnon energy bands. The associated Chern number is defined as the integration of the Berry curvature over the Brillouin zone (BZ), $$\begin{aligned}
\mathcal C_\alpha=\frac{1}{2\pi}\int_{BZ} d^2 k~\Omega_{\alpha}(\vec k),\end{aligned}$$ where $\alpha=1,2,3$ label the lower, middle, and upper magnon bands respectively.
In Fig. \[BC\] we have shown the evolution of the magnon bands and the Berry curvatures for varying light intensity. We can see that the lower and upper magnon bands and their corresponding Berry curvatures change with varying light intensity, whereas the middle magnon band remains unchanged. Consequently, the system changes from one topological magnon insulator with Chern numbers $(-1,0,1)$ to another one with Chern numbers $(1,0,-1)$ as shown in the photoinduced topological phase diagram in Fig. \[Phase\] (a). In other words, exposing a topological magnon insulator to a varying light intensity field redistribute the magnon band structures and subsequently leads to a topological phase transition between one topological magnon insulator to another with different Berry curvatures and Chern numbers.
A crucial consequence of topological magnon insulators is the thermal Hall effect [@alex5a; @alex1]. Theoretically, the thermal Hall effect is understood as a consequence of the Berry curvatures induced by the DM interaction [@alex2; @alex0; @alex2q]. If we focus on the regime in which the Bose distribution function is close to equilibrium, the same theoretical concept of undriven thermal Hall effect can be applied in the photoinduced system. The transverse component $\kappa_{xy}$ of the thermal Hall conductivity is given explicitly as [@alex2; @alex2q] $$\begin{aligned}
\kappa_{xy}=-k_B^2 T\int_{BZ} \frac{d^2k}{(2\pi)^2}~ \sum_{\alpha=1}^N c_2\lb n_\alpha\rb\Omega_{\alpha}(\vec k),
\label{thm}\end{aligned}$$ where $ n_\alpha=n\big[ \epsilon_{\alpha}(\vec k)\big]=1/ \big[e^{\epsilon_{\alpha}(\vec k)/k_BT}-1\big]$ is the Bose distribution function close to thermal equilibrium, $k_B$ is the Boltzmann constant, $T$ is the temperature, and $ c_2(x)=(1+x)\lb \ln \frac{1+x}{x}\rb^2-(\ln x)^2-2\text{Li}_2(-x)$, with $\text{Li}_2(x)$ being the dilogarithm. Indeed, the thermal Hall conductivity is the Berry curvature weighed by the $c_2$ function. Therefore, any change of the Berry curvature will affect the thermal Hall conductivity. Evidently, as shown in Fig. \[Phase\] (b), the two photoinduced phases of the intrinsic topological magnon insulator have different signs of the anomalous thermal Hall conductivity due to the sign change in the Berry curvatures. The elliptic ring in the topological phase diagram in Fig. \[Phase\] is an artifact of the kagomé lattice, together with circularly polarized light. It does not exist with linearly polarized light, and it is also not present on the honeycomb lattice.
{width="7in"}
**Photoinduced topological magnon semimetal.** The topological phase transitions in periodically driven intrinsic topological magnon insulators can also be extended to synthetic topological magnon semimetals with gapless magnon bands. As we mentioned above the photoinduced distorted interactions $t_0^{AB}\neq t_0^{BC}\neq t_0^{CA}$ can be controlled by the amplitude and the polarization of the light intensity, therefore there is a possibility to obtain new interesting magnon phases in periodically driven intrinsic topological magnon insulators. Let us consider three different limiting cases of the photoinduced distorted interactions.
(i): $t_0^{AB}=0;~t_0^{BC}\neq t_0^{CA}\neq 0$, which leads to the magnon bands $\epsilon_{\vec k}^0=t_z$ and $$\begin{aligned}
\epsilon_{ \vec k}^{\pm}&=t_z \pm \frac{1}{\sqrt{2}}\sqrt{\lb t_0^{BC}\rb^2\big[ 1+\cos(2k_1)\big]+\lb t_0^{CA}\rb^2\big[ 1+\cos(2k_3)\big]}\end{aligned}$$
(ii): $t_0^{BC}=0;~t_0^{AB}\neq t_0^{CA}\neq 0$. The magnon bands in this case are given by $\epsilon_{\vec k}^0=t_z$ and $$\begin{aligned}
\epsilon_{ \vec k}^{\pm}&=t_z \pm \frac{1}{\sqrt{2}}\sqrt{\lb t_0^{AB}\rb^2\big[ 1+\cos(2k_2)\big]+\lb t_0^{CA}\rb^2\big[ 1+\cos(2k_3)\big]}\end{aligned}$$
(iii): $t_0^{CA}=0;~t_0^{BC}\neq t_0^{CA}\neq 0$. In this case we have $\epsilon_{\vec k}^0=t_z$ and $$\begin{aligned}
\epsilon_{ \vec k}^{\pm}&=t_z \pm \frac{1}{\sqrt{2}}\sqrt{\lb t_0^{AB}\rb^2\big[ 1+\cos(2k_2)\big]+\lb t_0^{BC}\rb^2\big[ 1+\cos(2k_1)\big]}\end{aligned}$$ In each case there are three magnon bands featuring one flat magnon band and two dispersive magnon bands, similar to the undriven topological magnon insulator in Fig. d. However, in the present case there is a possibility to obtain other interesting magnon phases different from the gapped topological magnon bands in the undriven system. For instance, cases (i)–(iii) realize pseudospin-1 Dirac-Weyl magnon cones or three-component bosons at ${\bf K}_1=\lb \pm \pi/2,\mp \pi/2\sqrt{3}\rb$, ${\bf K}_2=\lb 0,\mp \pi/\sqrt{3}\rb$, and ${\bf K}_3=\lb \pm \pi/2,\pm \pi/2\sqrt{3}\rb$ respectively, as indicated by red and green dots in Fig. \[klat\](b). The pseudospin-1 Dirac-Weyl magnon cones occur at the energy of the flat band $\epsilon_{{\bf K}_i}=t_z$ as shown in Fig. . Expanding the Floquet-Bloch magnon Hamiltonian in the vicinity of ${\bf K}_1$ yields $$\begin{aligned}
\mathcal H^{0}({\bf K}_1 +{\vec q}) \simeq t_z \text{I}_{3\times 3} \pm v_xq_x\lambda_x \mp v_yq_y\lambda_y,\end{aligned}$$ where $v_x= t_0^{BC}$ and $v_y =t_0^{CA}$. The ${\lambda}'s$ are the pseudospin-$1$ representation of the SU(2) Lie algebra $[\lambda_i, \lambda_j]=i\epsilon_{ijk}\lambda_k$, where $$\begin{aligned}
\lambda_x &=
\begin{pmatrix}
0& 0& e^{i\varphi}\\
0&0&0\\
e^{-i\varphi}&0&0
\end{pmatrix},~
\lambda_y =
\begin{pmatrix}
0& 0& 0\\
0&0&e^{-i\varphi}\\
0&e^{i\varphi}&0
\end{pmatrix},~\lambda_z =
\begin{pmatrix}
0& ie^{2i\varphi}& 0\\
-ie^{-2i\varphi}&0&0\\
0&0&0
\end{pmatrix}.\end{aligned}$$ Similar pseudospin-$1$ linear Hamiltonian can be obtained for the Dirac-Weyl magnon cones around ${\bf K}_2$ and ${\bf K}_3$.
**Conclusion**
We have presented a study of photoinduced topological phase transitions in periodically driven intrinsic topological magnon insulators. The main result of this report is that intrinsic topological magnon insulators in the kagomé ferromagnets can be driven to different topological phases with different Berry curvatures using photo-irradiation. Therefore, each topological phase is associated with a different sign of the thermal Hall conductivity, which results in a sign reversal of the magnon heat photocurrent. These topological transitions require no external magnetic field. Interestingly, we observed that by varying the light intensity, the periodically driven intrinsic topological magnon insulators can also realize synthetic gapless topological magnon semimetals with pseudospin-$1$ Dirac-Weyl magnon cones. We believe that our results should also apply to 3D topological magnon insulators. In fact, a 3D topological magnon insulator should also have a Dirac magnon cone on its 2D surface, which can be photo-irradiated to engineer a 2D topological magnon insulator in analogy to electronic systems [@dik]. Here, we have studied the off-resonant regime, when the driving frequency $\omega$ is larger than the magnon bandwidth $\Delta$ of the undriven system. In this regime, the Floquet sidebands are decoupled and can be considered independently. By lowering the driving frequency below the magnon bandwidth, the Floquet sidebands overlap, which results in photon absorption. In this limit the system would have several overlapping topological phases depending on the polarization of the light. In general, we believe that the predicted results in this report are pertinent to experiments and will remarkably impact future research in topological magnon insulators and their potential practical applications to photo-magnonics [@benj] and magnon spintronics [@magn; @benja].
**Methods**
**Neutral particle with magnetic dipole moment in an external electromagnetic field.** Two-dimensional topological magnon insulators (or Dirac magnons) can be captured by massive (or massless) (2+1)-dimensional Dirac equation near $\pm {\bf K}$. In general, a massive neutral particle with mass ($m$) couples non-minimally to an external electromagnetic field (denoted by the tensor $ F_{\mu\nu}$) via its magnetic dipole moment ($\mu$). In (3+1) dimensions, the system is described by the Dirac-Pauli Lagrangian [@bjo] $$\begin{aligned}
\mathcal L=\bar\psi(x)(i\gamma^\mu\partial_\mu-\frac{\mu_m}{2}\sigma^{\mu\nu} F_{\mu\nu}-m)\psi(x),\end{aligned}$$ where $\hbar=c=1$ has been used. Here $x\equiv x^\mu=(x^0,\vec x)$, $\bar\psi(x)=\psi^\dg(x)\gamma^0$, and $\gamma^\mu=(\gamma^0,\vec\gamma)$ are the $4\times 4$ Dirac matrices that obey the algebra \^,\^=2g\^, g\^=(1,-1,-1,-1),and \^=\[\^,\^\]=i\^\^,().
For the purpose of our study in this report, we consider an electromagnetic field with only spatially uniform and time-varying electric field vector $\vec{E}(\tau)$ (however, the resulting AC phase is valid for a general electric field $\vec{E}(\tau,\vec r)$). In this case, the corresponding Hamiltonian is given by $$\begin{aligned}
\mathcal H=\int d^3 x ~\psi^\dg(x)\big[\vec{\alpha}\cdot\big(-i\vec{\nabla}-i\mu_m\beta\vec{E}(\tau)\big)+m\beta\big]\psi(x),
\label{eqn21}\end{aligned}$$ where $\vec{\alpha}=\gamma^0\vec \gamma$, and $\beta=\gamma^0$.
In (2+1) dimensions, the Hamiltonian corresponds to that of 2D topological magnon insulators (near $\pm {\bf K}$) with magnetic dipole moment $\vec \mu=\mu_m\hat z$. In this case, the Dirac matrices are simply Pauli matrices given by $$\begin{aligned}
\beta=\gamma^0=\sigma_z,~\gamma^1=i\sigma_y,~\gamma^2=-i\sigma_x.\end{aligned}$$ The corresponding momentum space Hamiltonian in (2+1) dimensions now takes the form $$\begin{aligned}
\mathcal H=\int \frac{d^2k}{(2\pi)^2}~\psi^\dg(\bo, \tau)\mathcal H(\bo, \tau)\psi(\bo, \tau),\end{aligned}$$ where $$\begin{aligned}
\mathcal H(\bo, \tau)=\vec{\sigma}\cdot\big[\bo+\mu_m\big(\vec{E}(\tau)\times \hat z\big)\big]+m\sigma_z,~\text{with}~\vec{\sigma}=(\sigma_x,\sigma_y).
\label{eqn24}\end{aligned}$$ We can clearly see the time-dependent AC phase from the Hamiltonian in Eq. \[eqn24\].
**Magnonic Floquet-Bloch theory.** Periodically driven quantum systems are best described by the Floquet-Bloch theory. The magnonic version describes the interaction of light with magnonic Bloch states in insulating magnets. In this section, we develop this theory for the time-dependent magnon Hamiltonian Eq. in momentum space. We consider the time-dependent Schrödinger equation for the system $$\begin{aligned}
i\hbar\frac{d\ket{\psi(\vec{k},\tau)}}{d\tau}=\mathcal{H}(\vec{k},\tau)\ket{\psi(\vec{k},\tau)},\end{aligned}$$ where $\ket{\psi(\vec{k},\tau)}$ is the driven wave function. Due to the periodicity of the vector potential $\vec A(\tau)$, the driven Hamiltonian $\mathcal H(\vec{k},\tau)$ is also periodic and can be expanded in Fourier space as $$\begin{aligned}
\mathcal{H}(\vec{k},\tau)= \mathcal{H}(\vec{k}, \tau+T)=\sum_{n=-\infty}^{\infty} e^{in\omega \tau}\mathcal{H}_n(\vec{k}),\end{aligned}$$ where $\mathcal{H}_n(\vec{k})=\frac{1}{T}\int_{0}^T e^{-in\omega \tau}\mathcal{H}(\vec{k}, \tau) d\tau=\mathcal{H}_{-n}^\dg(\vec{k})$ is the Fourier component. The ansatz for solution to the Schrödinger equation can be written as $$\begin{aligned}
\ket{\psi_\alpha(\vec{k}, \tau)}&=e^{-i \epsilon_\alpha(\vec{k}) \tau}\ket{\xi_\alpha(\vec{k}, \tau)}=e^{-i \epsilon_\alpha(\vec{k}) \tau}\sum_{n=-\infty}^{\infty} e^{in\omega \tau}\ket{\xi_{_\alpha,n}(\vec{k})}\end{aligned}$$ where $\ket{\xi_\alpha(\vec{k}, \tau)}$ is the time-periodic Floquet-Bloch wave function of magnons and $\epsilon_\alpha(\vec{k})$ are the magnon quasi-energies. The corresponding Floquet-Bloch eigenvalue equation is given by $\mathcal{H}_F(\vec{k},\tau)\ket{\xi_{\alpha}(\vec{k},\tau)}=\epsilon_\alpha(\vec{k})\ket{\xi_{\alpha}(\vec{k},\tau)}$, where $\mathcal{H}_F(\vec{k},\tau)=\mathcal{H}(\vec{k},\tau)-i\partial_\tau$ is the Floquet operator. This leads to a time-independent Floquet eigenvalue equation $$\begin{aligned}
\sum_m [\mathcal H^{n-m}(\vec k) + m\omega\delta_{n,m}]\xi_{\alpha}^m(\vec{k})= \epsilon_\alpha(\vec{k})\xi_{\alpha}^n(\vec{k}),\end{aligned}$$ where $\mathcal H^{p}(\vec k)=\frac{1}{T}\int_0^T d\tau e^{-ip\omega \tau}\mathcal H^{p}(\vec k, \tau)$. The associated Bessel function integral is given by
$$\begin{aligned}
\frac{1}{T}\int_0^T d\tau e^{-ip\omega \tau}e^{iz\sin(\omega\tau)}e^{iz^\prime\sin(\omega\tau+\phi)}=e^{ip\arctan \lb \frac{z^{\prime}\sin(\phi)}{z+z^{\prime}\cos(\phi)} \rb}\mathcal{J}_p\lb \sqrt{z^2+z^{\prime 2}+2zz^\prime\cos(\phi)}\rb,\end{aligned}$$
where $\mathcal J_p$ is the Bessel function of order $p$. In Eq. we consider the zeroth order approximation corresponding to $p=0$.
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**Acknowledgements**
Research at Perimeter Institute is supported by the Government of Canada through Industry Canada and by the Province of Ontario through the Ministry of Research and Innovation.
**Author Contributions**
S. A. Owerre conceived the idea, performed the calculations, discussed the results, and wrote the manuscript.
**Additional Information**
**Supplementary information** accompanies this paper.
**Competing Interests**. I declare that the author has no competing interests as defined by Nature Research, or other interests that might be perceived to influence the results and/or discussion reported in this paper.
|
---
abstract: 'We explore the low-$\ell$ likelihood of the angular spectrum $C_\ell$ of masked CMB temperature maps using an adaptive importance sampler. We find that, in spite of a partial sky coverage, the likelihood distribution of each $C_{\ell}$ closely follows an inverse gamma distribution. Our exploration is accurate enough to measure the inverse gamma parameters along with the correlation between multipoles. Those quantities are used to build an approximation of the joint posterior distribution of the low-$\ell$ likelihood. The accuracy of the proposed approximation is established using both statistical criteria and a mock cosmological parameter fit. When applied to the WMAP5 data set, this approximation yields cosmological parameter estimates at the same level of accuracy as the best current techniques but with very significant speed gains.'
author:
- |
K. Benabed$^1$[^1], J.-F. Cardoso$^{1,2}$, S. Prunet$^1$ & E. Hivon$^1$\
$^1$Institut d’Astrophysique de Paris, 98bis Bd Arago, 75014 Paris, France.\
$^2$Laboratoire de Traitement et Communication de l’Information, LTCI/CNRS 46, rue Barrault, 75013 Paris, France.
bibliography:
- 'lowly.bib'
date: 'Accepted —. Received —; in original form '
title: 'TEASING: a fast and accurate approximation for the low multipole likelihood of the Cosmic Microwave Background temperature'
---
\[firstpage\]
cosmic microwave background – methods: data analysis – methods: statistical
Introduction
============
The CMB angular spectrum $C=\{C_\ell\}$ is a central quantity for conducting statistical inference based on CMB observations [@Bond:1987MNRAS.226..655B]. The high resolution of available [@Hinshaw:2008p6414] and forthcoming CMB observations [@bluebook] makes it necessary (at least in the case of partial sky coverage) to adopt a processing scheme in which the low-$\ell$ and high-$\ell$ parts of the data are processed independently [@Efstathiou:2006p4428]. This paper addresses the large scale part of the problem: inference regarding low multipoles based on a partial low-resolution CMB map.
After defining the problem of low-$\ell$ pixel-based likelihood and introducing some notations (Sec. \[sec:likelihood\]), we first show how to build a (large) set of $N$ *importance samples* of the angular spectrum such that all integrals of interest for statistical inference can be approximated by Monte-Carlo estimates (Sec. \[sec:building-samples\]). Based on those results, we propose in Sec. \[sec:lapprox\] a new approximation to the likelihood for partially observed low-resolution CMB maps. This approximation was initially built as part of the importance sampler but it turns out to be so accurate that it is of independent interest. This paper and the recent reference [@Rudjord:2008p4864] are similar in spirit but differ in the sampling method and in the proposed likelihood approximation.
Likelihood {#sec:likelihood}
==========
We recall some well-known facts about the likelihood of the angular spectrum of a CMB temperature map.
In the ideal case of noise-free, beam-free, full-sky map (represented by the vector $\mathbf{x}$ of pixels), one has direct access to the harmonic coefficients $a_{\ell m}$ of the sky. Assuming an isotropic Gaussian field, the empirical angular spectrum $\widehat{C}_{\ell}=\frac{1}{2\ell+1}\sum_{m}|a_{\ell m}|^{2}$ is a sufficient statistic for the data and their probability distribution takes the factorized form [@BJK:2000ApJ...533...19B]: $$\label{eq:likely-factorized}
p(\mathbf{x}|C)
\propto
\prod_{\ell\geq0}\exp-\frac{2\ell+1}{2}
\left(\frac{\hat C_\ell }{C_{\ell}}+\log C_{\ell}\right)
.$$ In the case of a flat prior $p(C)$, expression (\[eq:likely-factorized\]) combined with Bayes rule $p(C|\mathbf{x})=p(\mathbf{x}|C)p(C)/p(\mathbf{x})$ reveals that, given $\mathbf{x}$, the angular spectrum $C$ is distributed as a product of inverse gamma densities: $$\begin{aligned}
\label{eq:factorpost}
p(C|\mathbf{x}) & = \prod_{\ell}i\Gamma(C_{\ell};\alpha_{\ell},\beta_{\ell})\\
i\Gamma(x;\alpha,\beta) & \equiv \frac{\beta^{\alpha}}{\Gamma(\alpha)}\ x^{-\alpha-1}\ e^{-\frac{\beta}{x}},
\label{eq:defigamma}\end{aligned}$$ with parameters $\alpha_{\ell}=(2\ell-1)/2$ and $\beta_{\ell}=(2\ell+1)\hat C_\ell/2$.
Such a factorization does not hold when only a fraction of the sky is observed (or has to be ignored because of excessive contamination by foregrounds), or when the stationary CMB is contaminated by non stationary noise [@Gorski:1994ApJ...430L..85G; @Tegmark:1997PhRvD..55.5895T]. However, for small sky masks and/or small deviations from stationarity, deviations from the factorized form (\[eq:likely-factorized\]) are expected to be small, suggesting the new likelihood approximation developed in Sec. \[sec:lapprox\].
**Pixel-based likelihood.** We turn to the actual case of interest: partial sky coverage, presence of independent additive Gaussian noise, low-pass effect of a beam. The data set, represented by an ${\ensuremath{{N_{\text{pix}}}}}\times 1$ vector $\mathbf{x}$ of pixel values, can no longer be losslessly compressed into a sufficient spectral statistic $\hat C_\ell$. Rather, one must use the plain Gaussian density: $$p(\mathbf{x}| \mathbf{R})
=
|2\pi\mathbf{R}|^{-1/2}e^{-\frac{1}{2}\mathbf{x}^{T}\mathbf{R}^{-1}\mathbf{x}}
\label{eq:likely-general}$$ where the covariance matrix $\mathbf{R}$ of $\mathbf{x}$ has contributions from the CMB signal and from noise. For two pixels $i$ and $j$ with angular separation $\theta_{ij}$, the CMB part of the covariance matrix has an $(i,j)$ entry given by [@BJK:2000ApJ...533...19B] $$\sum_\ell \frac{2\ell+1}{4\pi}W_{\ell}C_{\ell}P_{\ell}(\cos\theta_{ij})$$ where $P_{\ell}$ is the Legendre polynomial of order $\ell$ and where the window function $W_{\ell}$ can represent *e.g.* the spectral response of an azimuthally symmetric beam, or more generally the convolution of the signal with any azimuthally symmetric kernel. Hence, we ignore the complications due to an anisotropic beam as well as the presence of residual foreground contaminants.
The noise part of the covariance matrix could take any form but, in this work, it is taken to correspond to an isotropic noise with angular spectrum $N_\ell$. We can thus define a total angular spectrum $D_{\ell}$ $$\label{eq:CtoD}
D_{\ell} = W_{\ell}C_{\ell}+N_{\ell}$$ which is unambiguously related to $C_\ell$ since the beam $B_\ell$ and the noise spectrum $N_\ell$ are assumed to be known.
**Free parameters.** In practice, we consider a more restricted model for the covariance matrix of the observed pixels. First, the adjustable multipoles are restricted to a range ${\ensuremath{{\ell_{\text{min}}}}}\leq\ell\leq{\ensuremath{{\ell_{\text{max}}}}}$ while other multipoles are kept at constant values. Second, we only consider uncorrelated noise with zero mean and variance $\sigma^2$ per pixel. It contributes a term $\sigma^2\delta_{ij}$ to $\mathbf{R}$ and corresponds to a flat angular spectrum $N_\ell = \sigma^2\,/ \Omega_\mathrm{pix}$ if all pixels have the same area $\Omega_\mathrm{pix}$. Then, the covariance matrix of $\mathbf{x}$ as a function of $\mathbf{D}=\{D_\ell\}_{\ell={\ensuremath{{\ell_{\text{min}}}}}}^{\ell={\ensuremath{{\ell_{\text{max}}}}}}$ is spelled out as: $\mathbf{R}(\mathbf{D}) = \mathbf{R}^\text{var} (\mathbf{D}) + \mathbf{R}^\text{cst}$ with $$\begin{aligned}
\label{eq:covmatD}
\mathbf{R}_{ij}^\text{var}(\mathbf{D}) &=
\sum_{\ell={\ensuremath{{\ell_{\text{min}}}}}}^{\ell={\ensuremath{{\ell_{\text{max}}}}}} \frac{2\ell+1}{4\pi}(D_\ell-N_\ell)P_{\ell}(\cos\theta_{ij}) \\
\mathbf{R}_{ij}^\text{cst} &=
\sum_{\ell\ \mathrm{fixed}}
\frac{2\ell+1}{4\pi} W_\ell C_\ell P_{\ell}(\cos\theta_{ij}) \ +\ \sigma^2 \delta_{ij}
\label{eq:covmatDfix}\end{aligned}$$
**Priors and posterior distributions.** In all the following, the prior distribution on $\mathbf{D}$ is taken to be flat for $D_\ell\geq N_\ell$. At all angular frequencies such that $W_\ell C_\ell \gg N_\ell$ (figure \[fig:wlnlcl\] illustrates the values used in this paper), this is almost identical to a flat prior on the positive values of $C_\ell$. The posterior distribution of $\mathbf{D}$ given the data $\mathbf{x}$ is $$\pi(\mathbf{D})=
p(\mathbf{D}|\mathbf{x})
\propto
p(\mathbf{x}| \mathbf{R}(\mathbf{D})) \ \prod_{\ell={\ensuremath{{\ell_{\text{min}}}}}}^{\ell={\ensuremath{{\ell_{\text{max}}}}}} \mathbf{1}(D_\ell \geq N_\ell)$$ where $p(\mathbf{x}| \mathbf{R}(\mathbf{D}))$ is evaluated using eqs (\[eq:likely-general\]), (\[eq:covmatD\]) and (\[eq:covmatDfix\]).
**About noise and regularization.** On a cut sky, the CMB part of the covariance matrix may be poorly conditioned with a trough in its eigenvalue spectrum corresponding to those modes which are mostly localized in the cut. In this case, it is customary [@Eriksen:2006p3364; @Hinshaw:2007ApJS..170..288H] to add a very small amount of noise to the data and to add the corresponding contribution to the covariance matrix as in eq. (\[eq:covmatDfix\]). Another reason for adding uncorrelated noise is to cover spurious noise correlation possibly introduced when the observed sky map is downgraded and to simplify the noise structure [@Dunkley:2008p3305]. See figure \[fig:wlnlcl\] for the values used in our experiments.
![The WMAP best fit spectrum $C_\ell$ (black solid line), the noise spectrum $N_\ell$ for a variance of $\sigma^2=1\mu
K^2$/pixel (black dashed line), and the angular spectra $W_\ell
C_\ell$ (dot dashed) and $W_\ell C_\ell + N_\ell$ (solid) when $W_\ell$ is the window function $W_\ell$ of eq. (\[eq:archcos\]) (green) or the WMAP Gaussian beam (red). Spectra are rescaled by $\ell(\ell+1)/2\pi$ for clarity. []{data-label="fig:wlnlcl"}](figs/wlclnl.pdf){width="1\columnwidth"}
Another possibility is regularization by projection onto the most significant eigen-vectors of the covariance matrix [@BJK:2000ApJ...533...19B] but this possibility is not considered here.
Building a sample of the low-$\ell$ posterior with importance sampling {#sec:building-samples}
======================================================================
This section reports on the construction of *importance samples* of the $C_{\ell}$ under their joint posterior for two data sets. The principle of importance sampling is first briefly recalled in section \[sec:ISintro\]; our specific technique (an adaptive variant) is described in section \[sub:adapt:algo\] and applied to a synthetic CMB cut sky map (sec. \[sec:synthmap\]) and to the official WMAP5 low resolution map(sec. \[sec:samplewmap5\]).
Importance sampling {#sec:ISintro}
-------------------
Importance sampling is a well established technique to explore a probability distribution when no method for directly sampling from it is available (the well known VEGAS algorithm [@Lepage:1978p4691] for instance, is based on importance sampling). Consider estimating the expectation $E f(x)=\int f(x)\pi(x)dx$ of some function $f$ of $x$ when the random variable $x$ is distributed under $\pi$. If $x_i, i=1,N$ are $N$ samples of $x$, then $E f(x)$ can be estimated by the sample average $\frac1N \sum_i f(x_i)$. In contrast, importance sampling relies on samples $x_i$ distributed under a *proposal distribution* $g$ not necessarily equal to $\pi$. If the support of $g$ includes the support of $\pi$ then $$E f = \int f(x) \pi(x) dx = \int f(x)\frac{\pi(x)}{g(x)} g(x) dx$$ so that, if the samples $x_i$ are distributed under $g$, then $Ef$ is estimated without bias by $$\frac1N \sum_{i=1}^N w_i f(x_i)
\quad\text{where}\quad
w_i = w(x_i) \equiv \frac{\pi(x_i)}{g(x_i)}$$ The factors $w_i$ are called *importance weights*.
Monte-Carlo integration reaches its maximum efficiency when the samples are drawn independently under a proposal distribution $g$ which is identical to the target distribution $\pi$. While MCMC methods try to draw from the target distribution $\pi$, they do not build independent samples; in contrast, importance sampling (usually) relies on independent draws from an approximate distribution $g$ and corrects the discrepancy using importance weights $w_i$. Therefore, importance sampling should outperform MCMC methods whenever independent samples can be drawn from a proposal distribution which is “close enough” to the target.
The agreement between target and proposal distributions can be measured by the Kullback-Leibler divergence $$K(\pi|g)\equiv \int \log\frac{\pi(x)}{g(x)} \, \pi(x)\, dx ,$$ which is often remapped as the so-called *perplexity criterion*: $ {\ensuremath{\mathcal{P}}}(\pi|g)\equiv \exp - K(\pi|g) $ so that perfect agreement is reached when ${\ensuremath{\mathcal{P}}}=1$. Another criterion is the *effective sample size* ($ESS$) of an importance sample: $$ESS=\frac
{\left(\sum_i w_i\right)^2 }
{\sum_{i} w_i^2 }$$ If the proposal matches the target perfectly, then $ESS=N$, otherwise it is smaller than the number of importance samples. The effective sample size is directly related to the variance of the MC estimates.
Importance sampling is well fitted to the problem at hand for at least two reasons: ease of parallelization and availability of a good proposal distribution.
Parallelization is a strong requirement due to the high computational cost of CMB studies. We are planning to sample a 30- to 40-dimensional space, and the computation of the likelihood for a given angular spectrum costs about 5 seconds for ${\ensuremath{{\ell_{\text{max}}}}}=48$ and ${\ensuremath{{N_{\text{pix}}}}}=3072$ on a typical 2GHz CPU. Since importance sampling can be trivially parallelized, it makes it straightforward to take full advantage of CPU clusters. For instance, computing $10^5$ samples would take about 4 days on a single CPU but is reduced to mere hours on a cluster. The Markov-Chain Monte-Carlo algorithm cannot be parallelized as easily. Indeed, to be able to mix different parallel chains, one has to ensure that they have correctly converged [@rosenthal00], which can be a difficult task in 30 to 40 dimensions.
Regarding the proposal distribution, one can draw inspiration from the noise-free, full-sky case (\[eq:factorpost\]) since a mask hiding less than $20\%$ of the sky and a high signal to noise situation are expected to modify it only slightly[^2]. Indeed, as demonstrated below, a product of independent inverse gamma distributions turn out to be a very efficient proposal distribution, provided it is correctly tuned. Such a tuning is achieved via an *adaptive importance sampling*, as explained next.
An adaptive importance sampling algorithm {#sub:adapt:algo}
-----------------------------------------
Importance sampling is efficient only if the proposal distribution is close enough to the target, an objective which may be difficult to reach in large dimensions (sampling angular spectra in the range $0\leq\ell\leq 40$ qualifies as large problem). To tackle this complexity, we resort to *adaptive importance sampling* which consists in running a sequence of importance runs in which the proposal distribution is improved at each run based on the results of previous runs. A more detailed description of adaptive importance sampling (based upon the PMC algorithm from [@Cappe:2007p3640]) in the context of cosmology can be found in [@darren2009].
**General scheme.** The general scheme, based on a parametric family of proposal distributions $g(\mathbf{y} ;\theta)$, is as follows:
1. \[algo:init\] Start with the best available guess of $\theta$ for the parameters of the proposal distribution.
2. \[algo:debut\] Sample under $g(\mathbf{y}; \theta)$. Compute and store the importance weights.
3. \[algo:reest\] Re-estimate $\theta$ so that $g(\mathbf{y};
\theta)$ best matches the current sample set.
4. If the (estimated) perplexity ${\ensuremath{\mathcal{P}}}(\pi(\mathbf{y}) | g(\mathbf{y};
\theta))$ is high enough (*e.g.* above $0.5$) or if it has not changed significantly during the last iterations, exit to \[algo:fin\]. Otherwise, go to \[algo:debut\] for another importance run with the re-estimated parameters.
5. \[algo:fin\] Use the last value of $\theta$ for a large final importance sampling run.
**Sampling angular spectra.** In our experiments, we sample the total angular spectrum, that is, ${\mathbf{y}=\mathbf{D}} = \{ D_\ell\}_{\ell=\ell_\mathrm{min}}^{\ell=\ell_\mathrm{max}}$ and use independent inverse gamma distributions for the proposal: $$g( \mathbf{D} ; \theta ) = \prod_{\ell=\ell_\mathrm{min}}^{\ell=\ell_\mathrm{max}}
i\Gamma( D_\ell ;\alpha_\ell,\beta_\ell) .$$ Hence we must adapt a vector $\theta=\{\alpha_\ell,
\beta_\ell\}_{\ell=\ell_\mathrm{min}}^{\ell=\ell_\mathrm{max}}$ of $2(\ell_\mathrm{max}-\ell_\mathrm{min}+1)$ parameters. As a starting point at step \[algo:init\], we use $$\alpha_\ell =\frac{(2\ell+1)}{2}{\ensuremath{{f_{\text{sky}}}}}-1, \qquad
\beta_\ell =\frac{(2\ell+1)}{2}{\ensuremath{{f_{\text{sky}}}}}D_\ell^{ML}$$ where $D_{\ell}^{ML}$ is the maximum likelihood estimate of the angular spectrum. At step \[algo:reest\], parameters $\alpha_\ell$ and $\beta_\ell$ are re-estimated at their maximum likelihood values (see appendix).
The target density $\pi(\mathbf{D})$ is the posterior distribution of $\mathbf{D}$ when the prior distribution of $\mathbf{D}$ is flat. Hence, it is proportional to the likelihood.
In the two examples presented below, this iterative algorithm reached a perplexity above $0.6$ after the first step of 50k samples and a 500k samples set was produced during the final sampling phase.
Synthetic map {#sec:synthmap}
-------------
![The synthetic CMB map used at sec. \[sec:synthmap\].[]{data-label="fig:syntmap"}](figs/synt_map){width="1\columnwidth"}
We first describe the results of adaptive importance sampling runs on a synthetic CMB map. The map is prepared at resolution ${\ensuremath{{N_{\text{side}}}}}=16$ from the WMAP5 best fit power spectrum [@Dunkley:2008p3305] using HEALPix [@2005ApJ...622..759G]. To avoid aliasing small scale power into large scale modes, the map is smoothed prior to down-sampling using a synthetic window function $w_{\ell}$: $$\label{eq:archcos}
W_{\ell}=\left\{
\begin{array}{lc}
1 & 0\leq \ell \leq 40\\
\frac{1+\cos\left((\ell-40)\pi/8\right)}{2} & 40\leq \ell\leq 48 \\
0 & 48 \leq \ell
\end{array}\right.$$ which is used to explore the posterior of $C_{\ell}$ up to $\ell=40$.
The posterior of the power spectrum is given by the likelihood described in Eq. (\[eq:likely-general\]), with a flat prior. The Galactic region is excluded using the WMAP5 mask, hiding $18\%$ of the sky. The map is shown in figure \[fig:syntmap\]. A $1\mu$K/pixel noise is taken into account in the likelihood, but no noise is actually added to the map. This level should not affect our results as it is much lower than $\Omega_\mathrm{pix} C_{40}$ (see figure \[fig:wlnlcl\]).
{width="1\columnwidth"}{width="1\columnwidth"}\
{width="1\columnwidth"}{width="1\columnwidth"}
.
We build a sample of the posterior of the masked map using the adaptive importance sampling algorithm described above. We only explore $\ell=2$ to $40$, the other modes ($\ell=0,1$ and $41\leq\ell\leq 48$) being held constant to the ML estimate.
The initial proposal is given by the product of independent inverse gamma distributions, as described in \[sub:adapt:algo\], centered at $D_{\ell}^{ML}$ with a width given by an effective sky coverage equal to ${\ensuremath{{f_{\text{sky}}}}}\times0.98$ to ensure that the initial proposal is wide enough.
Only one adaptation step was needed. It took about $58$min on 80 2 GHz CPUs to produce the first 50k samples (about $6$sec for each likelihood evaluation, taking into account all overheads). The final 500k samples run took $6$ hours and $21$ minutes on 120 2 GHz CPU (about $5.5$sec for each likelihood evaluation, taking into account all overheads). The adaptive algorithm behaved very well: the first step reached ${\ensuremath{\mathcal{P}}}=0.68$ while the second run hit ${\ensuremath{\mathcal{P}}}=0.93$. This last run had an effective sample size $ESS=437029$, *i.e.* a ratio $ESS/N=0.874$.
Figures. (\[fig:cutsky:marginal\])-(\[fig:cutsky:cov\]) give an overview of the results. First, looking at the 1D marginal distributions, figure (\[fig:cutsky:marginal\]) shows a few marginals ($\pi_\ell$) and their best inverse gamma fits. The inverse gamma model is seen to account very well for both the tails and the mode of the distribution, in line with the high perplexity reached in the last iteration. This agreement validates *a posteriori* the adaptive approach. On this synthetic map, at least, the marginals follow closely an inverse gamma distribution.
The peaks of the marginals and an effective sky coverage at multipole $\ell$, denoted $f_{\ell}$, are obtained by inverting $$\begin{aligned}
\alpha_{\ell} & = & \frac{(2\ell+1)}{2}\ f_{\ell}-1\label{eq:alphaL}\\
\beta_{\ell} & = & \frac{(2\ell+1)}{2}\
f_{\ell}\left(W_{\ell}{{C_{\ell}^{\text{peak}}}}+N_{\ell}\right),\label{eq:betaL}\end{aligned}$$ Both quantities are shown in figure (\[fig:cutsky:clfl\]). The $C_{\ell}^{peak}$ and $C_{\ell}^{ML}$ discrepancy is small; it is below the percent order, albeit with a few modes disagreeing by at most $3\%$. The effective sky coverage, however, is quite different from ${\ensuremath{{f_{\text{sky}}}}}$. Its behaviour indicates a transition between scales that are not affected significantly by the cut, and scales that are smaller than the cut, so that their deficit of modes is given by ${\ensuremath{{f_{\text{sky}}}}}$. Our resolution is probably not good enough to reach this regime.
One would expect some discrepancy between the ${{C_{\ell}^{\text{peak}}}}$ and the ML estimate. Indeed, since the cut induces correlation between scales, there is no reason for the peak of the posterior to be identical to the peak of the marginals in each direction. The small discrepancy can only be explained by a low level of correlation between the $C_{\ell}$s, so that the peak of the marginals is close to the joint peak. As a first estimate of the correlation, figure. (\[fig:cutsky:cov\]) shows the correlation matrix measured on our sample $$\label{eq:V:def}
[V]_{\ell,\ell'} \equiv \mathrm{Corr} \left( C_{\ell}, C_{\ell'}\right) .$$ In this figure, the diagonal of the matrix is removed so as not to dominate the off diagonal terms. Those exhibit a pattern below the $6\%$ level. Most of the correlation is located around $\ell=12$, and the correlation seems to extend significantly for about $6$ modes off the diagonal.
Several checks can be performed to assess the accuracy of this matrix. First, the effective sample size allows us to estimate the error on the matrix measurement to be of the order of $0.15\%$, which is well below the observed correlation pattern. One can also measure the correlation matrix on the results of the first iteration of the adaptive algorithm, which provides us with an independent exploration of the posterior. The noise was much higher (with a level, according to the $ESS$ of this run of about $0.6\%$), but the pattern observed on figure (\[fig:cutsky:cov\]) is easily recovered. Finally, we checked on a *full sky* run that no correlation pattern is visible.
![Top panel: angular spectra. Blue dashed line: the power spectrum used to synthesize the map; red: the ML estimate $C_{\ell}^{ML}$; black dots: ${{C_{\ell}^{\text{peak}}}}$. The error bars are $68\%$ limits obtained from the inverse gamma fits for the marginals. Bottom panel: Sky coverage $f_{\ell}$. Black line: effective coverage $f_{\ell}$; the blue dashed line shows ${\ensuremath{{f_{\text{sky}}}}}=N_{mask} /{\ensuremath{{N_{\text{pix}}}}}$.[]{data-label="fig:cutsky:clfl"}](figs/synt_clandfl){width="1\columnwidth"}
![The correlation matrix $V$ for $C_{\ell}$ (see Eq. (\[eq:V:def\])) with the diagonal removed. Most of the correlation is located around $\ell=12$ and extends only to a few neighboring modes. The correlation is always below the $6\%$ level.[]{data-label="fig:cutsky:cov"}](figs/synt_corr){width="1\columnwidth"}
WMAP5 map {#sec:samplewmap5}
---------
We perform a similar experiment using the WMAP map distributed along with the five year WMAP likelihood code found on the Lambda website [^3]. The setting is slightly different, since the window function is a $9.18^{\circ}$ Gaussian beam, cutting much more high frequency power than the window function (\[eq:archcos\]) (see figure \[fig:wlnlcl\]). Therefore, only the range $2\leq \ell\leq 30$ is explored here, with the other multipole powers held constant at their ML values. As done in the WMAP likelihood code, a $1\mu$K/pixel noise is added to the data and to the model. We take care of adding the specific noise realization used in the likelihood code. Indeed, with the beam used, the signal to noise at $\ell=30$ is only $\sim14$ and our tests have shown a small dependency of the value of the higher $C_\ell$s on the noise realization.
As in the previous run, only one adaptation step turns out to be needed. It took $32$ minutes on 120 CPUS for 50k samples, while the second and final run produced 500k samples in $5$ hours and $19$ minutes. The first iteration reached ${\ensuremath{\mathcal{P}}}=0.48$, the second one ${\ensuremath{\mathcal{P}}}=0.96$ and an effective sample size $ESS=457600$ ($ESS/N=0.92$).
The results are generally similar to those reported in section \[sec:synthmap\]. We do not show more 1D marginal plots, but present the recovered ${{C_{\ell}^{\text{peak}}}}$ and $f_{\ell}$ (figure \[fig:WMAP5:clfl\]), as well as the correlation matrix (figure \[fig:WMAP5:cov\]). The ${{C_{\ell}^{\text{peak}}}}$ and the ML estimates are somewhat similar to the WMAP5 power spectrum, with a small discrepancy also observed by [@Eriksen:2006p3364] using Gibbs sampling and in [@Rudjord:2008p4864] (zooming on their figure 5). At any rate, the discrepancy is always within the $C_{\ell}$ error bars.
The effective coverage $f_{\ell}$ is similar to the one reported in section \[sec:synthmap\], with a transition from $1$ to ${\ensuremath{{f_{\text{sky}}}}}$ but differs in some details, indicating that it is not only a function of the mask, but also of the actual data set.
Finally, figure (\[fig:WMAP5:cov\]) shows the correlation matrix. It exhibits structures similar to those in figure (\[fig:cutsky:cov\]). As for the $f_{\ell}$, the differences between figures \[fig:WMAP5:cov\] and \[fig:cutsky:cov\] indicate that the correlation matrix does not depend only on the mask.
![Same as figure \[fig:cutsky:clfl\] for the WMAP5 data set. The dashed blue on the top panel of the top panel line now is the WMAP empirical spectrum.[]{data-label="fig:WMAP5:clfl"}](figs/WMAP_clandfl){width="1\columnwidth"}
![Same as figure \[fig:cutsky:cov\] for the WMAP5 data set.[]{data-label="fig:WMAP5:cov"}](figs/WMAP_corr){width="1\columnwidth"}
Approximating the low-$\ell$ likelihood {#sec:lapprox}
=======================================
For both data sets considered in previous section, the posterior distribution of the total angular spectrum $D_\ell$ revealed similar and striking features: the marginals are very well approximated by inverse gamma distributions and there is a weak correlation between multipoles (below the $10\%$ level). Since we used a flat prior, these findings suggest that a copula approximation to the likelihood should be quite accurate (in addition to being fast, by design). This approach is somewhat similar to what has been proposed by [@BJK:2000ApJ...533...19B] and implemented at low-$\ell$ in [@Rudjord:2008p4864] and at high-$\ell$ in [@Hamimeche:2008p3608]. It differs in that, instead of offset log normal (as in [@BJK:2000ApJ...533...19B]), spline approximation [@Rudjord:2008p4864] or Taylor expansion inspired approximation [@Hamimeche:2008p3608], we use inverse gamma cumulative functions for Gaussianization.
Copula approximation {#sec:copula-approximation}
--------------------
A good approximation formula must at least reproduce the inverse gamma marginals, and the observed level of correlation. A generic way of building multivariate distributions with specified marginals and some correlation is provided by *copula models* [@sklar59].
**The copula model.** Denote ${\ensuremath{{\cal{N}}}}^{(d)}(\cdot; \mu, M)$ the $d$-variate Gaussian density with mean $\mu$ and covariance matrix $M$. Consider a set of zero-mean unit-variance Gaussian variables $G_{\ell}$ with density ${\ensuremath{{\cal{N}}}}^{(d)}(G_{\ell};0, M_{G})$ where $M_G$ has only $1$’s on the diagonal and possibly non-zero off diagonal terms. Consider those transformed variables $D_\ell=D_\ell(G_\ell)$ which have an inverse gamma distribution with parameters $\alpha_\ell$ and $\beta_\ell$, that is, $G_\ell$ and $D_\ell$ are related by $$\label{eq:repar:gauss:integ}
{\ensuremath{{\cal{N}}}}(G_{\ell};0,1)\
dG_{\ell}=i\Gamma(D_{\ell};\alpha_{\ell},\beta_{\ell})\ dD_{\ell} .$$ The distribution of $D_\ell$ is then easily seen to be $$\label{eq:approx:def}
\tilde{\pi}(D_{\ell})
\equiv
\prod_{k}i\Gamma(D_{k};\alpha_{k},\beta_{k}).
\,
\frac
{ {\ensuremath{{\cal{N}}}}^{(d)} (G_{\ell};0, M_{G})}
{\prod_{k}{\ensuremath{{\cal{N}}}}^{(1)} (G_{k} ;0, 1)}
.$$ Distribution (\[eq:approx:def\]) is called the *copula approximation*. It belongs to a parametric model with $2d+d(d-1)/2$ parameters: each of the $d$ multipoles requires a pair $(\alpha_\ell,\beta_\ell)$ for the marginal distribution and the correlation matrix $M_G$ depends on $d(d-1)/2$ free parameters.
**Two properties.** Probability distributions of the form (\[eq:approx:def\]) enjoy two nice properties which readily follow from their construction. First, the marginal distribution of each $D_\ell$ remains an inverse Gamma regardless of the correlation level (which is independently controlled by the matrix $M_G$). Second, marginalization over any subset of $D_\ell$ is readily achieved by removing the corresponding rows and columns of matrix $M_G$.
**Gaussianization.** Evaluating the copula density (\[eq:approx:def\]) requires explicit Gaussianization, that is mapping $D_\ell$ to $G_\ell$. This is easy since relation (\[eq:repar:gauss:integ\]) implies that $$G_{\ell}
\equiv
\mbox{c}N^{-1}( \mbox{c}i\Gamma(D_{\ell};\alpha_{\ell},\beta_{\ell})),
\label{eq:repar:gauss:def}$$ where $\mbox{c}i\Gamma(\cdot; \alpha, \beta)$ denotes the cumulative distribution function (CDF) of the inverse gamma distribution and $\mbox{c}N^{-1}$ is the inverse CDF (or quantile function) of the standard normal distribution, sometimes called the *probit* function. The former is $$\mbox{c}i\Gamma(x;\alpha,\beta)\equiv\int_{0}^{x}i\Gamma(t;\alpha,\beta)\
\mbox{d}t=\Gamma\left(\alpha,\beta/x\right)/\Gamma(\alpha).$$ while the latter, if missing from a statistical library, can be computed as $\mbox{c}N(x)^{-1} = \sqrt{2} \mbox{erf}^{-1}(2x-1)$ with $\mbox{erf}(y) = \frac{2}{\sqrt{\pi}}\int_{0}^{y}\exp(-t^{2})\
\mbox{d}t$.
**Speed.** Copula evaluation is very fast. Using a custom code to compute the inverse error function, and the free GSL library[^4] for the gamma and cumulative gamma distribution, we can compute about $18000$ samples per second while the pixel-based likelihood needs about $5.5$ seconds per sample on the same computer within the same setting (*i.e.* same overheads). Moreover, one can also sample directly from the copula by first drawing $G_{\ell}$ under to their multivariate Gaussian distribution and then invert Eq. (\[eq:repar:gauss:def\]) to get the $D_{\ell}$ values.
**Learning the copula model.** Learning the $2d+d(d-1)/2$ parameters of a copula models from (importance) samples of $D_\ell$ is straightforward. In a first step, one estimates for each $\ell$, the inverse gamma parameters $(\alpha_\ell, \beta_\ell)$ by maximum likelihood (see appendix \[sec:mlegamma\]). In a second step, the samples are Gaussianized via eq. (\[eq:repar:gauss:def\]) using the estimated values of $(\alpha_\ell,\beta_\ell)$. Finally, matrix $M_G$ is plainly estimated as the sample correlation matrix of the Gaussianized samples.
**Significance of correlation.** Given a copula model $\tilde\pi$ with correlation matrix $M_G$, there is a simpler copula model with the same marginals but without correlation, that is, with $M_G=I$. This model is denoted $\tilde\pi_0$ and called the *uncorrelated model* which, of course, is not as accurate as $\tilde\pi$. Since $\tilde \pi_0$ and $\tilde \pi$ are Gaussian distributions, the loss can be quantified exactly thanks to a Pythagorean property of the KLD which yields $$\label{eq:pythasym}
K(\pi | \tilde \pi_0 ) = K(\pi | \tilde{\pi}) + K(\tilde{\pi} | \tilde\pi_0 ) .$$ It shows that the mismatch $K(\pi | \tilde \pi_0)$ of the uncorrelated approximation to the posterior is larger than the mismatch $K(\pi|\tilde \pi)$ of the regular copula by a positive term $K(\tilde{\pi}|\tilde\pi_0 )$. This term can be computed in closed form: $$\label{eq:kullcorr}
K(\tilde{\pi}|\tilde \pi_0)=-\frac12 \log\det M_G$$ which is positive unless $M_G=I$ and readily gives a measure of the price to pay for ignoring correlation.
Validation : first results
---------------------------
We first look at some self-consistency results when learning a copula model from the importance samples obtained from the WMAP data set discussed in section \[sec:samplewmap5\].
**High perplexity.** The first important thing to report is that, on the perplexity scale, the copula approximation is remarkably good: we reach ${\ensuremath{\mathcal{P}}}(\pi|\tilde{\pi})=0.99$ on a 500k simulation sample using estimates of the ${{C_{\ell}^{\text{peak}}}}$, $f_{\ell}$ and $M_G$ obtained on the same sample. As a simple cross-validation test, we split the 500k sample into two subsets of equal size, re-estimate the copula parameters on the first subset and compute the perplexity using the second subset. We find a negligible decrease in perplexity of about $5\times10^{-4}$.
Thus, the copula approximation appears to work extremely well on this data set. Still, one should look further than a single number. This section looks into more details of the approximation.
**Gaussianization.** Even though the marginals were found to be well approximated by inverse gamma distributions, the Gaussianized importance samples may show some small hints of…non Gaussianity. Indeed, for each $G_\ell$, we computed the skewness, the kurtosis and the Kullback divergence to a standard Gaussian. See figure \[fig:flcor\] for the WMAP data set (a similar plot can be obtained on the other data set). The plot shows a small deviation from Gaussianity showing that the target densities are not *exactly* inverse gamma distributed. In addition, those non Gaussian indicators degrade with $\ell$ and are correlated with $f_\ell$. Since the latter measures deviation from the full-sky case, this is not unexpected.
![$f_{\ell}$, cumulants and Kullback divergence of $G_{\ell}$ exhibit some correlation. From top to bottom, $f_\ell$ (and ${\ensuremath{{f_{\text{sky}}}}}$), skewness, kurtosis and Kullback divergence between the marginals and standard normal. Note that the Kullback divergence is estimated from an histogram. The last two panels have their ordinates downwards to better show the correlation. Error bars are measured on 500 Gaussian simulations of size $ESS$ $(=457600)$[]{data-label="fig:flcor"}](figs/gauss_div){width="1\columnwidth"}
**Correlation matrices.** By design, the copula correctly predicts the correlation matrix of the *Gaussianized* variables but it is not necessarily accurate as a predictor of the correlation matrix $V$ of $C_\ell$. Here, we check that $V$ is well predicted by the covariance matrix of the copula model, denoted $\tilde V$. Matrix $V$ is estimated as described before (based on an importance sample); matrix $\tilde V$ is obtained from the same importance samples, re-weighed by $\tilde \pi/\pi$. The results are displayed on figure \[fig:diff:corrBis\] and show an excellent agreement, with small and evenly distributed errors.
------------------------------------------------- ---------------------------------------------------------------- ------------------------------------------------------------------------
{width="0.63\columnwidth"} {width="0.63\columnwidth"} {width="0.63\columnwidth"}
------------------------------------------------- ---------------------------------------------------------------- ------------------------------------------------------------------------
Perplexities.
-------------
We briefly report on the relative perplexity and Kullback divergence between the posterior and its approximations on the WMAP5 data set. Some results are reported in table \[tab:perp\].
Approximation Perplexity Kullback ($\times 10^{-3}$)
----------------------------------- ------------ -----------------------------
Copula $\tilde\pi$ 0.991 $8.6$
Uncorrelated copula $\tilde\pi_0$ 0.965 $35.2$
Uncorrelated last run 0.956 $45.0$
Naive $\tilde\pi_\mathrm{naive}$ 0.779 $249.6$
LogNormal 0.191 $1655.3$
: Perplexities. See text.[]{data-label="tab:perp"}
Since the Gaussianized variables were found to be weakly correlated, it may be tempting simply to ignore this correlation and to resort to the *uncorrelated* approximation $\tilde{\pi}_{0}$ defined at sec. \[sec:copula-approximation\]. In this case, the fit is slightly degraded: we measure ${\ensuremath{\mathcal{P}}}(\pi|\tilde{\pi}_{0}) = 0.97$, in line with the perplexity obtained after the last step of the adaptive importance run (${\ensuremath{\mathcal{P}}}=0.96$, sec. \[sec:samplewmap5\]) showing that the determination of ${{C_{\ell}^{\text{peak}}}}$ and $f_{\ell}$ is only marginally improved by the 500k simulation. The contribution of correlation to the quality of the fit is given on the Kullback scale by the Pythagorean decomposition (\[eq:pythasym\]). Numerical evaluation by Monte Carlo integration gives, term-to-term: $$\label{eq:pythnum}
35.18 \ 10^{-3} \approx 8.61 \ 10^{-3} + 27.3 \ 10^{-3}.$$ This is only an approximate equality because of MC errors. The last term was also evaluated using eq. (\[eq:kullcorr\]), yielding $27.5\
10^{-3}$. These values show that correlation accounts for most part of the mismatch in the sense that $ K(\pi | \tilde{\pi}) \approx \frac13
K(\tilde{\pi} | \tilde\pi_0 )$.
Those results can be compared to the naive approximation used as the initial proposal in our adaptive importance sampling runs, that is, the copula approximation $\pi_\mathrm{naive}$ with $C_{\ell}^{ML}$, ${\ensuremath{{f_{\text{sky}}}}}$ and ignoring the correlation. It gives a perplexity of ${\ensuremath{\mathcal{P}}}(\pi|\tilde{\pi}_{\rm naive})=0.76$ corresponding to a huge increase in Kullback divergence.
Finally, we compute, as a comparison baseline, the perplexity of the classical offset log-normal approximation [@BJK:2000ApJ...533...19B]. The estimation of the curvature at the peak is easily derived from $f_{\ell}$. The perplexity goes down to ${\ensuremath{\mathcal{P}}}=0.2$ for that approximation.
Validation : pseudo-cosmological parameters
-------------------------------------------
We now compare several likelihood functions via their impact on estimation of (pseudo) cosmological parameters from WMAP data. Since only the low $\ell$ part of the spectrum is considered, only a few cosmological parameters can be fitted. We choose to perform our comparisons using a simple model with only two parameters, amplitude and spectral index, that is, we consider $$\tilde{C}_{\ell}
\equiv
C_{\ell}^{\rm ref}\times A\left(\frac{\ell}{\ell_{0}}\right)^{n},$$ where $C_{\ell}^{\rm ref}$ is a reference angular spectrum (here the WMAP1 best fit spectrum) and where the relative amplitude $A$ and the relative spectral index $n$ are our pseudo-cosmological parameters. The reference power spectrum being a fit on a broader range of multipoles, the posterior of $(A,n)$ is not centered at $(1,0)$.
Figure \[fig:anplots\] shows the $1,2$ and $3\,\sigma$ contours and the peak position for different likelihood approximations. The top panel presents a comparison between the WMAP5 likelihood code, used both in pixel based and Gibbs mode [@Dunkley:2008p3305], and copula approximations with or without correlations (i.e. $\tilde \pi$ and $\tilde \pi_0$). They all appear to be in remarkably good agreement. The small discrepancies in the contour curves (which are smaller than the grid step size) are much smaller than the width of the mode. The peaks of the copula approximations and of the Gibbs approximation are very slightly displaced compared to the official WMAP5 results, at a distance of the order of the step size of the grid on which likelihoods are evaluated. The bottom panel presents a comparison with the log-normal approximation described in previous chapter. As expected, the quality of that last approximation is poor, with a deviation of the best fit $(A,n)$ of the order of $\sigma/4$. Nonetheless, the areas of the $1,2$ and $3\sigma$ regions are similar, probably because these areas are mostly controlled by the values of $f_{\ell}$.
![Posterior distribution for $(A,n)$ using different likelihood approximations. Both panels: the dashed blue line shows official WMAP5 likelihood code and the black solid line shows the copula approximation $\tilde \pi$. Top panel: green dotted line is the Gibbs implementation included in the official code, the red dash-dotted line is the copula approximation ignoring correlations, $\tilde \pi_0$. Bottom panel: solid magenta line is the log-normal approximation. The colored symbols mark the peak of each posterior.[]{data-label="fig:anplots"}](figs/WMAP_pseudo "fig:"){width="1\columnwidth"}\
![Posterior distribution for $(A,n)$ using different likelihood approximations. Both panels: the dashed blue line shows official WMAP5 likelihood code and the black solid line shows the copula approximation $\tilde \pi$. Top panel: green dotted line is the Gibbs implementation included in the official code, the red dash-dotted line is the copula approximation ignoring correlations, $\tilde \pi_0$. Bottom panel: solid magenta line is the log-normal approximation. The colored symbols mark the peak of each posterior.[]{data-label="fig:anplots"}](figs/WMAP_pseudo_log "fig:"){width="1\columnwidth"}
Conclusion
==========
Using an adaptive importance sampling algorithm, we explored the low-$\ell$ posterior of partially observed CMB maps, both synthetic and real. From this exploration, we built a copula-based approximation for that posterior distribution. Numerical evaluation of that approximation is much faster than the pixel-based computation. We showed that the approximation is very close to the actual posterior with an accuracy which is probably sufficient for most cosmological applications. For example, on a simple two-parameter pseudo cosmological model, we found a discrepancy which is negligible with respect to the width of the posterior mode (figure \[fig:anplots\]).
The copula approximation uses two ingredients: a model of marginal distributions and a correlation matrix. The marginals are mostly distributed as inverse gammas, as in the full-sky case, but with different parameters. Maybe surprisingly, the correlations between (Gaussianized) multipoles are found to be quite low ($<10\%)$. Ignoring them in the toy cosmological model illustrated by figure \[fig:anplots\] does not change significantly the posterior. However, when considering the full joint distribution of the multipoles (as opposed to its *projection* onto the two-parameter toy model), the correlation is significant: the Kullback divergence from the true posterior to its copula approximation quadruples if the correlation is left out. In both cases however, the Kullback divergence remains small.
The main limitation of the proposed approximation is that it requires an exploration of the posterior to measure the parameters of the approximation. We used an adaptive importance sampling algorithm, but a MCMC algorithm, Gibbs-based [@2004PhRvD..70h3511W] or Hybrid MC-based [@Taylor:2007p3316] can also be used. Both methods exhibit good scaling properties thanks to a smart re-writing of the posterior and could, if convergence is well controlled, provide estimates at higher $\ell$. Indeed, a very recent work, published at the time we were finishing this paper follows a similar path and demonstrate a Gaussianization technique based on splines rather than on inverse gamma models [@Rudjord:2008p4864].
Another approach would be to determine the parameters of the marginals directly from the likelihood, without resorting to a sampling-based exploration. We are currently working on an analytical derivation of the approximation which would make it possible to build an approximation valid for higher $\ell$ at low computational cost. Being able to reach smaller scales is also important to explore the transition between low $\ell$ estimates and high-$\ell$ ones. Indeed, at very small scales, the problem becomes intractable and requires the use of asymptotic approximations to the likelihood [@Percival:2006MNRAS.372.1104P; @Smith:2006PhRvD..73b3517S].
Finally, it is not clear yet whether the same kind of approximation can be built for polarized fields. In the temperature case addressed here, we took advantage of a low correlation situation, thanks to a high signal to noise ratio and relatively small masked area. Polarized observations will be noisier and it remains to be seen if copula approximations are up to the task. This is the subject of current investigations.
Acknowledgments {#acknowledgments .unnumbered}
===============
We thank J. Dunkley for her detailed description of the large scale map used in the WMAP5 likelihood. The authors were greatly helped by the comments and remarks from F. Bouchet, H.K. Eriksen, members of the ECOSSTAT ANR project and the Planck CTP working group. The ANR grant ECOSSTAT (ANR-05-BLAN-0283-04) provided financial support for part of this work. We acknowledge the use of the HEALPix package[^5].
ML estimation of inverse gamma parameters {#sec:mlegamma}
=========================================
The log-likelihood $\log\mathcal{L}(\alpha,\beta)$ for a sample of $N$ independent realizations $X_i$ under an inverse gamma density is $$\log\mathcal{L}= \sum_i^N \left( \alpha\log\beta-\log\Gamma(\alpha) - (\alpha+1)\log X_i - \frac{\beta}{X_i} \right)$$ as seen from eq. (\[eq:defigamma\]). The ML estimate for $(\alpha,\beta)$ is the solution of $\frac{\partial\log\mathcal{L}}{\partial\alpha}=0$ and $\frac{\partial\log\mathcal{L}}{\partial\beta}=0$ leading to the two estimating equations: $$\log\beta-\psi(\alpha)=\frac{1}{N}\sum_i^N\log X_i
,
\qquad
\frac{\alpha}{\beta}-=\frac{1}{N}\sum_i^N\frac{1}{X_i}$$ where $\psi(u)$ is the log-derivative of the gamma function, also known as the digamma function. Using the last equation to express $\beta$ in terms of $\alpha$, the ML estimate can be obtained by solving $$\log\alpha-\psi(\alpha)=\frac{1}{N}\sum_i^N\log X_i - \log \left(\frac{1}{N}\sum_i^N\frac{1}{X_i} \right).$$ This is quickly done numerically in a few steps of a Newton algorithm; both the digamma function and its derivative being available in the GSL package.
\[lastpage\]
[^1]: E-mail: [[benabed@iap.fr]{}](mailto:benabed@iap.fr)
[^2]: This situation is representative of CMB data sets from satellites such as WMAP and Planck.
[^3]: <http://lambda.gsfc.nasa.gov/>
[^4]: <http://www.gnu.org/software/gsl/>
[^5]: <http://healpix.jpl.nasa.gov>
|
---
abstract: 'MagAO-X is an entirely new “extreme” adaptive optics system for the Magellan Clay 6.5 m telescope, funded by the NSF MRI program starting in Sep 2016. The key science goal of MagAO-X is high-contrast imaging of accreting protoplanets at H$\alpha$. With 2040 actuators operating at up to 3630 Hz, MagAO-X will deliver high Strehls ($>70$%), high resolution (19 mas), and high contrast ($< 1\times10^{-4}$) at H$\alpha$ (656 nm). We present an overview of the MagAO-X system, review the system design, and discuss the current project status.'
author:
- 'Jared R. Males'
- 'Laird M. Close'
- Kelsey Miller
- Lauren Schatz
- David Doelman
- Jennifer Lumbres
- Frans Snik
- Alex Rodack
- Justin Knight
- Kyle Van Gorkom
- 'Joseph D. Long'
- Alex Hedglen
- Maggie Kautz
- Nemanja Jovanovic
- Katie Morzinski
- Olivier Guyon
- Ewan Douglas
- 'Katherine B. Follette'
- Julien Lozi
- Chris Bohlman
- Olivier Durney
- Victor Gasho
- Phil Hinz
- Michael Ireland
- Madison Jean
- Christoph Keller
- Matt Kenworthy
- Ben Mazin
- Jamison Noenickx
- Dan Alfred
- Kevin Perez
- Anna Sanchez
- Corwynn Sauve
- Alycia Weinberger
- Al Conrad
bibliography:
- 'report.bib'
title: 'MagAO-X: project status and first laboratory results'
---
INTRODUCTION {#sec:intro}
============
AO systems are now in routine use at many telescopes in the world; however, nearly all work only in the infrared (IR, $\lambda>1$ $\mu$m) due to the challenges of working at shorter wavelengths. The Magellan AO (MagAO) system was the first to routinely produce visible-AO science on a large aperture telescope[@2013ApJ...774...94C; @close_2014_hd142527; @2014ApJ...786...32M; @rodigas_2015; @sallum_nature_2015] (see Close et al. in these proceedings [@laird_magao]). Other large telescopes with visible AO systems include the 5 m at Palomar [@dekany_2013] and ESO’s 8 m VLT with the ZIMPOL camera behind the Spectro-Polarimetric High-contrast Exoplanet REsearch (SPHERE) instrument [@roelfsema_2014].
MagAO-X is an *entirely new* visible-to-near-IR “extreme” AO (ExAO) system. When completed, MagAO-X will consist of: (1) a 2040 actuator deformable mirror (DM) controlled at (up to) 3.63 kHz by a pyramid wavefront sensor (PWFS); (2) cutting-edge coronagraphs to block a star’s light; and (3) a suite of focal plane instruments including imagers and spectrographs enabling high-contrast and high-resolution science.
MagAO-X will deliver high Strehls ($\gtrsim 70\%$ at H$\alpha$), high resolutions ($14-30$ mas), and high contrasts ($\lesssim 10^{-4}$) from $\sim$ 1 to 10 $\lambda/D$ . Among many compelling science cases, MagAO-X will revolutionize our understanding of the earliest stages of planet formation, enable high spectral-resolution imaging of stellar surfaces, and could take the first images of an exoplanet in reflected light.
Status
------
MagAO-X is funded by the NSF MRI program (award \#1625441), beginning in Sep, 2016. We passed a rigorous external preliminary design review (PDR), which was managed by the Magellan Observatory, in May, 2017. The detailed design was completed, and procurement is nearly complete. The last major item to be delivered is the MEMS deformable mirror. Integration is underway in the Extreme Wavefront Control Lab (EWCL) in Steward Observatory at the University of Arizona.
Our current plan is to hold the Pre-Ship Review, a Magellan Telescope requirement, in early 2019, and then ship to Las Campanas Observatory (LCO) in mid to late Feb, 2019. Technical first light would then occur in roughly March 2019. The instrument will be shipped back and forth from LCO to Tucson several times, at least until commissioning is complete in late 2020.
Here we provide a very abbreviated overview of the MagAO-X design. Reference is made to the many other presentations in this conference providing more detail. In addition, the complete PDR documentation is available at <https://magao-x.org/docs/pdr>.
SCIENCE JUSTIFICATION
=====================
MagAO-X will enable a wide range of astrophysical observations. Here we review a few of these.
A Survey of the Low Mass Distribution of Young Gas Giant Planets
----------------------------------------------------------------
The main science goal of the initial operations of MagAO-X is to conduct a survey of nearby T Tauri and Herbig Ae/Be stars for newly formed accreting planets in H$\alpha$.
### Proof-of-concept – Imaging LkCa 15 b at H$\alpha$
We used MagAO’s simultaneous differential imaging (SDI) mode [@close_2014_hd142527] to discover H$\alpha$ from a forming protoplanet (LkCa 15 b)[@sallum_nature_2015], detecting an accretion stream shock at 90 mas. Dynamical stability places the mass of LkCa 15 b at $2^{+3}_{-1.5}$ $M_{Jup}$. Correcting for extinction we found an accretion rate of $\dot{M} = 1.16\times10^{-9}$ $M_{\odot}/yr$ [@sallum_nature_2015]. Observed $H{\alpha}$ rate was $\sim0.5e/s$ at the peak pixel at Strehl $\sim$5%.
### MagAO-X $H\alpha$ Protoplanet Survey
While LkCa 15 (145 pc; 1–2 Myr) is fairly faint ($I$$\sim$$11$ mag) there are many brighter, closer similarly young accreting targets. A review of the populations of nearby young moving groups and clusters yields 160 accreting, $<$10 Myr old, D$<$150 pc, targets all with I$<$10 mag for this survey—bright enough for good AO correction. There are also 33 more stars with I$<$12 mag that will be excellent targets with moderate to good AO correction. Extrapolating from our initial ($3/10=30\%$) success rate for the young star GAPplanetS $H\alpha$ survey having accreting objects [@Fol_2018] the 193 stars yield $\sim$$59$ new protoplanet systems using just 5 nights per semester.
This survey of protoplanet systems will define the population of low-mass outer EGPs, and will help reveal where and how gas planets actually form and grow. As the main initial science case for MagAO-X, these observations define the performance requirements of our design. In Table \[tab:high\_reqs\] we detail the guide star statistics, and the corresponding system performance requirements needed to enable this survey.
[ccc||ccc]{} &\
I & d & Numb. & Sep & $\Delta$H$\alpha$ & Strehl$^1$\
mag & \[pc\] & & \[mas\] & mag & \[%\]\
5 & 225 & 6 & 75 & 12.0 & 70\
8 & 150 & 25 & 100 & 9.0 & 50\
10 & 150 & 129 &100 & 7.0 & 30\
12 & 150 & 44$^2$ & 100 & 5.0 & 20\
\
\
Other Science Cases
-------------------
Here we present a short summary of several additional science cases. These are generally spanned by the parameters of the H$\alpha$ survey in terms of guide star brightness and separations, and are less-demanding, and so we do not specifically derive requirements for these. Rather, they are presented to give an idea of the breadth of use-cases for this new instrument.
### Circumstellar Disks
Disk science is a challenging application of AO, with low surface brightness and characteristics similar to the uncorrected seeing halo, so high-Strehl high-contrast ExAO is critical. MagAO-X will push the two frontiers in circumstellar disk science. The first is detailed imaging of geometry, particularly in the 5-50 AU region analogous to the outer part of the solar system. Most disks sit at 50-150 pc, so reaching radii comparable to the giant planet region requires imaging at 50-120 mas. Existing systems push in to at best $\sim$150 mas. For some disks, an inner working angle of $\sim$100 mas will push to the exozodiacal light region for the first time. For example, in the well-known HR 4796A disk, SED fits show that the 8-20 $\mu$m flux cannot be fully explained by the outer, $\sim$100 K, ring, suggesting a ring at 3-7 AU [@wahhaj_2005]. MagAO-X has the potential to image this inner ring. The second frontier is multi-wavelength study of disks to derive the chemical make-up and dynamical state [@rodigas_2015; @stark_2014]. This requires a large wavelength grasp from visible through near-infrared so MagAO-X’s ability to image at $\sim$0.45 $\mu$m complements existing systems.
![\[fig:wt\_block\] The MagAO-X woofer-tweeter architecture. Left: in Phase-I we employ the vAPP coronagraph and SDI. Right: in Phase-II we will employ a PIAACMC and spectrographs.](wt_block_diag.png "fig:"){height="2in"} ![\[fig:wt\_block\] The MagAO-X woofer-tweeter architecture. Left: in Phase-I we employ the vAPP coronagraph and SDI. Right: in Phase-II we will employ a PIAACMC and spectrographs.](wt_block_diagII.png "fig:"){height="2in"}
### Fundamental Properties of Young Solar-System-like EGPs
Dedicated exoplanet-imagers GPI and SPHERE are in operation, and GPI has discovered the first planet of this new era: 51 Eri b is a 600-K $\sim$2 M$_\textrm{Jup}$ exoplanet imaged 13 AU from its 20-Myr-old, 30-pc-away F-type host star [@2015Sci...350...64M]. This planet is different from other exoplanets (whether imaged or analyzed by transit spectroscopy): its atmosphere is the closest analog yet to solar system atmospheres because of its Saturn-scale orbit, Jupiter-scale mass, and cool temperature such that CH$_4$ was detected in the GPI spectrum.
We have conducted a prototype experiment with existing MagAO using the exoplanet $\beta$ Pic b, which can be imaged with the current VisAO due to its brightness (youth and mass) and its 300-400 mas separation[@2014ApJ...786...32M]. We also demonstrated using such measurements to empirically measure the fundamental properties of this solar-system-scale exoplanet[@2015ApJ...815..108M]. MagAO-X will extend these observations to shorter wavelengths, and to smaller mass, smaller separation planets such as 51 Eri b. MagAO-X will also enable characterization of such planets with the DARKNESS and RHEA@MagAO-X spectrographs.
### Resolved Stellar Photospheres
High resolution spectroscopy will be enabled by the 3x3 single-mode fiber-fed IFS, RHEA@MagAO-X, with $R\sim$60,000 spectral resolution (PI: Mike Ireland). The combination of the ExAO resolution and contrast with high spectral resolution enables many exciting science cases. For instance: the largest resolvable non-Mira stars accessible from MagAO include Betelgeuse ($\sim$50 mas), Antares ($\sim$40 mas), Arcturus ($\sim$21 mas), Aldebaran ($\sim$20 mas) and $\alpha$ Boo ($\sim$19 mas). These stars lose mass through a complex process in an interplay between a hot ($\sim$10,000 K) corona and a cool ($\sim$2000 K), slow ($\sim$10 km/s) molecular wind. These states can not co-exist so asymmetries of some kind are expected. Resolving the photosphere in lines and molecular bands enables the multi-dimensional structure of these regions to be imaged. Upwelling and downwelling velocities on the surface are of order a few km/s, separable at sufficient resolution. A single image of a stellar photosphere would be the first ever direct measurement of convection in a star other than the Sun.
### Asteroids
MagAO-X will have resolutions of 14–21 mas in $g$-$r$ bands, which correspond to $\sim$20–30 km on a main-belt asteroid (MBA). On a typical night more than 80 MBAs brighter than I=13 (implying $\gtrsim50$ mas) will be resolvable by MagAO-X. This will provide true dimensions, avoiding degeneracies in light-curve analysis. MagAO-X will enable sensitive searches for and orbit determination of MBA satellites. In combination, these directly measure density and hence estimate composition [@britt_2002]. This will directly inform the theories of terrestrial planet formation [@mordasini_2011]. See Ref. for a proof of concept with existing VisAO.
OPTOMECHANICAL DESIGN
=====================
System Architecture
-------------------
[r]{}[3in]{} {width="3in"}
Our original concept for MagAO-X was to employ a cascaded control system, where the existing MagAO system provided an initial low-order low-speed correction, followed by a high order “afterburner” system. This is similar to the current SCExAO architecture[@2015PASP..127..890J; @julien_scexao]. During the preliminary design phase of the project, we opted to prioritize a self-contained woofer-tweeter system, using the static facility f/11 secondary at Clay. The main driver of this choice is calibration: full calibration and end-to-end testing of the system with the MagAO adaptive secondary mirror would require occupying the actual telescope with a retroreflector installed. A secondary driver is operational, as using the f/11 (which is normally installed) provides much more flexibility compared to requiring the f/16 ASM to be installed to support MagAO-X operations. With the self-contained woofer-tweeter architecture, MagAO-X is a completely separate AO system from existing MagAO.
Figure \[fig:wt\_block\] shows the high level architecture of MagAO-X. Light enters from the f/11 secondary (off the tertiary) and the pupil is relayed to an Alpao DM-97 woofer. Another pupil image is formed on the Boston Micromachines (BMC) 2k DM. A beamsplitter then splits light (either 50/50 or with dichroics) between the PWFS and the science channel. The PWFS signal is reconstructed and fed back to the DMs in closed loop. The science light is passed through the coronagraph. At left we show our “Phase I” plan, which includes a vector Apodizing Phase Plate (vAPP) coronagraph, followed by a simultaneous differential imaging system which makes use of dichroic beamsplitters and two EMCCD cameras. Light rejected by the coronagraph, and/or light contained in the vAPP leakage term is used for coronagraph low-order WFS (LOWFS). At right is the “Phase II” plan, which will employ a Phase Induced Amplitude Apodization Complex Mask Coronagraph (PIAACMC) with a Lyot-LOWFS[@2017PASP..129i5002S]. In this phase we will also deploy spectrographs including the DARKNESS MKIDS[@2018PASP..130f5001M; @alex_mec] array-IFU and the RHEA single-mode fiber-fed IFU.
Detailed Design
---------------
MagAO-X consists of an air-isolated optical table with an active height control and leveling system. It has two levels. The upper level is at the height of the telescope beam above the Nasmyth platform, and contains the woofer, or low-order DM (LODM), the tweeter, or high-order DM (HODM), the image re-rotator (K-mirror), and the atmospheric dispersion corrector (ADC). The lower level houses the PWFS, various filter wheels and beamsplitter selectors, the coronagraph optics, and the science cameras.
The design of MagAO-X is illustrated in Figure \[fig:magaox\_clay\], showing the MagAO-X optical bench on the Nasmyth platform of the Clay telescope, along with the electronics rack. Figure \[fig:magaox\_cover\] shows a closeup of the instrument. A dust cover will protect the instrument, with a slight positive pressure to prevent dust contamination. It is not under vacuum nor is it thermally controlled. For more information about the optomechanical design of MagAO-X see Close et al. in these proceedings[@laird_optomech].
![\[fig:magaox\_cover\] MagAO-X closeup. Left: the instrument with dust cover installed. Right: dust cover removed to show the details inside.](magaox_covered.png "fig:"){height="2.5in"} ![\[fig:magaox\_cover\] MagAO-X closeup. Left: the instrument with dust cover installed. Right: dust cover removed to show the details inside.](magaox_uncovered.png "fig:"){height="2.5in"}
Since MagAO-X will be deployed on the Nasmyth platform of an Alt-Az telescope, we require an image derotator to keep the pupil aligned to the WFS, DMs, and coronagraphs. For a description of the custom K-mirror used to accomplish this see Hedglen et al. in these proceedings[@alex_k]
Since MagAO-X is not thermally controlled, thermally stable optics mounts are critical to maintaining the fine alignment of the system. Custom mounts have been developed, starting with off the shelf components, which achieve our needed stability. The development and testing of these custom mounts is described in detail in Kautz et al., in these proceedings[@maggie_mount].
Pyramid WFS
-----------
The high order WFS (HOWFS) of MagAO-X is a modulated pyramid WFS, based largely on the LBT/MagAO design. Some subtle differences are that it is an all-reflective design, and the modulator is exactly in a pupil plane in collimated space. The pyramid itself is an exact copy of the double 4-sided pyramid prism in existing MagAO. The detector used is an OCAM-2K EMCCD from First Light Imaging. A detailed design study was carried out to optimize the placement and size of the 4 pupil images on the OCAM-2K in bin-2 mode, allowing us to run the system at speeds up to 3630 Hz. For a full description of this design and initial results from the PWFS in the lab see Schatz et al. in these proceedings[@lauren_pwfs].
Coronagraphs
------------
Our initial configuration is based on the vector Apodizing Phase Plate (vAPP) coronagraph. APP coronagraphs are pupil-plane only devices, utilizing a phase pattern in a pupil plane to change the distribution of intensity in the focal plane. See Snik et al. in these proceedings for a review of vAPP coronagraphy[@frans_vapp]. We have demonstrated vAPPs in the Clio camera on existing MagAO[@2017ApJ...834..175O]. A prototype MagAO-X version has been tested in our lab[@kelsey_lowfs], and the final design is nearly complete and will be fabricated in the coming months.
MagAO-X is designed to accommodate any Lyot-type coronagraph. In our final configuration we plan to implement the Phase Induced Amplitude Apodization Complex Mask Coronagraph (PIAACMC)[@2010ApJS..190..220G]. The preliminary design is underway, and we are investigating strategies for fabrication and the required tolerance of the focal plane complex mask zones. For more information about the steps being taken to perfect the fabrication and modeling of complex focal plane masks see Knight et al. in these proceedings[@justin_piaacmc].
OPTICAL SPECIFICATIONS
======================
The specifications for the optics of MagAO-X are set to control the static and non-common path (NCP) wavefront errors (WFEs). These sources of WFE affect the Strehl ratio and the post-coronagraph contrast. The requirements on the optical surface quality are derived from the high-level science requirements, in conjunction with the performance simulations and error budget. They are
- Requirement: Static/NCP WFE $<$ 45 nm rms. At H$\alpha$ this multiplies Strehl ratio by 0.83. This specification ensures that the bright star Strehl ratio requirement of 70% can be met.
- Goal: Static/NCP WFE $<$ 35 nm rms. Strehl ratio ($S$) due to static/NCP impacts exposure time as $t$$\propto$$S^3$. Hence we should seek maximum possible performance.
- Requirement: Static/NCP WFE contribution to contrast should be $<$ $1\times10^{-3}$ from 0 to 24 $\lambda/D$. This is after the effects of low-order WFS (LOWFS) and/or focal-plane WFS (FPWFS) are taken into account.
All of these requirements are derived for the science channel. The WFS channel requirement is simply that it should not be worse, as it has fewer optics. Here we briefly summarize the analysis. Please refer to the MagAO-X PDR documents (<https://magao-x.org/docs/pdr>, section 5.1) for an in-depth discussion.
The Clay Telescope
------------------
MagAO-X is designed for the 6.5 m Magellan Clay telescope. We collected the as-built measurements of the surfaces of the primary mirror (M1), the f/11 fixed secondary (M2), and the tertiary mirror (M3).
[**M1**]{}: The Magellan Clay primary mirror was cast and polished at the Steward Observatory Mirror Lab (SOML). From the post-polishing test data we found that the surface height has 12.52 nm rms across all spatial frequencies sampled by the map. The central obscuration is 29% (defined by the secondary baffle) and we will undersize the outer edge by 4.4% to mask the poorly polished edge. This annulus has a 12.52 nm rms, only negligibly reduced from the unmasked map.
[**M2**]{}: A post-fabrication measurement of the surface structure function of the Clay f/11 static secondary was available. We fit this to a power law with index $\alpha-2$, where $\alpha$ is the index of the spatial PSD. The measurements indicate that the surface of M2 has a residual of 12.7 nm rms.
[**M3**]{}: The manufacturer’s test report for the (now re-coated) tertiary mirror contained a surface map. From this, we find that M3 has a reported surface finish of 13.8 nm rms.
Deformable Mirrors
------------------
Here we describe the specifications of the deformable mirrors we will use for wavefront correction.
### High Order DM
The high order DM, or tweeter, is a Boston Micromachines Corp. (BMC) MEMS 2k. This is a 2040 actuator device, with 50 actuators across a circular aperture. The main optical characteristics are summarized in Table \[tab:bmc\]. In Figure \[fig:2kmodeopt\] we show gain maps for the Fourier basis, optimized using the framework developed in Ref. . The size of the modal basis in spatial frequency space is indicated for 1K and 2K DMs, demonstrating that the choice of a 2K DM is well justified by our science case and performance requirements.
![\[fig:2kmodeopt\] Optimum gain maps for the Fourier basis, using the framework developed in Ref. . The size of the modal basis in spatial frequency space is indicated for 1K and 2K DMs. This demonstrates that the choice of a 2K DM is well justified by our requirements.](2kmodeopt.png){width="6in"}
-------- ------- -------- -------- -------- ------ ---------- --------- --------
Total Linear Pitch Stroke CA
Act. Act. \[mm\] $\mu$m mm Max Spec Typical Actual
Value: 2040 50 0.4 3.5 19.7 20 13 11.3
-------- ------- -------- -------- -------- ------ ---------- --------- --------
: Boston Micromachines 2k Specifications. \[tab:bmc\]
Delivery of the MagAO-X DM is expected in July 2018. The device itself has been wired, coated, tested, and is undergoing final packaging. The chosen device has one major defect, a so-called “bump” which limits the stroke of the surface in one position affecting 4 actuators (which are all functional). This “bump” will be partially masked by a secondary support spider, and in the coronagraphs will be covered with a circle. Additionally there is one pair of coupled actuators, and two more minor “bumps” which are negligible in normal operations.
![\[fig:2kmap\] Surface maps of the 2040 actuator MEMS deformable mirror selected for MagAO-X. The flat was measured at BMC after a closed-loop flattening procedure, and we then fit and subtracted the first 9 Zernike polynomials to simulate the woofer. Left shows the complete surface, where the “bump” defect is clearly visible. The middle pane shows the surface masked by the spiders. At right is the coronagraphic pupil, with an oversized central obscuration, undersized outer diameter (to account for polishing errors at the edge), and spiders widened for alignment tolerances. The masked surface is 11.3 nm rms.](2kdm_triple.png){width="6in"}
BMC produced a flat map of the device. We then subtracted the first 9 Zernike polynomials from it and analyzed the result. With the “bump” masked and using our undersized coronagraphic pupil the DM flat surface has 11.3 nm rms residual. See Figure \[fig:2kmap\]. Once the device is delivered we will perform an extensive characterization[@kyle_dms].
### Low Order DM
We plan to use an Alpao DM97-15 for our low order DMs in two positions. The first serves as the woofer to minimize the stroke requirements on the BMC 2k tweeter. The second will be used in the science channel (after the HOWFS beamsplitter), under control by the LOWFS. This allows for correction of non-common path errors sensed by the LOWFS. The main optical characteristics are summarized in Table \[tab:alpao\].
-------- ------- -------- -------- ------ ----------------------------- ---------
Total Linear Pitch CA [Flat surface \[nm rms\]]{}
Act. Act. \[mm\] mm Max Spec Typical
Value: 97 11 1.5 13.5 7 4
-------- ------- -------- -------- ------ ----------------------------- ---------
: Alpao DM97-15 Specifications. \[tab:alpao\]
We have procured one of these devices and begun testing and characterization. For details see Van Gorkom et al.[@kyle_dms] in these proceedings.
MagAO-X Flats and OAPs
----------------------
All flats have been delivered and are undergoing final testing before integration. These optics are better than $\lambda/40$ PV in the lowest orders. Importantly, these are super-polished surfaces, so that at higher spatial frequencies and over the typical beam footprints in the system they perform equivalent to $\lambda/100$ PV surface or better. Surface roughness is better than 1 Angstrom rms over the ISO-10110-8 band.
The custom off-axis parabolas (OAPs) were also recently delivered and are being checked before integration. Two have been used to obtain initial WFS data[@lauren_pwfs] and demonstrated excellent PSF image quality[@laird_optomech]. These are polished to better than $\lambda/50$ rms reflected wavefront over the control band of MagAO-X, and have a surface roughness of better than 1 nm rms. The surfaces were specified using a PSD to control WFE in bands of interest due to their effect on contrast. See <https://magao-x.org/docs/pdr> section 5.1 for more detail.
Optics Error Budget
-------------------
We have conducted a detailed analysis of the MagAO-X design with the optical specifications just discussed, to ensure that the we will meet the requirements imposed by the science case. We analyzed each optic using a von Karman PSD with parameters characteristic of such optics. The effects of wavefront control were modeled assuming near-perfect correction of static WFEs from optical surfaces in the control band of the system. We categorize the resultant static and NCP errors as either uncorrectable high spatial frequency common path (i.e. “fitting error” of the DMs) or the post-LOWFS NCP errors. Together these make up the instrumental WFE budget, that is not including residual atmospheric turbulence. In-house verification of the delivered optics is ongoing. The instrument WFE budget based on manufacturer reports is summarized in Table \[tab:wfe\_budget\].
[l|c]{} & Total WFE\
Category & \[nm rms\]\
Uncorrectable CP & $\leq 35 nm$\
Post-LOWFS NCP & $\leq 20 nm$\
Total Instrumental & $\leq 40 nm$\
\
Fresnel Analysis
----------------
The above analysis only considered phase, treating the propagation as if such phase errors add in quadrature. For the purposes of calculating total WFE and Strehl ratio this is sufficient, however it does not address the impact of various errors on contrast. A more detailed Fresnel propagation analysis is needed to test whether the PSDs of the optical surfaces allow the system to meet the contrast requirements. For details of this analysis see Lumbres et al. in these proceedings[@jhen_fresnel], where we find that as-designed MagAO-X will deliver $6\times10^{-5}$ raw contrast at H$\alpha$ using the vAPP coronagraph.
System Throughput
-----------------
We analyzed the throughput of the as-designed system. This ensures that we can adequately analyze and simulate system performance. The key quantity is the photon rate (photons/sec) delivered to each of the detector planes in the system. These are the high-order wavefront sensor (HOWFS), the low-order WFS (LOWFS), and the two SDI science cameras.
The following coatings were assumed:
- Primary and Secondary: protected Aluminum
- Tertiary: a custom coating with enhanced reflectivity near 0.8 $\mu$m.
- OAPs and Flats: protected Silver, using a curve provided by Thor Labs.
- Alpao DMs: same protected Silver curve.
- BMC MEMS DM: unprotected gold
- Transmissive surfaces: a standard anti-reflective coating.
The quantum efficiency (QE) curves for the First Light Imaging OCAM-2K (HOWFS detector) and the Andor iXon 897 (LOWFS detector) and Princeton Instruments EMCCDs (Science detectors) were digitized from the manufacturer specification sheets. We include atmospheric transmission calculated using the BTRAM IDL code. We assumed 5.0 mm precipitable water vapor (PWV), and observing at zenith distance 30$^o$, or airmass 1.15.
We then created a notional set of filters designed for the H$\alpha$ science case:
- A dichroic beamsplitter which divides the light between the HOWFS and Science channels. Cuts-on at 0.68 $\mu$m, with $T$ and $R$ both 95%.
- A LOWFS filter which selects only in-band light of the vAPP leakage term, including the 5% leakage term.
- An H$\alpha$ SDI filter set, as quoted by a vendor (see Opto-Mechanical design for details).
Finally, the complete transmission curves for each of the planes in the H$\alpha$ configuration was calculated by multiplying the above curves. This the reflectance or transmittance for each optic in the system, the atmosphere, the detector QE, and the appropriate filter curves. We also included a 10% loss due to diffraction derived from the Fresnel propagation analysis. These final transmission curves are shown in Figure \[fig:trans\_final\]. The characteristics of this filter system are given in Table \[tab:char\_halpha\].
![Filter transmission curves for MagAO-X at each of the detector planes for H$\alpha$ SDI. Blue: PWFS, green: LOWFS, maroon & dark green: SDI cameras. \[fig:trans\_final\]](throughput.png){width="4in"}
------------------- ------------ ------------- ------------------ --------------------- -----------------
Plane Throughput $\lambda_0$ $\Delta \lambda$ $F_\gamma$(0) Notes
$\mu$m $\mu$m Photons/sec
Science H$\alpha$ 0.177 0.657 0.0082 $2.3\times 10^{9}$
Science Cont. 0.176 0.668 0.0083 $2.6\times 10^{9}$
LOWFS 0.010 0.662 0.033 $5.9\times 10^{8}$ 5% vAPP leakage
HOWFS 0.227 0.851 0.257 $7.6\times 10^{10}$
------------------- ------------ ------------- ------------------ --------------------- -----------------
: Filter characteristics for the H$\alpha$ configuration. \[tab:char\_halpha\]
Compute and Control
===================
The MagAO-X compute system hardware is entirely commercial off-the-shelf (COTS), utilizing standard components. There are three computers: the Instrument Control Computer (ICC), the Real-Time computer (RTC), and the AO Control Computer (AOC). The fundamentals of these computers are identical, meaning they use the same motherboard, processor, and RAM. This facilitates sparing. Because these are all common COTS components, it also allows for managing obsolescence by making it likely that replacements will be available for some time, and furthermore eventual upgrades should be simple to manage.
Each machine uses dual 16 core Intel Xeon processors. For additional computing power we employ COTS GPUs, for now relying on consumer grade GPUs rather than the much more expensive units intended for scientific and engineering use. Experience on SCExAO has shown this is adequate for MagAO-X closed loop operation at 3.63 kHz. The mixed CPU/GPU architecture can be scaled up to meet additional computing requirements required to support advanced AO operation modes, such as predictive control[@2018JATIS...4a9001M; @2017arXiv170700570G] and multi-sensor operation[@2017arXiv170700570G].
Our real time software is based on the “Compute and Control for Adaptive Optics” (CACAO) system, developed for SCExAO and now being refactored significantly to support broad usage in ExAO systems. See <https://github.com/cacao-org/cacao> for more information about CACAO.
Telemetry
---------
An important challenge in high-order and high-speed ExAO systems is management of telemetry. We estimate that, when fully operational, MagAO-X could generate as much as 10 TB of data per night. A key feature of our computing system is that it is designed to store all system telemetry all the time, and we plan to save this to facilitate on-line optimization, experiments in post-processing, and off-line system analysis. This is manageable on-site with COTS hardware, but this volume of data will require physical transport from LCO back to UofA. See <https://magao-x.org/docs/pdr> Section 3.2 for details.
Our motivations for recording system-wide telemetry for later use include:
- Continuous updates (on timescales of $\sim$ 1 minute) of machine learning predictive controllers[@2017arXiv170700570G]
- Analysis of system optimization
- Post-processing estimation of PSF residuals to assist PSF subtraction/calibration
Using such large volumes of information in post-processing will be very challenging. We have been working to develop computational systems capable of handling such large data sets[@2016SPIE.9913E..0FH]. When employing PSF estimation strategies such as KLIP, matrices with sizes set by the number of images must be decomposed. Long et al. in these proceedings[@joseph_sirius] describe work to use vAPP observations with existing MagAO[@2017ApJ...834..175O] to develop a pipeline capable of reducing $>10^4$ images at a time with advanced dimensionality reducing algorithms. Future work will explore using system telemetry, in addition to focal plane images, to improve detection limits.
WAVEFRONT CONTROL
=================
High Order Wavefront Control
----------------------------
The high order, meaning both high spatial and high temporal frequencies, is provided by the PWFS running at speeds up to 3.63 kHz in closed loop with the BMC 2040 actuator MEMS HODM. To ensure that the HODM is operated around its mid-bias region and to deal with aberrations outside its 3.5 $\mu$m stroke, a woofer is needed. We considered using the ASM, but have settle on the Alpao DM-97 as the woofer. This device can operate at up to 2 kHz.
We have considered three strategies for managing this split control system:
1. Cascaded control: The existing MagAO is used as an initial cleanup stage, and MagAO-X functions as an independent “afterburner”. This was our original concept, and is included here for comparison.
2. Split control: The woofer and tweeter are essentially treated as a single DM for the purposes of wavefront control. This necessitates operating the system no faster than the 2 kHz speed of the woofer.
3. Offloading: The tweeter receives all commands directly from the PWFS at full rate, but at some interval the average shape is offloaded to the woofer. This has been analyzed and found to be stable[@2006SPIE.6306E..0BB], and the BMC 2K and Alpao DM-97 have sufficient stroke to support it.
The first option, cascaded control, is by far the simplest to implement and is in use on SCExAO[@julien_scexao]. We show results from a suite of simulations as the blue curves in Fig \[fig:sim\_strehl\]. However, as we discuss above this option is operationally very complex due to the requirement to have the adaptive secondary on the telescope.
The second option is a conventional method for woofer-tweeter control. The main drawback here is the requirement to run the system at no more than 2 kHz due to the dynamic response of the Alpao DM-97. Simulations with this method are shown as the red curves in Fig. \[fig:sim\_strehl\].
The results in Fig \[fig:sim\_strehl\] show that the split woofer-tweeter control, even though limited to 2 kHz, will meet our requirements. This is true in 25 percentile conditions at LCO for all guide stars, and is true for most stars in median conditions. We have not yet implemented offloading in simulation, but because correction will occur at 3.63 kHz on bright stars we expect to exceed the requirement with the offloading control scheme.
![\[fig:sim\_strehl\] Simulated Strehl](sim_strehl.png){width="3in"}
Figure \[fig:sim\_psfs\] shows example PSFs and post-coronagraph science images for the vAPP coronagraph. See <http://magao-x.org/docs/pdr> Section 4.1 for more detailed discussion of these results.
![\[fig:sim\_psfs\] Simulated PSFs and post-coronagraph science images. Left Two Images: a 5th mag guide star in median conditions. Right Two Images: a 12th mag guide star in 25th percentile conditions.](psf_5th_50th.png "fig:"){width="3in"} ![\[fig:sim\_psfs\] Simulated PSFs and post-coronagraph science images. Left Two Images: a 5th mag guide star in median conditions. Right Two Images: a 12th mag guide star in 25th percentile conditions.](psf_12th_25th.png "fig:"){width="3in"}
FP/CLOWFS
---------
A key capability in any high-contrast imaging instrument is control of non-common-path (NCP) aberrations. These aberrations are not sensed by the PWFS, but do affect contrast in the final science focal planes. We are implementing several strategies for sensing and controlling these.
CLOWFS uses light rejected by the coronagraph to sense low order aberrations. MagAO-X is designed to feed light rejected from either the focal plane mask (FPM) or the Lyot mask of the coronagraph to an EMCCD camera. We plan to use already developed techniques[@2017PASP..129i5002S] for applying this in Lyot-type coronagraphs.
We are also developing several novel strategies for CLOWFS with the vAPP coronagraph. We plan to implement an FPM at an intermediate focal plane after the vAPP which reflects all light from the image except that in the dark hole and leakage terms. This light is then sent to the LOWFS camera. Standard CLOWFS strategies can be applied (i.e. using out of focus images). We are also testing the use of modal wavefront sensing spots encoded in the vAPP liquid crystals. The amount of light in these spots relates directly to the amplitude of the corresponding aberration. We also plan to use linear dark field control (LDFC)[@2017JATIS...3d9002M] to control mid-spatial frequencies. For a detailed description of our FP and CLOWFS strategies see Miller et al. in these proceedings[@kelsey_lowfs].
The LDFC technique relies on a dark hole being created using some other technique. In the case of the vAPP this is achieved by the coronagraph directly, though one must minimize low-order and mid-order aberrations to achieve the full potential. For our more aggressive PIAACMC architecture, we will need to implement post-coronagraphic wavefront control techniques. One avenue we are exploring is the use of high-rate focal plane images, combined with WFS telemetry, to infer the static and NCP aberrations. This was proposed by Frazin[@2013ApJ...767...21F] as a post-processing technique, and we are investigating extending it to a real-time control system. See Rodack et al. in these proceedings[@alex_rtfa] for a complete introduction to this concept.
CONCLUSION
==========
The MagAO-X project is now undergoing integration at Steward Observatory. The design and component specifications have been carefully developed to ensure that we meet the very demanding performance requirements based on high-contrast imaging of exoplanets at visible wavelengths. Assuming remaining procurement completes in short order, we expect to be on-sky for initial testing at the Magellan Clay telescope in the 2019A semester.
In addition to the key science goal of surveying the population of H$\alpha$ emitting accreting protoplanets, we view MagAO-X as a development platform for future ExAO systems on the next generation of Giant Segmented Mirror Telescopes (GSMTs). These large telescopes have great potential to revolutionize our knowledge of the universe. In particular, they will be able to detect and characterize planets in much larger numbers than currently possible. See Males et al.[@jared_gsmts] and Fitzgerald et al.[@fitz_psi], in these proceedings. A key goal of this research is to develop the ability for GSMTs to characterize potentially habitable planets around nearby late-type stars[@2012SPIE.8447E..1XG; @2014SPIE.9148E..20M]. Though our main priority is the science cases outlined in Section 2, MagAO-X will be an enabling platform for developing, testing, and proving technologies for the GSMT ExAO systems to come.
We are very grateful for support from the NSF MRI Award \#1625441 (MagAO-X).
|
---
abstract: |
For every countable group $G$, there are $2^{\omega}$ distinct classes of coarsely equivalent subsets of $G$.
[**MSC**]{} :54E15, 20F69
[**Keywords**]{} : ballean, coarse structure, asymorphism, coarse equivalence
address:
- 'Department of Computer Science and Cybernetics, Kyiv University, Volodymyrska 64, 01033, Kyiv, Ukraine'
- 'Department of Computer Science and Cybernetics, Kyiv University, Volodymyrska 64, 01033, Kyiv, Ukraine'
author:
- 'Igor Protasov, Ksenia Protasova'
title: Counting coarse subsets of a countable group
---
Introduction and results
========================
Following [@b5], [@b6], we say that a [*ball structure*]{} is a triple ${{\mathcal B}}=(X,P,B)$, where $X$, $P$ are non-empty sets, and for all $x\in X$ and $\alpha\in P$, $B(x, \alpha)$ is a subset of $X$ which is called a [*ball of radius*]{} $\alpha$ around $x$. It is supposed that $x\in B(x, \alpha)$ for all $x\in X$, $\alpha\in P$. The set $X$ is called the [*support*]{} of ${{\mathcal B}}$, $P$ is called the [*set of radii*]{}.
Given any $x\in X$, $A\subseteq X$, $\alpha\in P$, we set $$B^*(x,\alpha)=\{y\in X:x\in B(y,\alpha)\},\ B(A,\alpha)=\bigcup_{a\in A}B(a,\alpha),\ B^*(A,\alpha)=\bigcup_{a\in A}B^*(a,\alpha).$$
A ball structure $\mathcal{B}=(X,P,B)$ is called a [*ballean*]{} if
- for any $\alpha,\beta\in P$, there exist $\alpha',\beta'$ such that, for every $x\in X$, $$B(x,\alpha)\subseteq B^*(x,\alpha'),\ B^*(x,\beta)\subseteq B(x,\beta');$$
- for any $\alpha,\beta\in P$, there exists $\gamma\in P$ such that, for every $x\in X$, $$B(B(x,\alpha),\beta)\subseteq B(x,\gamma);$$
- for any $x,y\in X$, there exists $\alpha\in P$ such that $y\in B(x, \alpha)$.
We note that a ballean can be considered as an asymptotic counterpart of a uniform space, and could be defined [@b7] in terms of entourages of the diagonal $\Delta_X$ in $X\times X$. In this case a ballean is called a [*coarse structure*]{}. For categorical look at the balleans and coarse structures as “two faces of the same coin” see [@b2].
Let $\mathcal{B}=(X,P,B)$, $\mathcal{B'}=(X',P',B')$ be balleans. A mapping $f:X\to X'$ is called [*coarse*]{} if, for every $\alpha\in P$, there exists $\alpha'\in P'$ such that, for every $x\in X$, $f(B(x,\alpha))\subseteq B'(f(x),\alpha')$.
A bijection $f:X\rightarrow X'$ is called an [*asymorphism*]{} between ${{\mathcal B}}$ and ${{\mathcal B}}'$ if $f$ and $f^{-1}$ are coarse. In this case ${{\mathcal B}}$ and ${{\mathcal B}}'$ are called [*asymorphic*]{}.
Let ${{\mathcal B}}= (X,P,B)$ be a ballean. Each subset $Y$ of $X$ defines a [*subballean*]{} ${{\mathcal B}}_Y = (Y,P,B_Y)$, where $B_Y(y,\alpha) = Y \cap B(y, \alpha)$. A subset $Y$ of $X$ is called [*large*]{} if $X = B(Y, \alpha)$, for $\alpha \in P$. Two balleans ${{\mathcal B}}$ and ${{\mathcal B}}'$ with supports $X$ and $X'$ are called [*coarsely equivalent*]{} if there exist large subsets $Y\subseteq X$ and $Y' \subseteq X'$ such that the subballeans ${{\mathcal B}}_Y$ and ${{\mathcal B}}'_{Y'}$ are asymorphic.
Every infinite group $G$ can be considered as the ballean $(G, \mathfrak{F}_{G}, B)$, where $ \mathfrak{F}_{G} $ is the family of all finite subsets of $G$, $B(g,F)= Fg \bigcup \{g\}$.
We note that finitely generated groups are finitary coarsely equivalent if and only if $G$ and $H$ are quasi-isometric [@b3 Chapter 4].
A classification of countable locally finite groups (each finite subset generates finite subgroup) up to asymorphisms is obtained in [@b4](see also [@b5 p. 103]).
*Two countable locally finite groups $G_{1}$ and $G_{2}$ are asymorphic if and only if the following conditions hold:*
$(i)$ for every finite subgroup $F\subset G_{1}$, there exists a finite subgroup $H$ of $G_{2}$ such that $|F|$ is a divisor of $|H|$;
$(ii)$ for every finite subgroup $H$ of $G_{2}$, there exists a finite subgroup $F$ of $G_{1}$ such that $|F|$ is a divisor of $|F|$.
It follows that there are continuum many distinct types of countable locally finite groups and each group is asymorphic to some direct sum of finite cyclic groups.
The following coarse classification of countable Abelian groups is obtained in [@b1].
[*Two countable Abelian groups are coarsely equivalent if and only if the torsion-free ranks of $G$ and $H$ coincide and $G$ and $H$ are either both finitely generated or infinitely generated.*]{}
In particular, any two countable torsion Abelian groups are coarsely equivalent.
Given a group $G$, we consider each non-empty subsets as a subballean of $G$ and say that a class of all pairwise coarsely equivalent subsets is a [*coarse subset* ]{} of $G$.
For a countable group $G$, we prove that there as many coarse subsets of $G$ as possible by the cardinal arithmetic.
[**Theorem.**]{} [ *For a countable group $G$, there are $2^{\omega}$ coarse subsets of $G$.*]{}
Every countable group $G$ contains either countable finitely generated subgroup or countable locally finite subgroup, so we split the proof into corresponding cases.
Proof: finitely generated case
==============================
2.1. We take a finite system $S,$ $S=S^{-1}$ of generators of $G$ and consider the Cayley graph $\Gamma$ with the set of vertices $G$ and the set of edges $\{\{g,h\}: gh^{-1} \in S, \ g\neq h\}$. We denote by $\rho$ the path metric on $\Gamma$ and choose a geodesic ray $V=\{v_{n}: n\in\omega\}$, $v_{0}$ is the identity of $G$, $\rho(v_{n}, v_{m})=|n-m|$.
Then the subballean of $G$ with the support $V$ is asymorphic to the metric ballean $ \ (\mathbb{N}, \ \mathbb{N}\bigcup\{0\}, \ B)$, where $B(x,r)=\{y\in\mathbb{N}: d(x,y )\leq r\}, \ d(x,y)= |x-y|$. Thus, it suffices to find a family $\mathfrak{F}$, $|\mathfrak{F}|= 2^{\omega}$ of pairwise coarsely non-equivalent subsets of $\mathbb{N}$.
2.2. We choose a sequence $(I_{n})_{n\in\omega}$ of intervals of $\mathbb{N}$, $I_{n}=[a_{n}, b_{n}]$, $ \ b_{n}<a_{n+1}$ such that
$(1) \ \ \ \ b_{n}- a_{n} > n \ a_{n}.$
Then we take an almost disjoint family $\mathcal{A}$ of infinite subsets of $\omega$ such that $|\mathcal{A}|=2^{\omega}$. Recall that $\mathcal{A}$ is almost disjoint if $|W \bigcap W^{\prime}|<\omega$ for all distinct $W, \ W^{\prime} \in \mathcal{A}$.
For each $W\in\mathcal{A}$, we denote $I_{W}= \bigcup\{I_{n}: n\in W\}$. To show that $\mathfrak{F}=\{ I_{W}: W\in \mathcal{A}\}$ is the desired family of subsets of $\mathbb{N}$, we take distinct $W, W^{\prime}\in\mathcal{A}$ and assume that $I_{W}$, $I_{W^{\prime}}$ are coarsely equivalent. Then there exist large subsets $X, X^{\prime}$ of $I_{W}$, $I_{W^{\prime}}$, and an asymorphism $f: X\longrightarrow X^{\prime}$. We choose $r\in\mathbb{N}$ such that $$I_{W}\subseteq B(X, r), \ I_{W^{\prime}}\subseteq B(X^{\prime}, r)$$ and note that if an interval $I$ of length $2r$ is contained in $I_{W}$ then $I$ must contain at least one point of $X$, and the same holds for the pair $I_{W^{\prime}}, X^{\prime}$.
Since $f$ is an asymorphism, we can take $t\in\mathbb{N}$ such that, for all $x\in X$, $ \ x^{\prime}\in X^{\prime}$,
$(2) \ \ \ \ f(B_{X}(x, 2r+2)\subseteq B_{X^{\prime}}(f(x), t);$
$(3) \ \ \ \ f^{-1}(B_{X^{\prime}}(x^{\prime}, 2r+2)\subseteq B_{X}(f^{-1}(x^{\prime}), t).$
We use $(1)$ to choose $m\in W\backslash W^{\prime}$, $m> \max (W\bigcap W^{\prime})$ such that
$(4) \ \ \ \ b_{m}-a_{m} > 2r a_{m};$
$(5) \ \ \ \ b_{m}-a_{m} > 2t.$
2.3. We denote $Z=X\bigcap [a_{m}, b_{m}]$ and enumerate $Z$ in increasing order $Z=\{z_{0}, \ldots, z_{k}\}$. Then $d(z_{i}, z_{i+1})\leq 2r + 2$ because otherwise the interval $[z_{i}+1, z_{i+1}-1]$ of length $2r$ has no points of $X$.
If $f(z_{0})< a_{m}$ then, by $(2)$ and $(5)$, $f(Z)\subseteq [1, a_{m}-1]$. On the other hand, $k\geq(b_{m}-a_{m})/2r - 1$ and, by $(4)$, $ \ (b_{m}-a_{m})/2r > a_{m}$. Hence, $k >a_{m}-1$ contradicting $f(Z)\subseteq [1, a_{m}-1]$ because $f$ is a bijection.
If $f(z_{0})> b_{m}$ then we take $s\in W^{\prime}$ such that $f(z_{0})\in I_{s}$. Since $m> \max (W\wedge W^{\prime})$ and $s> m$, we have $s\in W^{\prime}\setminus W$, so we can repeat above argument for $f^{-1}$ and $I_{s}$ in place of $f$ and $I_{m}$ with usage $(3)$ instead of $(2)$.
Proof: locally finite case
===========================
3.1. Let $G$ be an arbitrary countable group and let $X$, $A$ be infinite subsets of $G$. Suppose that there exist an infinite subset $Y$ of $X$, a partition $A= B\bigcup C$ and $k, l\in\mathbb{N} $, $k < l$ such that
$(6) \ \ \ \ $ there exists $ H\in \mathfrak{F}_{G}$ such that, for every $y\in Y$, $$|B_{X}(y, H)|\geq k;$$
$(7) \ \ \ \ $ for every $ F \in \mathfrak{F}_{G}$, there exists $ Y^{\prime}\in \mathfrak{F}_{G}$ such that, for every $y\in Y\backslash Y^{\prime}$, $$|B_{X}(y, H)|\geq l;$$
$(8) \ \ \ \ $ there exists $ K\in \mathfrak{F}_{G}$ such that, for every $b\in B$, $$|B_{A}(b, K)|>l;$$
$(9) \ \ \ \ $ for every $ F \in \mathfrak{F}_{G}$, there exists $ C^{\prime}\in \mathfrak{F}_{G}$ such that, for every $c\in C\backslash C^{\prime}$, $$|B_{A}(c, F)|< k.$$
Then $X$ and $A$ are not asymorphic.
We suppose the contrary and let $f: X\longrightarrow A$ be an asymorphism. We take an infinite subset $I$ of $Y$ such that either $f(I)\subset C$ or $f(I)\subset B$.
Assume that $f(I)\subset C$ and choose $ F \in \mathfrak{F}_{G}$ such that, for every $x\in X$, $$f(B_{X} (x, H))\subseteq B_{A} (f(x), F).$$ For this $F$, we use $(9)$ to choose corresponding $C^{\prime}$. We take $y\in I$ such that $f(y)\in C\setminus C^{\prime}$. By $(6)$, $f(B(y,H))\geq k$. By $(9)$, $ B_{A}(f(y),F)< k$ and we get a contradiction because $f$ is a bijection.
If $f(I)\subset B$ then, by $(8)$, $B_{A}(b,K)>l$ for every $b\in f(I)$. Since $f^{-1}$ is coarse, there is $ F \in \mathfrak{F}_{G}$ such that, for every $a\in A$ $$f^{-1}(B_{A}(a), K)\subseteq B_{X} (f^{-1}(a), F).$$ For this $F$, we choose $Y^{\prime}$ satisfying $(7)$ and get a contradiction.
3.2. Now we assume that $G$ is locally finite and show a plan how to choose the desired family $\mathfrak{F}$, $|\mathfrak{F}| = 2^{\omega}$ of pairwise coarsely non-equivalent subsets of $G$.
We construct some special sequence $(Y_{n})_{n\in\omega}$ of pairwise disjoint subsets of $G$. Then we take a family $\mathcal{A}$ of almost disjoint infinite subset of $\omega$, $|\mathfrak{F}| = 2^{\omega}$, denote
$(10) \ \ \ \ X_{W} =\bigcup\{ Y_{n}: n\in W\}, \ \ W\in \mathcal{A},$
and get $\mathfrak{F}$ as $\{ X_{W}: W\in \mathcal{A}\}.$
3.3. We represent $G$ as the union of an increasing chain $\{F_{n}: n\in\omega\}$ of finite subgroups such that
$(11) \ \ \ \ |F_{n+1}| > |F_{n}|^{2}.$
Then we choose a double sequence $(g_{nm})_{n,m\in\omega}$ of elements of $G$ such that
$(12) \ \ \ \ F_{n} F_{m} g_{n m} \bigcap F_{i} F_{j} g_{ij}=\emptyset$ for all distinct $(n, m)$, $(i, j)$ from $\omega\times\omega$, and put $$Y_{n}=\bigcup\{ F_{m}g_{n m}:m\in\omega\}.$$
3.4. We take distinct $W, W^{\prime}\in\mathcal{A}$ and prove that $X_{W}$ and $X_{W^{\prime}}$ (see (10)) are not coarsely equivalent. We suppose the contrary and choose large asymorphic subsets $Z_{W}$ and $Z_{W^{\prime}}$ of $X_{W}$ and $X_{W^{\prime}}$. Then we take $t\in\omega$ such that $$X_{W}\subseteq F_{t} Z_{W}, \ X_{W^{\prime}}\subseteq F_{t} Z_{W^{\prime}}.$$ If $n>t$ and either $F_{n}g_{nm} \subset X_{W}$ or $F_{n}g_{nm}\subset X_{W^{\prime}}$ then
$(13) \ \ \ \ |F_{n} g_{n m} \bigcap Z_{W}|\geq \frac{|F_{n}|}{|F_{t}|}, \ \
|F_{n} g_{nm}\bigcap Z_{W^{\prime}}| \geq \frac{|F_{n}|}{|F_{t}|}.$
To apply 3.1, we choose $s\in W\setminus W^{\prime}, \ s>t $ and denote
$$X= Z_{W}, \ \ Y=Y_{s}\bigcap Z_{W}, \ \ A=Z_{W^{\prime}}, \ \ B=\bigcup\{Y_{i}: i\in W^{\prime}, \ \ i>s\},$$ $$C= \bigcup\{Y_{i}: i\in W^{\prime}, \ i<s \}, \ \ k=\frac{|F_{s}|}{|F_{t}|}, \ \ l=|F_{s}|.$$
By (13) with $s=n$, we get (6). By (12) with $s=n$, we get (7).
If $n>s$ then $|F_{n}|/|F_{t}|> |F_{n}|/|F_{s}|$. By (11), $|F_{n}|/|F_{s}|> |F_{s}|$, so $|F_{n}|/|F_{t}|> |F_{s}|$ and, by (13), we have (8).
If $n<s$ then $|F_{n}|<|F_{s}|/|F_{t}|$ and, by (12), we get (9).
Comments
=========
A subset $A$ of an infinite group $G$ is called
$\bullet$ [*thick*]{} if, for every $F\in\mathfrak{F}_{G}$ , there exists $g\in A$ such that $Fg\subset A$;
$\bullet$ [*small*]{} if $L\setminus A$ is large for every large subset $L$ of $G$;
$\bullet$ [*thin*]{} if, for every $F\in\mathfrak{F}_{G}$ , there exists $H\in \mathfrak{F}_{G}$ such that $B_{A} (g, F)= \{g\}$ for each $g\in A\setminus H$.
A subset $A$ is thick if and only if $L\bigcap A\neq\emptyset$ for every large subset $L$ of $G$. For a countable group $G$, in the proof of Theorem, we construct $2^{\omega}$ pairwise coarsely non-equivalent thick subsets of $G$.
Every large subset $L$ of $G$ is coarsely equivalent to $G$, so any two large subsets of $G$ are coarsely equivalent. If $G$ is countable then any two thin subset $S, T$ of $G$ are asymorphic: any bijection $f: S\longrightarrow T$ is an asymorphism. Every thin subset is small. But a small subset $S$ of $G$ could be asymorphic to $G$: we take a group $G$ containing a subgroup $S$ isomorphic to $G$ such that the index of $S$ in $G$ is infinite.
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CONTACT INFORMATION
|
---
abstract: 'We report on the realization and verification of quantum entanglement between an NV electron spin qubit and a telecom-band photonic qubit. First we generate entanglement between the spin qubit and a 637 nm photonic time-bin qubit, followed by photonic quantum frequency conversion that transfers the entanglement to a 1588 nm photon. We characterize the resulting state by correlation measurements in different bases and find a lower bound to the Bell state fidelity of $\geq 0.77 \pm 0.03$. This result presents an important step towards extending quantum networks via optical fiber infrastructure.'
author:
- Anna Tchebotareva
- 'Sophie L. N. Hermans'
- 'Peter C. Humphreys'
- Dirk Voigt
- 'Peter J. Harmsma'
- 'Lun K. Cheng'
- 'Ad L. Verlaan'
- Niels Dijkhuizen
- Wim de Jong
- Anaïs Dréau
- Ronald Hanson
title: ' Entanglement between a diamond spin qubit and a photonic time-bin qubit at telecom wavelength.'
---
[^1]
[^2]
Quantum networks connecting and entangling long-lived qubits via photonic channels [@Wehner2018] may enable new experiments in quantum science as well as a range of applications such as secure information exchange between multiple nodes, distributed quantum computing, clock synchronization, and quantum sensor networks [@BenOr2006; @Broadbent2009; @LiangJiang2009; @Ekert2014; @Cirac1999; @Gottesman2012; @Nickerson2014; @Komar2014; @Bancal2012]. A key building block for long-distance entanglement distribution via optical fibers is the generation of entanglement between a long-lived qubit and a photonic telecom-wavelength qubit (see Fig. 1a). Such building blocks are now actively explored for various qubit platforms [@Dudin2010; @Yamamoto2012; @Eschner2018; @deRiedmatten2018; @Keller2018; @Lanyon2019].
The NV center in diamond is a promising candidate to act as a node in such quantum networks thanks to a combination of long spin coherence and spin-selective optical transitions that allow for high fidelity initialization and single-shot read out [@Awshalom2018]. Moreover, memory qubits are provided in the form of surrounding carbon-13 nuclear spins. These have been employed for demonstrations of quantum error correction [@Cramer2016; @Waldherr2014; @Taminiau2014] and entanglement distillation [@Kalb2017]. Heralded entanglement between separate NV center spin qubits has been achieved by generating spin-photon entangled states followed by a joint measurement on the photons [@Bernien2013].
Extending such entanglement distribution over long distances is severely hindered by photon loss in the fibers. The wavelength at which the NV center emits resonant photons, the so-called zero-phonon-line (ZPL) at $637$ nm, exhibits high attenuation in optical glass fibers. Quantum-coherent frequency conversion to the telecom band can mitigate these losses by roughly $7$ orders of magnitude for a distance of $10$ km [@Miya1979; @Nagayama2002] and would enable the quantum network to optimally benefit from the existing telecom fiber infrastructure. Recently, we have realized the conversion of 637 nm NV photons to $1588$ nm (in the telecom L-band) using a difference frequency generation (DFG) process and shown that the intrinsic single-photon character is maintained during this process [@Dreau2018]. However, for entanglement distribution an additional critical requirement is that the quantum information encoded by the photon is preserved during the frequency conversion.
Here we demonstrate entanglement between an NV center spin qubit and a time-bin encoded frequency-converted photonic qubit at telecom wavelength. The concept of our experiment is depicted in Fig. 1b. We first generate spin-photon entanglement, then convert the photonic qubit to the telecom band, and finally characterize the resulting state through spin-photon correlation measurements in different bases.
We use two of the NV center electron spin-$1$ sublevels as our qubit subspace. We denote the $m_s = 0$ and $m_s = -1$ ground states as $\ket{0}$ and $\ket{1}$, respectively. To generate the desired spin-photon entangled state, we first initialize the spin in $\ket{0}$ and prepare the balanced superposition $\ket{\psi} = \frac{1}{\sqrt{2}}\left(\ket{0} + \ket{1}\right)$ using a microwave $\pi/2$-pulse. Then we apply a spin-selective optical $\pi$-pulse, such that the $\ket{0}$ state will be excited, followed by photon emission (lifetime of 12 ns). Next, we flip the spin state using a microwave $\pi$-pulse and apply the optical excitation for a second time. This generates the following spin-photon entangled state: $$\ket{\text{NV spin, photon}} = \frac{1}{\sqrt{2}}\ket{1,E} +\frac{1}{\sqrt{2}} \ket{0,L},
\label{eq:state}$$ where the basis states for the photonic qubit are given by the early ($\ket{E}$) and late ($\ket{L}$) time bins, which are separated in the experiment by 190 ns, limited by the state preparation time.
Next, the photon is converted to the telecom wavelength of 1588 nm using a difference frequency generation (DFG) process, by combining it with a strong pump laser inside a periodically poled lithium niobate (PPLN) crystal waveguide (Fig. 1c) [@Dreau2018]. The resulting spin-telecom photon state is characterized via correlation measurements. We read out the photonic qubit in the Z basis using time-resolved detection that discriminates between the early and late time bins. To access other photonic qubit bases we use an imbalanced interferometer [@Franson1989] with a tunable phase difference $\Delta\phi$ between the two arms. For each photonic qubit basis, we read out the spin state in the basis where maximum correlation is expected. From the measured correlations in three orthogonal bases we find the fidelity to the desired maximally entangled state.
The diamond sample containing the NV center is cooled to $\approx 4$ K. The optical setup is schematically depicted in Fig. 2a. Laser light at $637$ nm is used to apply the optical $\pi$-pulses. In the photon detection path, the emitted 637 nm photons are separated from reflected excitation light using a cross-polarization configuration and time filtering. The off-resonant phonon side band emission is separated by dichroic filtering and sent to a detector (D1) for spin readout. The 637 nm photons are combined with a strong pump laser (emission wavelength of $1064$ nm) and directed into the PPLN crystal for the DFG process. Afterwards, the remaining pump laser light is filtered out by a prism, a long-pass dielectric filter and a narrow-band fiber Bragg grating. The total conversion efficiency of the DFG setup is $\eta_c\approx 17\%$ [@Dreau2018]. To ensure the frequency and phase stability of the converted photons, both the NV excitation laser and the pump laser are locked to an external reference cavity (Stable Laser Systems).
Figure 2b shows the experimental sequence used in the experiments. Our protocol starts with checking whether the NV center is in the desired charge state and on resonance with the control lasers [@Robledo2010]. Once this test is passed, the spin-photon entangled state is generated. If a photon is detected, we read out the spin state in the appropriate basis and re-start the protocol. In case no photon is detected, we reinitialize the spin and again generate an entangled state. After 250 failed attempts to detect a photon, we re-start the protocol.
We first measure spin-photon correlations in the ZZ basis. To measure the photon in the Z basis, we send the frequency-converted photons directly to a superconducting nanowire detector (D2) that projects the photonic qubit in the time-bin basis, and, upon photon detection, we read out the spin qubit in the corresponding Z basis. Figure 2c shows the observed correlation data. The probability to measure the spin in $\ket{0}$ is plotted for photon detection events in the early and late time-bins. We have performed this measurement for both the 637 nm photons (red) and for the frequency-converted photons at 1588 nm (purple). For the unconverted photons we measure correlations that are perfect within measurement uncertainty (contrast of $E_{Z} = |P_E\left( \ket{0}\right) - P_L\left( \ket{0}\right)| = 0.997 \pm 0.018$). For the frequency converted photons we measure $P_E\left( \ket{0}\right) = 0.09 \pm 0.05$ for the early time bin and $P_L\left( \ket{0}\right) = 0.95 \pm 0.05$ for the late time bin, yielding a contrast of $E_{Z} = 0.86 \pm 0.07$. All data in this work are corrected for spin readout infidelity and dark counts of the detectors, both of which are determined independently.
The contrast for the telecom photons is lowered by noise coming from spontaneous parametric down converted (SPDC) photons and Raman scattering induced by the strong pump laser [@Dreau2018; @Fejer2010]. We characterize this noise contribution separately by blocking the incoming 637 nm path and find an expected signal to noise ratio (SNR) between $4.8$ and $7.7$. This SNR bounds the maximum observable contrast for the ZZ correlations to $0.85 \pm 0.03$, and thus fully explains our data. We use this SNR later to determine the different noise contributions for the correlation data in the other bases. Additionally, we conclude from the relative number of detection events in the early and late time bin (659 vs 642 events) that the amplitudes of the two parts of the spin-photon entangled state are well balanced.
To verify the spin-photon entanglement, we measure spin-photon correlations in two other spin-photon bases by sending the frequency-converted photons into the imbalanced fiber interferometer (see Fig. 3a). The fiber arm length difference is $\approx 40$ m, which corresponds to a photon travel time difference of $190$ ns between the two arms. In this way the early time bin taking the long arm overlaps at the second beam splitter with the late time bin taking the short arm, thus allowing us to access the phase relation between the two. To access a specific photon qubit basis, we introduce a tunable phase difference $\Delta \phi$ between the long and short arms of the interferometer. In particular, detection of a photon by the detector D3 projects the spin into the state $$\ket{\text{NV}}_{D3} = \frac{1}{\sqrt{2}}\left( \ket{0} + e^{i\left(\Delta\phi-\frac{\pi}{4} \right)}\ket{1}\right).$$ We use two orthogonal set points, labelled X and Y, with $\Delta\phi = \pi/4$ and $\Delta\phi = 3\pi/4$, respectively, as indicated in Fig. 3c. A key requirement for this experiment is that the interferometer is stable with respect to the frequency of the down-converted photons; any instabilities in the interferometer will reduce the interference contrast and prevent us from accessing the true spin-photon correlations. For this reason the interferometer is thermally and vibrationally isolated. Furthermore, we split the experiment into cycles of 1 second (see Fig. 4a), of which the first 100 ms is used to actively stabilize the phase setpoint of the interferometer. Within this 100 ms, we feed metrology light into the interferometer in the reverse direction via shutter S and a circulator. This metrology light is generated by a second DFG setup, using input from the excitation and pump lasers, thus ensuring a fixed frequency relation between the metrology light and the frequency-converted photons. By comparing the light intensities on detectors PD2 and PD3 with the values corresponding to the desired $\Delta\phi$ setpoint as determined from a visibility fringe (calibrated every $100$ s), an error signal is computed and feedback is applied to the fiber piezo stretcher (FPS). After this adjustment the light intensities are measured again. A histogram of the measured phases during the experiments relative to the setpoints is plotted in Fig. 4b. We note that one could also measure the spin-photon correlations at the second output of the interferometer, which for symmetric states as Eq.1 would yield the same correlations but with opposite sign; however, in the current experiment the slow ($\approx 1$ s) recovery of the detector after being blinded due to metrology light leakage through this output port prevented us from using the second output.
In the remaining 900 ms of each cycle spin-photon correlations are measured using the same protocol as for the ZZ basis (see Fig. 2b). To read out the NV spin state in the appropriate rotated basis, the eigenstates $\ket{\text{X}}$ ($\ket{\text{Y}}$) and the $\ket{\text{-X}}$ ($\ket{\text{-Y}}$) are mapped onto the $\ket{0}$ and $\ket{1}$ states, respectively, by applying an appropriate MW pulse before optical readout.
Figure 4c shows the measured spin-photon correlations in the X and Y basis (bottom), along with expected correlations for the ideal state (top). The letters indicate the spin and photon bases respectively, for example -XX indicates that the NV spin is measured along the -X axis on the Bloch sphere, while the photon is projected on +X. The measured contrast between the correlations and anti-correlations in the X basis is $E_{X} = 0.52 \pm 0.07$ and $E_{Y} = 0.69 \pm 0.07$ in the Y basis.
All data show clear (anti-)correlation between the NV spin qubit and the telecom photonic qubit. With the contrast data from all three orthogonal photon readout bases, we calculate the fidelity $\mathcal{F}$ of our produced state (conditioned on photon detection) to the maximally entangled state of Eq. 1 as $$\mathcal{F}= \frac{1}{4}\left(1+ E_{X} + E_{Y} + E_{Z}\right),
%\mathclap{
%\mathscr{F}\!=\! \frac{1}{4}\!\left(\! 1\!+\!\frac{\!|\langle\! \text{XX}\! \rangle\!-\!\langle\!-\text{XX}\!\rangle|}{2}\! +\!
% \frac{|\!\langle\!\text{YY}\!\rangle\!-\!\langle\!-\text{YY}\! \rangle |}{2}\! +\!
% \frac{|\!\langle\text{E}\! \rangle\!-\!\langle\text{L}\!\rangle|}{2}\!\right),
%}$$ yielding a fidelity of $\mathcal{F} = 0.77 \pm 0.03$. This value exceeds the classical boundary of $0.5$ by more than eight standard deviations, proving the generation of entanglement between the NV spin qubit and the frequency-converted photonic qubit. For comparison, reported fidelities for unconverted NV spin-photon entangled states range from $\approx 0.7$ [@Togan2010; @Trupke2018] to more than $ 0.9$ (estimated from an observed spin-spin entangled state fidelity of $\approx 0.9$ [@Hensen.Bell.test]).
The observed fidelity is reduced compared to the ideal value of 1 due to several factors. First, the initial spin-photon entangled state has imperfections, for instance due to photon emission and re-excitation of the NV center during the optical $\pi$-pulse [@Humphreys2018] and small frequency shifts due to spectral diffusion. In addition, the remaining frequency variations of the two locked lasers ($\sim$200 kHz) leads to phase uncertainty between the two terms in Eq.1. All these effects reduce the contrast of the XX and YY correlations, but not that of the ZZ correlations. Second, spontaneous parametric downconversion (SPDC) and Raman scattered photons, produced during the frequency conversion process, add noise to the state as described above and reduce correlations in all bases. Based on these factors, we expect a state fidelity in the range $0.82-0.87$.
The slight difference between the expected and measured state fidelity could be due to the inaccuracies and fluctuations in setting the interferometer phase setpoint. Imperfect interferometer settings result in measurement bases that slightly deviate from the expected X and Y bases, reducing the maximally observable correlations. Therefore, the obtained $\mathcal{F}\geq 0.77 \pm 0.03$ sets a lower bound on the true entangled state fidelity.
In conclusion, we demonstrated entanglement between an NV center spin qubit and a time-bin encoded photonic qubit at telecom wavelength, which is an essential step towards long-distance quantum networks based on remote entanglement between NV center nodes. In future experiments the observed state fidelity can be further increased in several ways. A more narrow band frequency filtering after the DFG1 setup would reduce the added noise in the frequency conversion, as the current narrow-band filter has a linewidth $\sim 10$ times larger than the NV-emitted resonant ZPL photons. The signal could be increased by improving the conversion efficiency. Finally, the emission rate of resonant photons and the collection efficiency can be increased by placing the NV center in an optical cavity [@Faraon2012; @Johnson2015; @Hunger2016; @Riedel2017; @Bogdanovic2017; @Englund2018].
[*Acknowledgements.*]{} We thank L. Schriek, E. Nieuwkoop, J. Lugtenburg and W. Peterse for experimental assistance, and M. J. A. de Dood and C. Osorio Tamayo for useful discussions. We acknowledge financial support from the Netherlands Organisation for Scientific Research (NWO) through a VICI grant and the Zwaartekracht Grant Quantum Software Consortium and the European Research Council through an ERC Consolidator Grant.\
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[^1]: These two authors contributed equally to this work.
[^2]: These two authors contributed equally to this work.
|
---
abstract: 'The Morse-Bott inequalities relate the topology of a closed manifold to the topology of the critical point set of a Morse-Bott function defined on it. The Morse-Bott inequalities are sometimes stated under incorrect orientation assumptions. We show that these assumptions are insufficient with an explicit counterexample and clarify the origin of the mistake.'
author:
- 'Thomas O. Rot'
bibliography:
- 'morse-bott.bib'
title: 'The Morse-Bott inequalities, orientations, and the Thom isomorphism in Morse homology.'
---
The Morse-Bott inequalities relate the topology of a closed manifold $M$ to the topology of the critical manifolds of a Morse-Bott function $f$ defined on it. Let $P_t(M)=\sum_i \mathrm{rank} \,H_i(M;\mZ) t^i$ denote the Poincaré polynomial with $\mZ$ coefficients, and let $MB_t(f)=\sum_j P_t(M_j)t^{| M_j|}$ be the Morse-Bott polynomial. Here the sum runs over the critical submanifolds of the function $f$, and $|M_j|$ denotes the index of the critical submanifold, i.e. the dimension of the negative normal bundle of $M_j$. We always take critical submanifolds to be connected components of the critical point set, which implies that the index is well defined. The Morse-Bott inequalities state that, under suitable orientation assumptions, there exists a polynomial $Q_t$ with non-negative coefficients such that $$\label{eq:morsebott}
MB_t(f)=P_t(M)+(1+t)Q_t.$$ The orientation assumptions differ from paper to paper: In [@jiang] no orientation assumptions are made. In [@banyaga] it is assumed that the critical submanifolds are orientable and in [@banyagadynamical; @hurtubise] it is additionally assumed that the ambient manifold is orientable. Below we show that these hypotheses are insufficient through an explicit counterexample, see also the errata [@Banyaga:QUC7Szzy; @Hurtubise:sz1P0qDm]. We construct a Morse-Bott function with orientable critical submanifolds on an orientable manifold that does not satisfy the Morse-Bott inequalities .
Correct orientation assumptions for the Morse-Bott inequalities to hold are well known: Either one defines the Morse-Bott polynomial using homology with local coefficients in the negative normal bundles of the critical submanifolds, cf. [@bottmorsebott; @Bismut:1986vx], or one requires that the negative normal bundles are orientable as is for example done in [@Nicolaescu:2011jl].
We now discuss the counterexample. Let $f\colon\thinspace\mathbb{R}P^5\rightarrow \mR$ be the function $$f([x_0,x_1,\ldots,x_5])=\frac{-x_4^2+x_5^2}{x_0^2+\ldots+x_5^2}.$$ The function is Morse-Bott with one minimum at $[0,0,0,0,1,0]$, one maximum at $[0,0,0,0,0,1]$, and one other critical submanifold $\mathbb{R}P^3=\{[x_0,x_1,x_2,x_3,0,0]\}$ of index $1$. We know that $P_t(\mR P^{2n+1})=1+t^{2n+1}$ and the Morse-Bott polynomial is $$MB_t(f)=P_t(\mathrm{min})t^0+P_t(\mR P^3)t^1+P_t(\mathrm{max})t^5=1+t+t^4+t^5.$$ There exists no polynomial $Q_t$ with non-negative coefficients such that $1+t+t^4+t^5=1+t^5+(1+t) Q_t$, hence the Morse-Bott inequalities are violated. Note that odd dimensional projective spaces are orientable and that the normal bundle of $\mR P^3$ in $\mR P^5$ is also orientable. This example satisfies the hypotheses of [@banyaga; @banyagadynamical; @hurtubise; @jiang], but violates Equation .
The origin of the mistake is that the Thom isomorphism theorem requires homology with local coefficients or an orientation assumption on the bundle. We explain this briefly and refer to the appendix of [@abbondandoloschwarz] for a more thorough discussion of the Thom isomorphism in Morse homology. Given a closed manifold $M$, a Morse function $f\colon\thinspace M\rightarrow \mR$, and vector bundles[^1] $p^\pm\colon\thinspace E^\pm\rightarrow M$, define a Morse function $F\colon\thinspace E^+\oplus E^-\rightarrow \mR$ by $$\label{eq:morsefunction}
F(x^++x^-)=f(p^+(x^+))+\norm{x^+}_{E^+}^2-\norm{x^-}_{E^-}^2.$$ The critical points and the unstable manifolds of $F$ and $f$ are related as follows: $x\in \mathrm{crit}\, F$ if and only if $p(x)\in \mathrm{crit}\, f$ and $T_xW^u(x;F)\cong T_{p(x)}W^u(x;f)\oplus E^-_{p(x)}$. The generators of the Morse complexes of $f$ and $F$ coincide, but the grading is shifted by the dimension of $E^-$. To define the differential in Morse homology one orients the unstable manifolds. If the fibers $E^-_{p(x)}$ are coherently oriented, i.e. $E^-$ is oriented, then the manifolds $W^u(x;F)\cap W^s(y;F)$ and $W^u(p(x);f)\cap W^s(p(y);f)$ have the same induced orientation. The differentials of both Morse complexes then agree up to the shift in grading, and the Morse homology of $F$ is the Morse homology of $f$ shifted by the dimension of $E^-$. In general the Morse homology of $F$ computes the homology of the disc bundle in $E^-$ modulo its boundary, the sphere bundle in $E^-$, i.e. $HM_*(F)\cong H_*(DE^-,SE^-;\mZ)$. If the bundle $E^-$ is not orientable this does not relate to the Morse homology of $f$ directly. This sign issue can be explicitly seen for the standard Morse function on $M=S^1$ with two critical points and $E^-$ the non-orientable bundle over $S^1$.
The failure of the Thom isomorphism to hold for non-orientable bundles comes up in Morse-Bott theory. To relate the homology of the critical submanifolds to the homology of the ambient manifold, the Morse-Bott function $f$ is perturbed to a Morse function $F=f+g$, where $g$ is sufficiently small and supported near the critical submanifolds. In a suitable tubular neighbourhood of a critical manifold the function $F$ has the form as in Equation . Assuming that all the critical submanifolds have different values, the Morse complex of $F$ is filtered by choices of regular values of $f$. This gives a spectral sequence converging to the Morse homology of $F$. The second page of the spectral sequence are the groups isomorphic to $E_{i,j}^2\cong H_i(DN^-M_j,SN^-M_j;\mZ)$. If the negative normal bundles are orientable, this homology is the sum of the homologies of $M_j$’s shifted by the dimensions of the negative normal bundles $N^-M_j$’s. Standard commutative algebra gives the Morse-Bott inequalities.
*Acknowledgement: The counterexample above is contained in the author’s thesis [@Rot:ww] written under supervision of Federica Pasquotto and Rob Vandervorst. The author thanks them, Hansjörg Geiges, David Hurtubise, and Silvia Sabatini for correspondence on the topic of this paper.*
[^1]: Here and below we implicitly put suitable Riemannian metrics on the various bundles which are not really relevant to the discussion.
|
---
abstract: 'We calculate the transverse momentum distribution for the production of massive lepton-pairs in longitudinally polarized proton-proton reactions at collider energies within the context of perturbative quantum chromodynamics. For values of the transverse momentum $Q_T$ greater than roughly half the pair mass $Q$, $Q_T > Q/2$, we show that the differential cross section is dominated by subprocesses initiated by incident gluons, provided that the polarized gluon density is not too small. Massive lepton-pair differential cross sections should be a good source of independent constraints on the polarized gluon density, free from the experimental and theoretical complications of photon isolation that beset studies of prompt photon production. We provide predictions for the spin-averaged and spin-dependent differential cross sections as a function of $Q_T$ at energies relevant for the Relativistic Heavy Ion Collider (RHIC) at Brookhaven, and we compare these with predictions for real prompt photon production.'
address: |
$^a$High Energy Physics Division, Argonne National Laboratory\
Argonne, Illinois 60439\
$^b$Jefferson Laboratory, Newport News, VA 23606\
$^c$Hampton University, Hampton, VA 23668
author:
- 'Edmond L. Berger$^a$, Lionel E. Gordon$^{b,c}$, and Michael Klasen$^a$'
date: 'September 20, 1999'
title: 'Spin Dependence of Massive Lepton Pair Production in Proton-Proton Collisions'
---
Introduction and Motivation {#sec:1}
===========================
Both massive lepton-pair production, $h_1 + h_2 \rightarrow \gamma^* + X;
\gamma^* \rightarrow l \bar{l}$, and prompt real photon production, $h_1 + h_2 \rightarrow \gamma + X$ are valuable probes of short-distance behavior in hadron reactions. The two reactions supply critical information on parton momentum densities, in addition to the opportunities they offer for tests of perturbative quantum chromodynamics (QCD). Spin-averaged parton momentum densities may be extracted from spin-averaged nucleon-nucleon reactions, and spin-dependent parton momentum densities from spin-dependent nucleon-nucleon reactions. An ambitious experimental program of measurements of spin-dependence in polarized proton-proton reactions will begin soon at Brookhaven’s Relativistic Heavy Ion Collider (RHIC) with kinematic coverage extending well into the regions of phase space in which perturbative quantum chromodynamics should yield reliable predictions.
Massive lepton-pair production, commonly referred to as the Drell-Yan process [@ref:DY], provided early confirmation of three colors and of the size of next-to-leading contributions to the cross section differential in the pair mass Q. The mass and longitudinal momentum (or rapidity) dependences of the cross section (integrated over the transverse momentum $Q_T$ of the pair) serve as laboratory for measurement of the [*antiquark*]{} momentum density, complementary to deep-inelastic lepton scattering from which one gains information of the sum of the quark and antiquark densities. Inclusive prompt real photon production is a source of essential information on the [*gluon*]{} momentum density. At lowest order in perturbation theory, the reaction is dominated at large values of the transverse momentum $p_T$ of the produced photon by the “Compton" subprocess, $q + g \rightarrow \gamma + q$. This dominance is preserved at higher orders, indicating that the experimental inclusive cross section differential in $p_T$ may be used to determine the density of gluons in the initial hadrons [@ref:BQ; @ref:Baer; @ref:Aurenche; @ref:GV].
In two previous papers [@ref:BGKDY], we addressed the production of massive lepton-pairs as a function of the transverse momentum $Q_T$ of the pair in unpolarized nucleon-nucleon reactions, $h_1 + h_2 \rightarrow
\gamma^* + X$, in the region where $Q_T$ is greater than roughly half of the mass of the pair, $Q_T > Q/2$. We demonstrated that the differential cross section in this region is dominated by subprocesses initiated by incident gluons. Correspondingly, massive lepton-pair differential cross sections in unpolarized nucleon-nucleon reactions are a valuable, heretofore overlooked, independent source of constraints on the spin-averaged gluon density.
Turning to longitudinally polarized proton-proton collisions in this paper, we study the potential advantages that the Drell-Yan process may offer for the determination of the spin-dependence of the gluon density. To be sure, the cross section for massive lepton-pair production is smaller than it is for prompt photon production. However, just as in the unpolarized case, massive lepton pair production is cleaner theoretically since long-range fragmentation contributions are absent as are the experimental and theoretical complications associated with isolation of the real photon. Moreover, the dynamics of spin-dependence in hard-scattering processes is a sufficiently complex topic, and its understanding at an early stage in its development, that several defensible approaches for extracting polarized parton densities deserve to be pursued with the expectation that consistent results must emerge.
There are notable similarities and differences in the theoretical analyses of massive lepton-pair production and prompt real photon production. At first-order in the strong coupling strength, $\alpha_s$, the Compton subprocess and the annihilation subprocess $q +\bar{q} \rightarrow
\gamma + g$ supply the transverse momentum of the [*directly*]{} produced prompt photons. Identical subprocesses, with the real $\gamma$ replaced by a virtual $\gamma^*$, are responsible at ${\cal O}(\alpha_s)$ for the transverse momentum of massive lepton-pairs. An important distinction, however, is that fragmentation subprocesses play a very important role in prompt real photon production at collider energies. In these long-distance fragmentation subprocesses, the photon emerges from the fragmentation of a final parton, e.g., $q + g \rightarrow q + g$, followed by $q \rightarrow \gamma + X$. The necessity to invoke phenomenological fragmentation functions and the infrared ambiguity [@ref:BGQ] of the isolated cross section in next-to-leading order raise questions about the extent to which isolated prompt photon data may be used for fully quantitative determinations of the gluon density. It is desirable to investigate other physical processes for extraction of the gluon density that are free from these systematic uncertainties. Fortunately, no isolation would seem necessary in the case of virtual photon production (and subsequent decay into a pair of muons) in typical collider or fixed target experiments. Muons are observed only after they have penetrated a substantial hadron absorber. Thus, any hadrons within a typical cone about the direction of the $\gamma^*$ will have been stopped, and the massive lepton-pair signal will be entirely inclusive.
Another significant distinction between massive lepton-pair production and prompt real photon production is that interest in $h_1 + h_2 \rightarrow
\gamma^* + X$ has been drawn most often to the domain in which the pair mass $Q$ is relatively large, justifying a perturbative treatment based on a small value of $\alpha_s(Q)$ and the neglect of inverse-power high-twist contributions (except near the edges of phase space). The focus in prompt real photon production is directed to the region of large values of $p_T$ where $\alpha_s(p_T)$ is small. Interest in the transverse momentum $Q_T$ dependence of the massive lepton-pair production cross section has tended to be limited to small values of $Q_T$ where the cross section is largest. Fixed-order perturbation theory [@ref:Reno] is applicable for large $Q_T$, but it is inadequate at small $Q_T$, and all-orders resummation methods [@ref:CSS; @ref:DWS; @ref:AEGM; @ref:AK; @ref:LY] have been developed to address the region $Q_T << Q$.
As long as $Q_T$ is large, the perturbative requirement of small $\alpha_s(Q_T)$ can be satisfied without a large value of $Q$. We therefore explore and advocate the potential advantages of studies of $d^2\sigma/dQ dQ_T$ as a function of $Q_T$ for modest values of $Q$, $Q \sim 2$ to 3 GeV, below the range of the traditional Drell-Yan region. There are various backgrounds with which to contend at small $Q$ such as the contributions to the event rate from prompt decays of heavy flavors, e.g., $h_1 + h_2 \rightarrow c + \bar{c} + X; c \rightarrow l + X$. These heavy flavor contributions may be estimated by direct computation [@ref:BerSop] and/or bounded through experimental measurement of the like-sign-lepton distributions.
In Sec. II, we review perturbative QCD calculations of the transverse momentum distribution for massive lepton-pair production in the case in which the initial nucleon spins are polarized as well as in the spin-average case. In Sec. III, we present next-to-leading order predictions for the transverse momentum dependence of the cross sections for massive lepton-pair and real prompt photon production in unpolarized proton-proton collisions at energies typical of the RHIC collider. Predictions for spin dependence are provided in Sec. IV. Our conclusions are summarized in Sec. V.
Massive Lepton Pair Production and Prompt Photon Production at Next-to-leading Order {#sec:2}
====================================================================================
In inclusive hadron interactions at collider energies, $h_1 + h_2 \rightarrow
\gamma^* + X$ with $\gamma^* \rightarrow l \bar{l}$, lepton pair production proceeds through partonic hard-scattering processes involving initial-state light quarks $q$ and gluons $g$. In lowest-order QCD, at ${\cal O}(\alpha_s^0)$, the only partonic subprocess is $q + \bar{q}
\rightarrow \gamma^*$. At ${\cal O}(\alpha_s)$, both $q + \bar{q} \rightarrow
\gamma^* + g$ and $q + g \rightarrow \gamma^* + q$ participate, with the recoil of the final parton balancing the transverse momentum of the lepton-pair. These processes are shown in Figs. 1(a) and 2(a). Calculations of the cross section at order ${\cal O}(\alpha_s^2)$ involve virtual loop corrections to these ${\cal O}(\alpha_s)$ subprocesses (Figs. 1(b) and 2(b)) as well as contributions from a wide range of $2 \rightarrow 3$ parton subprocesses (of which some examples are shown in Figs. 1(c) and 2(c)).
The physical cross section is obtained through the factorization theorem, $$\frac{d^2\sigma_{h_1h_2}^{\gamma^*}}{dQ_T^2dy} = \sum_{ij} \int dx_1 dx_2
f^i_{h_1}(x_1,\mu_f^2) f^j_{h_2}(x_2,\mu_f^2)
\frac{sd^2\hat{\sigma}_{ij}^{\gamma^*}}
{dtdu}(s,Q,Q_T,y;\mu_f^2).
\label{dy1}$$ It depends on the hadronic center-of-mass energy $S$ and on the mass $Q$, the transverse momentum $Q_T$, and the rapidity $y$ of the virtual photon; $\mu_f$ is the factorization scale of the scattering process. The usual Mandelstam invariants in the partonic system are defined by $s =
(p_1+p_2)^2,~t = (p_1-p_{\gamma^*})^2$, and $u = (p_2-p_{\gamma^*})^2$, where $p_1$ and $p_2$ are the momenta of the initial state partons and $p_{\gamma^*}$ is the momentum of the virtual photon. The indices $ij \in
\{q\bar{q},qg\}$ denote the initial parton channels whose contributions are added incoherently to yield the total physical cross section. Functions $f^j_{h}(x,\mu)$ denote the usual spin-averaged parton distribution functions.
The partonic cross section $\hat\sigma_{ij}^{\gamma^*}(s,Q,Q_T,y;\mu_f^2)$ is obtained commonly from fixed-order QCD calculations through $$\frac{d^2\hat{\sigma}_{ij}^{\gamma^*}}{dtdu} =
\alpha_s (\mu^2) \frac{d^2\hat{\sigma}_{ij}^{\gamma^*,(a)}}{dtdu}
+ \alpha_s^2(\mu^2) \frac{d^2\hat{\sigma}_{ij}^{\gamma^*,(b)}}{dtdu}
+ \alpha_s^2(\mu^2) \frac{d^2\hat{\sigma}_{ij}^{\gamma^*,(c)}}{dtdu}
+ {\cal O} (\alpha_s^3).
\label{dy2}$$ The tree, virtual loop, and real emission contributions are labeled (a), (b), and (c) as are the corresponding diagrams in Figs. 1 and 2. The parameter $\mu$ is the renormalization scale. It is set equal to the factorization scale $\mu_f = \sqrt{Q^2+Q_T^2}$ throughout this paper.
The cross section for $h_1 + h_2 \rightarrow \l\bar{l} + X$, differential in the invariant mass of the lepton pair $Q^2$ as well as its transverse momentum and rapidity, is obtained from Eq. (\[dy1\]) by the relation $$\frac{d^3\sigma_{h_1h_2}^{l\bar{l}}}{dQ^2dQ_T^2dy} = \left(
\frac{\alpha_{em}}{3\pi Q^2} \right) \frac{d^2\sigma_{h_1h_2}^{\gamma^*}}
{dQ_T^2dy}(S,Q,Q_T,y),
\label{dy3}$$ where $Q^2 = (p_l + p_{\bar l})^2$, and $p_l, p_{\bar l}$ are the four-momenta of the two final leptons. The Drell-Yan factor $\alpha_{em}/(3\pi Q^2)$ is included in all numerical results presented in this paper.
While the full next-to-leading order QCD calculation exists for massive lepton-pair production in the case of unpolarized initial nucleons, only a partial calculation is available in the polarized case [@ref:CCFG]. Correspondingly, we present spin-averaged differential cross sections at next-to-leading order, but we calculate spin asymmetries at leading order. Spin asymmetries are obtained by dividing the spin-dependent differential cross section by its spin-averaged counterpart. For prompt photon production, comparisons of asymmetries computed at next-to-leading order with those at leading order show only modest differences [@ref:Frixvogel], whereas the cross sections themselves are affected more significantly. Given the similarity of prompt photon production and massive lepton-pair production in the region of $Q_T$ of interest to us [@ref:BGKDY], we expect that the leading-order asymmetries will serve as a useful guide for massive lepton-pair production.
Rewriting Eq. (\[dy3\]) and integrating over an interval in $Q^2$, we calculate the spin-averaged differential cross section $E d^3\sigma^{l\bar{l}}_{h_1 h_2}/dp^3$ as $$\frac{Ed^3\sigma_{h_1 h_2}^{l\bar{l}}}{dp^3} =
\frac{\alpha_{em}}{3\pi^2 S}\sum_{ij}\int^{Q^2_{max}}_{Q^2_{min}}
\frac{dQ^2}{Q^2} \int^1_{x^{min}_1} \frac{dx_1}{x_1-\bar{x}_1}
f^i_{h_1}(x_1,\mu_f^2) f^j_{h_2}(x_2,\mu_f^2)
s \frac{d\hat{\sigma}_{ij}^{\gamma^*}}
{dt}.
\label{sdy1}$$ In Eq. (\[sdy1\]), $Q^2_{max}$ and $Q^2_{min}$ are the chosen upper and lower limits of integration for $Q^2$, and $x_1^{min}=(\bar{x}_1-\tau)/(1-\bar{x}_2)$. The value of $x_2$ is determined from $x_2=(x_1\bar{x}_2-\tau)/(x_1-\bar{x}_1)$, with $$\bar{x}_1=\frac{Q^2-U}{S}=\frac{1}{2}e^y\sqrt{x_T^2+4\tau} ,$$ and $$\bar{x}_2=\frac{Q^2-T}{S}=\frac{1}{2}e^{-y}\sqrt{x_T^2+4\tau}.$$ We use $P_1$ and $P_2$ to denote the four-vector momenta of the incident nucleons; $S = (P_1 + P_2)^2$. The invariants in the hadronic system, $T=(P_1-p_{\gamma^*})^2$ and $U=(P_2-p_{\gamma^*})^2$, are related to the partonic invariants by $$(t-Q^2)=x_1(T-Q^2)=-x_1\bar{x}_2S ,$$ and $$(u-Q^2)=x_2(U-Q^2)=-x_2\bar{x}_1S .$$ The scaled variables $x_T$ and $\tau$ are $$x_T=\frac{2 Q_T}{\sqrt{S}},\;\;\;\; \tau = \frac{Q^2}{S}.$$
When the initial nucleons are polarized longitudinally, we can compute the difference of cross sections $$\Delta \sigma = {\sigma_{++}-\sigma_{+-}} ,$$ where $+,-$ denote the helicities of the incident nucleons.
In analogy to Eq. (\[sdy1\]), we find $$\frac{Ed^3\Delta \sigma_{h_1 h_2}^{l\bar{l}}}{dp^3} =
\frac{\alpha_{em}}{3\pi^2 S}\sum_{ij}\int^{Q^2_{max}}_{Q^2_{min}}
\frac{dQ^2}{Q^2} \int^1_{x^{min}_1} \frac{dx_1}{x_1-\bar{x}_1}
\Delta f^i_{h_1}(x_1,\mu_f^2) \Delta f^j_{h_2}(x_2,\mu_f^2)
s \frac{d\Delta \hat{\sigma}_{ij}^{\gamma^*}}
{dt}.
\label{sdy2}$$ The functions $\Delta f^j_{h}(x,\mu)$ denote the spin-dependent parton distribution functions, defined by $$\Delta f^i_h(x,\mu_f)=f^i_{h,+}(x,\mu_f)-f^i_{h,-}(x,\mu_f) ;$$ $f^i_{h\pm}(x,\mu_f)$ is the distribution of partons of type $i$ with positive $(+)$ or negative $(-)$ helicity in hadron $h$. Likewise, the polarized partonic cross section $\Delta \hat {\sigma}^{\gamma^*}$ is defined by $$\Delta \hat{\sigma}^{\gamma^*}=\hat {\sigma}^{\gamma^*}(+,+)-
\hat {\sigma}^{\gamma^*}(+,-),$$ with $+,-$ denoting the helicities of the incoming partons.
The hard subprocess cross sections in leading order for the unpolarized and polarized cases are $$s\frac{d\hat{\sigma}_{q\bar{q}}}{dt}=-s\frac{d\Delta\hat{\sigma}_{q\bar{q}}}{dt}
=e_q^2\frac{2\pi\alpha_{em}C_F}{N_C}\frac{\alpha_s}{s}\left[\frac{u}{t}+
\frac{t}{u}+\frac{2Q^2(Q^2-u-t)}{u t}\right],$$
$$s\frac{d\sigma_{qg}}{dt}=-e^2_q\frac{\pi
\alpha_{em}}{N_C}\frac{\alpha_s}{s}\left[\frac{s}{t}+\frac{t}{s}+\frac{2Q^2
u}{s t}\right] ,$$
and $$s\frac{d\Delta \sigma_{qg}}{dt}=e^2_q\frac{\pi
\alpha_{em}}{N_C}\frac{\alpha_s}{s}\left[\frac{2u+s}{t}-\frac{2u+t}{s}\right] .$$
Our results on the longitudinal spin dependence are expressed in terms of the two-spin longitudinal asymmetry $A_{LL}$, defined by $$A_{LL} = \frac{\sigma^{\gamma^*}(+,+)-\sigma^{\gamma^*}(+,-)}
{\sigma^{\gamma^*}(+,+)+\sigma^{\gamma^*}(+,-)},$$ where $+,-$ denote the helicities of the incoming protons.
Unpolarized Cross Sections {#sec:3}
==========================
We turn in this Section to explicit evaluations of the differential cross sections as functions of $Q_T$ at collider energies. We work in the $\overline{\rm MS}$ renormalization scheme and set the renormalization and factorization scales equal. We employ the MRST set of spin-averaged parton densities [@ref:MRST] and a two-loop expression for the strong coupling strength $\alpha_s(\mu)$, with five flavors and appropriate threshold behavior at $\mu = m_b$; $\Lambda^{(4)} = 300$ MeV. The strong coupling strength $\alpha_s$ is evaluated at a hard scale $\mu = \sqrt{Q^2+Q_T^2}$. We present results for three values of the center-of-mass energy, $\sqrt S =$ 50, 200, and 500 GeV.
For $\sqrt S =$ 200 GeV, we present the invariant inclusive cross section $Ed^3\sigma/d p^3$ as a function of $Q_T$ in Fig. 3. Shown in this figure are the $q {\bar q}$ and $q g$ perturbative contributions to the cross section at leading order and at next-to-leading order. We average the invariant inclusive cross section over the rapidity range -1.0 $< y <$ 1.0 and over the mass interval 5 $<Q<$ 6 GeV. For $Q_T<$ 1.5 GeV, the $q {\bar q}$ contribution exceeds that of $q g$ channel. However, for values of $Q_T >$ 1.5 GeV, the $q g$ contribution becomes increasingly important. As shown in Fig. 4(a), the $q g$ contribution accounts for about 80 % of the rate once $Q_T \simeq Q$. The results in Fig. 4(a) also demonstrate that subprocesses other than those initiated by the $q {\bar q}$ and $q g$ initial channels contribute negligibly.
In Fig. 4(b), we display the fractional contributions to the cross section as a function of $Q_T$ for a larger value of Q: 11 $<Q<$ 12 GeV. In this case, the fraction of the rate attributable to $qg$ initiated subprocesses again increases with $Q_T$. It becomes 80 % for $Q_T \simeq Q$.
For the calculations reported in Figs. 3 and 4(a,b), we chose values of Q in the traditional range for studies of massive lepton-pair production, viz., above the interval of the $J/\psi$ and $\psi'$ states and either below or above the interval of the $\Upsilon's$.
For Fig. 4(c), we select the interval 2.0 $<Q<$ 3.0 GeV. In this region, one would be inclined to doubt the reliability of leading-twist perturbative descriptions of the cross section $d\sigma/dQ$, [*integrated*]{} over all $Q_T$. However for values of $Q_T$ that are large enough, a perturbative description of the $Q_T$ dependence of $d^2\sigma/dQdQ_T$ ought to be justified. The results presented in Fig. 4(c) demonstrate that, as at higher masses, the $qg$ incident subprocesses dominate the cross section for $Q_T \simeq Q$.
The calculations presented in Figs. 4 show convincingly that data on the transverse momentum dependence of the cross section for massive lepton-pair production at RHIC collider energies should be a valuable independent source of information on the spin-averaged gluon density.
In Fig. 5, we provide next-to-leading order predictions of the differential cross section as a function of $Q_T$ for three values of the center-of-mass energy and two intervals of mass $Q$. Taking $Ed^3\sigma/dp^3 = 10^{-3} \rm{pb/GeV}^2$ as the minimum accessible cross section, we may use the curves in Fig. 5 to establish that the massive lepton-pair cross section may be measured to $Q_T =$ 7.5, 14, and 18.5 GeV at $\sqrt S =$ 50, 200, and 500 GeV, respectively, when 2 $< Q <$ 3 GeV, and to $Q_T =$ 6, 11.5, and 15 GeV when 5 $< Q <$ 6 GeV. In terms of reach in the fractional momentum $x_{gluon}$ carried by the gluon, these values of $Q_T$ may be converted to $x_{gluon} \simeq x_T = 2 Q_T/\sqrt S =$ 0.3, 0.14, and 0.075 at $\sqrt S =$ 50, 200, and 500 GeV when 2 $< Q <$ 3 GeV, and to $x_{gluon} \simeq$ 0.24, 0.115, and 0.06 when 5 $< Q <$ 6 GeV. On the face of it, the smallest value of $\sqrt S$ provides the greatest reach in $x_{gluon}$. However, the reliability of fixed-order perturbative QCD as well as dominance of the $qg$ subprocess improve with greater $Q_T$. The maximum value $Q_T \simeq $ 7.5 GeV attainable at $\sqrt S = 50$ GeV argues for a larger $\sqrt S$.
It is instructive to compare our results with those expected for prompt real photon production. In Fig. 6, we present the predicted differential cross section for prompt photon production for three center-of-mass energies. We display the result with full fragmentation taken into consideration (upper line) and with no fragmentation contributions included (lower line). Comparing the magnitudes of the prompt photon and massive lepton pair production cross sections in Figs. 5 and 6, we note that the inclusive prompt photon cross section is a factor of 1000 to 4000 greater than the massive lepton-pair cross section integrated over the mass interval 2.0 $< Q <$ 3.0 GeV, depending on the value of $Q_T$. This factor is attributable in large measure to the factor $\alpha_{em}/(3 \pi Q^2)$ associated with the decay of the virtual photon to $\mu^+ \mu^- $. Again taking $Ed^3\sigma/dp^3 = 10^{-3} \rm{pb/GeV}^2$ as the minimum accessible cross section, we may use the curves in Fig. 6 to establish that the real photon cross section may be measured to $p_T =$ 14, 33, and 52 GeV at $\sqrt S =$ 50, 200, and 500 GeV, respectively. The corresponding reach in $x_T = 2 p_T/\sqrt S =$ 0.56, 0.33, and 0.21 at $\sqrt S =$ 50, 200, and 500 GeV is two to three times that of the massive lepton-pair case.
The breakdown of the real photon direct cross section at $\sqrt S =$ 200 GeV into its $q {\bar q}$ and $qg$ components is presented in Fig. 7. As may be appreciated from a comparison of Figs. 4 and 7, dominance of the $qg$ contribution in the massive lepton-pair case is as strong as in the prompt photon case. The significantly smaller cross section in the case of massive lepton-pair production means that the reach in $x_{gluon}$ is restricted to about a factor of two to three less, depending on $\sqrt S$ and $Q$, than that potentially accessible with prompt photons in the same sample of data. Nevertheless, it is valuable to be able to investigate the gluon density with a process that has reduced experimental and theoretical systematic uncertainties from those of the prompt photon case.
In our previous papers [@ref:BGKDY] we compared our spin-averaged cross sections with available fixed-target and collider data on massive lepton-pair production at large values of $Q_T$, and we were able to establish that fixed-order perturbative calculations, without resummation, should be reliable for $Q_T > Q/2$. The region of small $Q_T$ and the matching region of intermediate $Q_T$ are complicated by some level of phenomenological ambiguity. Within the resummation approach, phenomenological non-perturbative functions play a key role in fixing the shape of the $Q_T$ spectrum at very small $Q_T$, and matching methods in the intermediate region are hardly unique. For the goals we have in mind, it would appear best to restrict attention to the region $Q_T \geq Q/2$.
Predictions for Spin Dependence {#sec:4}
===============================
Given theoretical expressions derived in Sec. II that relate the spin-dependent cross section at the hadron level to spin-dependent partonic hard-scattering matrix elements and polarized parton densities, we must adopt models for spin-dependent parton densities in order to obtain illustrative numerical expectations. For the spin-dependent densities that we need, we use the three different parametrizations suggested by Gehrmann and Stirling (GS) [@GS]. We have verified that the positivity requirement $\left | \Delta f^j_{h}(x,\mu_f)/f^j_{h}(x,\mu_f) \right | \le 1$ is satisfied.
The current deep inelastic scattering data do not constrain the polarized gluon density tightly, and most groups present more than one plausible parametrization. Gehrmann and Stirling [@GS] present three such parametrizations, labelled GSA, GSB, and GSC. In the GSA and GSB sets, $\Delta
G(x,\mu_o)$ is positive for all $x$, whereas in the GSC set $\Delta G(x,\mu_o)$ changes sign. After evolution to $\mu_f^2 = 100$ GeV$^2$, $\Delta G(x,\mu_f)$ remains positive for essentially all $x$ in all three sets, but its magnitude is small in the GSB and GSC sets.
In this Section, we present two-spin longitudinal asymmetries for massive lepton-pair production as a function of transverse momentum. Results are displayed for $pp$ collisions at the center-of-mass energies $\sqrt{S}=$ 50, 200, and 500 GeV typical of the Brookhaven RHIC collider.
In Figs. 8(a-c), we present the two-spin longitudinal asymmetries, $A_{LL}$, as a function of $Q_T$. As noted earlier, these asymmetries are computed in leading-order. More specifically, we use leading-order partonic subprocess cross sections $\hat{\sigma}$ and $\Delta \hat{\sigma}$ with next-to-leading order spin-averaged and spin-dependent parton densities and a two-loop expression for $\alpha_s$. The choice of a leading-order expression for $\Delta \hat{\sigma}$ is required because the full next-to-leading order derivation of $\Delta \hat{\sigma}$ has not been completed for massive lepton-pair production. Experience with prompt photon production indicates that the leading-order and next-to-leading order results for the asymmetry are similar so long as both are dominated by the $qg$ subprocess. Results are shown for three choices of the polarized gluon density. The asymmetry becomes sizable for large enough $Q_T$ for the GSA and GSB parton sets but not in the GSC case. Comparing the three figures, we note that $A_{LL}$ is nearly independent of the pair mass $Q$ as long as $Q_T$ is not too small. This feature should be helpful for the accumulation of statistics; small bin-widths in mass are not necessary, but the $J/\psi$ and $\Upsilon$ resonance regions should be excluded.
As noted above the $qg$ subprocess dominates the [*[spin-averaged]{}*]{} cross section. It is interesting and important to inquire whether this dominance persists in the spin-dependent situation. In Figs. 9 and 10, we compare the contribution to the asymmetry from the polarized $qg$ subprocess with the complete answer for all three sets of parton densities. The $qg$ contribution is more positive than the full answer for values of $Q_T$ that are not too small; the full answer is reduced by the negative contribution from the $q \bar{q}$ subprocess for which the parton-level asymmetry ${\hat{a}}_{LL} = -1$. At small $Q_T$, the net asymmetry may be driven negative by the $q \bar{q}$ contribution, and based on our experience with other calculations [@ref:BergerGordon], from processes such as $gg$ that contribute in next-to-leading order. For the GSA and GSB sets, we see that once it becomes sizable (e.g., 5% or more), the total asymmetry from all subprocesses is dominated by the large contribution from the $qg$ subprocess.
As a general rule in studies of polarization phenomena, many subprocesses can contribute small and conflicting asymmetries. Asymmetries are readily interpretable only in situations where the basic dynamics is dominated by one major subprocess and the overall asymmetry is sufficiently large. In the case of massive lepton-pair production that is the topic of this paper, when the overall asymmetry $A_{LL}$ itself is small, the contribution from the $qg$ subprocess cannot be said to dominate the answer. However, if a large asymmetry is measured, similar to that expected in the GSA case at the larger values of $Q_T$, Figs. 9 and 10 show that the answer is dominated by the $qg$ contribution, and data will serve to constrain $\Delta G(x,\mu_f)$. If $\Delta G(x,\mu_f)$ is small and a small asymmetry is measured, such as for the GSC parton set, or at small $Q_T$ for all parton sets, one will not be able to conclude which of the subprocesses is principally responsible, and no information could be adduced about $\Delta G(x,\mu_f)$, except that it is small.
In Figs. 10 (a) and (b), we examine the energy dependence of our predictions for two different intervals of mass $Q$. For $Q_T$ not too small, we observe that $A_{LL}$ in massive lepton pair production is well described by a scaling function of $x_T = 2Q_T/\sqrt S$, $A_{LL}(\sqrt S,Q_T) \simeq h_{\gamma^*}(x_T)$. In our discussion of the spin-averaged cross sections, we took $Ed^3\sigma/dp^3 = 10^{-3} \rm{pb/GeV}^2$ as the minimum accessible cross. Combining the results in Fig. 5 with those in Fig. 10, we see that longitudinal asymmetries $A_{LL} = 20 \%,\, 7.5 \%,\; \rm{and}\, 3 \%$ are predicted at this level of cross section at $\sqrt{S}=$ 50, 200, and 500 GeV when 2 $< Q <$ 3 GeV, and $A_{LL} = 11 \%,\, 5 \%,\, \rm{and}\; 2 \%$ when 5 $< Q <$ 6 GeV. For a given value of $Q_T$, smaller values of $\sqrt S$ result in greater asymmetries because $\Delta G(x)/G(x)$ grows with $x$.
The predicted cross sections in Fig. 5 and the predicted asymmetries in Fig. 10 should make it possible to optimize the choice of center-of-mass energy at which measurements might be carried out. At $\sqrt{S}=$ 500 GeV, asymmetries are not appreciable in the interval of $Q_T$ in which event rates are appreciable. At the other extreme, the choice of $\sqrt{S}=$ 50 GeV does not allow a sufficient range in $Q_T$. Accelerator physics considerations favor higher energies since the instantaneous luminosity increases with $\sqrt{S}$. Investigations in the energy interval $\sqrt{S}=$ 150 to 200 GeV would seem preferred.
In Fig. 11, we display predictions for $A_{LL}$ in prompt real photon production for three values of the center-of-mass energy. These calculations are done at next-to-leading order in QCD. Dominance of the $qg$ contribution is again evident as long as $A_{LL}$ is not too small. So long as $Q_T \ge Q$, we note that the asymmetry in massive lepton-pair production is about the same size as that in prompt real photon production, as might be expected from the strong similarity of the production dynamics in the two cases. As in massive lepton-pair production, $A_{LL}$ in prompt photon production is well described by a scaling function of $x_T = 2p_T/\sqrt S$, $A_{LL}(\sqrt S, p_T) \simeq h_{\gamma}(x_T)$. For $Ed^3\sigma/dp^3 = 10^{-3} \rm{pb/GeV}^2$, we predict longitudinal asymmetries $A_{LL} = 31 \%,\, 17 \%,\, \rm{and}\; 10 \%$ in real prompt photon production at $\sqrt S =$ 50, 200, and 500 GeV.
Discussion and Conclusions {#sec:5}
==========================
In this paper we focus on the $Q_T$ distribution for $p + p \rightarrow
\gamma^* + X$. We present and discuss calculations carried out in QCD at RHIC collider energies. We show that the differential cross section in the region $Q_T \geq Q/2$ is dominated by subprocesses initiated by incident gluons. Dominance of the $qg$ contribution in the massive lepton-pair case is as strong as in the prompt photon case, $p + p \rightarrow \gamma + X$. As our calculations demonstrate, the $Q_T$ distribution of massive lepton pair production offers a valuable additional method for direct measurement of the gluon momentum distribution. The method is similar in principle to the approach based on prompt photon production, but it avoids the experimental and theoretical complications of photon isolation that beset studies of prompt photon production.
As long $Q_T$ is large, the perturbative requirement of small $\alpha_s(Q_T)$ can be satisfied without a large value of $Q$. We therefore explore and advocate the potential advantages of studies of $d^2\sigma/dQ dQ_T$ as a function of $Q_T$ for modest values of $Q$, $Q \sim 2$ GeV, below the traditional Drell-Yan region.
For the goals we have in mind, it would appear best to restrict attention to the region in $Q_T$ above the value at which the resummed result falls below the fixed-order perturbative expectation. A rough rule-of-thumb based on our calculations is $Q_T \geq Q/2$. Uncertainties associated with resummation make it impossible to use data on the $Q_T$ distribution at small $Q_T$ to extract precise information on parton densities.
In this paper we also present a calculation of the longitudinal spin-dependence of massive lepton-pair production at large values of transverse momentum. We provide polarization asymmetries as functions of transverse momenta that may be useful for estimating the feasibility of measurements of spin-dependent cross sections in future experiments at RHIC collider energies. The Compton subprocess dominates the dynamics in longitudinally polarized proton-proton reactions as long as the polarized gluon density $\Delta G(x,\mu_f)$ is not too small. As a result, two-spin measurements of inclusive prompt photon production in polarized $pp$ scattering should constrain the size, [*[sign]{}*]{}, and Bjorken $x$ dependence of $\Delta G(x,\mu_f)$. Significant values of $A_{LL}$ (i.e., greater than 5 %) may be expected for $x_T = 2Q_T/ \sqrt S > 0.10$ if the polarized gluon density $\Delta G(x,\mu_f)$ is as large as that in the GSA set of polarized parton densities. If so, the data could be used to determine the polarization of the gluon density in the nucleon. On the other hand, for small $\Delta G(x,\mu_f)$, dominance of the $qg$ subprocess is lost, and $\Delta G(x,\mu_f)$ is inaccessible.
Acknowledgments {#acknowledgments .unnumbered}
===============
Work in the High Energy Physics Division at Argonne National Laboratory is supported by the U.S. Department of Energy, Division of High Energy Physics, Contract W-31-109-ENG-38. This work was supported in part by DOE contract DE-AC05-84ER40150 under which Southeastern Universities Research Association operates the Thomas Jefferson National Accelerator Facility.
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---
abstract: |
We describe two general mechanisms for producing pairing bijections (bijective functions defined from $\N^2 \to \N$). The first mechanism, using $n$-adic valuations results in parameterized algorithms generating a countable family of distinct pairing bijections. The second mechanism, using characteristic functions of subsets of $\N$ provides $2^\N$ distinct pairing bijections. Mechanisms to combine such pairing functions and their application to generate families of permutations of $\N$ are also described. The paper uses a small subset of the functional language Haskell to provide type checked executable specifications of all the functions defined in a [*literate programming*]{} style. The self-contained Haskell code extracted from the paper is available at <http://logic.cse.unt.edu/tarau/research/2012/infpair.hs> .
[**Keywords:**]{} [*pairing / unpairing functions, data type isomorphisms, infinite data objects, lazy evaluation, functional programming.* ]{}
author:
- '\'
bibliography:
- 'INCLUDES/theory.bib'
- 'go/tarau.bib'
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title: On Two Infinite Families of Pairing Bijections
---
=1
module InfPair where import Visuals import Data.Bits –import Pi
Introduction
============
A [*pairing bijection*]{} is a bijection $f:\N \times \N \to \N$. Its inverse $f^{-1}$ is called an [*unpairing*]{} bijection.
We are emphasizing here the fact that these functions are bijections as the name [*pairing function*]{} is sometime used in the literature to indicate injective functions from $\N \times \N$ to $\N$.
Pairing bijections have been used in the first half of 19-th century by Cauchy as a mechanism to express duble summations as simple summations in series. They have been made famous by their uses in the second half of the 19-th century by Cantor’s work on foundations of set theory. Their most well known application is to show that infinite sets like $\N$ and $\N \times \N$ have the same cardinality. A classic use in the theory of recursive functions is to reduce functions on multiple arguments to single argument functions. Reasons on why they are an interesting object of study in terms of practical applications ranging from multi-dimensional dynamic arrays to proximity search using space filling curves are described in [@DBLP:conf/ipps/Rosenberg02a; @lawder99; @lawder:2000; @faloutsos:2001].
Like in the case of Cantor’s original function $f(x,y)={1\over2}(x+y)(x+y+1)+y$, pairing bijections have been usually hand-crafted by putting to work geometric or arithmetic intuitions.
While it is easy to prove (non-constructively) that there is an uncountable family of distinct pairing bijections, we have not seen in the literature general mechanisms for building families of pairing bijections indexed by $\N$ or $2^\N$. It is even easier to generate (constructively) a countable family of pairing functions simply by modifying its result of a fixed pairing function with a reversible operation (e.g XOR with a natural number, seen as the index of the family).
This paper introduces two general mechanisms for generating such families, using $n$-adic valuations (section \[nadic\]) and characteristic functions of subsets of $\N$ (section \[char\]), followed by a discussion of related work (section \[rel\]) and our conclusions (section \[concl\]).
We will give here a glimpse of why our arguably more complex pairing bijections are interesting.
The $n$-adic valuation based pairing functions will provide a general mechanism for designing strongly asymmetric pairing functions, where changes in one of the arguments have an exponential impact on the result.
The characteristic-function mechanism, while intuitively obvious, opens the doors, in combination with a framework providing bijections between them and arbitrary data-types [@everything], to custom-build arbitrarily intricate pairing functions associated to for instance to “interesting” sequences of natural numbers or binary expansions of [@intseq] real numbers.
We will use a subset of the non-strict functional language Haskell (seen as an equational notation for typed $\lambda$-calculus) to provide executable definitions of mathematical functions on $\N$, pairs in $\N \times \N$, subsets of $\N$, and sequences of natural numbers. We mention, for the benefit of the reader unfamiliar with the language, that a notation like [f x y]{} stands for $f(x,y)$, represents sequences of type [t]{} and a type declaration like [f :: s -> t -> u]{} stands for a function $f: s \times t \to u$ (modulo Haskell’s “currying” operation, given the isomorphism between the function spaces ${s \times t} \to u$ and ${s \to t} \to u$). Our Haskell functions are always represented as sets of recursive equations guided by pattern matching, conditional to constraints (simple arithmetic relations following `|` and before the `=` symbol). Locally scoped helper functions are defined in Haskell after the [where]{} keyword, using the same equational style. The composition of functions [f]{} and [g]{} is denoted [f . g]{}. It is also customary in Haskell, when defining functions in an equational style (using [=]{}) to write $f=g$ instead of $f~x=g~x$ (“point-free” notation). The use of Haskell’s “call-by-need” evaluation allows us to work with infinite sequences, like the infinite list notation, corresponding to the set $\N$ itself.
Deriving Pairing Bijections from $n$-adic valuations {#nadic}
====================================================
We first overview a mechanism for deriving pairing bijections from one-solution Diophantine equations. Let us observe that
$\forall z \in \N^+=\N-\{0\}$ the Diophantine equation $$\label{dio}
2^x(2y+1)=z$$ has exactly one solution $x,y \in \N$.
This follows immediately from the unicity of the decomposition of a natural number as a multiset of prime factors. Note that a slight modification of equation \[dio\] results in the pairing bijection originally introduced in [@pepis; @kalmar1], seen as a mapping between the pair $(x,y)$ and $z$. $$\label{diopair}
2^x(2y+1)-1=z$$
We will generalize this mechanism to obtain a family of bijections between $\N \times \N$ and $\N^+$ (and the corresponding pairing bijections between $\N \times \N$ and $\N$) by choosing an arbitrary base $b$ instead of $2$.
Given a number $n\in \N,~n>1$, the $n$-adic valuation of a natural number $m$ is the largest exponent $k$ of $n$, such that $n^k$ divides m. It is denoted $\nu_n(m)$.
Note that the solution $x$ of the equation (\[dio\]) is actually $\nu_2(z)$. This suggest deriving similar Diophantine equations for an arbitrary $n$-adic valuation. We start by observing that the following holds:
\[qm\] $\forall b \in \N, b>1, \forall y \in \N$ if $\exists q, m$ such that $b>m>0, y=bq+m$, then there’s exactly one pair $(y', m')$, $b-1>m'\geq 0$ such that $y'=(b-1)q+m'$ and the function associating $(y',m')$ to $(y,m)$ is a bijection.
$y=bq+m, b>m>0$ can be rewritten as $y-q-1=bq-q+m-1, b>m>0$, or equivalently $y-q-1=(b-1)q+(m-1), b>m>0$ from where it follows that setting $y'=y-q-1$ and $m'=m-1$ ensures the existence and unicity of y’ and m’ such that $y'=(b-1)q+m'$ and $b-1>m'>0$. We can therefore define a function $f$ that transforms a pair $(y,m)$, such that $y=bq+m$ with $b>m>0$, into a pair $(y',m')$, such that $y'=q(b-1)+m'$ with $b-1>m' \geq 0$. Note that the transformation works also in the opposite direction with $y'=y-q-1$ giving $y=y'+q+1$, and with $m'=m-1$ giving $m=m'+1$. Therefore $f$ is a bijection.
$\forall b \in \N,b>1, ~\forall z \in \N,z>0$ the system of Diophantine equations and inequations $$b^x*(y'+q+1)=z$$ $$y'=(b-1)q+m'$$ $$b-1>m'\geq 0$$ has exactly one solution $x,y' \in \N$.
Let $f^{-1}$ be the inverse of the bijection $f$ defined in Proposition \[qm\]. Then $f^{-1}$ provides the desired unique mapping, that gives $y=y'+q+1$ and $m=m'-1$ such that $b>m>0$. Therefore $y \equiv m~(mod~b)$ with $m>0$. And as $y$ is not divisible with $b$, we can determine uniquely $x$ as the largest power of $b$ dividing $z$, $x=\nu_b(z)$.
We implement, for and arbitrary $b \in \N$, the Haskell code corresponding to these bijections as the functions [nAdicCons b]{} and [nAdicDeCons b]{}, defined between $\N \times \N$ and $N^{+}$.
nAdicCons :: N->(N,N)->N nAdicCons b (x,y’) | b>1 = (b\^x)\*y where q = y’ ‘div‘ (b-1) y = y’+q+1
nAdicDeCons :: N->N->(N,N) nAdicDeCons b z | b>1 && z>0 = (x,y’) where hd n = if n ‘mod‘ b > 0 then 0 else 1+hd (n ‘div‘ b) x = hd z y = z ‘div‘ (b\^x) q = y ‘div‘ b y’ = y-q-1
Using [nAdicDeCons]{} we define the head and tail projection functions [nAdicHead]{} and [nAdicTail]{}:
nAdicHead, nAdicTail :: N->N->N nAdicHead b = fst . nAdicDeCons b nAdicTail b = snd . nAdicDeCons b
The following examples illustrate the operations for base [3]{}:
\*InfPair> nAdicCons 3 (10,20) 1830519 \*InfPair> nAdicHead 3 1830519 10 \*InfPair> nAdicTail 3 1830519 20
Note that [nAdicHead n x]{} computes the $n$-adic valuation of x, $\nu_n(x)$ while the tail corresponds to the “information content” extracted from the remainder, after division by $\nu_n(x)$.
We call the natural number computed by [nAdicHead n x]{} the $n$-adic head of $x \in \N^{+}$, by [nAdicTail n x]{} the $n$-adic tail of $x \in \N^{+}$ and the natural number in $\N^+$ computed by [nAdicCons n (x,y)]{} the $n$-adic cons of $x,y \in \N$.
By generalizing the mechanism shown for the equations \[dio\] and \[diopair\] we derive from [nAdicDeCons]{} and [nAdicCons]{} the corresponding [*pairing*]{} and [*unpairing*]{} bijections [nAdicPair]{} and [nAdicUnPair]{}:
nAdicUnPair :: N->N->(N,N) nAdicUnPair b n = nAdicDeCons b (n+1)
nAdicPair :: N->(N,N)->N nAdicPair b xy = (nAdicCons b xy)-1
One can see that we obtain a countable family of bijections $f_b: \N \times \N \rightarrow \N$ indexed by $b \in \N$, $b>1$.
The following examples illustrate the work of these bijections for $b=3$. Note the use of Haskell’s higher-order function “[map]{}”, that applies the function [nAdicUnPair 3]{} to a list of elements and collects the results to a list, and the special value “[it]{}”, standing for the previously computed result.
\*InfPair> map (nAdicUnPair 3) \[0..7\] \[(0,0),(0,1),(1,0),(0,2),(0,3),(1,1),(0,4),(0,5)\] \*InfPair> map (nAdicPair 3) it \[0,1,2,3,4,5,6,7\]
### Deriving bijections between $\N$ and $[\N]$
For each base [b>1]{}, we can also obtain a pair of bijections between natural numbers and lists of natural numbers in terms of [nAdicHead]{}, [nAdicTail]{} and [nAdicCons]{}:
nat2nats :: N->N->\[N\] nat2nats \_ 0 = \[\] nat2nats b n | n>0 = nAdicHead b n : nat2nats b (nAdicTail b n)
nats2nat :: N->\[N\]->N nats2nat \_ \[\] = 0 nats2nat b (x:xs) = nAdicCons b (x,nats2nat b xs)
The following example illustrate how they work:
\*InfPair> nat2nats 3 2012 \[0,2,2,0,0,0,0\] \*InfPair> nats2nat 3 it 2012
Using the framework introduced in [@calc09fiso; @everything] and summarized in the [**Appendix**]{}, we can “reify” these bijections as [Encoders]{} between natural numbers and sequences of natural numbers (parameterized by the first argument of [nAdicHead]{} and [nAdicTail]{}). Such Encoders can now be “morphed”, by using the bijections provided by the framework, into various data types sharing the same “information content” (e.g. lists, sets, multisets).
nAdicNat :: N->Encoder N nAdicNat k = Iso (nat2nats k) (nats2nat k)
In particular, for $k=2$, we obtain the [Encoder]{} corresponding to the Diophantine equation (\[dio\])
nat :: Encoder N nat = nAdicNat 2
The following examples illustrate these operations, lifted through the framework defining bijections between datatypes, given in [**Appendix**]{}.
\*InfPair> as (nAdicNat 3) list \[2,0,1,2\] 873 \*InfPair> as (nAdicNat 7) list \[2,0,1,2\] 27146 \*InfPair> as nat list \[2,0,1,2\] 300 \*InfPair> as list nat it \[2,0,1,2\]
### Deriving new families of Encoders and Permutations of $\N$
For each $l,k \in \N$ one can generate a family of permutations (bijections $f:\N\rightarrow \N$), parameterized by the pair [(l,k)]{}, by composing [nat2nats l]{} and [nats2nat k]{}.
nAdicBij :: N -> N -> N -> N nAdicBij k l = (nats2nat l) . (nat2nats k)
The following example illustrates their work on the initial segment of $\N$:
\*InfPair> map (nAdicBij 2 3) \[0..31\] \[0,1,3,2,9,5,6,4,27,14,15,8,18,10,12,7,81,41,42, 22,45,23,24,13,54,28,30,16,36,19,21,11\] \*InfPair> map (nAdicBij 3 2) \[0..31\] \[0,1,3,2,7,5,6,15,11,4,13,31,14,23,9,10,27,63, 12,29,47,30,19,21,22,55,127,8,25,59,26,95\]
It is easy to see that the following holds:
$$(\mathit{nAdicBij}~k~l) \circ (\mathit{nAdicBij}~l~k) \equiv \mathit{id}$$
As a side note, such bijections might have applications to cryptography, provided that a method is devised to generate “interesting” pairs [(k,l)]{} defining the encoding. We can derive [Encoders]{} representing functions between $\N$ and sequences of natural numbers, parameterized by a (possibly infinite) list of [nAdicHead / nAdicTail]{} bases, by repeatedly applying the $n$-adic head, tail and cons operation parameterized by the (assumed infinite) sequence [ks]{}:
nAdicNats :: \[N\]->Encoder N nAdicNats ks = Iso (nat2nAdicNats ks) (nAdicNats2nat ks)
nat2nAdicNats :: \[N\]->N->\[N\] nat2nAdicNats \_ 0 = \[\] nat2nAdicNats (k:ks) n | n>0 = nAdicHead k n : nat2nAdicNats ks (nAdicTail k n)
nAdicNats2nat :: \[N\]->\[N\]->N nAdicNats2nat \_ \[\] = 0 nAdicNats2nat (k:ks) (x:xs) = nAdicCons k (x,nAdicNats2nat ks xs)
For instance, the Encoder [nat’]{} corresponds to [ks]{} defined as the infinite sequence starting at [2]{}.
nat’ :: Encoder N nat’ = nAdicNats \[2..\]
The following examples illustrate the mechanism:
\*InfPair> as nat’ list \[2,0,1,2\] 1644 \*InfPair> as list nat’ it \[2,0,1,2\] \*InfPair> map (as nat’ nat) \[0..15\] \[0,1,2,3,4,7,6,5,8,19,14,15,12,13,10,9\] \*InfPair> map (as nat’ nat) \[0..15\] \[0,1,2,3,4,7,6,5,8,19,14,15,12,13,10,9\]
Note that functions like [as nat’ nat]{} illustrate another general mechanism for defining permutations of $\N$.
Pairing bijections derived from characteristic functions of subsets of $\N$ {#char}
===========================================================================
We start by connecting the bitstring representation of characteristic functions to our bijective data transformation framework (overviewed in the [**Appendix**]{}).
The bijection between lists and characteristic functions of sets
----------------------------------------------------------------
The function [list2bins]{} converts a sequence of natural numbers into a characteristic function of a subset of $\N$ represented as a string of binary digits. The algorithm interprets each element of the list as the number of [0]{} digits before the next [1]{} digit. Note that infinite sequences are handled as well, resulting in infinite bitstrings.
list2bins :: \[N\]->\[N\]
list2bins \[\] = \[0\] list2bins ns = f ns where f \[\] = \[\] f (x:xs) = (repl x 0) ++ (1:f xs) where repl n a | n <= 0 = \[\] repl n a = a:repl (pred n) a
The function [bin2list]{} converts a characteristic function represented as bitstrings back to a list of natural numbers.
bins2list :: \[N\] -> \[N\] bins2list xs = f xs 0 where f \[\] \_ = \[\] f (0:xs) k = f xs (k+1) f (1:xs) k = k : f xs 0
Together they provide the Encoder [bins]{}, that we will use to connect characteristic functions to various data types.
bins :: Encoder \[N\] bins = Iso bins2list list2bins
The following examples (where the Haskell library function [take]{} is used to restrict execution to an initial segment of an infinite list) illustrate their use:
\*InfPair> list2bins \[2,0,1,2\] \[0,0,1,1,0,1,0,0,1\] \*InfPair> bins2list it \[2,0,1,2\]
\*InfPair> take 20 (list2bins \[0,2..\]) \[1,0,0,1,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0\] \*InfPair> bins2list it \[0,2,4,6\]
The following holds:
If $M$ is a subset of $\N$, the bijection [as bins set]{} returns the bitstring associated to $M$ and its inverse is the bijection [as set bins]{}.
Observe that the transformations are the composition of bijections between bitstrings and lists and bijections between lists and sets.
The following example illustrates this correspondence:
\*InfPair> as bins set \[0,2,4,5,7,8,9\] \[1,0,1,0,1,1,0,1,1,1\] \*InfPair> as set bins it \[0,2,4,5,7,8,9\]
Note that, for convenient use on finite sets, the functions do not add the infinite stream of [0]{} digits indicating its infinite stream of non-members, but we will add it as needed when the semantics of the code requires it for representing accurately operations on infinite sequences. We will use the same convention through the paper.
Splitting and merging bitstrings with a characteristic function
---------------------------------------------------------------
Guided by the characteristic function of a subset of $\N$, represented as a bitstring, the function [bsplit]{} separates a (possibly infinite) sequence of numbers into two lists: members and non-members.
bsplit :: \[N\] -> \[N\] -> (\[N\], \[N\]) bsplit \_ \[\] = (\[\],\[\]) bsplit \[\] (n:ns) = error (“bspilt provides no guidance at: ”++(show n)) bsplit (0:bs) (n:ns) = (xs,n:ys) where (xs,ys) = bsplit bs ns bsplit (1:bs) (n:ns) = (n:xs,ys) where (xs,ys) = bsplit bs ns
Guided by the characteristic function of a subset of $\N$, represented as a bitstring, the function [bmerge]{} merges two lists of natural numbers into one, by interpreting each [1]{} in the characteristic function as a request to extract an element of the first list and each [0]{} as a request to extract an element of the second list.
bmerge :: \[N\] -> (\[N\], \[N\]) -> \[N\] bmerge \_ (\[\],\[\]) = \[\] bmerge bs (\[\],\[y\]) = \[y\] bmerge bs (\[x\],\[\]) = \[x\] bmerge bs (\[\],ys) = bmerge bs (\[0\],ys) bmerge bs (xs,\[\]) = bmerge bs (xs,\[0\]) bmerge (0:bs) (xs,y:ys) = y : bmerge bs (xs,ys) bmerge (1:bs) (x:xs,ys) = x : bmerge bs (xs,ys)
The following examples (trimmed to finite lists) illustrate their use:
\*InfPair> bsplit \[0,1,0,1,0,1\] \[10,20,30,40,50,60\] (\[20,40,60\],\[10,30,50\]) \*InfPair> bmerge \[0,1,0,1,0,1\] it \[10,20,30,40,50,60\]
Defining pairing bijections, generically
----------------------------------------
We design a generic mechanism to derive pairing functions by combining the data type transformation operation [as]{} with the [bsplit]{} and [bmerge]{} functions that apply a characteristic function encoded as a list of bits.
genericUnpair :: Encoder t -> t -> N -> (N, N) genericUnpair xEncoder xs n = (l,r) where bs = as bins xEncoder xs ns = as bins nat n (ls,rs) = bsplit bs ns l = as nat bins ls r = as nat bins rs
genericPair :: Encoder t -> t -> (N, N) -> N genericPair xEncoder xs (l,r) = n where bs = as bins xEncoder xs ls = as bins nat l rs = as bins nat r ns = bmerge bs (ls,rs) n = as nat bins ns
Let us observe first that for termination of this functions depends on termination of the calls to [bsplit]{} and [bmerge]{}, as illustrated by the following examples:
\*InfPair> genericPair bins (cycle \[0\]) (10,20) \^CInterrupted. \*InfPair> genericUnpair bins (cycle \[1\]) 42 (\^CInterrupted.
In this case, the characteristic functions given by [cycle \[0\]]{} or [cycle \[1\]]{} would trigger an infinite search for a non-existing first [1]{} or [0]{} in [bsplit]{} and [bmerge]{}.
Clearly, this suggests restrictions on the acceptable characteristic functions.
We will now give sufficient conditions ensuring that the functions [genericUnpair]{} and [genericPair]{} terminate for any values of their last arguments. Such restrictions, will enable them to define families of pairing functions parameterized by characteristic functions derived from various data types.
We call [*bloc*]{} of digits occurring in a characteristic function any (finite or infinite) contiguous sequence of digits.
Note that an infinite bloc made entirely of $0$ (or $1$) digits can only occur at the end of the sequence defining the characteristic function, i.e. only if it exists a number $n$ such that the index of each member of the bloc is larger than $n$.
\[infset\] If $\{a_n\}_{n \in \N}$ is an infinite sequence of bits containing only finite blocks of [0]{} and [1]{} digits, [genericPair bins]{} and [genericUnPair bins]{} define a family of pairing bijections parameterized by $\{a_n\}_{n \in \N}$.
Having an alternation of finite blocks of $1$s and $0$s, ensures that, when called from [genericPair]{} and [genericUnPair]{}, the functions [bmerge]{} and [bsplit]{} terminate.
For instance, [*Morton*]{} codes [@lawder:2000] are derived by using a stream of alternating [1]{} and [0]{} digits (provided by the Haskell library function [cycle]{})
bunpair2 = genericUnpair bins (cycle \[1,0\]) bpair2 = genericPair bins (cycle \[1,0\])
and working as follows:
\*InfPair> map bunpair2 \[0..10\] \[(0,0),(1,0),(0,1),(1,1),(2,0), (3,0),(2,1),(3,1),(0,2),(1,2),(0,3)\] \*InfPair> map bpair2 it \[0,1,2,3,4,5,6,7,8,9,10\]
\[infset\] If $\{a_n\}_{n \in \N}$ is an infinite sequence of non-decreasing natural numbers, the functions [genericPair set]{} and [genericUnPair set]{} define a family of pairing bijections parameterized by $\{a_n\}_{n \in \N}$.
Given that the sequence is non-decreasing, it represents canonically an infinite set such that its complement is also infinite, represented as a non-decreasing sequence. Therefore, the associated characteristic function will have an alternation of finite blocks of [1]{} and [0]{} digits, inducing a pairing/unpairing bijection.
The bijection [bpair k]{} and its inverse [bunpair k]{} are derived from a [set]{} representation (implicitly morphed into a characteristic function).
bpair k = genericPair set \[0,k..\] bunpair k = genericUnpair set \[0,k..\]
Note that for [k = 2]{} we obtain exactly the bijections [bpair2]{} and [bunpair2]{} derived previously, as illustrated by the following example:
\*InfPair> map (bunpair 2) \[0..10\] \[(0,0),(1,0),(0,1),(1,1),(2,0),(3,0), (2,1),(3,1),(0,2),(1,2),(0,3)\] \*InfPair> map (bpair 2) it \[0,1,2,3,4,5,6,7,8,9,10\]
We conclude with a similar result for lists:
If $\{a_n\}_{n \in \N}$ is an infinite sequence of natural numbers only containing finite blocks of 0s, the functions [genericPair list]{} and [genericUnPair list]{} define a family of pairing bijections parameterized by $\{a_n\}_{n \in \N}$.
It follows from Prop. \[infset\] by observing that such sequences are transformed into infinite sets represented as non-decreasing sequences.
The [**Appendix**]{} discusses a few more examples of such pairing functions and visualizes a few space-filling curves associated to them.
There are $2^\N$ pairing functions defined using characteristic functions of sets of $\N$.
Observe that a characteristic function corresponding to a subset of $\N$ containing an infinite bloc of [0]{} or [1]{} digits necessarily ends with the bloc. Therefore, by erasing the bloc we can put such functions in a bijection with a finite subset of $\N$. Given that there are only a countable number of finite subsets of $\N$, the cardinality of the set of the remaining subsets’ characteristic functions is $2^\N$.
Related Work {#rel}
============
Pairing functions have been used in work on decision problems as early as [@pepis; @kalmar1; @robinson50; @robinson55; @robinson68a; @robinsons68b]. There are about [19200]{} Google documents referring to the original “Cantor pairing function” among which we mention the surprising result that, together with the successor function it defines a decidable subset of arithmetic [@DBLP:journals/tcs/CegielskiR01]. An extensive study of various pairing functions and their computational properties is presented in [@ceg99; @DBLP:conf/ipps/Rosenberg02a]. They are also related to 2D-space filling curves (Z-order, Gray-code and Hilbert curves) [@lawder99; @lawder:2000; @Lawder:2001; @faloutsos:2001]. Such curves are obtained by connecting pairs of coordinates corresponding to successive natural numbers (obtained by applying unpairing operations). They have applications to spatial and multi-dimensional database indexing [@lawder99; @lawder:2000; @Lawder:2001; @faloutsos:2001] and symbolic arbitrary length arithmetic computations [@sac12]. Note also that [bpair 2]{} and [bunpair 2]{} are the same as the functions defined in [@pigeon] and also known as Morton-codes, with uses in indexing of spatial databases [@lawder99].
Conclusion {#concl}
==========
We have described mechanisms for generating countable and uncountable families of pairing / unpairing bijections. The mechanism involving $n$-adic valuations is definitely novel, and we have high confidence (despite of their obviousness) that the characteristic function-based mechanisms are novel as well, at least in terms of their connections to list, set or multiset representations provided by the implicit use of our bijective data transformation framework [@calc09fiso].
Given the space constraints, we have not explored the natural extensions to more general tupling / untupling bijections (defined between $\N^k$ and $\N$) as well as bijections between finite lists, sets and multisets that can be derived quite easily, using the data transformation framework given in the [**Appendix**]{}. For the same reasons we have not discussed specific applications of these families of pairing functions, but we foresee interesting connections with possible cryptographic uses (e.g “one time pads” generated through intricate combinations of members of these families).
The ability to associate such pairing functions to arbitrary characteristic functions as well as to their equivalent set, multiset, list representations provides convenient tools for inventing and customizing pairing / unpairing bijections, as well as the related tupling / untupling bijections and those defined between natural numbers and sequences, sets and multisets of natural numbers.
We hope that our adoption of the non-strict functional language Haskell (freely available from [haskell.org](haskell.org)), as a complement to conventional mathematical notation, enables the empirically curious reader to instantly validate our claims and encourage her/him to independently explore their premises and their consequences.
Appendix {#appendix .unnumbered}
========
An Embedded Data Transformation Language {#an-embedded-data-transformation-language .unnumbered}
----------------------------------------
We will describe briefly the embedded data transformation language used in this paper as a set of operations on a groupoid of isomorphisms. We refer to ([@calc09fiso; @arxiv:fISO]) for details.
### The Groupoid of Isomorphisms {#the-groupoid-of-isomorphisms .unnumbered}
We implement an isomorphism between two objects X and Y as a Haskell data type encapsulating a bijection $f$ and its inverse $g$.
\[5\] (14,11)[$X$]{} (39,11)[$Y$]{} (14,12)(39,12)[$f=g^{-1}$]{} (39,10)(14,10)[$g=f^{-1}$]{}
We will call the [*from*]{} function the first component (a [*section*]{} in category theory parlance) and the [*to*]{} function the second component (a [*retraction*]{}) defining the isomorphism. The isomorphisms are naturally organized as a [*groupoid*]{}.
data Iso a b = Iso (a->b) (b->a)
from (Iso f \_) = f
to (Iso \_ g) = g
compose :: Iso a b -> Iso b c -> Iso a c compose (Iso f g) (Iso f’ g’) = Iso (f’ . f) (g . g’)
itself = Iso id id
invert (Iso f g) = Iso g f
Assuming that for any pair of type [Iso a b]{}, $f \circ g = id_b$ and $g
\circ f=id_a$, we can now formulate [*laws*]{} about these isomorphisms.
[*The data type [Iso]{} has a groupoid structure, i.e. the [*compose*]{} operation, when defined, is associative, [*itself*]{} acts as an identity element and [*invert*]{} computes the inverse of an isomorphism.* ]{}
### The Hub: Sequences of Natural Numbers {#the-hub-sequences-of-natural-numbers .unnumbered}
To avoid defining $\frac{n(n-1)}{2}$ isomorphisms between $n$ objects, we choose a [*Hub*]{} object to/from which we will actually implement isomorphisms.
Choosing a [*Hub*]{} object is somewhat arbitrary, but it makes sense to pick a representation that is relatively easy convertible to various others and scalable to accommodate large objects up to the runtime system’s actual memory limits.
We will choose as our [Hub]{} object [*sequences of natural numbers*]{}. We will represent them as lists i.e. their Haskell type is .
type N = Integer type Hub = \[N\]
We can now define an [Encoder]{} as an isomorphism connecting an object to [*Hub*]{}
type Encoder a = Iso a Hub
together with the combinator “[as]{}”, providing an [*embedded transformation language*]{} for routing isomorphisms through two [Encoders]{}.
as :: Encoder a -> Encoder b -> b -> a as that this x = g x where Iso \_ g = compose that (invert this)
The combinator “[as]{}” adds a convenient syntax such that converters between [A]{} and [B]{} can be designed as:
a2b x = as B A x b2a x = as A B x
0.30cm
\[5\] (26,0)[$Hub$]{} (14,11)[$A$]{} (39,11)[$B$]{}
(26,0)(39,10)[$b$]{} (26,0)(14,10)[$a^{-1}$]{} (39,10)(26,0)[$b^{-1}$]{} (14,10)(26,0)[$a$]{} (14,12)(39,12)[$a2b=as~B~A$]{} (39,10)(14,10)[$b2a=as~A~B$]{} 0.30cm
Given that has been chosen as the root, we will define our sequence data type [*list*]{} simply as the identity isomorphism on sequences in .
list :: Encoder \[N\] list = itself
The [Encoder]{} [mset]{} for multisets of natural numbers is defined as:
mset :: Encoder \[N\] mset = Iso mset2list list2mset
mset2list, list2mset :: \[N\]->\[N\] mset2list xs = zipWith (-) (xs) (0:xs) list2mset ns = tail (scanl (+) 0 ns)
The [Encoder]{} [set]{} for sets of natural numbers is defined as:
set :: Encoder \[N\] set = Iso set2list list2set
set2list, list2set :: \[N\]->\[N\] list2set = (map pred) . list2mset . (map succ) set2list = (map pred) . mset2list . (map succ)
Note that these converters between lists, multisets and sets make no assumption about finiteness of their arguments and therefore they can used in a non-strict language like Haskell on infinite objects as well.
Examples of pairing functions derived from characteristic functions {#examples-of-pairing-functions-derived-from-characteristic-functions .unnumbered}
-------------------------------------------------------------------
The function [syracuse]{} is used in an equivalent formulation of the Collatz conjecture. Interestingly, it can be computed using the [nAdicTail]{} which results after dividing a number $n$ with $\mu_2(n)$. Note that we derive our pairing function directly from the [list]{} representation of the range of this function as [genericPair]{} and [genericUnpair]{} implicitly construct the associated characteristic function.
syracuse :: N->N syracuse n = nAdicTail 2 (6\*n+4)
nsyr 0 = \[0\] nsyr n = n : nsyr (syracuse n)
syrnats = map syracuse \[0..\]
syrpair = genericPair list syrnats syrunpair = genericUnpair list syrnats
Figures \[bunpair2\] and \[bunpair3\] show the “[*Z-order*]{}” (Morton code) path connecting successive values in the range of the function [bunpair 2]{} and [bunpair 3]{}. Figures \[syrUnpair\] and \[piUnpair\] show the path connecting the values in the range of unpairing functions associated, respectively to the Syracuse function and the binary digits of $\pi$. Interestingly, at a first glance, some regular patterns emerge even in the case of such notoriously irregular characteristic functions.
sqpair = genericPair set (map (\^2) \[0..\]) squnpair = genericUnpair set (map (\^2) \[0..\])
npair = genericPair list \[0..\] nunpair = genericUnpair list \[0..\]
bnats = concatMap (as bins nat) \[0..\]
bnatpair = genericPair bins bnats bnatunpair = genericUnpair bins bnats
powunpair = genericUnpair set (map (2\^) \[0..\]) powpair = genericPair set (map (2\^) \[0..\])
- – add “import Pi” to the top of this file – using PI as a source of a bitstream bin\_pi = as bins nat (machin\_pi (2\^12))
pi\_pair = genericPair bins bin\_pi pi\_unpair = genericUnpair bins bin\_pi -
infs u n = (bsize n) - s where (a,b) = u n s = (bsize a)+(bsize b)
bsize 0 = 0 bsize n | n>0 = 1 + (bsize (drop2digit n)) where drop2digit n = (shiftR n 1)+(1 .&. n)-1
xunp u x n = u (n ‘xor‘ x)
xp p x (a,b) = (p (a,b)) ‘xor‘ x
xtest p u (x,n) = xp p x (xunp u x n)
xplot = pplot (xunp (bunpair2) 1) 7
xtest1 = map (xunp (bunpair2) 7) \[0..15\]
|
---
author:
- 'Morgan T. Hibberd'
- 'Alisa L. Healy'
- 'Daniel S. Lake'
- Vasileios Georgiadis
- 'Elliott J. H. Smith'
- 'Oliver J. Finlay'
- 'Thomas H. Pacey'
- 'James K. Jones'
- Yuri Saveliev
- 'David A. Walsh'
- 'Edward W. Snedden'
- 'Robert B. Appleby'
- Graeme Burt
- 'Darren M. Graham'
- 'Steven P. Jamison[^1]'
title: 'Acceleration of relativistic beams using laser-generated terahertz pulses'
---
Laser-driven acceleration in dielectric structures is a well-established approach [@Peralta2013; @Breuer2013; @Naranjo2012; @Niedermayer2018] that may hold the key to overcoming the technological limitations of conventional particle accelerators. However, injecting sub-femtosecond particle bunches into optical-frequency accelerating structures to achieve whole-bunch acceleration remains a significant challenge. A promising solution is to down-convert the laser excitation into the terahertz (THz) frequency regime, where THz pulses with electric fields exceeding 1GV/m [@Liao2019] have recently been reported. With experimental demonstrations of THz-driven acceleration, compression and streaking with low-energy (sub-100keV) electron beams [@Nanni2015; @Kealhofer2016; @Huang2016; @Zhang2018], operation at relativistic beam energies is now essential to realize the full potential of THz-driven structures. Here we present the first THz-driven acceleration of a relativistic 35MeV electron beam at the CLARA research facility at Daresbury Laboratory, exploiting the collinear excitation of a dielectric-lined waveguide (DLW) driven by the longitudinal electric field component of polarization-tailored, narrowband THz pulses. These results pave the way to unprecedented control over relativistic electron beams, providing bunch compression for ultrafast electron diffraction, energy manipulation for bunch diagnostics, and ultimately delivering high-field gradients for compact THz-driven particle acceleration.
The need to overcome the electrical breakdown threshold currently limiting the achievable accelerating field gradients of radio-frequency (RF)-based accelerators has led to the development of both laser-driven [@England2014] and beam-driven [@OShea2016] acceleration schemes. With the laser-based schemes also capable of exploiting the femtosecond timing synchronization to laser-generated electron bunches, this has resulted in developments in the optical-infrared regime exploring free-space acceleration [@Carbajo2016; @Thevenet2016], laser-driven plasma wakefield acceleration [@Faure2004; @Leemans2014; @Guenot2017] and dielectric laser acceleration (DLA) in scalable, phase-matched dielectric microstructures [@Peralta2013; @Breuer2013; @Naranjo2012; @Niedermayer2018]. DLA schemes face significant challenges using sub-micron optical wavelengths to drive the structures, which put extreme tolerances on fabrication and bunch timing jitter, while also limiting the amount of bunch charge that can be supported.
Laser-generated THz radiation exists in the ideal millimeter-scale wavelength regime, making structure fabrication simpler but most importantly providing picosecond pulse cycle lengths well-suited for flexible manipulation of sub-picosecond electron bunches with pC-level charge. In recent years, numerous THz-electron interactions have been explored, including acceleration using a dielectric-lined waveguide [@Nanni2015], electron emission off metal nanotips [@Li2016], streaking and bunch compression both with metallic resonators [@Kealhofer2016] and by driving dielectric tubes with circularly polarized THz pulses [@Zhao2019], and the development of novel phase-matching schemes for increased interaction length based on segmented waveguides [@Zhang2018], near-field travelling waves [@Walsh2017] and inverse free-electron laser (IFEL) coupling [@Curry2018]. The wide scope of work demonstrates the drive towards understanding and exploiting the unique capabilities of THz pulses through structure-mediated interactions for unparalleled control over the spatial, temporal and energy properties of electron beams.
{width="\textwidth"}
Here we demonstrate purely longitudinal, linear acceleration of relativistic electron beams driven by laser-generated THz radiation. In contrast to acceleration of sub-relativistic beams, or transversely-coupled IFEL-driven acceleration [@Curry2018], the resonance frequency for the interaction and the acceleration efficiency remain constant with electron energy gain, offering a viable route to future large-scale, high-energy accelerators.
Narrowband and frequency-tunable THz pulses were phase-velocity matched to 35MeV electron bunches () using a longitudinal accelerating mode in a DLW structure for collinear interaction, with an accelerating gradient of 2MV/m achieved using modest THz pulse energies. The resulting peak THz-driven acceleration of long-duration (6ps FWHM) bunches was determined from the modulation of the energy spectrum, which also revealed quantitatively the time-energy phase-space of the electron bunch. Electron bunches with short duration (2ps FWHM) comparable to the period of the THz pulse, are shown to undergo preferential acceleration or deceleration dependent on the timing of electron bunch injection relative to the phase of the THz pulse, demonstrating the route to whole-bunch acceleration of sub-picosecond relativistic particle beams.
The THz-electron beam interaction arrangement is shown schematically in Fig.\[fig:Setup\]. Relativistic 35MeV electron bunches with configurable bunch duration and chirp were delivered to the experiment by the CLARA accelerator [@Clarke2014]. Magnetic quadrupole triplets provided electron bunch transverse control for coupling into the DLW and for optimizing the energy resolution of the dipole spectrometer. Frequency-tunable, narrowband (100GHz FWHM) THz pulses with approximately 2$\mu$J energy were generated through chirped-pulse beating [@Chen2011; @Uchida2015] in a lithium niobate crystal (see Methods). A quasi-TEM$_{01}$ mode was generated by a phase-shifter plate providing a $\lambda/2$ shear and an effective polarity inversion [@Cliffe2016; @Hibberd2019] along the horizontal midline of the THz beam, which is revealed by the electro-optic sampling measurements in Fig.\[fig:Setup\]a. The DLW structure was a rectangular copper waveguide lined at the top and bottom with 60$\mu$m-thick fused-quartz, leaving a vertical 575$\mu$m-thick free-space aperture for electron beam propagation. An integrated linearly-tapered horn was used to couple the quasi-TEM$_{01}$ mode of the THz beam into the accelerating longitudinal section magnetic (LSM$_{11}$) mode [@Healy2018] of the DLW, which was designed for phase-velocity matching with the electron bunch at an operating frequency of 0.4THz (see Supplementary Information).
{width="\textwidth"}
The effect of THz-driven acceleration on an approximately linear chirped, 6ps FWHM electron bunch is demonstrated in Fig.\[fig:Modulation\], with single-shot energy spectra shown for THz off (Fig.\[fig:Modulation\]a) and THz on (Fig.\[fig:Modulation\]b) following temporal overlap of the THz pulse with an electron bunch in the DLW. The difference between the energy spectra in Fig.\[fig:Modulation\]c reveals an energy modulation shown in Fig.\[fig:Modulation\]d, with up to 90% modulation strength, defined as the peak-to-peak amplitude of the modulation normalized to the amplitude of the unmodulated (THz off) spectrum. The modulation arises from the temporally sinusoidal energy gain induced by the multi-cycle THz pulse on the initial longitudinal phase-space distribution of the electron bunch. With sufficiently large energy gain (and loss), it was possible to impose a significant change in the temporally localized chirp of the bunch, resulting in a non-sinusoidal spectral density. This behavior can be observed from the measured modulation in Fig.\[fig:Modulation\]d, with the peaks occurring where the THz-induced modulation flattened the local chirp to zero.
The peak THz acceleration ($\delta E_{\rm THz}$) was determined from comparison of the measured single-shot modulation with a calculation of a sinusoidal THz-driven modulation imposed on a model electron bunch. The spatial resolution limitation of the spectrometer, arising from beam emittance and uncorrelated time-slice energy spread served to reduce the observed modulation strength, and were included in the modelling as an effective energy spread ($\delta E_{\rm eff}$) (see Methods). Simultaneous optimization of peak THz acceleration, effective energy spread and chirp (both linear and cubic) provided the modelled modulation shown in Fig.\[fig:Modulation\]d, with a final value of obtained. With an effective interaction length limited to approximately 4.3mm by group-velocity walk-off and the THz pulse duration (see Supplementary Information), an average acceleration gradient of 2MV/m was determined. The model also predicted a value of , and both linear and cubic chirp components of 47.3keV/ps and 0.4keV/ps$^3$, respectively.
To further support the above calculation of the acceleration gradient, an independent and direct measurement of the phase-space distribution was obtained from the known THz period (0.4THz) and the measured energy modulation period $\Delta E(E)$. The experimentally determined chirp of the electron beam ($\Delta E(E)/\tau$) shown in Fig.\[fig:Modulation\]f, was obtained from 100 successive shots using the same electron bunch configuration from . The quadratic dependence indicates the presence of a third-order chirp component, in good agreement with the modelled chirp (solid blue line in Fig.\[fig:Modulation\]f), obtained independently from the single-shot analysis of Fig.\[fig:Modulation\]d. The measured linear chirp (47.5keV/ps) was of comparable magnitude with the value (53keV/ps) predicted by beam dynamics simulations. However, the degree of phase-space curvature implied by the cubic chirp, revealed from the experimental and modelled data in Fig.\[fig:Modulation\]f, was not expected from the space-charge dominated beam dynamics of the CLARA electron gun. This direct experimental measurement of the time-energy phase-space distribution of a relativistic electron beam is often inaccessible, requiring RF-driven transverse deflecting structures and dedicated electron beam optics [@Rohrs2009]. These measurements therefore demonstrate the advantage of THz-driven accelerating structures for both manipulating and exploring the picosecond temporal structure of high-energy electron beams.
![**Phase-velocity matching.** Measurement of both the modulation strength with supporting model obtained from the calculated DLW dispersion, and modulation period, as a function of THz driving frequency. A long-bunch accelerator configuration with increased bunch chirp was used for these measurements.[]{data-label="fig:Phase_Matching"}](Fig3.jpg){width="\columnwidth"}
The role of the DLW structure in matching the THz phase-velocity ($v_{\rm p}$) and electron bunch velocity ($v_{\rm e}$) was explored through the frequency-dependent interaction strength, as shown in Fig.\[fig:Phase\_Matching\]. The DLW was designed with $v_{\rm p} = v_{\rm e} = 0.9999c$ at 0.40THz, and correspondingly the maximum interaction (modulation strength) was observed at this frequency. The length-integrated interaction strength was modelled using the calculated DLW dispersion for a 100GHz FWHM bandwidth THz pulse. The width of the resonant frequency response was attributable to both the spectral width of the THz pulse and the temporal walk-off between the THz pulse envelope and the electron bunch. For the 7ps FWHM duration THz pulses used here, the group-velocity walk-off limited the effective interaction length to 4.3mm (see Supplementary Information). Over the THz frequency range investigated, the modulation period imposed on the bunch was dominated by the interaction efficiency and was observed to remain approximately constant at the value (165$\pm$10keV) obtained at the resonant frequency of 0.4THz, as shown in Fig.\[fig:Phase\_Matching\].
{width="\textwidth"}
Compared with optical-frequency DLA structures, THz-driven acceleration offers the potential for injection of an electron bunch into a single half-cycle (or accelerating bucket) of the THz electric field. To demonstrate this, the CLARA accelerator was configured to provide an electron beam with low time-integrated energy spread and an independently measured bunch duration of approximately 2ps FWHM (see Methods). With a duration comparable to the THz period, preferential acceleration or deceleration of the bunch was achievable, dependent on the bunch injection timing relative to the phase of the THz pulse. Single-shot electron spectra measured in this configuration are shown in Fig.\[fig:Bunch\_Spreading\]a, with observed spectral profiles ranging from symmetric energy spreading to either asymmetric acceleration or deceleration. The maximum measured energy spread FWHM doubled from 21keV (THz off) to 42keV (THz on) for the symmetric spreading observed at $\pi$ injection phase. This symmetric modulation is similar to that observed in optical-frequency DLA structures [@Peralta2013], where the bunch duration covers many cycles of the optical excitation pulse. However, the asymmetric energy spread observed in Fig.\[fig:Bunch\_Spreading\]a was only achievable through a whole-bunch charge-density asymmetry with respect to the THz oscillation period. A model distribution (see Fig.\[fig:Bunch\_Spreading\]d) of the low energy spread, short-duration electron bunch was based on particle tracking simulations, with a small residual chirp expected from this accelerator configuration (see Methods). This distribution was then modulated at the THz frequency (see Fig.\[fig:Bunch\_Spreading\]e) over the full 2$\pi$ range of injection phases to give the modelled energy spectra in Fig.\[fig:Bunch\_Spreading\]b and c. Optimal matching to the measured spectra was achieved using an effective energy spread (see Methods) of , residual linear chirp of 7.8keV/ps and a peak THz-driven acceleration of , consistent with the value obtained from the multi-cycle modulation analysis in Fig.\[fig:Modulation\]. The experimentally observed and modelled asymmetric acceleration is unambiguous evidence of predominant injection into a single half-cycle of the THz pulse.
In summary, we have successfully demonstrated the linear acceleration of relativistic electron beams using a dielectric-lined waveguide driven by narrowband, frequency-tunable, polarization-tailored THz pulses. We were able to exploit multi-cycle THz-driven modulation to manipulate the electron energy spectra and determine the bunch chirp, while also observe preferential acceleration of bunches with duration comparable to a single cycle of the THz pulse. We demonstrated an accelerating gradient of 2MV/m using modest $\mu$J-level THz pulses; increasing the gradient by orders of magnitude will be possible with the implementation of high-energy THz sources [@Liao2019]. Our results establish the operation of dielectric structures driven by laser-generated THz pulses in the unexplored relativistic energy regime and pave the way to future acceleration and manipulation of high-energy particles, as well as compact relativistic beam diagnostics.
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Acknowledgements {#acknowledgements .unnumbered}
================
We wish to acknowledge the CLARA technical and scientific teams for their support and considerable help on all aspects of the operation of the accelerator. We also wish to acknowledge Peter G. Huggard and Mat Beardsley from Rutherford Appleton Laboratory (RAL)-Space for the manufacture of the dielectric-lined waveguide structure and for the provision of a THz Schottky diode used for THz-electron beam synchronization. This work was supported by the United Kingdom Science and Technology Facilities Council (Grant Nos. ST/N00308X/1, ST/N003063/1 and ST/P002056/1).
Author contributions {#author-contributions .unnumbered}
====================
All authors participated in the experiment and contributed to data analysis. M.T.H., D.S.L., D.A.W., V.G., O.J.F. and D.M.G. developed the THz source. A.L.H., G.B. and S.P.J. designed the DLW. A.L.H., E.J.H.S., O.J.F., R.B.A. and S.P.J modelled the electron energy spectra and performed the longitudinal phase-space calculations. V.G. characterized the DLW and developed the data acquisition software. T.H.P., J.K.J. and Y.S. analyzed the beam dynamics of the CLARA accelerator. M.T.H., D.M.G. and S.P.J. wrote the manuscript with contributions from all. E.W.S., R.B.A., G.B., D.M.G. and S.P.J. managed the project.
Competing interests {#competing-interests .unnumbered}
===================
The authors declare no competing interests.
Additional information {#additional-information .unnumbered}
======================
**Supplementary information** is available for this paper at: https://doi.org/xx.xxxx/xxxxxx.\
**Correspondence and requests for materials** should be addressed to S.P.J.
Methods {#methods .unnumbered}
=======
**Electron beam.** This experiment was carried out at the CLARA research facility at Daresbury Laboratory. Electron bunches were generated from a copper cathode by 266nm, 2ps FWHM photoinjector laser pulses at a repetition rate of 10 Hz. The bunches were initially accelerated in a 3GHz RF gun to 5MeV and then transported to a linac for further acceleration up to 35MeV. The relativistic electron bunches were transported to an experimental user station using a magnetic dog-leg, which in combination with the linac, allowed for manipulation of the longitudinal phase-space of the beam. Two main accelerator configurations were used for this experiment; a long bunch with large energy spread (Fig.\[fig:Modulation\]), and a shorter bunch with low energy spread (Fig.\[fig:Bunch\_Spreading\]). For the long-bunch configuration, the bunch duration and chirp were determined directly from the THz-driven modulation data (Fig.\[fig:Modulation\] and main text) and were broadly consistent with beam dynamics simulations performed using the code Elegant. For the short-bunch configuration, a bunch duration of 2ps FWHM, measured by the method of RF zero-phasing, was used directly in the model analysis of Fig.\[fig:Bunch\_Spreading\]. The linear chirp of 7.8keV/ps was in line with accelerator tuning and space-charge expectations. In the experimental user station, the electron beam was transversely focused into the DLW by an upstream quadrupole triplet (see Fig.\[fig:Setup\]) to a RMS transverse size of 100$\mu$m. After transmission through the DLW a downstream quadrupole triplet was used for matching the beam into the energy spectrometer and minimizing the horizontal $\beta$-function.
**Laser systems.** The experiment was performed using a customized terawatt (TW) laser system [@Priebe2008], capable of producing up to 800mJ laser pulses at 10Hz with a center wavelength of 800nm and Fourier-limited pulse duration of 60fs. The TW system comprised of a Ti:sapphire oscillator (Micra, Coherent) producing 800nm, 30fs, 4nJ pulses at 83MHz, which were stretched to approximately 150ps and used to seed a Ti:sapphire regenerative amplifier (Legend, Coherent) providing 800nm, 50fs (Fourier-limited), 1mJ pulses at 1kHz. The uncompressed output of the regenerative amplifier was routed through a multi-pass Ti:sapphire amplifier (MPA), where two frequency-doubled Nd:YAG lasers (Powerlite II, Continuum) producing 532nm, 10ns, 1.5J pulses at 10Hz provided the pump energy for amplification, which following re-compression to 60fs resulted in TW peak powers. For the experiment, a laser pulse energy of approximately 50mJ was delivered to the electron beam experimental user station through an evacuated beam transport line.
**Terahertz generation.** Narrowband THz pulses were generated using a chirped-pulse beating scheme combined with tilted pulse-front pumping of a 0.6% MgO-doped stoichiometric lithium niobate (LiNbO$_3$) crystal. Chirped-pulse beating was achieved by adjusting the TW grating-pair compressor to chirp the input laser pulse and a Michelson interferometer was used to generate THz radiation at the beat frequency set by the interferometric combination of the two chirped laser pulses in the LiNbO$_3$ crystal. With an input chirped pulse duration of approximately 12ps FWHM, THz pulses with centre frequency of 0.40THz and bandwidth of 100GHz were generated. The tilted pulse-front was achieved using a diffraction grating with groove density of 1700mm$^{-1}$, incident angle of 31.2$^{\circ}$ and diffracted angle of 57.5$^{\circ}$. A 4f-lens configuration consisting of two cylindrical lenses of 130mm and 75mm focal length was used to image the pulse-front tilt in the LiNbO$_3$ crystal. The resulting pump 1/*e*$^2$ spot size on the LiNbO$_3$ crystal was and with a pump pulse energy of , THz pulses with an approximate energy of 2.1$\mu$J were measured at the crystal surface using a pyroelectric detector (THZ-I-BNC, Gentec). The THz radiation was collected by a 90$^{\circ}$ off-axis parabolic mirror (OPM) of 152.4mm focal length and routed by silver mirrors into a vacuum chamber through a quartz window. Focusing of the THz radiation was achieved using a 228.6mm focal length OPM combined with a 100mm focal length TPX lens, resulting in a 1/*e*$^2$ spot size of approximately 3mm at the waveguide coupler entrance and a measured THz energy of approximately 0.79$\mu$J. A small fraction (5%) of the chirped 12ps laser beam was re-compressed back to 60fs FWHM by a second grating-pair compressor and focused on to a 500$\mu$m-thick, (110)-cut ZnTe crystal in a back-reflection geometry for electro-optic sampling measurements of the THz electric field at the entrance of the waveguide. For generating the quasi-TEM$_{01}$ THz mode, a 40mm-diameter polytetrafluoroethylene (PTFE) phase-shifter plate was made with a thickness difference between the top and bottom halves of approximately 800$\mu$m.
**Synchronization.** The ultrafast 83MHz Ti:sapphire oscillator (at the front end of the TW laser system) was synchronized to a reference frequency from the CLARA master clock using a commercial phase-lock loop electronics module (Synchrolock, Coherent). The master clock was derived from the Ti:sapphire oscillator (Element, Spectra-Physics) integrated into the CLARA photoinjector laser system. The RMS timing jitter between the two oscillators was measured to be approximately 100fs. Temporal overlap of the THz pulse with the electron bunch at the DLW entrance was achieved using a Schottky diode, which detected both the driving THz pulse directly and the coherent transition radiation (CTR) generated by the 35MeV electron beam incident on a vacuum-metal interface inserted into the beam path. Phase adjustments of the commercial phase-lock loop electronics module were used for coarse control, after which a mechanical delay stage (with range of 1ns and step-size of 33fs) on the THz beam line was used for fine control of the temporal overlap.
**Waveguide structure.** The waveguide design was a hollow, rectangular copper structure lined at the top and bottom with 60$\mu$m-thick fused quartz (using a glycol phthalate adhesive), leaving a 575$\mu$m-thick free-space aperture of 30mm length and 1.2mm width for electron beam propagation (see Supplementary Information). To maximize coupling of the THz radiation into the accelerating mode of the DLW, a tapered horn structure was fabricated with an entrance aperture of 3.25mm by 3.18mm and length of 23mm. A 45$^\circ$ aluminium mirror with a 400$\mu$m aperture aligned to the DLW was used to spatially overlap the incident reflected THz radiation with the transmitted electron beam. The overall DLW/coupler/mirror structure was located on a motorized 5-axis translation stage providing fine control over the positioning and tilt angle for optimization of the THz and electron beam transmission.
**Modelling.** An effective energy spread ($\delta E_{\rm eff}$) was used in the model analyses to describe the spatial resolution ($\sigma_{\rm rms}$) of the electron spectrometer and was related to the spectrometer dispersion ($D$), uncorrelated time-slice energy spread ($\delta E_{\rm uncorr}$), beam emittance ($\epsilon$) and beam optical $\beta$-function by , where $\sigma_{\rm rms}$ was determined to be 270$\mu$m and 200$\mu$m for the beam conditions used in the measurements of Fig.\[fig:Modulation\] and Fig.\[fig:Bunch\_Spreading\], respectively.
The modelled frequency-dependent interaction strength shown in Fig.\[fig:Phase\_Matching\] was calculated using for particles injected with phase-offset $z_0$. The propagating field $E_{\rm z}(z,t)$ at the particle position was calculated from the frequency domain spectrum of the THz pulse and the frequency-dependent propagation wavevector for the DLW.\
**Data availability.** The data associated with the paper are openly available from the Mendeley Data Repository at: http://dx.doi.org/xxxxxxxxxxxx.
[^1]: s.jamison@lancaster.ac.uk
|
---
abstract: 'We have been using the 0.76-m Katzman Automatic Imaging Telescope (KAIT) at Lick Observatory to optically monitor a sample of 157 blazars that are bright in gamma rays, being detected with high significance ($\ge 10\sigma$) in one year by the Large Area Telescope (LAT) on the [*Fermi Gamma-ray Space Telescope*]{}. We attempt to observe each source on a 3-day cadence with KAIT, subject to weather and seasonal visibility. The gamma-ray coverage is essentially continuous. KAIT observations extend over much of the 5-year [*Fermi*]{} mission for several objects, and most have $>100$ optical measurements spanning the last three years. These blazars (flat-spectrum radio quasars and BL Lac objects) exhibit a wide range of flaring behavior. Using the discrete correlation function (DCF), here we search for temporal relationships between optical and gamma-ray light curves in the 40 brightest sources in hopes of placing constraints on blazar acceleration and emission zones. We find strong optical–gamma-ray correlation in many of these sources at time delays of $\sim 1$ to $\sim 10$ days, ranging between $-40$ and +30 days. A stacked average DCF of the 40 sources verifies this correlation trend, with a peak above 99% significance indicating a characteristic time delay consistent with 0 days. These findings strongly support the widely accepted leptonic models of blazar emission. However, we also find examples of apparently uncorrelated flares (optical flares with no gamma-ray counterpart and gamma-ray flares with no optical counterpart) that challenge simple, one-zone models of blazar emission. Moreover, we find that flat-spectrum radio quasars tend to have gamma rays leading the optical, while intermediate and high synchrotron peak blazars with the most significant peaks have smaller lags/leads. It is clear that long-term monitoring at high cadence is necessary to reveal the underlying physical correlation.'
author:
- 'Daniel P. Cohen, Roger W. Romani, Alexei V. Filippenko, S. Bradley Cenko, Benoit Lott, WeiKang Zheng, and Weidong Li'
title: 'Temporal Correlations between Optical and Gamma-ray Activity in Blazars'
---
Introduction
============
Blazars make up a class of radio-loud active galactic nuclei (AGNs) that have a relativistic jet pointing very nearly along Earth’s line of sight. These sources are generally extremely bright and highly variable from radio to gamma-ray wavebands [e.g., @bk79; @up95]. The spectral energy distributions (SEDs) of blazars are characterized by two dominant peaks, one near radio to ultraviolet wavelengths and the other at higher, X-ray/gamma-ray energies. Optical to ultraviolet emission in blazars is widely accepted to be caused by synchrotron emission from electrons in the jet. Higher energy, hard-X-ray–GeV–TeV emission is attributed to inverse-Compton scattering (ICS) of seed photons by the synchrotron-emitting electrons (the ones responsible for the lower-energy emission), or the alternative hadronic processes based in jet proton interactions [e.g., @jon74; @kon81; @mb92].
In the favored leptonic models of blazar emission, synchrotron radiation and ICS both occur along the jet and derive from the same population of electrons, yielding a strong correlation between low- and high-energy wavebands. [e.g., @sik94]. Observed flares derive from events, commonly modeled as propagating shocks (or collisions of shocks), that occur in the jet at subparsec to parsec distances from the central engine and accelerate the jet electrons to high energies [e.g., @spa01]. While synchrotron photons are emitted near the shock front in the jet, the origin of the seed photons for ICS is not clear. These seed photons could be produced in the synchrotron-emitting jet itself (synchrotron self-Compton \[SSC\]) or from an external source (external Compton \[EC\]) such as the accretion disk, broad-line region (BLR), or dusty infrared torus (hot-dust region \[HDR\]) [e.g., @jon74; @sik94].
Multi-wavelength correlation studies of blazars can thus help to place constraints on the dominant mechanisms driving variability and identify the relationship between emission zones. For example, the leptonic models predict a strong correlation between synchrotron-produced optical and ICS-produced gamma-ray emission. Lags or leads of high significance between flares in these wavebands may help place constraints on the location of the ICS seed photons relative to the synchrotron-emitting shock in the jet and discern between the SSC and EC processes. Alternatively, observations of a flare in one waveband with no correlated flare in the other might suggest multiple zones of emission or support hadronic models of blazar emission. With the advent and success of the [*Fermi Gamma-ray Space Telescope*]{} and its primary scientific instrument the Large Area Telescope ([*Fermi*]{}/LAT), multi-wavelength studies have been extended into this MeV–GeV energy range and these goals are being realized. For example, @fur14 and @max14 both present fascinating investigations of radio–gamma-ray correlations in blazars utilizing the discrete correlation function (DCF) – the former using cm to sub-mm data from the F-GAMMA monitoring project and the latter using 15 GHz data from the Owens Valley 40-m telescope.
In this study, we use the DCF to investigate correlations between optical and gamma-ray light curves of blazars. Optical data were collected with the robotic 0.76-m Katzman Automatic Imaging Telescope (KAIT) at Lick Observatory, which has been monitoring sources detected by LAT for much of the [*Fermi*]{} mission. Here we present results from computing the DCFs between optical and gamma-ray light curves in the 40 brightest sources out of the 157 monitored blazars. A future paper will report the results for the other sources.
This paper is organized as follows. In §\[obs\] we describe data collection and production of the light curves. Section \[lags\] and §\[stacks\] present the results and interpretation for DCFs of individual sources and for stacked DCFs of subsets of sources, respectively. We conclude in §\[disc\] with a brief discussion of our findings.
Observations {#obs}
============
KAIT
----
Since August 2009, we have been using KAIT to monitor gamma-ray bright blazars. The base sample consists of the blazars at Galactic latitude $|b|>10^{\circ}$ in the KAIT declination band $-25^{\circ}<\delta<70^{\circ}$ detected in the first-year LAT blazar catalog [@1LAC] at a significance $>10\sigma$ and in the historical optical (POSS) at $R<18$ mag. There were 140 such sources. A few additional optical/gamma-bright blazars have been added to bring the monitored sample to 157. Every available night we attempt unfiltered observations, with effective color close to that of the $R$ band [@li03], of 30–50 sources. Light curves are produced through a pipeline that utilizes aperture photometry and performs brightness calibrations using USNO B1.0 catalog stars in each source field. Currently, KAIT light curves typically contain at least 100 data points with an average in-season cadence of $\sim 3$ days, extending over much of the full continuous 5-year [*Fermi*]{} coverage. All KAIT AGN light curves are made publicly available on the web at <http://hercules.berkeley.edu/kait-agn> and are updated in nearly real time.
[*Fermi*]{}/LAT
---------------
The [*Fermi*]{}/LAT data were collected over the first 63 months of the mission from 2008 August 4 to 2013 November 4. Time intervals during which the rocking angle of the LAT was greater than 52$^{\circ}$ were excluded and a cut on the zenith angle of gamma rays of $100^{\circ}$ was applied. The Pass 7\_V6 Clean event class was used, with photon energies between 100 MeV and 200 GeV.
The LAT light curves were produced from variable-width bins, generated by the adaptive binning method (ABM), to obtain nearly constant relative-flux uncertainties of 25% in each bin [@lott12]. The systematic uncertainties are negligible relative to the statistical uncertainties in these light curves. In the following, we use the “optimum energy,” defined by @lott12 as the lower limit of the integral fluxes shown in the light curves. For the optimum energy, the accumulation times (i.e., bin widths) needed to fulfill the condition on the relative-flux uncertainty with the ABM are the shortest (on average) relative to other choices of lower energy limit. Because the sources are variable and the optimum energy value depends on the flux, we compute the optimum energy with the average flux over the first two years of LAT operation reported in the 2FGL catalog [@2FGL].
The photon index was set fixed to the value reported in the 2FGL catalog when assessing the time intervals with the ABM. For heavily confused sources, the fluxes of the main (maximum 3) neighboring sources were fitted as well in the ABM. Once the time intervals were determined with the ABM, the final analysis was performed with the unbinned likelihood method implemented in the [*pyLikelihood*]{} library of the Science Tools[^1] (v9r32p5). The spectra were modeled with single power-law functions, with both fluxes and photon indices set free for the source of interest as well as the sources found variable in the 2LAC [@2LAC] and located within $10^\circ$ of the source of interest.
Thus far, ABM light curves with coverage until the end of June 2013 have been produced for the 40 brightest sources in our sample of 157 blazars, selected according to detection significance after 2 years of [*Fermi*]{}/LAT operation. These sources are listed in Table \[tab\] with optical classification, SED classification, and redshift obtained from the 2LAC along with the optimum energy for each source. Also given is the duration of overlap for the KAIT and [*Fermi*]{}/LAT light curves — that is, the fraction of the LAT months containing at least one KAIT observation. Our sample includes 20 BL Lac objects and 20 flat-spectrum radio quasars (FSRQs); for the BL Lacs there are 6 low synchrotron peak (LSP), 8 intermediate synchrotron peak (ISP), and 6 high synchrotron peak (HSP) as defined in the 2LAC.
Individual Time Lags and Significances, Flaring Behaviors {#lags}
=========================================================
We compute the DCFs [@ek88], using the local normalization as in @wel99, between the KAIT and [*Fermi*]{}/LAT light curves for the blazars listed in Table \[tab\]. With the local normalization, the points in the DCF are bounded between $-1$ and 1, and they straightforwardly represent the linear correlation coefficient for each lag bin (see @whi94 for more on this technique). Levels of significance for each source are derived from the distribution of DCF points in each lag bin for false-match source pairs: we compute the DCF of the [*Fermi*]{}/LAT light curve of one source with the KAIT light curve of a [*different*]{} source for all possible false-pair matches (typically a few thousand pairs) using the 40 ${\it Fermi}$/LAT and full 157 KAIT light curves, yielding a distribution of correlation points at each lag bin. For this calculation, we use KAIT light curves within a right ascension of $\pm 3$ hr of the actual source of interest (thereby using sources with similar optical coverage to the actual source of interest). The levels of significance calculated are then expected to contain uncertainties from systematics, uncorrelated flaring events, and possibly window functions of the light curves.
In all plots showing a DCF in this paper, dashed blue, green, and orange lines represent 68%, 90%, and 99% significance levels, respectively. Centroid lags and uncertainties are derived from weighted least-squares Gaussian fits to DCF points in the peak. Peak significance values and errors are also derived from the Gaussian fits, converting the amplitude values to probability values using the false-match correlation distribution. Individual DCF points are occasionally higher, but within the individual point measurement errors. Wide DCF peaks may indicate either a range of characteristic timescales in the correlated response, or simple measurement limitations. Note that there may be unstudied correlations in the bins near the DCF peaks, especially since the adaptive binning light curves produce large time bins in low states which may spread the effective correlation among a range of lags. However, since the DCF peaks are dominated by LAT flaring events, we believe that this does not cause undue smoothing when the sources are relatively bright and the time bins are small.
The time delay in all DCF plots (and throughout this paper) is defined so that positive time delay $\tau > 0$ corresponds to gamma rays leading optical emission. Peak significance values and centroid lags derived from the DCF for each source are given in Table \[tab\]. For some of the sources the DCF was either flat or highly scattered, with no clear peak to fit; such centroid lags and peak significances are marked with “—” in Table \[tab\]. Examples of DCFs are shown in Figures \[0050\]–\[2232\].
From the light curves, perhaps the clearest correlation among these sources (although not the most significant) is in the FSRQ 4C +28.07, with nearly simultaneous gamma-ray and optical flares at MJD $\approx$ 55,900. The DCF peak for 4C +28.07 indicates a gamma-ray lead of $\tau = 4.1 \pm 1.3$ days. Visually, the good correlation in the brightest flares supports this short time scale, but the DCF peak width at half maximum is $\sim 80$ days. This likely represents the characteristic width of the flaring episodes, but may also describe a variation in the true lead/lag. Long time series with roughly 1-day sampling would be needed to distinguish these cases. Nevertheless, our fit time scale does provide an estimate for the typical lead/lag time, indicating relatively tight optical–gamma-ray correlation. Similar considerations apply to CTA 102, another FSRQ, where we see nearly simultaneous gamma-ray and optical flares at MJD $\approx$ 56,200. This results in a correlation peak indicating a gamma-ray lead of $\tau= 11.4 \pm 0.7$ days. The strong dominant peak limits the range of $\tau$ contributions to $-10$ to +25 days. The gamma-ray lead of the FSRQ 3C 279 by $19.7\pm 3.4$ days is less clear from its light curve, as the correlation peak in this source derives from somewhat smaller-scale variability (no large flares with overlapping optical and gamma-ray coverage). The BL Lacs PKS 0048$-$09 and 4C +01.28 are strongly correlated at $\tau = -5.3 \pm 3.1$ and $\tau = -13.8 \pm 3.1$ days, respectively. PKS 0048$-$09 owes its correlation to a pair of correlated optical and gamma-ray flares at MJD $\approx$ 55,500 and MJD $\approx$ 55,800, and 4C +01.28 to an early flare at MJD $\approx$ 55,600.
A few of the sources in our sample have been the subjects of other multi-wavelength studies. Most recently, in @ack14, the DCF for the FSRQ 4C +21.35 suggested gamma rays leading the optical by $\sim 35$ days during a flare in 2010, while the DCF we computed indicates a gamma-ray lead of $\tau = 8.6 \pm 1.5$ days. @hay12 studied 3C 279 and found gamma rays to lead the optical by $\sim 10$ days for a flare just before KAIT coverage began. The FSRQ 3C 454.3 was shown by @bon09 to be correlated at $\sim 0$ days for a flare that occurred before KAIT coverage began; unfortunately, for this source there is no overlap between the KAIT and [*Fermi*]{}/LAT light curves, so the DCF could not be computed. Discrepancies in DCFs between this and other studies of the same sources are caused by different observations of different flaring activity.
Of the 40 sources in our study, 8 are found to have DCF peaks above 90% significance. We find in general that these sources have strong optical–gamma-ray correlation with timescales on the order of days to tens of days, in agreement with similar studies of other sources [e.g., @abd10a; @abd10c; @ack12; @hay12; @bon09; @bon12]. In order to further visualize and understand the distribution of lags, we include a scatter plot in Figure \[scatter\] that shows the centroid lags and uncertainties versus the peak significances (in terms of Gaussian probability $\sigma$). For clarity, data points with high certainty of centroid lag value are shown to be larger than those with uncertain lags. The FSRQs, with several exceptions, tend to have gamma rays leading the optical by 0–20 days, while the BL Lacs are widely scattered with no clear trend toward lead or lag. This behavior apparently supports the current models that suggest EC is dominant in FSRQs while SSC is dominant in BL Lacs [e.g., @bot13]. However, with a small sample size this result should be treated with caution.
Models for SSC and those for EC do predict time delays between the optical and gamma-ray bands. For EC, the HDR and BLR are predicted to dominate the contribution from external radiation fields [@sik09]. Under the assumption that flares are produced by outbursts propagating down the jet, optical–gamma-ray leads and lags of roughly day timescales are predicted by the strong stratification of the radiation field, and its mismatch with the decreasing magnetic energy of the jet. Applying the EC model presented by @jan12, the preference of FSRQs (dominated by EC) to have gamma rays leading the optical by these timescales suggests that, for FSRQs, the locations of source activity (i.e., flare burst events) occur more often downstream but still within the radiation field of the BLR, or well downstream but still within the radiation field of the HDR. SSC models predict time delays, again of both signs, but generally of smaller magnitude [e.g., @sok04]. With strong optical–gamma-ray correlations for many sources, our results support leptonic single-zone models (both SSC and EC) of blazar emission; in the hadronic models, the low- and high-energy SED peaks vary independently, and strong correlation between optical and gamma-ray emission is not expected.
Uncorrelated Flares {#uncflares}
-------------------
There are a few sources among our 40 that exhibit peculiar flaring behavior — large gamma-ray flares with little or no optical counterpart or optical flares with no gamma-ray counterpart. The former are commonly called “orphan” flares, and have been attributed to hadronic processes [e.g., @bot07] or alternatively to contamination in the optical by accretion-disk emission [@ack14], but the origin of these flares remains uncertain. We see several such flares, such as in the light curve of the FSRQ PKS 0454$-$234 in Figure \[0457\], at MJD $\approx$ 55,850, where several strong gamma-ray flares show little optical activity nearby in time. The next large optical flare peaks at MJD $\approx$ 56,000; if it is the counterpart of the gamma-ray activity, it suggests a unique $\sim 150$ day delay (the alternative is that this optical flare is also an orphan). A similar large delay between X-rays/gamma-rays was detected (in the DCF) for 3C 279 by @hay12; however, a causal connection was ruled out owing to the lack of accompanying radio flares and the temporal structure of the X-ray and gamma-ray flares. In our case of PKS 0454$-$234 and others like it, substantially longer light curves or more multi-wavelength data are needed to confirm or dismiss a causal origin of such large delays.
In the sources BZQ J0850$-$1213, OP 313, and S4 1849+67, we also observe clear optical variability with no correlated gamma-ray activity. For the FSRQ S4 1849+67, shown in Figure \[1849\], the nearly simultaneous optical and gamma-ray peaks at MJD $\approx$ 55,750 indicate a small characteristic delay; the lack of gamma-ray activity near the second broad optical flare at MJD $\approx$ 56,100 is thus particularly significant. Optical flares with no gamma-ray counterpart have also been noted by other authors in various sources [e.g., @smi11; @dam13]. Such behavior has been attributed to separate emission zones for the flare outbursts or synchrotron emission modulation via magnetic-field changes, with minimal effect on the seed photon numbers, and thus relatively steady EC emission [e.g., @cha13]. Thus, orphan events argue for multi-zone synchrotron sites, with decoupled Compton emission.
Stacked Correlations {#stacks}
====================
In order to assess correlation trends between optical and gamma-ray emission in blazars, we stack, or average, DCFs for our 40 sources and for subsets of this sample. Significance levels for stacked DCFs are again derived from false-match correlations, but now each point in the false-match distribution is an average, using the same subclass, for as many sources as are used in the true stacked DCF. For example, if there are 20 FSRQs into the stacked DCF, then each point in the false-match distribution is an average of 20 false-match pair correlations, where the false-match pairs are drawn from subsets of only FSRQs. There are typically $\sim 100$ points in the false-match distribution in any given bin that are then used to calculate significance levels for stacked sources (compared with a few thousand for individual source false-match distributions).
The stacked DCF for the full set of 40 sources is shown in Figure \[stackfull\]. The peak is much higher than any one of the $\sim 100$ stacked false-pair points, indicating a significance above 99%. The centroid lag is $\tau = -2.8 \pm 1.8$ days (the lag value and its uncertainty for the stacked DCFs are again determined through a Gaussian fit). This very strong correlation signal indicates that, on average, blazars have strongly coupled optical–gamma-ray emission with a characteristic lag of roughly 1 day, most easily understood for leptonic models. The peak width implies apparently significant correlation from about $-60$ to +40 days, consistent with the range of characteristic lags from individual DCFs (from about $-40$ to +30 days). This dispersion may alternatively be dominated by the typical duration of the flare events. Indeed, we find that this stacked DCF is not much broader than the best individual DCFs, suggesting that $|\tau| \approx 1$ day delays are typical of our monitored sample. The peak fit indicates a small optical lead, which might be caused by typically larger (and more uncertain) characteristic time delays found for optical leads while significant gamma-ray leads tend to cluster around smaller values of $\tau$ (Fig. \[scatter\]). The $\tau < 0$ day characteristic lag for the stacked DCF peak might also reflect larger characteristic widths for the optical events: longer optical fall times would bias the correlation overlaps to negative $\tau$, especially for weaker flares.
The stacked DCFs for the 20 FSRQs and 20 BL Lacs in our sample are shown in Figure \[stackopt\]. There are a few interesting features to note from these average correlations. First, although both peaks are well above any one of the stacked false-pair distribution points, the peak for FSRQs appears less significant than that for BL Lacs. If BL Lacs are indeed dominated by SSC and FSRQs by EC, then this trend is expected, as the correlations for EC are somewhat weaker than for SSC (quadratic for SSC and linear for EC). Second, the stacked DCF peak for FSRQs is a factor of $\sim 2$ narrower than that for BL Lacs. The relative tightness of this stacked correlation peak is likely caused by FSRQs having higher typical variability than BL Lacs [e.g., @abd10d; @2LAC], on average leading to better determined correlation peaks and more tightly constrained lags. The larger spread in BL Lacs may then be caused by a wider distribution of effective lags between the sources or a lower amplitude and longer timescale for the typical flaring event. If the significance of individual source DCF peaks were to improve with better coverage of overlapping flares, we might expect the combined BL Lac DCF to settle down to the mean lag expected for SSC emission.
The location of the synchrotron peak in the SEDs might also correlate with DCF peak properties. FSRQs are LSP blazars, while BL Lacs have a wider distribution of synchrotron peak energy, including ISP and HSP BL Lacs. One might expect the latter to be increasingly SSC dominated. The stacked DCFs for the 6 LSP, 8 ISP, and 6 HSP BL Lacs are shown in Figure \[stacksed\]. At present, we have too few objects to infer real trends; the ISP BL Lacs exhibit the only well-defined peak, at $\tau = -1.6 \pm 3.9$ day.
Discussion {#disc}
==========
We have been monitoring 157 gamma-ray-bright sources detected by [*Fermi*]{}/LAT with KAIT at Lick Observatory, and here present a study of optical–gamma-ray correlations in 40 sources selected based on [*Fermi*]{}/LAT detection significance. Overall, optical and gamma-ray emission were found to be highly correlated in these sources, at time delays of roughly days. An average DCF for all 40 sources was found to have a peak significance above 99%, confirming the strength in correlation between the two wavebands. Such strong correlations support the leptonic models of ICS gamma-ray emission, in which seed photons are upscattered by relativistic jet electrons responsible for synchrotron optical emission.
Whether the seed photons are dominated by the synchrotron radiation produced by these electrons (SSC) or by external radiation (EC) is difficult to discern based on lags and correlation strengths of these sources. However, we find that the well-measured FSRQs tend to have positive lags (gamma rays leading the optical) while the best-measured BL Lacs show no clear trend toward lag or lead. This supports models with EC being dominant in FSRQs and SSC dominant in BL Lacs. Stacked DCFs of LSP, ISP, and HSP BL Lacs are consistent with increasing SSC dominance as synchrotron peak energy increases; ISP and HSP BL Lacs are found to have average DCF peak lags closer to 0 days than LSP BL Lacs. However, a larger sample is required to make definitive claims. We plan on performing a similar study with [*Fermi*]{}/LAT light curves for the full set of 157 blazars being monitored in order to verify these findings based on the 40 brightest sources.
Recently, optical–gamma-ray correlations in blazars have been investigated through modulation indices (rather than DCFs and lags between wavebands) by @hov14. With a very large sample size, this study found that HSP BL Lacs were most strongly and tightly correlated, supporting the notion that SSC becomes more prevalent in ISP and HSP sources while EC is more dominant in LSP sources. Based on the 40 sources in our study, we also find a stronger average correlation in BL Lacs (but narrower average peak for FSRQs, likely owing to the high variability of FSRQs in our sample). Observationally, this trend is somewhat surprising, as the strongest correlations in individual sources have been found for FSRQs [e.g., @hay12]. That BL Lacs are on average more strongly correlated than FSRQs will need further testing, although our findings and those of @hov14 support this claim.
We have shown that strongly flaring, well-measured FSRQs tend to show gamma-ray leads, supporting the conclusions of other variability studies. However, the situation for the BL Lacs is evidently more complex. Multi-year, multi-wavelength monitoring at high cadence is clearly needed to probe the mechanisms driving variability in these sources and possible connections to the central engines. With further KAIT coverage and the continued success of [*Fermi*]{}/LAT, we can start to break down this complexity and understand these objects at their most fundamental level.
The *Fermi* LAT Collaboration acknowledges generous ongoing support from a number of agencies and institutes that have supported both the development and the operation of the LAT as well as scientific data analysis. These include the National Aeronautics and Space Administration (NASA) and the Department of Energy in the United States, the Commissariat à l’Energie Atomique and the Centre National de la Recherche Scientifique/Institut National de Physique Nucléaire et de Physique des Particules in France, the Agenzia Spaziale Italiana and the Istituto Nazionale di Fisica Nucleare in Italy, the Ministry of Education, Culture, Sports, Science, and Technology (MEXT), High Energy Accelerator Research Organization (KEK), and Japan Aerospace Exploration Agency (JAXA) in Japan, and the K. A. Wallenberg Foundation, the Swedish Research Council, and the Swedish National Space Board in Sweden. Additional support for science analysis during the operations phase is gratefully acknowledged from the Istituto Nazionale di Astrofisica in Italy and the Centre National d’Études Spatiales in France.
The work was financed in part by NASA grants NNX10AU09G, GO-31089, NNX12AF12G, and NAS5-00147. We are also grateful for support from Gary and Cynthia Bengier, the Richard and Rhoda Goldman Fund, the Christopher R. Redlich Fund, the TABASGO Foundation, and NSF grant AST-1211916. KAIT and its ongoing operation were made possible by donations from Sun Microsystems, Inc., the Hewlett-Packard Company, AutoScope Corporation, Lick Observatory, the NSF, the University of California, the Sylvia and Jim Katzman Foundation, and the TABASGO Foundation. We dedicate this paper to the memory of our dear friend and collaborator, Weidong Li, whose unfailing devotion to KAIT was of pivotal importance for this work; his premature, tragic passing has deeply saddened us.
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[llccccccc]{}
PKS 0048$-$09 & J0050.6$-$0929 & BL Lac & ISP & 0.635 & 313 & $-5.3\pm3.1$ & $99.0_{-1.7}^{+0.7}$ & 42/54\
S2 0109+22 & J0112.1+2245 & BL Lac & ISP & 0.265 & 297 & $7.2\pm3.1$ & $81.8_{-6.7}^{+5.8}$ & 32/54\
B3 0133+388 & J0136.5+3905 & BL Lac & HSP & — & 546 & — & — & 33/54\
3C 66A & J0222.6+4302 & BL Lac & ISP & — & 293 & — & — & 31/54\
4C +28.07 & J0237.8+2846 & FSRQ & LSP & 1.206 & 283 & $4.1\pm1.3$ & $90.8_{-1.8}^{+1.5}$ & 27/48\
PKS 0301$-$243 & J0303.4$-$2407 & BL Lac & HSP & 0.260 & 337 & $7.9\pm3.3$ & $88.6_{-6.7}^{+5.2}$ & 31/62\
PKS 0420$-$01 & J0423.2$-$0120 & FSRQ & LSP & 0.916 & 235 & $7.5\pm3.9$ & $88.6_{-6.4}^{+5.4}$ & 32/54\
PKS 0454$-$234 & J0457.0$-$2325 & FSRQ & - & 1.003 & 220 & — & — & 32/54\
4C +14.23 & J0725.3+1426 & FSRQ & LSP & 1.038 & 293 & $13.5\pm3.8$ & $92.6_{-4.3}^{+1.9}$ & 23/50\
PKS 0805$-$07 & J0808.2$-$0750 & FSRQ & LSP & 1.837 & 322 & — & — & 27/53\
PKS 0829+046 & J0831.9+0429 & BL Lac & LSP & 0.174 & 319 & $-28.8\pm7.2$ & $93.5_{-7.5}^{+3.5}$ & 29/53\
BZQ J0850$-$1213 & J0850.2$-$1212 & FSRQ & LSP & 0.566 & 335 & — & — & 28/56\
S4 0917+44 & J0920.9+4441 & FSRQ & LSP & 2.189 & 237 & $8.0\pm3.0$ & $77.2_{-10.9}^{+8.0}$ & 33/62\
4C +55.17 & J0957.7+5522 & FSRQ & LSP & 0.899 & 320 & — & — & 31/56\
1H 1013+498 & J1015.1+4925 & BL Lac & HSP & 0.212 & 390 & $-2.6\pm5.5$ & $69.1_{-8.4}^{+5.9}$ & 32/56\
4C +01.28 & J1058.4+0133 & BL Lac & LSP & 0.888 & 264 & $-13.8\pm3.1$ & $97.7_{-2.4}^{+0.8}$ & 30/57\
TXS 1055+567 & J1058.6+5628 & BL Lac & ISP & 0.143 & 367 & $2.5\pm3.3$ & $86.1_{-10.4}^{+6.9}$ & 33/55\
Ton 599 & J1159.5+2914 & FSRQ & LSP & 0.725 & 225 & $-43.5\pm3.0$ & $74.0_{-7.1}^{+6.6}$ & 30/55\
4C +21.35 & J1224.9+2122 & FSRQ & LSP & 0.434 & 194 & $8.6\pm1.5$ & $85.3_{-4.3}^{+4.0}$ & 33/63\
PG 1246+586 & J1248.2+5820 & BL Lac & ISP & — & 373 & $-32.3\pm6.6$ & $86.0_{-3.6}^{+3.9}$ & 45/56\
S4 1250+53 & J1253.1+5302 & BL Lac & LSP & — & 373 & $-16.0\pm7.3$ & $64.3_{-7.7}^{+7.4}$ & 45/52\
3C 279 & J1256.1-0547 & FSRQ & LSP & 0.536 & 192 & $19.7\pm3.4$ & $92.3_{-2.9}^{+2.6}$ & 20/70\
OP 313 & J1310.6+3222 & FSRQ & LSP & 0.997 & 279 & $6.3\pm7.7$ & $44.1_{-19.1}^{+14.1}$ & 34/58\
PKS 1424+240 & J1427.0+2347 & BL Lac & ISP & — & 386 & $-26.4\pm3.8$ & $68.4_{-9.0}^{+6.9}$ & 36/57\
GB6 J1542+6129 & J1542.9+6129 & BL Lac & ISP & — & 330 & $-3.7\pm6.4$ & $64.0_{-9.3}^{+8.7}$ & 48/57\
PKS 1551+130 & J1553.5+1255 & FSRQ & - & 1.308 & 362 & — & — & 38/62\
PG 1553+113 & J1555.7+1111 & BL Lac & HSP & — & 421 & $-37.0\pm5.8$ & $89.4_{-1.4}^{+1.5}$ & 37/62\
4C +38.41 & J1635.2+3810 & FSRQ & LSP & 1.813 & 199 & $5.9\pm0.8$ & $87.1_{-2.2}^{+1.7}$ & 37/62\
S5 1803+784 & J1800.5+7829 & BL Lac & LSP & 0.680 & 269 & $-32.0\pm3.0$ & $57.4_{-28.4}^{+20.5}$ & 17/40\
S4 1849+67 & J1849.4+6706 & FSRQ & LSP & 0.657 & 284 & $-18.3\pm2.4$ & $56.9_{-10.3}^{+8.5}$ & 37/60\
1ES 1959+650 & J2000.0+6509 & BL Lac & HSP & 0.047 & 439 & $-10.1\pm3.9$ & $50.4_{-9.5}^{+8.6}$ & 23/59\
PKS 2144+092 & J2147.3+0930 & FSRQ & LSP & 1.113 & 203 & $-38.6\pm7.3$ & $93.2_{-6.6}^{+4.9}$ & 33/63\
BL Lacertae & J2202.8+4216 & BL Lac & ISP & 0.069 & 286 & $-2.6\pm0.7$ & $78.3_{-6.7}^{+5.0}$ & 33/62\
PKS 2201+171 & J2203.4+1726 & FSRQ & LSP & 1.076 & 292 & — & — & 37/60\
PKS 2227$-$08 & J2229.7$-$0832 & FSRQ & LSP & 1.560 & 205 & $-24.5\pm2.2$ & $82.4_{-7.2}^{+5.5}$ & 38/61\
CTA 102 & J2232.4+1143 & FSRQ & LSP & 1.037 & 244 & $11.4\pm0.7$ & $92.3_{-2.8}^{+2.0}$ & 30/57\
B2 2234+28A & J2236.4+2828 & BL Lac & LSP & 0.795 & 322 & — & — & 36/54\
PKS 2233$-$148 & J2236.5$-$1431 & BL Lac & LSP & — & 275 & $30.7\pm1.2$ & $83.9_{-9.2}^{+7.0}$ & 37/57\
RGB J2243+203 & J2243.9+2021 & BL Lac & HSP & — & 543 & $6.2\pm3.9$ & $65.1_{-19.0}^{+13.8}$ & 49/59\
[^1]: http://fermi.gsfc.nasa.gov/ssc/data/analysis/\
documentation/Cicerone/
|
---
abstract: |
We report on the serendipitous discovery in the Blanco Cosmology Survey (BCS) imaging data of a $z = 0.9057$ galaxy that is being strongly lensed by a massive galaxy cluster at a redshift of $z=0.3838$. The lens (BCS J2352-5452) was discovered while examining $i$- and $z$-band images being acquired in October 2006 during a BCS observing run. Follow-up spectroscopic observations with the GMOS instrument on the Gemini South 8m telescope confirmed the lensing nature of this system. Using weak plus strong lensing, velocity dispersion, cluster richness $N_{200}$, and fitting to an NFW cluster mass density profile, we have made three independent estimates of the mass $M_{200}$ which are all very consistent with each other. The combination of the results from the three methods gives $M_{200} = (5.1 \pm 1.3)
\times 10^{14} M_\Sun$, which is fully consistent with the individual measurements. The final NFW concentration $c_{200}$ from the combined fit is $c_{200} = 5.4^{+1.4}_{-1.1}$. We have compared our measurements of $M_{200}$ and $c_{200}$ with predictions for (a) clusters from $\Lambda$CDM simulations, (b) lensing selected clusters from simulations, and (c) a real sample of cluster lenses. We find that we are most compatible with the predictions for $\Lambda$CDM simulations for lensing clusters, and we see no evidence based on this one system for an increased concentration compared to $\Lambda$CDM. Finally, using the flux measured from the \[OII\]3727 line we have determined the star formation rate (SFR) of the source galaxy and find it to be rather modest given the assumed lens magnification.
author:
- |
E. J. Buckley-Geer, H. Lin, E. R. Drabek, S. S. Allam, D. L. Tucker, R. Armstrong, W. A. Barkhouse, E. Bertin, M. Brodwin, S. Desai, J. A. Frieman, S. M. Hansen, F. W. High, J. J. Mohr, Y.-T. Lin, C.-C. Ngeow, A. Rest, R. C. Smith,\
J. Song, A. Zenteno,
title: 'The serendipitous observation of a gravitationally lensed galaxy at z = 0.9057 from the Blanco Cosmology Survey: The Elliot Arc'
---
Introduction
============
Strong gravitational lenses offer unique opportunities to study cosmology, dark matter, galactic structure, and galaxy evolution. They also provide a sample of galaxies, namely the lenses themselves, that are selected based on total mass rather than luminosity or surface brightness. The majority of lenses discovered in the past decade were found through dedicated surveys using a variety of techniques. For example, the Sloan Digital Sky Survey (SDSS) data have been used to effectively select lens candidates from rich clusters [@hennawi08] through intermediate scale clusters [@allam07; @lin09] to individual galaxies [@bolton08; @willis06]. Other searches using the CFHTLS [@cabanac07] and COSMOS fields [@faure08; @jackson08] have yielded 40 and 70 lens candidates respectively. These searches cover the range of giant arcs with Einstein radii $\theta_{EIN}> 10\arcsec$ all the way to small arcs produced by single lens galaxies with $\theta_{EIN}<
3\arcsec$.
In this paper we report on the serendipitous discovery of a strongly lensed $z=0.9057$ galaxy in the Blanco Cosmology Survey (BCS) imaging data. The lens is a rich cluster containing a prominent central brightest cluster galaxy (BCG) and has a redshift of $z=0.3838$. Cluster-scale lenses are particularly useful as they allow us to study the effects of strong lensing in the core of the cluster and weak lensing in the outer regions. Strong lensing provides constraints on the mass contained within the Einstein radius of the arcs whereas weak lensing provides information on the mass profiles in the outer reaches of the cluster. Combining the two measurements allows us to make tighter constraints on the mass $M_{200}$ and the concentration $c_{200}$, of an NFW [@nfw95] model of the cluster mass density profile, over a wider range of radii than would be possible with either method alone [@natarajan98; @natarajan02; @bradac06; @bradac08a; @bradac08b; @diego07; @limousin07; @hicks07; @deb08; @merten09; @oguri09]. In addition, if one has spectroscopic redshifts for the member galaxies one can determine the cluster velocity dispersion, assuming the cluster is virialized, and hence obtain an independent estimate for $M_{200}$ [@becker07]. Finally one can also derive an $M_{200}$ estimate from the maxBCG cluster richness $N_{200}$ [@hansen05; @johnston07]. These three different methods, strong plus weak lensing, cluster velocity dispersion, and optical richness, provide independent estimates of $M_{200}$ ($M_{200}$ is defined as the mass within a sphere of overdensity 200 times the critical density at the redshift $z$) and can then be combined to obtain improved constraints on $M_{200}$ and $c_{200}$. Measurements of the concentration from strong lensing clusters is of particular interest as recent publications suggest that they may be more concentrated than one would expect from $\Lambda$CDM models [@broadhurst_barkana08; @oguri_blandford09].
The paper is organized as follows. In § \[sec:bcs\_survey\] we describe the Blanco Cosmology Survey. Then in § \[sec:discovery\] we discuss the initial discovery and the spectroscopic follow-up that led to confirmation of the system as a gravitational lens, the data reduction, the properties of the cluster, the extraction of the redshifts, and finally the measurement of the cluster velocity dispersion and estimate of the cluster mass. In § \[sec:lens\_modeling\] we summarize the strong lensing features of the system. In § \[sec:weak\_lens\] we describe the weak lensing measurements. In § \[sec:combined\_constraints\] we present the results of combining of the strong and weak lensing results and the final mass constraints derived from combining the lensing results with the velocity dispersion and richness measurements. We describe the source galaxy star formation rate measurements in § \[sec:SFR\] and finally in § \[sec:conclusions\] we conclude. We assume a flat cosmology with $\Omega_{\rm M}= 0.3$, $\Omega_{\Lambda}=0.7$, and $H_0=70$ km s$^{-1}$ Mpc$^{-1}$, unless otherwise noted.
The BCS Survey {#sec:bcs_survey}
==============
The Blanco Cosmology Survey (BCS) is a 60-night NOAO imaging survey program (2005-2008), using the Mosaic-II camera on the Blanco 4m telescope at CTIO, that has uniformly imaged $75 \deg^2$ of the sky in the SDSS $griz$ bands in preparation for cluster finding with the South Pole Telescope (SPT) [@vanderlinde10] and other millimeter-wave experiments. The depths in each band were chosen to allow the estimation of photometric redshifts for $L\ge
L_*$ galaxies out to a redshift of $z = 1$ and to detect galaxies to $0.5L_*$ at $5\sigma$ to these same redshifts. The survey was divided into two fields to allow efficient use of the allotted nights between October and December. Both fields lie near $\delta = -55^{\circ}$ which allows for overlap with the SPT. One field is centered near $\alpha=23.5$ hr and the other is at $\alpha=5.5$ hr. In addition to the large science fields, BCS also covers 7 small fields that overlap large spectroscopic surveys so that photometric redshifts (photo-z’s) using BCS data can be trained and tested using a sample of over 5,000 galaxies.
Discovery of the lens and spectroscopic follow-up {#sec:discovery}
=================================================
The lens BCS J2351-5452 was discovered serendipitously while examining $i$- and $z$-band images being acquired in October 2006 during the yearly BCS observing run. The discoverer (EJB-G) decided to name it “The Elliot Arc” in honor of her then eight-year old nephew. Table \[table\_obslog\] lists the observed images along with seeing conditions. Fig. \[color\_coadd\] shows a $gri$ color image of the source, lens and surrounding environment (the pixel scale is $0.268\arcsec$ per pixel). The source forms a purple ring-like structure of radius $\sim 7.5\arcsec$ with multiple distinct bright regions. The lens is the BCG at the center of a large galaxy cluster. Photometric measurements estimated the redshift of the cluster at $z\sim 0.4$, using the expected $g-r$ and $r-i$ red sequence colors, and also provided a photo-z for the source of $z\sim 0.7$, as described below. We note that this cluster was first reported as SCSO J235138$-$545253 in an independent analysis of the BCS data by @menanteau10 where its remarkable lens was noted and they estimated a photometric redshift of $z=0.33$ for the cluster.
We obtained Gemini Multi-Object Spectrograph (GMOS) spectra of the source and a number of the neighboring galaxies [@lin07]. We targeted the regions of the source labeled A1-A4 in Fig. \[knot\_targets\], and photometric properties of these bright knots are summarized in Table \[table\_knots\]. In addition we selected 51 more objects for a total of 55 spectra. The additional objects were selected using their colors in order to pick out likely cluster member galaxies. Fig. \[color-color\] shows the $r-i$ versus $i$ color-magnitude diagram (top plot) and the $g-r$ vs. $r-i$ color-color diagram (bottom plot) of the field. The blue squares in the bottom panel of Fig. \[color-color\] show the four targeted knots in the lensed arcs. The green curve is an Scd galaxy model [@cww] with the green circles indicating a photometric redshift for the arc of $z \sim 0.7$. Note this is not a detailed photo-z fit, but is just a rough estimate meant to show that the arc is likely at a redshift higher than the cluster redshift. Highest target priority was given to the arc knots and to the BCG. Then cluster red sequence galaxy targets were selected using the simple color cuts $1.55 \leq g-r \leq 1.9$ and $0.6 \leq r-i \leq 0.73$ (also shown in the bottom panel of Fig. \[color-color\]), which approximate the more detailed final cluster membership criteria described below in §\[sec:cluster\_properties\]. Red sequence galaxies with $i < 21.6$ ($3\arcsec$-diameter SExtractor aperture magnitudes) were selected, with higher priority given to brighter galaxies with $i(3\arcsec) \leq 21$. Additional non-cluster targets lying outside the cluster color selection box were added at lowest priority.
We used the GMOS R150 grating + the GG455 filter in order obtain spectra with about 4600 – 9000 Å wavelength coverage. This was designed to cover the \[OII\] 3727 emission line expected at $\sim 6300$ Å, given the photo-z estimate of $\sim 0.7$ for the arcs as well as the Mg absorption features at $\sim 7000$ Å (and the 4000 Å break at $\sim 5600$ Å) for the $z \sim 0.4$ cluster elliptical galaxies.
We used 2 MOS masks in order to fully target these cluster galaxies (along with the arcs) for spectroscopy. Each mask had a 3600 second exposure time split into 4 900-second exposures for cosmic ray removal. We also took standard Cu-Ar lamp spectra for wavelength calibrations and standard star spectra for flux calibrations. All data were taken in queue observing mode. A summary of the observations is given in Table \[table\_obslog\].
Data Reduction {#sec:reduction}
--------------
The BCS imaging data were processed using the Dark Energy Survey data management system (DESDM V3) which is under development at UIUC/NCSA/Fermilab [@mohr08; @ngeow06; @zenteno11]. The images are corrected for instrumental effects which include crosstalk correction, pupil ghost correction, overscan correction, trimming, bias subtraction, flat fielding and illumination correction. The images are then astrometrically calibrated and remapped for later coaddition. For photometric data, a photometric calibration is applied to the single-epoch and coadd object photometry. The Astr$\cal O$matic software[^1] SExtractor [@bertin96], SCAMP [@bertin06] and SWarp [@bertin02] are used for cataloging, astrometric refinement and remapping for coaddition over each image. We have used the coadded images in the $griz$ bands for this analysis. The spectroscopic data were processed using the standard data reduction package provided by Gemini that runs in the IRAF framework[^2]. We used version 1.9.1. This produced flux- and wavelength-calibrated 1-D spectra for all the objects. Additional processing for the source spectra was done using the IRAF task [apall]{}.
Cluster properties {#sec:cluster_properties}
------------------
We adopt the procedure used by the maxBCG cluster finder [@koester07a; @koester07b] to determine cluster membership and cluster richness and to derive a richness-based cluster mass estimate. We first measure $N_{gal}$, the number of cluster red sequence galaxies, within a radius $1~h^{-1}$ Mpc ($= 4.55\arcmin$) of the BCG, that are also brighter than $0.4 L_*$ at the cluster redshift $z = 0.38$. From [@koester07a], $0.4 L_*$ corresponds to an $i$-band absolute magnitude $M = -20.25 + 5 \log h$ at $z = 0$, while at $z = 0.38$, $0.4 L_*$ corresponds to an apparent magnitude $i = 20.5$ (specific value provided by J. Annis & J. Kubo, private communication), after accounting for both K-correction and evolution [also as described in @koester07a]. We apply this magnitude cut using the SExtractor $i$-band [MAG\_AUTO]{} magnitude, which provides a measure of a galaxy’s total light. (Note the $3\arcsec$-diameter aperture magnitude used earlier for target selection in general measures less light cf. [MAG\_AUTO]{}, but is better suited for roughly approximating the light entering a GMOS slit.) We set the red sequence membership cuts to be $g-r$ and $r-i$ color both within $2\sigma$ of their respective central values $(g-r)_0 = 1.77$ and $(r-i)_0 = 0.65$, where the latter are determined empirically based on the peaks of the color histograms of galaxies within $1~h^{-1}$ Mpc of the BCG. In applying the color cuts we use the colors defined by SExtractor $3\arcsec$-diameter aperture magnitudes (this provides higher S/N colors compared to using [MAG\_AUTO]{}), and for the uncertainty we define $\sigma = \sqrt{\sigma_{color}^2 + \sigma_{intrinsic}^2}$, where $\sigma_{color}$ is the color measurement error derived from the SExtractor aperture magnitude errors, and $\sigma_{intrinsic}$ is the intrinsic red sequence color width, taken to be 0.05 for $g-r$ and 0.06 for $r-i$ [@koester07a].
Carrying out the above magnitude and color cuts, we obtain an initial richness estimate $N_{gal} = 44$. Then, as discussed in [@hansen05], we define another radius $r_{200}^{gal} = 0.156~N_{gal}^{0.6}~h^{-1}~{\rm Mpc} = 1.51~h^{-1}~{\rm Mpc}$ ($= 6.88\arcmin$), and repeat the same cuts within $r_{200}^{gal}$ of the BCG to obtain a final richness estimate $N_{200} = 55$. Finally, using the weak lensing mass calibration of [@johnston07] for maxBCG clusters, we obtain a mass estimate $M_{200} = (8.794 \times 10^{13}) \times (N_{200}/20)^{1.28}~h^{-1}~M_\Sun =
(4.6 \pm 2.1) \times 10^{14}~M_\Sun \ (h = 0.7)$, where we have also adopted the fractional error of 0.45 derived by [@rozo09] for this $N_{200}$-based estimate of $M_{200}$ for maxBCG clusters.
We note that [@rozo10] apply a factor of 1.18 to correct the [@johnston07] cluster masses upward, in order to account for a photo-z bias effect that is detailed in [@mandelbaum08]. We have not applied this correction as it makes only a 0.4$\sigma$ difference, although we remark that the resulting mass $M_{200} = 5.4 \times 10^{14}~M_\Sun$ does appear to improve the (already good) agreement with our other mass estimates below (see §\[sec:veldisp\] and §\[sec:combined\]).
Fig. \[color-color\] shows color-magnitude and color-color plots of all galaxies that have $i < 21$ (SExtractor [MAG\_AUTO]{}) and that are within a radius $r_{200}^{gal} = 1.51~h^{-1}~{\rm Mpc}$ ($= 6.88\arcmin$) of the BCG. Note we have extended the magnitude limit here down to $i = 21$, to match the effective magnitude limit of our spectroscopic redshift sample (§\[sec:redshifts\] below) In particular, we find 86 maxBCG cluster members for $i < 21$, compared to the earlier $N_{200} = 55$ for $i < 20.5$ (corresponding to $0.4 L_*$). These member galaxies are shown using red symbols in Fig. \[color-color\] and their properties are given in Table \[cluster\_galaxies\].
Redshift determinations {#sec:redshifts}
-----------------------
The redshift extraction was carried out using the [xcsao]{} and [emsao]{} routines in the IRAF external package [rvsao]{} [@kurtz98]. We obtained spectra for the 55 objects that were targeted. Four of these spectra were of the source. Out of the remaining 51 spectra we had sufficient signal-to-noise in 42 of them to determine a redshift. Thirty of the objects with redshifts between 0.377 and 0.393 constitute our spectroscopic sample of cluster galaxies. Fig. \[all\_targets\] shows the spatial distribution of galaxies within a $6\arcmin \times 6\arcmin$ box centered on the BCG, with maxBCG cluster members, arc knots, and objects with spectroscopic redshifts indicated by different colors and symbols. Table \[cluster\_galaxies\] summarizes the properties of the 30 cluster member galaxies with redshifts, and Table \[other\_galaxies\] summarizes the properties of the remaining 12 spectroscopic non-member galaxies. In Fig. \[cluster\_spectra\] we show four examples of the flux-calibrated cluster member spectra including the BCG.
Examination of Table \[cluster\_galaxies\] and Table \[other\_galaxies\] shows that our spectroscopic sample is effectively limited at $i \approx 21$, as 39 of the 42 non-arc redshifts have $i < 21$. Note that of the 30 spectroscopically defined cluster members, 22 are also maxBCG members, while another 7 lie close to the maxBCG color selection boundaries. Also, of the 12 spectroscopic non-members, none meets the maxBCG criteria except the faintest one (with $i = 21.58$).
The redshift of the source was determined from a single emission line at 7100Å which is present with varying signal-to-noise in each of the knots that were observed. We take this line to be the \[OII\]3727Å line which yields a redshift of $0.9057\pm 0.0005$. The four flux-calibrated source spectra are shown in Fig. \[arc\_spectra\]. Knot A2 was observed under seeing conditions that were a factor of two worse than for the other three knots (see Table \[table\_obslog\]).
Velocity dispersion and cluster mass measurement {#sec:veldisp}
------------------------------------------------
We used the 30 cluster galaxies to estimate the redshift and velocity dispersion of the cluster using the biweight estimators of @beers90. We first use the biweight location estimator to determine the best estimate for [*cz*]{}. This yields a value of $cz = 115151.1 \pm 241.1 \ {\rm km \ s^{-1}}$ which translates to a redshift of $z_c = 0.3838\pm 0.0008$. We then use this estimate of the cluster redshift to determine the peculiar velocity $v_p$ for each cluster member relative to the cluster center of mass using $$v_p = \frac{(cz - cz_c)}{(1+z_c)} \label{eqn_vp}$$ We determine the biweight estimate of scale for $v_p$ which is equal to the velocity dispersion of the cluster. We find a value for the velocity dispersion of $\sigma_c = 855^{+108}_{-96} \ {\rm km \
s^{-1}}$. The uncertainties are obtained by doing a jackknife resampling. The redshift distribution is shown in Fig. \[vel\_disp\]. The overlaid Gaussian has a mean of $z_c$ and a width of $\sigma_c\times (1+z_c)$. The lines represent the individual peculiar velocities $v_p$ of the cluster members.
We can use the estimated velocity dispersion to derive an estimate for the cluster mass. We use the results of [@evrard08] [see also @becker07] which relates $M_{200}$ to the dark matter velocity dispersion $$M_{200} = 10^{15}\ M_{\sun} \frac{1}{h(z)}
\left(\frac{\sigma_{DM}}{\sigma_{15}} \right) ^{1/\alpha} \ , \label{eqn_m1}$$ where $h(z) = H(z)/100\ {\rm km\ s^{-1} Mpc^{-1}}$ is the dimensionless Hubble parameter. The values of the parameters were found to be $\sigma_{15} = 1082.9\pm 4\ {\rm km \ s^{-1}}$ and $\alpha = 0.3361\pm
0.0026$ [@evrard08]. Using the standard definition of velocity bias $b_v =
\sigma_{gal}/\sigma_{DM}$, where $\sigma_{gal}$ is the galaxy cluster velocity dispersion, we can rewrite Equation \[eqn\_m1\] as $$b_v^{1/\alpha} M_{200} = 10^{15}\ M_{\sun} \frac{1}{h(z)}
\left(\frac{\sigma_{gal}}{\sigma_{15}}\right) ^{1/\alpha} \ , \label{eqn_m2}$$ where the quantity $b_v^{1/\alpha} M_{200}$ parameterizes our lack of knowledge about velocity bias. Substituting in the measured values for $\sigma_{gal}$ we obtain $b_v^{1/\alpha} M_{200} = 5.79^{+2.22}_{-1.99}
\times 10^{14} M_{\sun}$.
@bayliss11 [and references therein] discuss an “orientation bias” effect which causes an upward bias in the measured velocity dispersions of lensing-selected clusters, due to the higher likelihood of the alignment along the line of sight of the major axes of the cluster halos, which are in general triaxial. [@bayliss11] estimate that on average this will result in the dynamical mass estimate being biased high by 19-20%, using the same relation between $M_{200}$ and velocity dispersion as we have used [Eqn. \[eqn\_m1\] above; @evrard08]. Correcting for this orientation bias effect would result in $b_v^{1/\alpha} M_{200} = 4.8 \times 10^{14} M_{\sun}$, which is not a significant difference, as the change is well under $1\sigma$. We therefore do not apply this correction, but we do note that it would improve the already good agreement with our other mass estimates in §\[sec:cluster\_properties\] and §\[sec:combined\] (assuming no velocity bias, $b_v = 1$.)
Strong Lensing Properties {#sec:lens_modeling}
=========================
We use the coadded $r$-band image shown in Fig. \[arc-r-band\] to study the strong lensing features of the system as it has the best seeing and hence shows the most detail. To remove the contribution to the arc fluxes from nearby objects we used GALFIT [@peng02] to model the profiles of these objects (galaxies and stars) and then subtracted the model from the image. This was done for all four bands $griz$. These subtracted images are used for all determinations of arc fluxes and positions. A number of individual knots can be observed in the system along with the more elongated features. For example it appears that knot A1 is actually composed of two individual bright regions which are resolved by the Sextractor object extraction described below. Knot A2 also appears to have two components although these are not resolved by the object extraction so we treat them as one in the modeling. Even though the cluster is fairly massive we do not see evidence for additional arc-like features outside of the central circular feature. In this case we expect the mass of the lens to be well constrained by the image positions.
We use the criteria that to obtain multiple images the average surface mass density within the tangential critical curve must equal the critical surface mass density $\Sigma_{crit}$. The tangentially oriented arcs occur at approximately the tangential critical curves and so the radius of the circle $\theta_{arc}$ traced by the arcs provides a measurement of the Einstein radius $\theta_{EIN}$ [@bartelmann96]. The mass $M_{EIN}$ enclosed with the Einstein radius is therfore given by $$M_{EIN} = \Sigma_{crit}\pi (D_l \theta_{EIN})^{2} \label{eqn_M}$$ Substituting for $\Sigma_{crit}$ gives $$M_{EIN} = \frac{c^2}{4G} \frac{D_l D_s}{D_{ls}}\theta^2_{EIN} \label{eqn_SIS}$$ where $D_{s}$ is the angular diameter distance to the source, $D_{l}$ the angular diameter distance to the lens, and $D_{ls}$ the angular diameter distance between the lens and the source. These values are $D_{s} = 1610$ Mpc, $D_{l} = 1081$ Mpc and $D_{ls} = 825$ Mpc.
To determine the Einstein radius we ran Sextractor [@bertin96] on the $r$-band image. This identified eight distinct objects in the image. We used the coordinates of those eight objects and fit them to a circle. The radius of the circle gives us a measure of the Einstein radius. The Einstein radius we measure is $\theta_{EIN}=7.53\pm 0.25\arcsec$ which translates to $39.5\pm 1.3$ kpc. This yields a mass estimate of $(1.5\pm 0.1) \times 10^{13}M_{\sun}$ and a corresponding velocity dispersion (assuming an isothermal model for the mass distribution) of $\sigma = 694\pm 12 \ {\rm km \
s^{-1}}$.
The magnification of the lens $f_{lens}$ can be roughly estimated under the assumption that the 1/2-light radius of a source at redshift $z\sim 0.9$ is about $0.46\arcsec$ (derived from the mock galaxy catalog described in [@jouvel09]). The ratio of the area subtended by the ring to that subtended by the source is $\sim 0.6 \times (4R/\delta r)$, where $R$ is the ring radius and $\delta r$ is the 1/2-light radius of the source. The $0.6$ factor accounts for the fraction of the ring that actually contains images. This gives a magnification of $f_{lens} = 39$.
To obtain a more quantitative value for the magnification we have used the [PixeLens]{}[^3] program [@saha04] to model the lens. [PixeLens]{} is a parametric modeling program that reconstructs a pixelated mass map of the lens. It uses as input the coordinates of the extracted image positions and their parities along with the lens and source redshifts. It samples the solution space using a Markov Chain Monte Carlo method and generates an ensemble of mass models that reproduce the image positions. We used the Sextractor image positions obtained above and assigned the parities according to the prescription given in [@read07]. In [@saha04] they note that if one uses pixels that are too large then the mass distribution is poorly resolved and not enough steep mass models are allowed. We have chosen a pixel size such that this should not be a problem.
It is well known (see for example [@saha06]) that changing the slope of the mass profile changes the overall magnification, in particular a steeper slope produces a smaller magnification but does not change the image positions. Therefore the quoted magnification should be taken as a representative example rather than a definitive answer. The magnification quoted is the sum over the average values of the magnification for each image position for 100 models. We obtain a value of $f_{lens} = 141\pm 39$ where the error is the quadrature sum of the RMS spreads of the individual image magnifications. [PixeLens]{} can also determine the enclosed mass within a given radius. For the 100 models we obtain $M_{EIN} = (1.4\pm 0.02) \times 10^{13} M_{\sun}$ which is within $1\sigma$ of the mass obtained from the circle fit described above.
In order to combine the strong lensing mass with the mass estimate from the weak lensing analysis (in §\[sec:combined\] below) we will need to estimate the mass within $\theta_{EIN}$ that is due to dark matter alone ($M_{DM}$). To do this we will need to subtract estimates of the stellar mass ($M_S$) and the hot gas mass ($M_G$) from the total mass $M_{EIN}$. To determine $M_S$ we use the GALAXEV [@bc03] evolutionary stellar population synthesis code to fit galaxy spectral energy distribution models to the $griz$ magnitudes of the BCG within the Einstein radius. The BCG photometric data are taken from the GALFIT modeling described above, and we sum up the light of the PSF-deconvolved GALFIT model inside the Einstein radius. The GALAXEV models considered are simple stellar population (SSP) models which have an initial, instantaneous burst of star formation; such models provide good fits to early-type galaxies, such as those in clusters. In particular we find a good fit to the BCG, using a SSP model with solar metallicity, a [@chabrier03] stellar initial mass function (IMF), and an age 9.25 Gyr (this age provided the best $\chi^2$ over the range we considered, from 1 Gyr to 9.3 Gyr, the latter being the age of the universe for our cosmology at the cluster redshift $z = 0.38$). The resulting stellar mass (more precisely the total stellar mass integrated over the IMF) is $M_S = 1.7 \times 10^{12} M_{\sun}$.
To estimate the gas mass $M_G$ we have looked at estimates of hot gas fraction $f_{gas}$ in cluster cores from X-ray observations. Typical $f_{gas}$ measurements are of order 10% [@maughan04; @pointecouteau04] which give us an $M_G$ estimate of $1.5\times 10^{12} M_{\sun}$.
Finally we calculate the [*total*]{} $M/L$ ratio within $\theta_{EIN}$ for the $i$-band. This yields a value of $(M/L)_i = 33.7\pm 4.4
\ (M/L)_{\sun}$.
Weak Lensing Measurements {#sec:weak_lens}
=========================
Adaptive Moments {#sec:adaptive_moments}
----------------
We used the program Ellipto [@smith01] to compute adaptive moments [@bernstein02; @hirata04] of an object’s light distribution, i.e., moments optimized for signal-to-noise via weighting by an elliptical Gaussian function self-consistently matched to the object’s size. Ellipto computes adaptive moments using an iterative method and runs off of an existing object catalog produced by SExtractor for the given image. Ellipto is also a forerunner of the adaptive moments measurement code used in the SDSS photometric processing pipeline Photo.
We ran Ellipto on our coadded BCS images and corresponding SExtractor catalogs, doing so independently in each of the $griz$ filters to obtain four separate catalogs of adaptive second moments: $$\begin{aligned}
Q_{xx} & = & \int x^2 \ w(x,y) I(x,y) \ dx dy
\left/ \int w(x,y) I(x,y) \ dx dy \right. \\
Q_{yy} & = & \int y^2 \ w(x,y) I(x,y) \ dx dy
\left/ \int w(x,y) I(x,y) \ dx dy \right. \\
Q_{xy} & = & \int x y \ w(x,y) I(x,y) \ dx dy
\left/ \int w(x,y) I(x,y) \ dx dy \right. \ ,\end{aligned}$$ where $I(x,y)$ denotes the measured counts of an object at position $x,y$ on the CCD image, and $w(x,y)$ is the elliptical Gaussian weighting function determined by Ellipto. The images are oriented with the usual convention that North is up and East is to the left, i.e., right ascension increases along the $-x$ direction and declination increases along the $+y$ direction. We then computed the ellipticity components $e_1$ and $e_2$ of each object using one of the standard definitions $$\begin{aligned}
e_1 & = & (Q_{xx} - Q_{yy}) / (Q_{xx} + Q_{yy}) \\
e_2 & = & 2 Q_{xy} / (Q_{xx} + Q_{yy}) \ .\end{aligned}$$
PSF Modeling {#sec:psf_modeling}
------------
For each filter, we then identified a set of bright but unsaturated stars to use for PSF fitting. We chose the stars from the stellar locus on a plot of the size measure $Q_{xx}+Q_{yy}$ from Ellipto vs. the magnitude [MAG\_AUTO]{} from SExtractor, using simple cuts on size and magnitude to define the set of PSF stars. We then derived fits of the ellipticities $e_1, e_2$ and the size $Q_{xx}+Q_{yy}$ of the stars vs. CCD $x$ and $y$ position, using polynomial functions of cubic order in $x$ and $y$ (i.e., the highest order terms are $x^3, x^2 y, xy^2$, and $y^3$). On each image, these fits were done separately in each of 8 rectangular regions, defined by splitting the image area into 2 parts along the $x$ direction and into 4 parts along the $y$ direction, corresponding to the distribution of the 8 Mosaic-II CCDs over the image. This partitioning procedure was needed in order to account for discontinuities in the PSF ellipticity and/or size as we cross CCD boundaries in the Mosaic-II camera. Also note that the individual exposures comprising the final coadded image in each filter were only slightly dithered, so that the CCD boundaries were basically preserved in the coadd. To illustrate the PSF variation in our images, we present in Figure \[fig\_psf\_whiskers\] “whisker plots” that show the spatial variation of the magnitude and orientation of the PSF ellipticity across our $i$- and $r$-band images . In addition, we also show the residuals in the PSF whiskers remaining after our fitting procedure, showing that the fits have done a good job of modeling the spatial variations of the PSF in our data.
We next used our PSF model to correct our galaxy sizes and ellipticities for the effects of PSF convolution. Specifically, for the size measure $Q_{xx}+Q_{yy}$ we used the simple relation [cf. @hirata03] $$Q_{xx,true}+Q_{yy,true} = (Q_{xx,observed}+Q_{yy,observed})
- (Q_{xx,PSF}+Q_{yy,PSF})$$ to estimate the true size $Q_{xx,true}+Q_{yy,true}$ of a galaxy from its observed size $Q_{xx,observed}+Q_{yy,observed}$, where $Q_{xx,PSF}+Q_{yy,PSF}$ is obtained from the PSF model evaluated at the $x,y$ position of the galaxy. For the ellipticities we similarly used the related expressions $$\begin{aligned}
e_{i,true} & = & \frac{e_{i,observed}}{R_2}
+ \left(1 - \frac{1}{R_2} \right) e_{i,PSF} \ , \ i=1,2 \label{eqn_etrue} \\
R_2 & \equiv & 1 - \frac{Q_{xx,PSF}+Q_{yy,PSF}}
{Q_{xx,observed}+Q_{yy,observed}}\end{aligned}$$ The relations used in this simple correction procedure strictly hold only for unweighted second moments, or for adaptive moments in the special case when both the galaxy and the PSF are Gaussians. We have therefore also checked the results using the more sophisticated “linear PSF correction” procedure of [@hirata03], which uses additional fourth order adaptive moment measurements (also provided here by Ellipto) in the PSF correction procedure. In particular, the linear PSF correction method is typically applied in weak lensing analyses of SDSS data. However, we found nearly indistinguishable tangential shear profiles from applying the two PSF correction methods, and we therefore adopted the simpler correction method for our final results.
Shear Profiles and Mass Measurements {#sec:shear_profiles}
------------------------------------
Given the estimates of the true galaxy ellipticities from Equation (\[eqn\_etrue\]), we then computed the tangential ($e_T$) and B-mode or cross ($e_\times$) ellipticity components, in a local reference frame defined for each galaxy relative to the BCG: $$\begin{aligned}
e_T & = & e_1 \cos(2\phi) - e_2 \sin(2\phi) \\
e_\times & = & e_1 \sin(2\phi) + e_2 \cos(2\phi)\end{aligned}$$ where $\phi$ is the position angle (defined West of North) of a vector connecting the BCG to the galaxy in question. Here we have dropped the subscript [*true*]{} for brevity. The ellipticities were then converted to shears $\gamma$ using $\gamma = e / R$, where $R$ is the responsivity, for which we adopted the value $R = 2(1-\sigma^2_{SN}) = 1.73$, using $\sigma_{SN} = 0.37$ as the intrinsic galaxy shape noise as done in previous SDSS cluster weak lensing analyses [e.g., @kubo07; @kubo09].
We then fit our galaxy shear measurements to an NFW profile by minimizing the following expression for $\chi^2$: $$\chi^2 = \sum_{i=1}^{N} \frac{[\gamma_i - \gamma_{NFW}(r_i; M_{200},c_{200})]^2}
{\sigma_\gamma^2} \label{eqn_chi2}$$ where the index $i$ refers to each of the $N$ galaxies in a given sample, $r_i$ is a galaxy’s projected physical radius from the BCG (at the redshift of the cluster), $\sigma_\gamma$ is the measured standard deviation of the galaxy shears, and $\gamma_{NFW}$ is the shear given by Equations (14-16) of [@wright00] for an NFW profile with mass $M_{200}$ and concentration $c_{200}$. We used a standard Levenberg-Marquardt nonlinear least-squares routine to minimize $\chi^2$ and obtain best-fitting values and errors for the parameters $M_{200}$ and $c_{200}$ of the NFW profile. Similar fits of the weak lensing radial shear profile to a parameterized NFW model have often been used to constrain the mass distributions of galaxy clusters [e.g., @king01; @clowe01; @kubo09; @oguri09; @okabe10]. Note that we chose the above expression for $\chi^2$ since it does [*not*]{} require us to do any binning in radius, but for presentation purposes below we will have to show binned radial shear profiles compared to the NFW shear profiles obtained from our binning-independent fitting method.
For the shear fitting analysis, we defined galaxy samples separately in each of the four $griz$ filters using cuts on the magnitude [MAG\_AUTO]{} and on the size $Q_{xx,observed}+Q_{yy,observed}$, as detailed in Table \[table\_weak\_lensing\]. The bright magnitude cut was chosen to exclude brighter galaxies which would tend to lie in the foreground of the cluster and hence not be lensed, while the faint magnitude cuts were set to the photometric completeness limit in each filter, as defined by the turnover magnitude in the histogram of SExtractor [MAG\_AUTO]{} values. For the size cut, we set it so that only galaxies larger than about 1.5 times the PSF size would be used, as has been typically done in SDSS cluster weak lensing analyses [e.g., @kubo07; @kubo09]. Note that in order to properly normalize the NFW shear profile to the measurements, we also need to calculate the critical surface mass density $\Sigma_{crit}$, which depends on the redshifts of the lensed source galaxies as well as the redshift of the lensing cluster; see Equations (9,14) of [@wright00]. To do this, we did not use any individual redshift estimates for the source galaxies in our analysis, but instead we calculated an effective value of $1/\Sigma_{crit}$ via an integral over the source galaxy redshift distribution published for the Canada-France-Hawaii Telescope Legacy Survey [CFHTLS; @ilbert06], as appropriate to the magnitude cuts we applied in each of the $griz$ filters.
Our NFW fitting results are shown in Figures \[nfw\_fit\_ir\]-\[nfw\_fit\_zg\] and detailed in Table \[table\_weak\_lensing\]. We show results for both the tangential and B-mode shear components. As lensing does not produce a B-mode shear signal, these results provide a check on systematic errors and should be consistent with zero in the absence of significant systematics. For all of our filters, our B-mode shear results are indeed consistent with no detected mass, as the best-fit $M_{200}$ is within about 1$\sigma$ of zero. On the other hand, for the tangential shear results in the $r$, $i$, and $z$ filters, we do indeed obtain detections of non-zero $M_{200}$ at the better than $1.5\sigma$ level. In the $g$ filter we do not detect a non-zero $M_{200}$. Comparing the weak lensing results from the different filters serves as a useful check of the robustness of our lensing-based cluster mass measurement, in particular as the images in the different filters are subject to quite different PSF patterns, as shown earlier in Fig. \[fig\_psf\_whiskers\]. Though the mass errors are large, the $M_{200}$ values from the $r$-, $i$-, and $z$-band weak lensing NFW fits are nonetheless consistent with each other and with the masses derived earlier from the velocity dispersion and maxBCG richness analyses. Moreover, independent of the NFW fits, we have also derived probabilities (of exceeding the observed $\chi^2$) that our [*binned*]{} shear profiles are consistent with the null hypothesis of zero shear. As shown in Table \[table\_weak\_lensing\], we see that the B-mode profiles are in all cases consistent with zero, as expected, but that the tangential profiles for the $r$ and $i$ filters are not consistent with the null hypothesis at about the $2\sigma$ level (probabilities $\approx 0.06$), thus providing model-independent evidence for a weak lensing detection of the cluster mass.
Combining Weak Lensing Constraints from Different Filters {#sec:combined_filters}
---------------------------------------------------------
Here we will combine the weak lensing shear profile information from the different filters $griz$ in order to improve the constraints on the NFW parameters, in particular on $M_{200}$. The main complication here is that although the ellipticity measurement errors are independent among the different filters, the most important error for the shear measurement is the intrinsic galaxy shape noise, which is correlated among filters because a subset of the galaxies is common to two or more filters, and for these galaxies we expect their shapes to be fairly similar in the different filters. In particular we find that the covariance of the true galaxy ellipticities between filters is large, for example, the covariance of $e_1$ between the $i$ and $r$ filters, ${\rm Cov}(e_{1,i},e_{1,r}) = \frac{1}{N} \sum (e_{1,i}-\bar{e}_{1,i})
(e_{1,r}-\bar{e}_{1,r})$, is about 0.9 times the variance of $e_1$ in the $i$ and $r$ filters individually. The same holds true for $e_2$ and for the other filters as well. We will not attempt to use a full covariance matrix approach to deal with the galaxy shape correlations when we combine the data from two or more filters. Instead, we take a simpler approach of scaling the measured standard deviation of the shear (the $\sigma_\gamma$ used to calculate $\chi^2$ in Equation \[eqn\_chi2\]) by $\sqrt{N/N_{unique}}$, where $N$ is the total number of galaxies in a given multi-filter sample, and $N_{unique}$ is the number of unique galaxies in the same sample. This is equivalent to rescaling $\chi^2$ in the NFW fit to correspond to $N_{unique}$ degrees of freedom instead of $N$. We have verified using least-squares fits to Monte Carlo simulations of NFW shear profiles that this simple approach gives the correct fit uncertainties on $M_{200}$ and $c_{200}$ when the mock galaxy data contain duplicate galaxies, with identical $e_1$ and $e_2$ values, simulating the case of [*completely*]{} correlated intrinsic galaxy shapes among filters. Note that our approach is conservative and will slightly overestimate the errors, because the galaxy shapes in the real data are about 90% correlated, not fully correlated, among filters.
Before fitting the combined shear data from multiple filters, we make one additional multiplicative rescaling of the shear values, so that all filters will have the same effective value of $1/\Sigma_{crit}$, corresponding to a fiducial effective source redshift $z_{crit} = 0.7$. This correction is small, with the largest being a factor of 1.18 for the $z$-band data. The results of the NFW fits for the multi-filter samples are given in Table \[table\_weak\_lensing\], where we have tried the filter combinations $i+r$, $i+r+z$, and $i+r+z+g$. We see that these multi-filter samples all provide better fractional errors on $M_{200}$ compared to those from the single-filter data. Also, as expected, the B-mode results in all cases are consistent with no detected $M_{200}$ and zero shear. For our final weak lensing results, we adopt the NFW parameters from the $i+r+z$ sample, as it provides the best fractional error ($\sigma_{M_{200}}/M_{200} \approx 0.5$) on $M_{200}$; we obtain $M_{200} = 5.0^{+2.9}_{-2.3} \times 10^{14} M_\Sun$, and $c_{200} = 4.9^{+3.9}_{-2.2}$. Figure \[nfw\_fit\_irz\_sl\] shows the shear profile data and best fit results for the $i+r+z$ sample. This final weak lensing value for $M_{200}$ agrees well with the earlier values of $M_{200}$ derived from the cluster galaxy velocity dispersion (assuming no velocity bias) and from the cluster richness $N_{200}$.
Combined Constraints on Cluster Mass and Concentration {#sec:combined_constraints}
======================================================
Combining Strong and Weak Lensing {#sec:combined}
---------------------------------
In this section we combine the strong lensing and weak lensing information together in order to further improve our constraints on the NFW profile parameters, in particular on the concentration parameter $c_{200}$. The addition of the strong lensing information provides constraints on the mass within the Einstein radius, close to the cluster center, thereby allowing us to better measure the central concentration of the NFW profile and improve the uncertainties on the concentration $c_{200}$. [@oguri09] incorporated the strong lensing information in the form of a constraint on the Einstein radius due to just the dark matter distribution of the cluster, and they specifically excluded the contribution of (stellar) baryons to the Einstein radius. Their intent, as well as ours in this paper (§ \[sec:all\_combined\]), is to compare the observed cluster NFW concentration to that predicted from dark-matter-[*only*]{} simulations. Thus the contribution of baryonic matter should be removed, most importantly in the central region within the Einstein radius, where baryonic effects are the largest due in particular to the presence of the BCG. In practice with the present data we can do this separation of the baryonic contribution only for the strong lensing constraint, and strictly speaking the weak lensing profile results from the total mass distribution rather than from dark matter alone.
Here we combine the strong and weak lensing data using an analogous but somewhat simpler method compared to that of [@oguri09], specifically by adding a second term to $\chi^2$ (Equation \[eqn\_chi2\]) that describes the constraint on the dark matter (only) mass within the observed Einstein radius: $$\chi^2 = \sum_{i=1}^{N} \frac{[\gamma_i - \gamma_{NFW}(r_i; M_{200},c_{200})]^2}
{\sigma_\gamma^2}
+ \frac{[M_{DM}(< \theta_E) - M_{NFW}(< \theta_E; M_{200},c_{200})]^2}
{\sigma_{M_{DM}(< \theta_E)}^2} \label{eqn_chi2_wl_sl}$$ where $\theta_E = 7.53\arcsec$ is the observed Einstein radius due to the [*total*]{} cluster mass distribution, $M_{DM}(< \theta_E)$ is the dark matter (only) mass within $\theta_E$, and $M_{NFW}(< \theta_E; M_{200},c_{200})$ is the mass within $\theta_E$ of an NFW profile with mass $M_{200}$, concentration $c_{200}$, redshift $z = 0.38$, and source redshift $z = 0.9057$. $M_{NFW}(< \theta_E; M_{200},c_{200})$ is derived based on Equation (13) of [@wright00]. As obtained earlier in §\[sec:lens\_modeling\], we estimate $M_{DM}(< \theta_E)$ by subtracting estimates of the stellar mass and hot gas mass from the total mass within $\theta_E$, obtaining $M_{DM}(< \theta_E) = (1.18 \pm 0.2) \times 10^{13} M_\Sun$ when subtracting off both stellar and gas mass, or $M_{DM}(< \theta_E) = (1.33 \pm 0.2) \times 10^{13} M_\Sun$ when subtracting off only stellar mass. The former is our best estimate of $M_{DM}(< \theta_E)$, while the latter serves as an upper limit on $M_{DM}(< \theta_E)$ and hence on the best-fit concentration $c_{200}$. We also conservatively estimate the error on $M_{DM}(< \theta_E)$ to be one of the stellar mass/gas mass components added in quadrature to the uncertainty on the total $M_{EIN}$ from §\[sec:lens\_modeling\].
We apply the combined strong plus weak lensing analysis to our best weak lensing sample, the multi-filter $i+r+z$ data set. The fit results are given in Table \[table\_weak\_lensing\] and shown in Figure \[nfw\_fit\_irz\_sl\]. We find $M_{200} = 4.9^{+2.9}_{-2.2} \times 10^{14}$ solar masses, nearly identical to the final weak lensing result. We also get a concentration $c_{200} = 5.5^{+2.7}_{-1.6}$, again consistent with the final weak lensing fit, but with a 30% improvement in the error on $c_{200}$, demonstrating the usefulness of adding the strong lensing information to constrain the NFW concentration. Using the upper limit $M_{DM}(< \theta_E)$ value (with only stellar mass subtracted) gives nearly the same $M_{200} = 4.8^{+2.8}_{-2.2} \times 10^{14} M_\Sun$, while the resulting NFW concentration is higher, as expected, with $c_{200} = 6.2^{+3.2}_{-1.7}$, but still consistent with the fit using our best estimate of $M_{DM}(< \theta_E)$.
Combining Lensing, Velocity Dispersion and Richness Constraints {#sec:all_combined}
---------------------------------------------------------------
In the above sections we have obtained quite consistent constraints on the cluster mass $M_{200}$ using three independent techniques: (1) $M_{200}({\rm lensing}) = 4.9^{+2.9}_{-2.2} \times 10^{14} M_\Sun$ from combined weak + strong lensing (§\[sec:combined\]); (2) $M_{200}(\sigma_c) = 5.79^{+2.22}_{-1.99} \times 10^{14} M_\Sun$ from the cluster galaxy velocity dispersion $\sigma_c$ (§\[sec:veldisp\]; assuming no velocity bias, $b_v = 1$); and (3) $M_{200}(N_{200}) = (4.6 \pm 2.1) \times 10^{14} M_\Sun$ from the maxBCG-defined cluster richness $N_{200}$ (§\[sec:cluster\_properties\]). We note that these methods are subject to different assumptions and systematic errors. For example, the velocity dispersion based mass estimate assumes the cluster is virialized, an assumption supported by the Gaussian-shaped velocity distribution of the cluster members shown in Fig. \[vel\_disp\]. Also, the richness based mass estimate relies on the $N_{200}$-$M_{200}$ calibration [@johnston07] obtained for SDSS maxBCG clusters at lower redshifts $z = 0.1-0.3$ and assumes that this calibration remains valid for our cluster at $z = 0.38$. It is encouraging that we are obtaining a cluster mass measurement that appears to be robust to these disparate assumptions and that shows good agreement among multiple independent methods.
We will therefore combine the results from the different techniques in order to obtain final constraints on $M_{200}$ and concentration $c_{200}$ that are significantly improved over what any one technique permits. Specifically, we can add the $M_{200}$ constraints from the velocity dispersion and richness measurements as additional terms to the weak + strong lensing $\chi^2$ (Equation \[eqn\_chi2\_wl\_sl\]): $$\begin{aligned}
\chi^2 & = & \sum_{i=1}^{N} \frac{[\gamma_i - \gamma_{NFW}(r_i; M_{200},c_{200})]^2}
{\sigma_\gamma^2}
+ \frac{[M_{DM}(< \theta_E) - M_{NFW}(< \theta_E; M_{200},c_{200})]^2}
{\sigma_{M_{DM}(< \theta_E)}^2} \nonumber \\
& + & \frac{[M_{200}(\sigma_c)-M_{200}]^2}{\sigma_{M_{200}(\sigma_c)}^2}
+ \frac{[M_{200}(N_{200})-M_{200}]^2}{\sigma_{M_{200}(N_{200})}^2}
\label{eqn_chi2_all}\end{aligned}$$ Minimizing this overall $\chi^2$ results in the final best-fitting NFW parameters $M_{200} = 5.1^{+1.3}_{-1.3} \times 10^{14} M_\Sun$ and $c_{200} = 5.4^{+1.4}_{-1.1}$. These results are consistent with the final lensing-based values $M_{200}({\rm lensing}) = 4.9^{+2.9}_{-2.2} \times 10^{14} M_\Sun$ and $c_{200}({\rm lensing}) = 5.5^{+2.7}_{-1.6}$, but have errors nearly a factor of two smaller. Note these quoted errors are 1-parameter, $1\sigma$ uncertainties; we plot the joint 2-parameter, $1\sigma$ and $2\sigma$ contours in Fig. \[contours\_M200\_c\].
We also note that for the three methods weak lensing, velocity dispersion, and cluster richness, the corresponding NFW parameters result from the [*total*]{} mass distribution, consisting of both dark matter and baryonic (stellar plus hot gas) components. Dark matter is dominant over the bulk of the cluster, while baryons can have a significant effect in the cluster core [e.g., @oguri09]. As described earlier (§ \[sec:combined\]), we have thus subtracted off the baryonic contribution to the strong lensing constraint as the intent is to compare (see below) our cluster concentration value against those from dark-matter-only simulations. Note that we have not isolated the dark matter contribution for the other three methods and cannot easily do so. For weak lensing, the shear profile is sensitive to the total mass distribution, not just to dark matter. For the velocity dispersion method, the galaxies act as test particles in the overall cluster potential, which is due, again, to both dark matter and baryons. For the cluster richness method, the [@johnston07] $N_{200}$-$M_{200}$ relation we use was derived from stacked cluster weak lensing shear profile fits, including a BCG contribution but otherwise no other baryonic components; thus again the $M_{200}$ value is essentially for the total mass distribution. Nonetheless, the bulk of the baryonic contribution is in the cluster core and is accounted for via the strong lensing constraint, so we expect the comparison below of our cluster concentration value to those of dark matter simulations to be a reasonable exercise.
Recent analyses [e.g., @oguri09; @broadhurst_barkana08] of strong lensing clusters have indicated that these clusters are more concentrated than would be expected from $\Lambda$CDM predictions, though others have argued that no discrepancy exists if baryonic effects are accounted for [@richard10]. In the former case, [@oguri09] found a concentration $c_{\rm vir} \approx 9$ for the 10 strong lensing clusters in their analysis sample, compared to a value of $c_{\rm vir} \approx 6$ expected for strong-lensing-selected clusters or $c_{\rm vir} \approx 4$ for clusters overall [e.g., @broadhurst_barkana08; @oguri_blandford09]. We illustrate these different concentration values in Fig. \[contours\_M200\_c\]. We use Eqn. (17) of [@oguri09], $\bar{c}_{\rm vir}({\rm sim}) = \frac{7.85}{(1+z)^{0.71}}(M_{\rm vir}/2.78 \times 10^{12} M_\Sun)^{-0.081}$, which comes from the $\Lambda$CDM N-body simulations of [@duffy08], to show the typical concentration of clusters overall, and multiply by a factor of 1.5 [@oguri09] to show the higher concentration expected for lensing selected clusters. We also use Eqn. (18) of [@oguri09], $\bar{c}_{\rm vir}({\rm fit}) = \frac{12.4}{(1+z)^{0.71}}(M_{\rm vir}/10^{15} M_\Sun)^{-0.081}$, to show the fit results for their cluster sample. In these relations, we set $z = 0.4$ to match the redshift of our cluster. Moreover, we convert from the $M_{\rm vir}, c_{\rm vir}$ convention used by [@oguri09] to our $M_{200}, c_{200}$ convention, using the detailed relations found in Appendix C of [@hu03] or in the Appendix of [@johnston07]. For the plotted $M_{200}$ range, it turns out that $c_{200} \approx 0.83\ c_{\rm vir}$. From Fig. \[contours\_M200\_c\], we see that our best-fit value of $c_{200} = 5.4^{+1.4}_{-1.1}$ is most consistent with the nominal $\Lambda$CDM concentration value for lensing-selected clusters, and does not suggest the need for a concentration excess in this particular case. It’s likely that larger strong lensing cluster samples will be needed to more robustly compare the distribution of concentration values with the predictions of $\Lambda$CDM models.
Source Galaxy Star Formation Rate {#sec:SFR}
=================================
We can use the \[OII\]3727 line in the calibrated spectra described in § \[sec:redshifts\] to estimate the star formation rate (SFR). As noted by @kennicutt98 the luminosities of forbidden lines like \[OII\]3727 are not directly coupled to the ionizing luminosity and their excitation is also sensitive to abundance and the ionization state of the gas. However the excitation of \[OII\] is well behaved enough that it can be calibrated through H$\alpha$ as an SFR tracer. This indirect calibration is very useful for studies of distant galaxies because \[OII\]3727 can be observed out to redshifts $z\approx 1.6$ and it has been measured in several large samples of faint galaxies (see references in @kennicutt98). If we know the \[OII\] luminosity then we can use equation 3 from @kennicutt98 to determine a star formation rate for the galaxy $${\rm SFR} (M_\sun\ {\rm yr}^{-1}) = (1.4 \pm 0.4) \times 10^{-41} (L{\rm [OII]})({\rm ergs\ s}^{-1})\label{eqn_SFR}$$ where the uncertainty reflects the range between blue emission-line galaxies (lower limit) and more luminous spiral and irregular galaxies (upper limit).
As noted above, in order to extract the SFR we need to determine the total source flux from the \[OII\] line. We determine this using $$f(\nu)_{[OII]} = \frac{f(\nu)_L}{f(\nu)_S}\times
f(\nu)_I \label{eqn_SFR_flux}$$ where $f(\nu)_{[OII]}$ is the total flux emitted by the source in the \[OII\] line, $f(\nu)_L$ is the flux measured in the \[OII\] line in each spectrum, $f(\nu)_S$ is the flux in the knot spectrum contained within the [*i*]{}-band filter band pass and $f(\nu)_I$ is the flux from the source in the [*i*]{}-band.
Using the GALFIT-subtracted [*i*]{}-band image we determine $f(\nu)_I$ by summing the flux in an annulus of width 3 that encompasses the arcs. The flux $f(\nu)_L$ is measured by fitting a gaussian plus a continuum to the \[OII\] line in each spectrum and integrating the flux under the gaussian fit. The flux $f(\nu)_S$ is calculated as follows. For each spectrum we first fit the continuum level, we then add the fitted continuum plus the \[OII\] line flux and convolve it with the filter response curve for the SDSS [*i*]{}-band filter and integrate the convolved spectrum.
We have determined $f(\nu)_{[OII]}$ separately for each knot that was targeted for spectra. The fluxes are listed in Table \[table\_sfr\] for each knot. We convert $f(\nu)_{[OII]}$ into an \[OII\] luminosity and then use Equation \[eqn\_SFR\] to determine a star formation rate for each knot. This rate is the raw rate which must be scaled by the lens magnification $f_{lens}$ to determine the true rate. We quote the SFR for the two values of $f_{lens}$ that were determined in §\[sec:lens\_modeling\]. We assume one magnitude of extinction [@kennicutt98] and have corrected the measured \[OII\] luminosity to account for this. This yields the star formation rates listed in Table \[table\_sfr\] for the two values of $f_{lens}$. The rate for knot A3 is higher by a factor of 2 compared to the others because it has a small $f(\nu)_S$ compared to the other knots but the value of $f(\nu)_L$ is quite similar to the other knots. This can clearly be seen in Figure \[arc\_spectra\]. We can combine the measurements for the four knots using a simple average to quote an overall SFR. This yields values of ${\rm SFR}(f_{lens}=49) = 4.6 \pm 0.7$ and ${\rm SFR}(f_{lens}=141) = 1.3\pm0.2$.
These rates are significantly smaller that those obtained for the 8 o’clock arc [@allam07] and the Clone [@lin09] which were $229 M_\sun\ {\rm yr}^{-1}$ and $45 M_\sun\ {\rm yr}^{-1}$ respectively (after converting to our chosen cosmology). Both these systems were at much higher redshift (2.72 and 2.0 respectively) so one would potentially expect higher rates from these systems. They also had smaller values of $f_{lens}$. We can compare our result to blue galaxies at similar redshift from the DEEP2 survey [@cooper08]. Using Figure 18 of @cooper08 we obtain a median SFR of about $34 M_\sun\ {\rm yr}^{-1}$ for a redshift $z=0.9$ galaxy which is also higher than our measurement. Other measurements using the AEGIS field [@noeske07] give a median SFR ranging from $10 M_\sun\ {\rm yr}^{-1}$ to $40 M_\sun\ {\rm yr}^{-1}$ depending weakly on the galaxy mass, which is unknown in our case. Our measurement can be compared to the far-right plot of Figure 1 in @noeske07 and we fall on the low side of the measured data. Note that these conclusions are dependent on the magnification values used, for example smaller values such as those obtained for the Clone or the 8 o’clock arc would yield larger values for the SFR.
Conclusions {#sec:conclusions}
===========
We have reported on the discovery of a star-forming galaxy at a redshift of $z=0.9057$ that is being strongly lensed by a massive galaxy cluster at a redshift of $z=0.3838$.
The Einstein radius determined from the lensing features is $\theta_{EIN}=7.53\pm 0.25\arcsec$ and the enclosed mass is $(1.5\pm 0.1) \times 10^{13}M_{\sun}$, with a corresponding SIS velocity dispersion of $\sigma = 694 \pm 12 \ {\rm km \
s^{-1}}$.
Using GMOS spectroscopic redshifts measured for 30 cluster member galaxies, we obtained a velocity dispersion $\sigma_c = 855^{+108}_{-96} \ {\rm km \
s^{-1}}$ for the lensing cluster.
We have derived estimates of $M_{200}$ from measurements of (1) weak lensing, (2) weak + strong lensing, (3) velocity dispersion $\sigma_c$, and (4) cluster richness $N_{200} = 55$. We obtained the following results for $M_{200}$: (1) $M_{200}({\rm weak \ lensing}) = 5.0^{+2.9}_{-2.3} \times 10^{14} M_\Sun$, (2) $M_{200}({\rm lensing}) = 4.9^{+2.9}_{-2.2} \times 10^{14} M_\Sun$, (3) $M_{200}(\sigma_c) = 5.79^{+2.22}_{-1.99} \times 10^{14} M_\Sun$ (assuming no velocity bias, $b_v = 1$), and (4) $M_{200}(N_{200}) = (4.6 \pm 2.1) \times 10^{14} M_\Sun$. These results are all very consistent with each other. The combination of the results from methods 2, 3 and 4 give $M_{200} = 5.1^{+1.3}_{-1.3}
\times 10^{14} M_\Sun$, which is fully consistent with the individual measurements but with an error that is smaller by a factor of nearly two. The final NFW concentration from the combined fit is $c_{200} = 5.4^{+1.4}_{-1.1}$, which is also consistent with the lensing-based value but again with a smaller error.
We have compared our measurements of $M_{200}$ and $c_{200}$ with predictions for (a) clusters from $\Lambda$CDM simulations, (b) lensing selected clusters from simulations, and (c) a real sample of cluster lenses from @oguri09. We find that we are most compatible with the predictions from $\Lambda$CDM simulations for lensing clusters, and we see no evidence that an increased concentration is needed for this one system. We are studying this further using other lensing clusters we observed from the SDSS [@diehl09]. These clusters will be the subject of a future paper.
Finally, we have estimated the star forming rate (SFR) to be between 1.3 to 4.6 $M_\sun \ {\rm yr}^{-1} $, depending on magnification. These are small star-forming rates when compared to some of our previously reported systems, and are also small when compared with rates found for other galaxies at similar redshifts. However we caution that this conclusion is entirely dependent on the derived lens magnification.
We thank the anonymous reviewer for helpful comments which have improved the paper.
Based on observations obtained at the Gemini Observatory, which is operated by the Association of Universities for Research in Astronomy, Inc., under a cooperative agreement with the NSF on behalf of the Gemini partnership: the National Science Foundation (United States), the Science and Technology Facilities Council (United Kingdom), the National Research Council (Canada), CONICYT (Chile), the Australian Research Council (Australia), Ministério da Ciência e Tecnologia (Brazil) and Ministerio de Ciencia, Tecnología e Innovación Productiva (Argentina). Gemini Proposal ID GS-2007B-Q-228.
The Blanco Cosmology Survey was performed on the Blanco 4m telescope located at the Cerro Tololo Inter-American Observatory (National Optical Astronomy Observatory) which is operated by the Association of Universities for Research in Astronomy, under contract with the National Science Foundation.
S. S. Allam acknowledges support from an HST Grant. Support of program no. 11167 was provided by NASA through a grant from the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS5-26555.
AZ and JM acknowledge the support of the Excellence Cluster Universe in Garching. Support for M.B. was provided by the W. M. Keck Foundation.
Thanks to Jeff Kubo for providing a copy of the Ellipto adaptive moments code. Thanks also go to Quarknet students Liana Nicklaus, Gina Castelvecchi, Braven Leung, Nick Gebbia, Alex Fitch, and their advisor Patrick Swanson, who worked with HL during summer 2009 on a weak lensing mass analysis of this cluster with different codes than used here.
Fermilab is operated by Fermi Research Alliance, LLC under Contract No. DE-AC02-07CH11359 with the United States Department of Energy.
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![(Top) $r-i$ vs. $i$([MAG\_AUTO]{}) color-magnitude diagram for all galaxies (black points) with $i < 21$ and within a radius $r_{200}^{gal} = 1.51~h^{-1}~{\rm Mpc}$ ($= 6.88\arcmin$) of the BCG. Colors are measured using $3\arcsec$-diameter aperture magnitudes. Galaxies meeting the maxBCG cluster color selection criteria (see §\[sec:cluster\_properties\]) are plotted in red, with red circles indicating cluster members brighter than $i = 20.5$, and red squares indicating fainter cluster members. (Bottom) $g-r$ vs. $r-i$ color-color diagram for the same galaxies as in the top panel. Red circles and squares again indicate brighter and fainter maxBCG cluster members, while the black rectangle indicates the color selection box (approximating the more detailed maxBCG color criteria) used to select likely cluster galaxies for GMOS spectroscopy (see §\[sec:discovery\]). In addition, the 4 bright knots A1-A4 (Fig. \[knot\_targets\]) in the lensed arcs are shown by the blue squares. The green curve is an Scd galaxy model [@cww] at redshifts $z = 0-2$, with green circles highlighting the redshift range $z = 0.65-0.75$, indicating an approximate photometric redshift $z \sim 0.7$ for the arc knots. \[color-color\]](f3.eps)
[ccccl]{} $g$ & 14 Dec 2006 & 2$\times 125$ sec & $1.44\arcsec$ &\
$r$ & 14 Dec 2006 & 2$\times 300$ sec & $1.29\arcsec$ &\
$g$ & 11 Nov 2008 & 2$\times 125$ sec & $1.03\arcsec$ &\
$r$ & 11 Nov 2008 & 2$\times 300$ sec & $0.88\arcsec$ &\
$i$ & 30 Oct 2006 & 3$\times 450$ sec & $1.18\arcsec$ &\
$z$ & 30 Oct 2006 & 3$\times 450$ sec & $1.31\arcsec$ &\
GG455 & 4 Aug 2007 & 4$\times 900$ sec & $0.56\arcsec$ & Mask 1 includes knots A1,A3,A4\
GG455 & 4 Aug 2007 & 4$\times 900$ sec & $1.14\arcsec$ & Mask 2 includes BCG and knot A2\
GG455 & 4 Aug 2007 & 1$\times 5$ sec & - & Cu-Ar Mask 1\
GG455 & 4 Aug 2007 & 1$\times 5$ sec & - & Cu-Ar Mask 2\
GG455 & 14 Aug 2007 & 1$\times 5$ sec & - & $1.5\arcsec$ slit\
GG455 & 14 Aug 2007 & 1$\times 90$sec & $0.95\arcsec$ & Standard star EG21\
[cccccc]{} A1 & 357.912477 & -54.881691 & 21.94 & 0.85 & 0.77\
A2 & 357.911467 & -54.882801 & 21.49 & 0.81 & 0.69\
A3 & 357.906225 & -54.883464 & 22.30 & 0.84 & 0.74\
A4 & 357.907100 & -54.879967 & 21.46 & 0.91 & 0.77\
[ccccccc]{} 15173 (BCG) & 357.908555 & -54.881611 & $ 17.36 \pm 0.00 $ & $ 1.86 \pm 0.01 $ & $ 0.68 \pm 0.00 $ & $ 0.3805 \pm 0.0003 $\
16097 & 357.972190 & -54.856522 & $ 18.58 \pm 0.00 $ & $ 1.87 \pm 0.02 $ & $ 0.71 \pm 0.01 $ &\
16926 & 358.069064 & -54.838013 & $ 18.67 \pm 0.01 $ & $ 1.72 \pm 0.02 $ & $ 0.75 \pm 0.01 $ &\
14954 & 357.990606 & -54.881805 & $ 18.70 \pm 0.01 $ & $ 1.78 \pm 0.02 $ & $ 0.69 \pm 0.01 $ &\
14458 & 357.922935 & -54.891348 & $ 19.05 \pm 0.00 $ & $ 1.77 \pm 0.02 $ & $ 0.66 \pm 0.01 $ & $ 0.3844 \pm 0.0002 $\
15111 & 357.911305 & -54.879770 & $ 19.10 \pm 0.01 $ & $ 1.76 \pm 0.03 $ & $ 0.64 \pm 0.01 $ &\
13772 & 357.854114 & -54.908062 & $ 19.21 \pm 0.01 $ & $ 1.76 \pm 0.02 $ & $ 0.65 \pm 0.01 $ &\
14873 & 357.913389 & -54.883120 & $ 19.22 \pm 0.01 $ & $ 1.78 \pm 0.02 $ & $ 0.66 \pm 0.01 $ &\
15204 & 357.917968 & -54.874795 & $ 19.22 \pm 0.01 $ & $ 1.81 \pm 0.02 $ & $ 0.64 \pm 0.01 $ & $ 0.3827 \pm 0.0002 $\
15305 & 357.929749 & -54.874524 & $ 19.32 \pm 0.01 $ & $ 1.66 \pm 0.02 $ & $ 0.66 \pm 0.01 $ &\
15124 & 357.915316 & -54.877257 & $ 19.39 \pm 0.01 $ & $ 1.72 \pm 0.03 $ & $ 0.65 \pm 0.01 $ & $ 0.3929 \pm 0.0005 $\
11813 & 357.856326 & -54.957697 & $ 19.40 \pm 0.01 $ & $ 1.70 \pm 0.03 $ & $ 0.63 \pm 0.01 $ &\
13629 & 357.781583 & -54.911494 & $ 19.57 \pm 0.01 $ & $ 1.67 \pm 0.03 $ & $ 0.64 \pm 0.01 $ &\
16084 & 357.932492 & -54.855191 & $ 19.62 \pm 0.01 $ & $ 1.81 \pm 0.03 $ & $ 0.67 \pm 0.01 $ & $ 0.3864 \pm 0.0003 $\
14828 & 357.858498 & -54.884039 & $ 19.69 \pm 0.01 $ & $ 1.76 \pm 0.03 $ & $ 0.67 \pm 0.01 $ &\
15056 & 357.929263 & -54.878393 & $ 19.69 \pm 0.01 $ & $ 1.83 \pm 0.03 $ & $ 0.71 \pm 0.01 $ &\
13028 & 357.836914 & -54.923702 & $ 19.69 \pm 0.01 $ & $ 1.70 \pm 0.03 $ & $ 0.65 \pm 0.01 $ &\
14267 & 357.742239 & -54.897565 & $ 19.70 \pm 0.01 $ & $ 1.66 \pm 0.04 $ & $ 0.73 \pm 0.01 $ &\
13939 & 357.743518 & -54.903441 & $ 19.72 \pm 0.01 $ & $ 1.78 \pm 0.04 $ & $ 0.70 \pm 0.01 $ &\
14892 & 357.917045 & -54.881040 & $ 19.75 \pm 0.01 $ & $ 1.67 \pm 0.03 $ & $ 0.58 \pm 0.01 $ &\
17276 & 358.061857 & -54.827190 & $ 19.83 \pm 0.01 $ & $ 1.68 \pm 0.03 $ & $ 0.67 \pm 0.01 $ &\
12997 & 357.992988 & -54.925261 & $ 19.85 \pm 0.01 $ & $ 1.79 \pm 0.04 $ & $ 0.70 \pm 0.01 $ &\
14685 & 357.948912 & -54.885316 & $ 19.85 \pm 0.01 $ & $ 1.77 \pm 0.03 $ & $ 0.70 \pm 0.01 $ &\
14727 & 357.914364 & -54.884540 & $ 19.86 \pm 0.01 $ & $ 1.74 \pm 0.04 $ & $ 0.64 \pm 0.01 $ &\
12907 & 357.971817 & -54.926860 & $ 19.88 \pm 0.01 $ & $ 1.73 \pm 0.03 $ & $ 0.71 \pm 0.01 $ &\
15525 & 357.891439 & -54.867148 & $ 19.95 \pm 0.01 $ & $ 1.85 \pm 0.04 $ & $ 0.67 \pm 0.01 $ & $ 0.3802 \pm 0.0005 $\
13874 & 357.767222 & -54.904760 & $ 19.98 \pm 0.01 $ & $ 1.76 \pm 0.04 $ & $ 0.68 \pm 0.01 $ &\
14875 & 357.896375 & -54.880676 & $ 20.00 \pm 0.01 $ & $ 1.81 \pm 0.04 $ & $ 0.67 \pm 0.01 $ &\
14827 & 357.956282 & -54.883059 & $ 20.02 \pm 0.01 $ & $ 1.73 \pm 0.04 $ & $ 0.69 \pm 0.01 $ &\
14169 & 357.942454 & -54.896963 & $ 20.05 \pm 0.01 $ & $ 1.69 \pm 0.04 $ & $ 0.67 \pm 0.01 $ &\
14620 & 357.906482 & -54.885446 & $ 20.09 \pm 0.01 $ & $ 1.65 \pm 0.03 $ & $ 0.66 \pm 0.01 $ & $ 0.3822 \pm 0.0004 $\
11254 & 357.792343 & -54.968633 & $ 20.11 \pm 0.01 $ & $ 1.76 \pm 0.03 $ & $ 0.63 \pm 0.01 $ &\
19279 & 357.988291 & -54.784591 & $ 20.12 \pm 0.01 $ & $ 1.67 \pm 0.04 $ & $ 0.67 \pm 0.01 $ &\
15027 & 357.880749 & -54.878200 & $ 20.16 \pm 0.01 $ & $ 1.76 \pm 0.04 $ & $ 0.63 \pm 0.01 $ & $ 0.3876 \pm 0.0006 $\
12805 & 357.947638 & -54.929596 & $ 20.18 \pm 0.01 $ & $ 1.70 \pm 0.04 $ & $ 0.76 \pm 0.01 $ &\
13899 & 358.003818 & -54.902294 & $ 20.19 \pm 0.01 $ & $ 1.84 \pm 0.04 $ & $ 0.66 \pm 0.01 $ &\
14741 & 357.943783 & -54.884299 & $ 20.21 \pm 0.01 $ & $ 1.85 \pm 0.05 $ & $ 0.69 \pm 0.01 $ &\
12671 & 358.055413 & -54.931583 & $ 20.22 \pm 0.01 $ & $ 1.71 \pm 0.04 $ & $ 0.59 \pm 0.01 $ &\
14843 & 357.901141 & -54.880772 & $ 20.23 \pm 0.01 $ & $ 1.72 \pm 0.04 $ & $ 0.61 \pm 0.01 $ &\
14088 & 357.936003 & -54.898050 & $ 20.27 \pm 0.01 $ & $ 1.79 \pm 0.05 $ & $ 0.69 \pm 0.01 $ & $ 0.3816 \pm 0.0005 $\
14969 & 357.910388 & -54.878452 & $ 20.29 \pm 0.01 $ & $ 1.76 \pm 0.05 $ & $ 0.64 \pm 0.01 $ &\
12875 & 357.935888 & -54.927451 & $ 20.30 \pm 0.01 $ & $ 1.80 \pm 0.04 $ & $ 0.64 \pm 0.01 $ &\
13537 & 357.937414 & -54.911304 & $ 20.31 \pm 0.01 $ & $ 1.75 \pm 0.04 $ & $ 0.68 \pm 0.01 $ & $ 0.3849 \pm 0.0003 $\
15314 & 357.902668 & -54.872019 & $ 20.34 \pm 0.01 $ & $ 1.70 \pm 0.05 $ & $ 0.60 \pm 0.01 $ & $ 0.3841 \pm 0.0004 $\
14669 & 357.916522 & -54.885626 & $ 20.36 \pm 0.01 $ & $ 1.89 \pm 0.06 $ & $ 0.73 \pm 0.02 $ & $ 0.3862 \pm 0.0003 $\
14639 & 357.954384 & -54.885672 & $ 20.36 \pm 0.01 $ & $ 1.77 \pm 0.05 $ & $ 0.67 \pm 0.01 $ &\
14232 & 357.904683 & -54.895183 & $ 20.38 \pm 0.01 $ & $ 1.76 \pm 0.04 $ & $ 0.64 \pm 0.01 $ & $ 0.3882 \pm 0.0003 $\
14703 & 357.865075 & -54.884639 & $ 20.41 \pm 0.01 $ & $ 1.81 \pm 0.05 $ & $ 0.62 \pm 0.02 $ &\
14690 & 357.914016 & -54.883857 & $ 20.42 \pm 0.01 $ & $ 1.68 \pm 0.04 $ & $ 0.64 \pm 0.01 $ &\
15463 & 357.882797 & -54.868348 & $ 20.43 \pm 0.01 $ & $ 1.71 \pm 0.05 $ & $ 0.63 \pm 0.01 $ & $ 0.3877 \pm 0.0004 $\
16005 & 357.991736 & -54.856356 & $ 20.44 \pm 0.01 $ & $ 1.70 \pm 0.05 $ & $ 0.66 \pm 0.01 $ &\
15333 & 357.975175 & -54.872090 & $ 20.44 \pm 0.01 $ & $ 1.69 \pm 0.05 $ & $ 0.63 \pm 0.01 $ &\
14972 & 357.909534 & -54.878210 & $ 20.45 \pm 0.01 $ & $ 1.87 \pm 0.06 $ & $ 0.68 \pm 0.02 $ &\
14086 & 357.829155 & -54.897455 & $ 20.48 \pm 0.01 $ & $ 1.68 \pm 0.04 $ & $ 0.59 \pm 0.01 $ &\
18418 & 357.768333 & -54.801860 & $ 20.49 \pm 0.01 $ & $ 1.65 \pm 0.04 $ & $ 0.59 \pm 0.01 $ &\
13764 & 357.905657 & -54.906287 & $ 20.50 \pm 0.01 $ & $ 1.73 \pm 0.05 $ & $ 0.65 \pm 0.01 $ &\
10692 & 357.819108 & -54.981725 & $ 20.53 \pm 0.02 $ & $ 1.71 \pm 0.06 $ & $ 0.64 \pm 0.02 $ &\
15516 & 357.931958 & -54.867137 & $ 20.56 \pm 0.01 $ & $ 1.68 \pm 0.05 $ & $ 0.61 \pm 0.01 $ & $ 0.3785 \pm 0.0001 $\
19588 & 357.961069 & -54.776725 & $ 20.57 \pm 0.02 $ & $ 1.64 \pm 0.05 $ & $ 0.64 \pm 0.02 $ &\
15002 & 357.897311 & -54.877856 & $ 20.58 \pm 0.01 $ & $ 1.70 \pm 0.05 $ & $ 0.67 \pm 0.02 $ & $ 0.3838 \pm 0.0004 $\
14800 & 357.901708 & -54.880859 & $ 20.59 \pm 0.01 $ & $ 1.73 \pm 0.05 $ & $ 0.64 \pm 0.01 $ &\
15788 & 357.914426 & -54.860804 & $ 20.61 \pm 0.01 $ & $ 1.75 \pm 0.05 $ & $ 0.64 \pm 0.02 $ & $ 0.3821 \pm 0.0002 $\
15373 & 357.965957 & -54.874282 & $ 20.61 \pm 0.01 $ & $ 1.66 \pm 0.05 $ & $ 0.64 \pm 0.02 $ & $ 0.3856 \pm 0.0005 $\
13697 & 357.905543 & -54.907479 & $ 20.64 \pm 0.01 $ & $ 1.81 \pm 0.06 $ & $ 0.68 \pm 0.02 $ & $ 0.3782 \pm 0.0005 $\
15187 & 357.874035 & -54.873868 & $ 20.65 \pm 0.01 $ & $ 1.89 \pm 0.06 $ & $ 0.64 \pm 0.02 $ & $ 0.3868 \pm 0.0006 $\
18026 & 358.056024 & -54.810258 & $ 20.66 \pm 0.01 $ & $ 1.74 \pm 0.05 $ & $ 0.70 \pm 0.02 $ &\
14378 & 357.875627 & -54.892067 & $ 20.71 \pm 0.02 $ & $ 1.66 \pm 0.05 $ & $ 0.56 \pm 0.02 $ &\
14844 & 357.899766 & -54.880469 & $ 20.72 \pm 0.04 $ & $ 1.75 \pm 0.31 $ & $ 0.72 \pm 0.09 $ &\
17455 & 357.997121 & -54.822676 & $ 20.72 \pm 0.04 $ & $ 1.54 \pm 0.40 $ & $ 0.88 \pm 0.12 $ &\
17729 & 357.875358 & -54.816995 & $ 20.74 \pm 0.01 $ & $ 1.78 \pm 0.06 $ & $ 0.64 \pm 0.02 $ &\
15068 & 358.094352 & -54.876409 & $ 20.75 \pm 0.02 $ & $ 1.64 \pm 0.07 $ & $ 0.68 \pm 0.02 $ &\
15994 & 357.763211 & -54.855887 & $ 20.82 \pm 0.02 $ & $ 1.80 \pm 0.06 $ & $ 0.64 \pm 0.02 $ &\
12892 & 358.014272 & -54.926461 & $ 20.86 \pm 0.02 $ & $ 1.65 \pm 0.07 $ & $ 0.68 \pm 0.02 $ &\
15697 & 357.897819 & -54.862968 & $ 20.86 \pm 0.02 $ & $ 1.73 \pm 0.06 $ & $ 0.55 \pm 0.02 $ & $ 0.3789 \pm 0.0003 $\
12589 & 357.785585 & -54.933352 & $ 20.87 \pm 0.03 $ & $ 1.66 \pm 0.10 $ & $ 0.63 \pm 0.03 $ &\
11976 & 357.893073 & -54.951394 & $ 20.88 \pm 0.02 $ & $ 1.62 \pm 0.06 $ & $ 0.61 \pm 0.02 $ &\
14664 & 357.901167 & -54.885034 & $ 20.89 \pm 0.02 $ & $ 1.84 \pm 0.08 $ & $ 0.68 \pm 0.02 $ &\
13901 & 357.824131 & -54.902094 & $ 20.90 \pm 0.02 $ & $ 1.69 \pm 0.06 $ & $ 0.64 \pm 0.02 $ &\
14595 & 357.914211 & -54.886290 & $ 20.93 \pm 0.03 $ & $ 1.80 \pm 0.13 $ & $ 0.66 \pm 0.03 $ &\
12863 & 357.874307 & -54.926847 & $ 20.95 \pm 0.02 $ & $ 1.66 \pm 0.07 $ & $ 0.62 \pm 0.02 $ &\
15156 & 357.891357 & -54.874854 & $ 20.95 \pm 0.03 $ & $ 1.59 \pm 0.09 $ & $ 0.56 \pm 0.03 $ &\
14825 & 357.922988 & -54.880749 & $ 20.95 \pm 0.02 $ & $ 1.63 \pm 0.07 $ & $ 0.64 \pm 0.02 $ &\
12650 & 357.958048 & -54.931820 & $ 20.96 \pm 0.02 $ & $ 1.77 \pm 0.08 $ & $ 0.64 \pm 0.02 $ &\
14407 & 357.917097 & -54.891402 & $ 20.97 \pm 0.02 $ & $ 1.75 \pm 0.07 $ & $ 0.63 \pm 0.02 $ & $ 0.3799 \pm 0.0004 $\
14939 & 357.872276 & -54.878410 & $ 20.97 \pm 0.02 $ & $ 1.75 \pm 0.07 $ & $ 0.62 \pm 0.02 $ &\
14944 & 357.913836 & -54.877816 & $ 20.98 \pm 0.03 $ & $ 2.17 \pm 0.23 $ & $ 0.64 \pm 0.05 $ &\
14271 & 357.899268 & -54.896523 & $ 19.13 \pm 0.01 $ & $ 1.90 \pm 0.02 $ & $ 0.71 \pm 0.01 $ & $ 0.3814 \pm 0.0002 $\
15403 & 357.908847 & -54.870362 & $ 19.52 \pm 0.01 $ & $ 1.58 \pm 0.02 $ & $ 0.67 \pm 0.01 $ & $ 0.3860 \pm 0.0003 $\
15827 & 357.957606 & -54.860595 & $ 19.64 \pm 0.01 $ & $ 1.11 \pm 0.02 $ & $ 0.45 \pm 0.01 $ & $ 0.3900 \pm 0.0004 $\
15400 & 357.866255 & -54.870713 & $ 19.69 \pm 0.01 $ & $ 1.62 \pm 0.03 $ & $ 0.65 \pm 0.01 $ & $ 0.3768 \pm 0.0003 $\
14466 & 357.909595 & -54.890780 & $ 20.47 \pm 0.02 $ & $ 1.57 \pm 0.05 $ & $ 0.60 \pm 0.02 $ & $ 0.3827 \pm 0.0002 $\
14492 & 357.914762 & -54.888447 & $ 20.88 \pm 0.02 $ & $ 1.58 \pm 0.06 $ & $ 0.64 \pm 0.02 $ & $ 0.3899 \pm 0.0004 $\
13372 & 357.917399 & -54.914403 & $ 20.98 \pm 0.02 $ & $ 1.60 \pm 0.06 $ & $ 0.65 \pm 0.02 $ & $ 0.3803 \pm 0.0003 $\
14505 & 357.870029 & -54.888283 & $ 21.27 \pm 0.02 $ & $ 1.63 \pm 0.08 $ & $ 0.66 \pm 0.03 $ & $ 0.3860 \pm 0.0003 $\
[ccccccc]{} 14193 & 357.895093 & -54.901998 & $ 17.99 \pm 0.00 $ & $ 1.59 \pm 0.01 $ & $ 0.58 \pm 0.00 $ & $ 0.2970 \pm 0.0003 $\
15313 & 357.893518 & -54.875789 & $ 18.75 \pm 0.00 $ & $ 1.11 \pm 0.01 $ & $ 0.47 \pm 0.01 $ & $ 0.2486 \pm 0.0002 $\
16682 & 357.903652 & -54.840904 & $ 19.97 \pm 0.01 $ & $ 1.40 \pm 0.03 $ & $ 0.57 \pm 0.01 $ & $ 0.3259 \pm 0.0002 $\
13520 & 357.887606 & -54.911328 & $ 20.10 \pm 0.01 $ & $ 0.51 \pm 0.01 $ & $ 0.27 \pm 0.01 $ & $ 0.0649 \pm 0.0001 $\
19352 & 357.902529 & -54.852090 & $20.18 \pm 0.01$ & $1.01 \pm 0.02$ & $0.25 \pm 0.01$ & $0.4214 \pm 0.0003$\
15509 & 357.941448 & -54.869154 & $ 20.29 \pm 0.01 $ & $ 1.57 \pm 0.05 $ & $ 0.62 \pm 0.02 $ & $ 0.4178 \pm 0.0005 $\
16409 & 357.890133 & -54.846245 & $ 20.30 \pm 0.01 $ & $ 0.89 \pm 0.02 $ & $ 0.31 \pm 0.01 $ & $ 0.3251 \pm 0.0002 $\
16570 & 357.911876 & -54.843091 & $ 20.42 \pm 0.01 $ & $ 0.77 \pm 0.03 $ & $ 0.58 \pm 0.02 $ & $ 0.1277 \pm 0.0002 $\
13423 & 357.960803 & -54.913826 & $ 20.63 \pm 0.02 $ & $ 1.01 \pm 0.03 $ & $ 0.68 \pm 0.02 $ & $ 0.2524 \pm 0.0002 $\
13620 & 357.889293 & -54.909472 & $ 20.86 \pm 0.02 $ & $ 1.66 \pm 0.08 $ & $ 0.90 \pm 0.02 $ & $ 0.5354 \pm 0.0004 $\
19257 & 357.902437 & -54.851578 & $21.45 \pm 0.03$ & $0.98 \pm 0.03$ & $-0.06 \pm 0.02$ & $0.2970 \pm 0.0004$\
16562 & 357.948746 & -54.841891 & $ 21.58 \pm 0.03 $ & $ 1.90 \pm 0.12 $ & $ 0.61 \pm 0.03 $ & $ 0.3595 \pm 0.0002 $\
[ccccccccll]{} $g$ & 1883 & 22.5 & 24.0 & 18.75 & 0.68 & $0.1^{+0.4}_{-0.1}$ & $> 45$ & 0.83 & 0.68\
$r$ & 7013 & 22.0 & 24.0 & 12.0 & 0.70 & $3.9^{+2.9}_{-2.1}$ & $6.5^{+5.3}_{-3.0}$ & 1.54 & 0.059\
$i$ & 3296 & 22.0 & 23.5 & 12.0 & 0.71 & $5.9^{+5.3}_{-3.8}$ & $3.7^{+13.1}_{-2.6}$ & 1.55 & 0.055\
$z$ & 2300 & 20.5 & 22.5 & 12.0 & 0.62 & $11.0^{+11.9}_{-7.1}$ & $1.8^{+3.6}_{-1.8}$ & 0.89 & 0.60\
\
$i$+$r$ & 7995 & & & & 0.70 & $4.2^{+2.8}_{-2.1}$ & $6.1^{+4.9}_{-3.0}$ & 1.58 & 0.048\
$i$+$r$+$z$ & 8996 & & & & 0.70 & $5.0^{+2.9}_{-2.3}$ & $4.9^{+3.9}_{-2.2}$ & 1.48 & 0.077\
$i$+$r$+$z$+$g$ & 9424 & & & & 0.70 & $4.3^{+2.8}_{-2.2}$ & $5.2^{+5.4}_{-2.5}$ & 1.50 & 0.069\
$i$+$r$+$z$+SL(s) & 8996 & & & & 0.70 & $4.8^{+2.8}_{-2.2}$ & $6.2^{+3.2}_{-1.7}$ & 1.48 & 0.077\
$i$+$r$+$z$+SL(sg) & 8996 & & & & 0.70 & $4.9^{+2.9}_{-2.2}$ & $5.5^{+2.7}_{-1.6}$ & 1.48 & 0.077\
WL+SL+$\sigma_c$+$N_{200}$ & 8996 & & & & 0.70 & $5.1^{+1.3}_{-1.3}$ & $5.4^{+1.4}_{-1.1}$ & 1.48 & 0.077\
$g$ & 1883 & 22.5 & 24.0 & 18.75 & 0.68 & $1.6^{+3.1}_{-1.5}$ & $6.5^{+10.1}_{-5.4}$ & 1.04 & 0.41\
$r$ & 7013 & 22.0 & 24.0 & 12.0 & 0.70 & $0.1^{+0.1}_{-0.1}$ & $> 63$ & 1.19 & 0.25\
$i$ & 3296 & 22.0 & 23.5 & 12.0 & 0.71 & $0.1^{+0.4}_{-0.1}$ & $> 0$ & 0.91 & 0.58\
$z$ & 2300 & 20.5 & 22.5 & 12.0 & 0.62 & $5.5^{+10.7}_{-5.2}$ & $0.3^{+1.2}_{-0.3}$ & 0.61 & 0.91\
\
$i$+$r$ & 7995 & & & & 0.70 & $0.1^{+0.1}_{-0.1}$ & $> 51$ & 1.10 & 0.34\
$i$+$r$+$z$ & 8996 & & & & 0.70 & $0.1^{+0.1}_{-0.1}$ & $> 27$ & 0.80 & 0.71\
$i$+$r$+$z$+$g$ & 9424 & & & & 0.70 & $0.1^{+0.6}_{-0.1}$ & $> 0$ & 0.77 & 0.75\
[lccccc]{} A1 & $1.06\pm 0.04\times 10^{-15}$ & $1.71\pm 0.06\times 10^{-16}$ & $1.36\pm 0.02\times 10^{-28}$ & $3.9\pm 1.1$ & $1.1\pm 0.4$\
A2 & $0.84\pm 0.04\times 10^{-15}$ & $1.43\pm 0.06\times 10^{-16}$ & $1.43\pm 0.02\times 10^{-28}$ & $3.1\pm 0.9$ & $0.85\pm 0.4$\
A3 & $2.09\pm 0.10 \times 10^{-15}$ & $1.28\pm 0.06\times 10^{-16}$ & $0.51 \pm 0.01\times 10^{-28}$ & $7.7\pm 2.2$ & $2.1\pm 0.4$\
A4 & $1.02\pm 0.02\times 10^{-15}$ & $2.83\pm 0.06\times 10^{-16}$ & $2.33\pm 0.02\times 10^{-28}$ & $3.7\pm 1.1$ & $1.0\pm 0.4$\
[^1]: http://www.astromatic.net
[^2]: http://www.gemini.edu/sciops/data-and-results/processing-software
[^3]: Version 2.17: http://www.qgd.uzh.ch/programs/pixelens/
|
---
abstract: 'Justification logics are epistemic logics that explicitly include justifications for the agents’ knowledge. We develop a multi-agent justification logic with evidence terms for individual agents as well as for common knowledge. We define a Kripke-style semantics that is similar to Fitting’s semantics for the Logic of Proofs ${\mathsf{LP}}$. We show the soundness, completeness, and finite model property of our multi-agent justification logic with respect to this Kripke-style semantics. We demonstrate that our logic is a conservative extension of Yavorskaya’s minimal bimodal explicit evidence logic, which is a two-agent version of ${\mathsf{LP}}$. We discuss the relationship of our logic to the multi-agent modal logic ${\mathsf{S4}}$ with common knowledge. Finally, we give a brief analysis of the coordinated attack problem in the newly developed language of our logic.'
author:
- 'Samuel Bucheli, Roman Kuznets,[[^1]]{}and Thomas Studer'
bibliography:
- 'bibliography.bib'
- 'JLBibliography.bib'
title: |
Explicit Evidence Systems\
with Common Knowledge
---
=1
Introduction {#sec:intro}
============
*Justification logics* [@Art08RSL] are epistemic logics that explicitly include justifications for the agents’ knowledge. The first logic of this kind, the *Logic of Proofs ${\mathsf{LP}}$*, was developed by Artemov [@Art95TR; @Art01BSL] to provide the modal logic ${\mathsf{S4}}$ with provability semantics. The language of justification logics has also been used to create a new approach to the logical omniscience problem [@ArtKuz09TARK] and to study self-referential proofs [@Kuz10TOCSnonote].
Instead of statements *$A$ is known*, denoted $\Box A$, justification logics reason about justifications for knowledge by using the construct ${\left[t\right]\!}A$ to formalize statements *$t$ is a justification for $A$*, where *evidence term $t$* can be viewed as an informal justification or a formal mathematical proof depending on the application. Evidence terms are built by means of operations that correspond to the axioms of ${\mathsf{S4}}$, as is illustrated in Fig. \[fig:LPaxioms\].
------------------------------------------------------------- ------------------------------------------------------------------------------------------------------------ ---------------
$\Box(A \rightarrow B)\rightarrow(\Box A\rightarrow\Box B)$ ${\left[t\right]\!}(A\rightarrow B)\rightarrow({\left[s\right]\!} A\rightarrow{\left[t\cdot s\right]\!}B)$ (application)
$\Box A \rightarrow A$ ${\left[t\right]\!} A \rightarrow A$ (reflexivity)
$\Box A \rightarrow \Box\Box A$ ${\left[t\right]\!} A \rightarrow {\left[!t\right]\!}{\left[t\right]\!} A$ (inspection)
${\left[t\right]\!} A \vee {\left[s\right]\!} A \rightarrow {\left[t+s\right]\!} A$ (sum)
------------------------------------------------------------- ------------------------------------------------------------------------------------------------------------ ---------------
Artemov [@Art01BSL] has shown that the Logic of Proofs ${\mathsf{LP}}$ is an *explicit counterpart* of the modal logic ${\mathsf{S4}}$ in the following formal sense: each theorem of ${\mathsf{LP}}$ becomes a theorem of ${\mathsf{S4}}$ if all terms are replaced with the modality $\Box$; and, vice versa, each theorem of ${\mathsf{S4}}$ can be transformed into a theorem of ${\mathsf{LP}}$ if occurrences of modality are replaced with suitable evidence terms. The latter process is called *realization*, and the statement of correspondence is called a *realization theorem*. Note that the operation $+$ introduced by the sum axiom in Fig. \[fig:LPaxioms\] does not have a modal analog, but it is an essential part of the proof of the realization theorem in [@Art01BSL]. Explicit counterparts for many normal modal logics between $\textsf{K}$ and $\textsf{S5}$ have been developed (see a recent survey in [@Art08RSL] and a uniform proof of realization theorems for all single-agent justification logics forthcoming in [@bgk10]).
The notion of *common knowledge* is essential in the area of multi-agent systems, where coordination among agents is a central issue. The standard textbooks [@FHMV95; @HM95] provide excellent introductions to epistemic logics in general and common knowledge in particular. Informally, common knowledge of $A$ is defined as the infinitary conjunction *everybody knows $A$ and everybody knows that everybody knows $A$ and so on*. This is equivalent to saying that common knowledge of $A$ is the greatest fixed point of $$\label{eq:operator}
\lambda X.(\text{everybody knows~$A$ and everybody knows~$X$})
\rlap{\enspace.}$$
Artemov [@Art06TCS] has created an explicit counterpart of McCarthy’s *any fool knows* common knowledge modality [@McCarSatHayIga78TR], where common knowledge of $A$ is defined as an arbitrary fixed point of . The relationship between the traditional common knowledge from [@FHMV95; @HM95] and McCarthy’s version is studied in [@Ant07LFCS].=-1
In this paper, we present a multi-agent justification logic with evidence terms for individual agents as well as for common knowledge, with the intention to provide an explicit counterpart of the $h$-agent modal logic of traditional common knowledge ${\mathsf{S4}}_{h}^{\mathsf{C}}$.
Multi-agent justification logics with evidence terms for each agent have been considered in [@TYav08TOCSnonote; @Ren09TARK; @Art10TR], although common knowledge is not present in any of them. Artemov’s interest [@Art10TR] lies mostly in exploring a case of two agents with unequal epistemic powers, e.g., Artemov’s Observer has sufficient evidence to reproduce his Object Agent’s thinking, but not vice versa. Yavorskaya [@TYav08TOCSnonote] studies various operations of evidence transfer between agents. Among their systems, Yavorskaya’s minimal[^2] bimodal explicit evidence logic, which is an explicit counterpart of ${\mathsf{S4}}_2$, is the closest to our system. We will show that in the case of two agents our system is its conservative extension. Finally, Renne’s system [@Ren09TARK] combines features of modal and dynamic epistemic logics, and hence cannot be directly compared to our system.
An epistemic semantics for ${\mathsf{LP}}$, *F-models*, was created by Fitting in [@Fit05APAL] by augmenting Kripke models with an *evidence function* that specifies which formulae are evidenced by a term at a given world. It is easily extended to the whole family of single-agent justification logics (for details, see [@Art08RSL]). In [@Art06TCS] Artemov extends F-models to justification terms for McCarthy’s common knowledge modality in the presence of several ordinary modalities, creating the most general type of epistemic models, sometimes called *AF-models*, where common evidence terms are given their own accessibility relation not directly dependent on the accessibility relations for individual modalities. Yavorskaya in [@TYav08TOCSnonote] proves a stronger completeness theorem with respect to singleton F-models, independently introduced by Mkrtychev [@Mkr97LFCS] and now known as *M-models*, where the role of the accessibility relation is completely taken over by the evidence function.
The paper is organized as follows. In Sect. \[sect:synt\], we introduce the language and give the axiomatization of a family of multi-agent justification logics with common knowledge. In Sect. \[sect:basprop\], we prove their basic properties including the internalization property, which is characteristic of all justification logics. In Sect. \[sect:soundcomp\], we give a Fitting-style semantics similar to AF-models and prove soundness and completeness with respect to this semantics as well as with respect to singleton models, thereby demonstrating the finite model property. In Sect. \[sect:conserv\], we show that for the two-agent case, our logic is a conservative extension of Yavorskaya’s minimal bimodal explicit evidence logic. In Sect. \[sect:real\], we show how our logic is related to the modal logic of traditional common knowledge and discuss the problem of realization. Finally, in Sect. \[s:attack:1\], we provide an analysis of the coordinated attack problem in our logic.
Syntax {#sect:synt}
======
To create an explicit counterpart of the modal logic of common knowledge ${{{\mathsf{S4}}_{h}}^{{\mathsf{C}}}}$, we use its axiomatization via the induction axiom from [@HM95] rather than via the induction rule to facilitate the proof of the internalization property for the resulting justification logic. We supply each agent with its own copy of terms from the Logic of Proofs, while terms for common and mutual knowledge employ additional operations. As motivated in [@BucKuzStu09M4M], a proof of ${\mathsf{C}}A$ can be thought of as an infinite list of proofs of the conjuncts ${\mathsf{E}}^m A$ in the representation of common knowledge through an infinite conjunction. To generate a finite representation of this infinite list, we use an explicit counterpart of the induction axiom $$A \wedge {\left[t\right]\!}_{\mathsf{C}}(A \rightarrow {\left[s\right]\!}_{\mathsf{E}}A) \rightarrow {\left[{\mathsf{ind}(t,s)}\right]\!}_{\mathsf{C}}A$$ with a binary operation ${\mathsf{ind}(\cdot,\cdot)}$. To access the elements of the list, explicit counterparts of the co-closure axiom provide evidence terms that can be seen as splitting the infinite list into its head and tail, $${\left[t\right]\!}_{\mathsf{C}}A \rightarrow {\left[{{\mathsf{ccl}}_1(t)}\right]\!}_{\mathsf{E}}A
\rlap{\enspace,}
\qquad
{\left[t\right]\!}_{\mathsf{C}}A \rightarrow {\left[{{\mathsf{ccl}}_2(t)}\right]\!}_{\mathsf{E}}{\left[t\right]\!}_{\mathsf{C}}A
\rlap{\enspace,}$$ by means of two unary co-closure operations ${{\mathsf{ccl}}_1(\cdot)}$ and ${{\mathsf{ccl}}_2(\cdot)}$. Evidence terms for mutual knowledge are represented as tuples of the individual agents’ evidence terms with the standard operation of tupling and with $h$ unary projections. While only two of the three operations on ${\mathsf{LP}}$ terms are adopted for common knowledge evidence and none for mutual knowledge evidence, it will be shown in Sect. \[sect:basprop\] that most remaining operations are definable with the notable exception of inspection for mutual knowledge.
We consider a system of ${h}$ agents. Throughout the paper, ${i}$ always denotes an element of $\{ 1, \dots, {h}\}$, ${*}$ always denotes an element of $\{ 1, \dots, {h}, {\mathsf{C}}\}$, and ${\circledast}$ always denotes an element of $\{ 1, \dots, {h}, {\mathsf{E}}, {\mathsf{C}}\}$.
Let ${\textnormal{Cons}}_{\circledast}\colonequals \{ c^{\circledast}_1, c^{\circledast}_2, \dots \}$ and ${\textnormal{Var}}_{\circledast}\colonequals \{ x^{\circledast}_1, x^{\circledast}_2, \dots \}$ be countable sets of *proof constants* and *proof variables* respectively for each ${\circledast}$. The sets ${\textnormal{Tm}}_1$, …, ${\textnormal{Tm}}_{h}$, ${\textnormal{Tm}}_{\mathsf{E}}$, and ${\textnormal{Tm}}_{\mathsf{C}}$ of *evidence terms for individual agents* and for *mutual* and *common knowledge* respectively are inductively defined as follows:
1. ${\textnormal{Cons}}_{\circledast}\subseteq {\textnormal{Tm}}_{\circledast}$;
2. ${\textnormal{Var}}_{\circledast}\subseteq {\textnormal{Tm}}_{\circledast}$;
3. $!_{i}t \in {\textnormal{Tm}}_{i}$ for any $t \in {\textnormal{Tm}}_{i}$;
4. $t +_{*}s \in {\textnormal{Tm}}_{*}$ and $t \cdot_{*}s \in {\textnormal{Tm}}_{*}$ for any $t, s \in {\textnormal{Tm}}_{*}$;
5. ${\left\langle t_1, \dots, t_{h}\right\rangle} \in {\textnormal{Tm}}_{\mathsf{E}}$ for any $t_1 \in {\textnormal{Tm}}_1$, …, $t_{h}\in {\textnormal{Tm}}_{h}$;
6. ${\pi}_it \in {\textnormal{Tm}}_i$ for any $t \in {\textnormal{Tm}}_{\mathsf{E}}$;
7. ${{\mathsf{ccl}}_1(t)} \in {\textnormal{Tm}}_{\mathsf{E}}$ and ${{\mathsf{ccl}}_2(t)} \in {\textnormal{Tm}}_{\mathsf{E}}$ for any $t \in {\textnormal{Tm}}_{\mathsf{C}}$;
8. ${\mathsf{ind}(t,s)} \in {\textnormal{Tm}}_{\mathsf{C}}$ for any $t \in {\textnormal{Tm}}_{\mathsf{C}}$ and any $s \in {\textnormal{Tm}}_{\mathsf{E}}$.
${\textnormal{Tm}}\colonequals {\textnormal{Tm}}_1 \cup \dots \cup {\textnormal{Tm}}_{h}\cup {\textnormal{Tm}}_{\mathsf{E}}\cup {\textnormal{Tm}}_{\mathsf{C}}$ denotes the set of all evidence terms. The indices of the operations $!$, $+$, and $\cdot$ will usually be omitted if they can be inferred from the context.
Let ${\textnormal{Prop}}\colonequals \{ P_1, P_2, \dots \}$ be a countable set of *propositional variables*. *Formulae* are denoted by $A$, $B$, $C$, etc. and defined by the following grammar $$A \coloncolonequals P_j {\mathrel{\mid}}\neg A {\mathrel{\mid}}(A \wedge A) {\mathrel{\mid}}(A \vee A) {\mathrel{\mid}}(A \rightarrow A) {\mathrel{\mid}}{\left[t\right]\!}_{\circledast}A
\rlap{\enspace,}$$ where $t \in {\textnormal{Tm}}_{\circledast}$. The set of all formulae is denoted by ${{\textnormal{Fm}}_{{{{\mathsf{LP}}_{h}}^{{\mathsf{C}}}}}}$. We adopt the following convention: whenever a formula ${\left[t\right]\!}_{\circledast}A$ is used, it is assumed to be well-formed, i.e., it is implicitly assumed that term $t \in {\textnormal{Tm}}_{\circledast}$. This enables us to omit the explicit typification of terms.
**Axioms of ${{{\mathsf{LP}}_{h}}^{{\mathsf{C}}}}$:**
1. all propositional tautologies
2. ${\left[t\right]\!}_{*}(A \rightarrow B) \rightarrow ({\left[s\right]\!}_{*}A \rightarrow {\left[t \cdot s\right]\!}_{*}B)$ (application)
3. ${\left[t\right]\!}_{*}A \rightarrow {\left[t + s\right]\!}_{*}A$, ${\left[s\right]\!}_{*}A \rightarrow {\left[t + s\right]\!}_{*}A$ (sum)
4. ${\left[t\right]\!}_{i}A \rightarrow A$ (reflexivity)
5. ${\left[t\right]\!}_{i}A \rightarrow {\left[! t\right]\!}_{i}{\left[t\right]\!}_{i}A$ (inspection)
6. ${\left[t_1\right]\!}_1 A \wedge \dots \wedge {\left[t_{h}\right]\!}_{h}A \rightarrow {\left[{\left\langle t_1, \dots, t_{h}\right\rangle}\right]\!}_{\mathsf{E}}A$ (tupling)
7. ${\left[t\right]\!}_{\mathsf{E}}A \rightarrow {\left[{\pi}_i t\right]\!}_i A$ (projection)
8. ${\left[t\right]\!}_{\mathsf{C}}A \rightarrow {\left[{{\mathsf{ccl}}_1(t)}\right]\!}_{\mathsf{E}}A$, ${\left[t\right]\!}_{\mathsf{C}}A \rightarrow {\left[{{\mathsf{ccl}}_2(t)}\right]\!}_{\mathsf{E}}{\left[t\right]\!}_{\mathsf{C}}A$ (co-closure)
9. $A \wedge {\left[t\right]\!}_{\mathsf{C}}(A \rightarrow {\left[s\right]\!}_{\mathsf{E}}A) \rightarrow {\left[{\mathsf{ind}(t,s)}\right]\!}_{\mathsf{C}}A$ (induction)
A *constant specification* ${\mathcal{CS}}$ is any subset $${\mathcal{CS}}\subseteq \bigcup_{{\circledast}\in \{ 1, \dots, {h}, E, C \}} \left\{ {\left[c\right]\!}_{\circledast}A \; : \; c \in {\textnormal{Cons}}_{\circledast}\text{ and } A \text{ is an axiom of~${{{\mathsf{LP}}_{h}}^{{\mathsf{C}}}}$} \right\}
\rlap{\enspace.}$$ A constant specification ${\mathcal{CS}}$ is called *${\mathsf{C}}$-axiomatically appropriate* if for each axiom $A$, there is a proof constant $c \in {\textnormal{Cons}}_{\mathsf{C}}$ such that ${\left[c\right]\!}_{\mathsf{C}}A \in {\mathcal{CS}}$. A constant specification ${\mathcal{CS}}$ is called *pure*, if ${\mathcal{CS}}\subseteq \left\{ {\left[c\right]\!}_{\circledast}A \; : \; c \in {\textnormal{Cons}}_{\circledast}\text{ and } A \text{ is an axiom} \right\}$ for some fixed ${\circledast}$, i.e., if for all ${\left[c\right]\!}_{\circledast}A \in {\mathcal{CS}}$, the constants $c$ are of the same type.
Let ${\mathcal{CS}}$ be a constant specification. The deductive system ${{{\mathsf{LP}}_{h}}^{{\mathsf{C}}}}({\mathcal{CS}})$ is the Hilbert system given by the axioms of ${{{\mathsf{LP}}_{h}}^{{\mathsf{C}}}}$ above and rules modus ponens and axiom necessitation: $${\displaystyle{\frac{A \quad A \rightarrow B}{B}}} \enspace, \qquad \qquad {\displaystyle{\frac{}{{\left[c\right]\!}_{\circledast}A}}} \enspace \text{, where ${\left[c\right]\!}_{\circledast}A \in {\mathcal{CS}}$.}$$ By ${{{\mathsf{LP}}_{h}}^{{\mathsf{C}}}}$ we denote the system ${{{\mathsf{LP}}_{h}}^{{\mathsf{C}}}}({\mathcal{CS}})$ with $$\label{eq:maxCS}
{\mathcal{CS}}= \left\{ {\left[c\right]\!}_{\mathsf{C}}A \; : \; c \in {\textnormal{Cons}}_{\mathsf{C}}\text{ and } A \text{ is an axiom of~${{{\mathsf{LP}}_{h}}^{{\mathsf{C}}}}$} \right\}
\rlap{\enspace.}$$
For an arbitrary ${\mathcal{CS}}$, we write $\Delta {\vdash}_{\mathcal{CS}}A$ to state that $A$ is derivable from $\Delta$ in ${{{\mathsf{LP}}_{h}}^{{\mathsf{C}}}}({\mathcal{CS}})$ and omit the mention of ${\mathcal{CS}}$ when working with the constant specification from by writing $\Delta {\vdash}A$. We use $\Delta, A$ to mean $\Delta \cup \{ A \}$.
Basic Properties {#sect:basprop}
================
In this section, we show that our logic possesses the standard properties expected of any justification logic. In addition, we show that the operations on terms introduced in the previous section are sufficient to express the operations of sum and application for mutual knowledge evidence and the operation of inspection for common knowledge evidence. This is the reason why $+_{\mathsf{E}}$, $\cdot_{\mathsf{E}}$, and $!_{\mathsf{C}}$ are not primitive connectives in the language. It should be noted that no inspection operation for mutual evidence terms can be defined, which follows from Lemma \[l:fp:1\] in Sect. \[sect:real\] and the fact that ${\mathsf{E}}A \rightarrow {\mathsf{E}}{\mathsf{E}}A$ is not a valid modal formula.
We begin with the following observation:
\[lem:basicproperties1\] For any constant specification ${\mathcal{CS}}$ and any formulae $A$ and $B$:
1. \[lem:basicproperties1:Erefl\] ${\vdash}_{\mathcal{CS}}{\left[t\right]\!}_{\mathsf{E}}A \rightarrow A$ for all $t \in {\textnormal{Tm}}_{\mathsf{E}}$; $({\mathsf{E}}\text{-reflexivity})$
2. for any $t, s \in {\textnormal{Tm}}_{\mathsf{E}}$, there is a term $t \cdot_{\mathsf{E}}s \in {\textnormal{Tm}}_{\mathsf{E}}$ such that\
${\vdash}_{\mathcal{CS}}{\left[t\right]\!}_{\mathsf{E}}(A \rightarrow B) \rightarrow ({\left[s\right]\!}_{\mathsf{E}}A \rightarrow {\left[t \cdot_{\mathsf{E}}s\right]\!}_{\mathsf{E}}B)$; $({\mathsf{E}}\text{-application})$
3. for any $t, s \in {\textnormal{Tm}}_{\mathsf{E}}$, there is a term $t +_{\mathsf{E}}s \in {\textnormal{Tm}}_{\mathsf{E}}$ such that\
${\vdash}_{\mathcal{CS}}{\left[t\right]\!}_{\mathsf{E}}A \rightarrow {\left[t +_{\mathsf{E}}s\right]\!}_{\mathsf{E}}A$ and ${\vdash}_{\mathcal{CS}}{\left[s\right]\!}_{\mathsf{E}}A \rightarrow {\left[t +_{\mathsf{E}}s\right]\!}_{\mathsf{E}}A$; $({\mathsf{E}}\text{-sum})$
4. \[i-conversion\] for any $t \in {\textnormal{Tm}}_{\mathsf{C}}$ and any $i \in \{ 1, \dots, {h}\}$, there is a term ${\downarrow\!}_i t \in {\textnormal{Tm}}_i$ such that\
${\vdash}_{\mathcal{CS}}{\left[t\right]\!}_{\mathsf{C}}A \rightarrow {\left[{\downarrow\!}_i t\right]\!}_i A$; $(i \text{-conversion})$
5. ${\vdash}_{\mathcal{CS}}{\left[t\right]\!}_{\mathsf{C}}A \rightarrow A$ for all $t \in {\textnormal{Tm}}_{\mathsf{C}}$. $({\mathsf{C}}\text{-reflexivity})$
<!-- -->
1. Immediate by the projection and reflexivity axioms.
2. Set $t \cdot_{\mathsf{E}}s \colonequals {\left\langle {\pi}_1 t \cdot_1 {\pi}_1 s, \dots, {\pi}_{h}t \cdot_{h}{\pi}_{h}s \right\rangle}$.
3. Set $t +_{\mathsf{E}}s \colonequals {\left\langle {\pi}_1 t +_1 {\pi}_1 s, \dots, {\pi}_{h}t +_{h}{\pi}_{h}s \right\rangle}$.
4. Set ${\downarrow\!}_i t \colonequals {\pi}_i {{\mathsf{ccl}}_1(t)}$.
5. Immediate by \[i-conversion\]. and the reflexivity axiom.
Unlike Lemma \[lem:basicproperties1\], the next lemma requires that a constant specification ${\mathcal{CS}}$ be ${\mathsf{C}}$-axiomatically appropriate.
Let ${\mathcal{CS}}$ be ${\mathsf{C}}$-axiomatically appropriate and $A$ be a formula.
1. For any $t \in {\textnormal{Tm}}_{\mathsf{C}}$, there is a term $!_{\mathsf{C}}t \in {\textnormal{Tm}}_{\mathsf{C}}$ such that\
${\vdash}_{\mathcal{CS}}{\left[t\right]\!}_{\mathsf{C}}A \rightarrow {\left[!_{\mathsf{C}}t\right]\!}_{\mathsf{C}}{\left[t\right]\!}_{\mathsf{C}}A$. $({\mathsf{C}}\text{-inspection})$
2. For any $t \in {\textnormal{Tm}}_{\mathsf{C}}$, there is a term ${\Lleftarrow}t \in {\textnormal{Tm}}_{\mathsf{C}}$ such that\
${\vdash}_{\mathcal{CS}}{\left[t\right]\!}_{\mathsf{C}}A \rightarrow {\left[{\Lleftarrow}t\right]\!}_{\mathsf{C}}{\left[{{\mathsf{ccl}}_1(t)}\right]\!}_{\mathsf{E}}A$. $({\mathsf{C}}\text{-shift})$
<!-- -->
1. Set $!_{\mathsf{C}}t \colonequals {\mathsf{ind}(c, {{\mathsf{ccl}}_2(t)})}$, where ${\left[c\right]\!}_{\mathsf{C}}({\left[t\right]\!}_{\mathsf{C}}A \rightarrow {\left[{{\mathsf{ccl}}_2(t)}\right]\!}_{\mathsf{E}}{\left[t\right]\!}_{\mathsf{C}}A) \in {\mathcal{CS}}$.
2. Set ${\Lleftarrow}t \colonequals c \, \cdot_{\mathsf{C}}\, (!_{\mathsf{C}}t)$, where ${\left[c\right]\!}_{\mathsf{C}}({\left[t\right]\!}_{\mathsf{C}}A \rightarrow {\left[{{\mathsf{ccl}}_1(t)}\right]\!}_{\mathsf{E}}A) \in {\mathcal{CS}}$.
The following two theorems are standard in justification logics. Their proofs can be taken almost word for word from [@Art01BSL] and are, therefore, omitted here.
Let ${\mathcal{CS}}$ be a constant specification and $\Delta \cup \{ A, B \} \subseteq {{\textnormal{Fm}}_{{{{\mathsf{LP}}_{h}}^{{\mathsf{C}}}}}}$. Then $\Delta, A {\vdash}_{\mathcal{CS}}B \text{ if and only if } \Delta {\vdash}_{\mathcal{CS}}A \rightarrow B$.
For any constant specification ${\mathcal{CS}}$, any propositional variable $P$, any $\Delta \cup \{ A, B \} \subseteq {{\textnormal{Fm}}_{{{{\mathsf{LP}}_{h}}^{{\mathsf{C}}}}}}$, any $x \in {\textnormal{Var}}_{\circledast}$, and any $t \in {\textnormal{Tm}}_{\circledast}$, $$\text{if } \Delta {\vdash}_{\mathcal{CS}}A, \text{ then } \Delta(x/t, P/B) {\vdash}_{{\mathcal{CS}}(x/t, P/B)} A(x/t, P/B)
\rlap{\enspace,}$$ where $A(x/t, P/B)$ denotes the formula obtained by simultaneously replacing all occurrences of $x$ in $A$ with $t$ and all occurrences of $P$ in $A$ with $B$, accordingly for $\Delta(x/t, P/B)$ and ${\mathcal{CS}}(x/t, P/B)$.
The following lemma states that our logic can internalize its own proofs, which is an important property of justification logics.
Let ${\mathcal{CS}}$ be a pure ${\mathsf{C}}$-axiomatically appropriate constant specification. If $${\left[s_1\right]\!}_{\mathsf{C}}B_1, \dots, {\left[s_n\right]\!}_{\mathsf{C}}B_n, C_1, \dots, C_m {\vdash}_{\mathcal{CS}}A
\rlap{\enspace,}$$ then for each ${\circledast}$, there is a term $t_{\circledast}(x_1, \dots, x_n, y_1, \dots, y_m) \in {\textnormal{Tm}}_{\circledast}$ such that $${\left[s_1\right]\!}_{\mathsf{C}}B_1, \dots, {\left[s_n\right]\!}_{\mathsf{C}}B_n, {\left[y_1\right]\!}_{\circledast}C_1, \dots, {\left[y_m\right]\!}_{\circledast}C_m {\vdash}_{\mathcal{CS}}{\left[t_{\circledast}(s_1, \dots, s_n, y_1, \dots, y_m)\right]\!}_{\circledast}A$$ for fresh variables $y_1, \dots, y_m \in {\textnormal{Tm}}_{\circledast}$.
We proceed by induction on the derivation of $A$.
If $A$ is an axiom, there is a constant $c \in {\textnormal{Tm}}_{\mathsf{C}}$ such that ${\left[c\right]\!}_{\mathsf{C}}A \in {\mathcal{CS}}$ because ${\mathcal{CS}}$ is ${\mathsf{C}}$-axiomatically appropriate. Then take $$t_{\mathsf{C}}\colonequals c,
\qquad
t_i \colonequals {\downarrow\!}_i c,
\qquad
t_{\mathsf{E}}\colonequals {{\mathsf{ccl}}_1(c)}$$ and use axiom necessitation, axiom necessitation and $i$-conversion, or axiom necessitation and the co-closure axiom respectively.
For $A = {\left[s_j\right]\!}_{\mathsf{C}}B_j$, $1 \leq j \leq n$, take $$t_{\mathsf{C}}\colonequals !_{\mathsf{C}}s_j,
\qquad
t_i \colonequals {\downarrow\!}_i !_{\mathsf{C}}s_j,
\qquad
t_{\mathsf{E}}\colonequals {{\mathsf{ccl}}_2(s_j)}$$ and use ${\mathsf{C}}$-inspection, ${\mathsf{C}}$-inspection and $i$-conversion, or the co-closure axiom respectively.
For $A = C_j$, $1 \leq j \leq m$, take $t_{\circledast}\colonequals y_j \in {\textnormal{Var}}_{\circledast}$ for a fresh variable $y_j$.
For $A$ derived by modus ponens from $D \rightarrow A$ and $D$, by induction hypothesis there are terms $r_{\circledast}, s_{\circledast}\in {\textnormal{Tm}}_{\circledast}$ such that ${\left[r_{\circledast}\right]\!}_{\circledast}(D \rightarrow A)$ and ${\left[s_{\circledast}\right]\!}_{\circledast}D$ are provable. Take $t_{\circledast}\colonequals r_{\circledast}\cdot_{\circledast}s_{\circledast}$ and use ${\circledast}$-application, which is an axiom for ${\circledast}= i$ and for ${\circledast}= {\mathsf{C}}$ or follows from Lemma \[lem:basicproperties1\] for ${\circledast}= {\mathsf{E}}$.
For $A = {\left[c\right]\!}_{\mathsf{C}}E \in {\mathcal{CS}}$ derived by axiom necessitation, take $$t_{\mathsf{C}}\colonequals !_{\mathsf{C}}c,
\qquad
t_i \colonequals {\downarrow\!}_i !_{\mathsf{C}}c,
\qquad
t_{\mathsf{E}}\colonequals {{\mathsf{ccl}}_2(c)}$$ and use ${\mathsf{C}}$-inspection, ${\mathsf{C}}$-inspection and $i$-conversion, or the co-closure axiom respectively.
\[constructivenecessitation\] Let ${\mathcal{CS}}$ be a pure ${\mathsf{C}}$-axiomatically appropriate constant specification. For any formula $A$, if ${\vdash}_{\mathcal{CS}}A$, then for each ${\circledast}$, there is a ground term $t \in {\textnormal{Tm}}_{\circledast}$ such that ${\vdash}_{\mathcal{CS}}{\left[t\right]\!}_{\circledast}A$.
The following two lemmas show that our system ${{{\mathsf{LP}}_{h}}^{{\mathsf{C}}}}$ can internalize versions of the induction rule used in various axiomatizations of ${{{\mathsf{S4}}_{h}}^{{\mathsf{C}}}}$ (see [@BucKuzStu09M4M] for a discussion of several axiomatizations of this kind).
\[cor:2\] Let ${\mathcal{CS}}$ be a pure ${\mathsf{C}}$-axiomatically appropriate constant specification. For any formula $A$, if ${\vdash}_{\mathcal{CS}}A \rightarrow {\left[s\right]\!}_{\mathsf{E}}A$, there is a term $t \in {\textnormal{Tm}}_{\mathsf{C}}$ such that ${\vdash}_{\mathcal{CS}}A \rightarrow {\left[{\mathsf{ind}(t,s)}\right]\!}_{\mathsf{C}}A$.
By constructive necessitation, there exists a term $t \in {\textnormal{Tm}}_{\mathsf{C}}$ such that ${\vdash}_{\mathcal{CS}}{\left[t\right]\!}_{\mathsf{C}}(A \rightarrow {\left[s\right]\!}_{\mathsf{E}}A)$. It remains to use the induction axiom and propositional reasoning.
Let ${\mathcal{CS}}$ be a pure ${\mathsf{C}}$-axiomatically appropriate constant specification. For any formulae $A$ and $B$, if we have ${\vdash}_{\mathcal{CS}}B \rightarrow {\left[s\right]\!}_{\mathsf{E}}(A \land B)$, then there exist a term $t \in {\textnormal{Tm}}_{\mathsf{C}}$ and a constant $c \in {\textnormal{Tm}}_{\mathsf{C}}$ such that ${\vdash}_{\mathcal{CS}}B \rightarrow {\left[c \cdot {\mathsf{ind}(t,s)}\right]\!}_{\mathsf{C}}A$, where ${\left[c\right]\!}_{\mathsf{C}}(A \land B \to A) \in {\mathcal{CS}}$.
Assume $$\label{l:ir:1}
{\vdash}_{\mathcal{CS}}B \rightarrow {\left[s\right]\!}_{\mathsf{E}}(A \land B)
\rlap{\enspace.}$$ From this we immediately get ${\vdash}_{\mathcal{CS}}A \land B \rightarrow {\left[s\right]\!}_{\mathsf{E}}(A \land B)$. Thus, by Lemma \[cor:2\], there is a $t \in {\textnormal{Tm}}_{\mathsf{C}}$ with $$\label{l:ir:2}
{\vdash}_{\mathcal{CS}}A \land B \rightarrow {\left[{\mathsf{ind}(t,s)}\right]\!}_{\mathsf{C}}(A \land B)
\rlap{\enspace.}$$ Since ${\mathcal{CS}}$ is ${\mathsf{C}}$-axiomatically appropriate, there is a constant $c \in {\textnormal{Tm}}_{\mathsf{C}}$ such that $$\label{l:ir:3}
{\vdash}_{\mathcal{CS}}{\left[c\right]\!}_{\mathsf{C}}(A \land B \to A)
\rlap{\enspace.}$$ Making use of ${\mathsf{C}}$-application, we find by and that $$\label{l:ir:4}
{\vdash}_{\mathcal{CS}}A \land B \rightarrow {\left[c \cdot {\mathsf{ind}(t,s)}\right]\!}_{\mathsf{C}}(A)
\rlap{\enspace.}$$ From we get by ${\mathsf{E}}$ that ${\vdash}_{\mathcal{CS}}B \rightarrow A \land B$. This, together with , finally yields ${\vdash}_{\mathcal{CS}}B \rightarrow {\left[c \cdot {\mathsf{ind}(t,s)}\right]\!}_{\mathsf{C}}(A)$.
Soundness and Completeness {#sect:soundcomp}
==========================
An *AF-model meeting a constant specification ${\mathcal{CS}}$* is a structure $\mathcal{M} = (W, R, {\mathcal{E}}, {\nu})$, where $(W, R, {\nu})$ is a Kripke model for ${\mathsf{S4}}_{h}$ with a *set of possible worlds* $W \ne \varnothing$, with a function $R \colon \{ 1, \dots, {h}\} \to {\mathcal{P}}(W \times W)$ that assigns a reflexive and transitive *accessibility relation* on $W$ to each agent $i \in \{ 1, \dots, {h}\}$, and with a *truth valuation* ${\nu}\colon {\textnormal{Prop}}\to {\mathcal{P}}(W)$. We always write $R_i$ instead of $R(i)$ and define the accessibility relations for mutual and common knowledge in the standard way: $R_{\mathsf{E}}\colonequals R_1 \cup \dots \cup R_{h}$ and $R_{\mathsf{C}}\colonequals \bigcup_{n = 1}^{\infty} (R_{\mathsf{E}})^n$.
An *evidence function* ${\mathcal{E}}\colon W \times {\textnormal{Tm}}\to {\mathcal{P}}\left({{\textnormal{Fm}}_{{{{\mathsf{LP}}_{h}}^{{\mathsf{C}}}}}}\right)$ determines the formulae evidenced by a term at a world. We define ${\mathcal{E}}_{\circledast}\colonequals {\mathcal{E}}\upharpoonright (W \times {\textnormal{Tm}}_{\circledast})$. Note that whenever $A \in {\mathcal{E}}_{\circledast}(w, t)$, it follows that $t \in {\textnormal{Tm}}_{\circledast}$. The evidence function ${\mathcal{E}}$ must satisfy the following closure conditions: for any worlds $w, v \in W$,
1. ${\mathcal{E}}_{*}(w, t) \subseteq {\mathcal{E}}_{*}(v, t)$ whenever $(w, v) \in R_{*}$; $(\text{monotonicity})$
2. if ${\left[c\right]\!}_{\circledast}A \in {\mathcal{CS}}$, then $A \in {\mathcal{E}}_{\circledast}(w, c)$; $(\text{constant specification})$
3. if $(A \rightarrow B) \in {\mathcal{E}}_{*}(w, t)$ and $A \in {\mathcal{E}}_{*}(w, s)$, then $B \in {\mathcal{E}}_{*}(w, t \cdot s)$; $(\text{application})$
4. ${\mathcal{E}}_{*}(w, s) \cup {\mathcal{E}}_{*}(w, t) \subseteq {\mathcal{E}}_{*}(w, s + t)$; $(\text{sum})$
5. if $A \in {\mathcal{E}}_{i}(w, t)$, then ${\left[t\right]\!}_{i}A \in {\mathcal{E}}_{i}(w, ! t)$; $(\text{inspection})$
6. if $A \in {\mathcal{E}}_i (w, t_i)$ for all $1 \leq i \leq {h}$, then $A \in {\mathcal{E}}_{\mathsf{E}}(w, {\left\langle t_1, \dots, t_{h}\right\rangle})$; $(\text{tupling})$
7. if $A \in {\mathcal{E}}_{\mathsf{E}}(w, t)$, then $A \in {\mathcal{E}}_i (w, {\pi}_i t)$; $(\text{projection})$
8. if $A \in {\mathcal{E}}_{\mathsf{C}}(w, t)$, then $A \in {\mathcal{E}}_{\mathsf{E}}(w, {{\mathsf{ccl}}_1(t)})$ and ${\left[t\right]\!}_{\mathsf{C}}A \in {\mathcal{E}}_{\mathsf{E}}(w, {{\mathsf{ccl}}_2(t)})$; $(\text{co-closure})$
9. if $A \in {\mathcal{E}}_{\mathsf{E}}(w, s)$ and $(A \rightarrow {\left[s\right]\!}_{\mathsf{E}}A) \in {\mathcal{E}}_{\mathsf{C}}(w, t)$,\
then $A \in {\mathcal{E}}_{\mathsf{C}}(w, {\mathsf{ind}(t,s)})$. $(\text{induction})$
When the model is clear from the context, we will directly refer to $R_1, \dots, R_{h}$, $R_{\mathsf{E}}$, $R_{\mathsf{C}}$, ${\mathcal{E}}_1, \dots, {\mathcal{E}}_{h}$, ${\mathcal{E}}_{\mathsf{E}}$, ${\mathcal{E}}_{\mathsf{C}}$, $W$, and ${\nu}$.
A ternary relation $\mathcal{M}, w \Vdash A$ for *formula $A$ being satisfied at a world $w \in W$ in an AF-model $\mathcal{M} = (W, R, {\mathcal{E}}, {\nu})$* is defined by induction on the structure of the formula $A$:
1. $\mathcal{M}, w \Vdash P$ if and only if $w \in {\nu}(P)$;
2. $\Vdash$ behaves classically with respect to the propositional connectives;
3. $\mathcal{M}, w \Vdash {\left[t\right]\!}_{\circledast}A$ if and only if 1) $A \in {\mathcal{E}}_{\circledast}(w, t)$ and 2) $\mathcal{M}, v \Vdash A$ for all $v \in W$ with $(w, v) \in R_{\circledast}$.
We write $\mathcal{M} \Vdash A$ if $\mathcal{M}, w \Vdash A$ for all $w \in W$. We write $\Vdash_{\mathcal{CS}}A$ and say that formula $A$ is *valid with respect to ${\mathcal{CS}}$* if $\mathcal{M} \Vdash A$ for all AF-models $\mathcal{M}$ meeting ${\mathcal{CS}}$.
\[soundness\] Provable formulae are valid: ${\vdash}_{\mathcal{CS}}A$ implies $\Vdash_{\mathcal{CS}}A$.
Let $\mathcal{M} = (W, R, {\mathcal{E}}, {\nu})$ be an AF-model meeting ${\mathcal{CS}}$ and let $w \in W$. We show soundness by induction on the derivation of $A$. The cases for propositional tautologies, for the application, sum, reflexivity, and inspection axioms, and for modus ponens rule are the same as for the single-agent case in [@Fit05APAL] and are, therefore, omitted. We show the remaining five cases:
(tupling)
: Assume $\mathcal{M}, w \Vdash {\left[t_i\right]\!}_i A$ for all $1 \leq i \leq {h}$. Then for all $1\leq i \leq {h}$, we have 1) $\mathcal{M}, v \Vdash A$ for all $v \in W$ with $(w, v) \in R_i$ and 2) $A \in {\mathcal{E}}_i (w, t_i)$. So, by the tupling closure condition, $A \in {\mathcal{E}}_{\mathsf{E}}(w, {\left\langle t_1, \dots, t_{h}\right\rangle})$ from 2). Since by definition $R_{\mathsf{E}}= \bigcup_{i=1}^{h}R_i$, it follows from 1) that $\mathcal{M}, v \Vdash A$ for all $v \in W$ with $(w, v)\in R_{\mathsf{E}}$. Hence, $\mathcal{M}, w \Vdash {\left[{\left\langle t_1, \dots, t_{h}\right\rangle}\right]\!}_{\mathsf{E}}A$.
(projection)
: Assume $\mathcal{M}, w \Vdash {\left[t\right]\!}_{\mathsf{E}}A$. Then 1) $\mathcal{M}, v \Vdash A$ for all $v \in W$ with $(w, v) \in R_{\mathsf{E}}$ and 2) $A \in {\mathcal{E}}_{\mathsf{E}}(w, t)$. By the projection closure condition, it follows from 2) that $A \in {\mathcal{E}}_i (w, {\pi}_i t)$. In addition, since $R_{\mathsf{E}}= \bigcup_{i=1}^{h}R_i$, we get $\mathcal{M}, v \Vdash A$ for all $v \in W$ with $(w, v) \in R_i$ by 1). Thus, $\mathcal{M}, w \Vdash {\left[{\pi}_i t\right]\!}_i A$.
(co-closure)
: Assume $\mathcal{M}, w \Vdash {\left[t\right]\!}_{\mathsf{C}}A$. Then 1) $\mathcal{M}, v \Vdash A$ for all $v \in W$ with $(w, v) \in R_{\mathsf{C}}$ and 2) $A \in {\mathcal{E}}_{\mathsf{C}}(w, t)$. It follows from 1) that for all $v' \in W$ with $(w, v') \in R_{\mathsf{E}}$, we have $\mathcal{M}, v' \Vdash A$ since $R_{\mathsf{E}}\subseteq R_{\mathsf{C}}$; also, due to the monotonicity closure condition, $\mathcal{M}, v' \Vdash {\left[t\right]\!}_{\mathsf{C}}A$ since $R_{\mathsf{E}}\circ R_{\mathsf{C}}\subseteq R_{\mathsf{C}}$. From 2), by the co-closure closure condition, $A \in {\mathcal{E}}_{\mathsf{E}}(w, {{\mathsf{ccl}}_1(t)})$ and ${\left[t\right]\!}_{\mathsf{C}}A \in {\mathcal{E}}_{\mathsf{E}}(w, {{\mathsf{ccl}}_2(t)})$. Hence, $\mathcal{M}, w \Vdash {\left[{{\mathsf{ccl}}_1(t)}\right]\!}_{\mathsf{E}}A$ and $\mathcal{M}, w \Vdash {\left[{{\mathsf{ccl}}_2(t)}\right]\!}_{\mathsf{E}}{\left[t\right]\!}_{\mathsf{C}}A$.
(induction)
: Assume $\mathcal{M}, w \Vdash A$ and $\mathcal{M}, w \Vdash {\left[t\right]\!}_{\mathsf{C}}(A \rightarrow {\left[s\right]\!}_{\mathsf{E}}A)$. From the second assumption and the reflexivity of $R_{\mathsf{C}}$, we get $\mathcal{M}, w \Vdash A \rightarrow {\left[s\right]\!}_{\mathsf{E}}A$; thus, $\mathcal{M}, w \Vdash {\left[s\right]\!}_{\mathsf{E}}A$ by the first assumption. So $A \in {\mathcal{E}}_{\mathsf{E}}(w, s)$ and, by the second assumption, $A \rightarrow {\left[s\right]\!}_{\mathsf{E}}A \in {\mathcal{E}}_{\mathsf{C}}(w, t)$. By the induction closure condition, we have $A \in {\mathcal{E}}_{\mathsf{C}}(w, {\mathsf{ind}(t,s)})$. To show $\mathcal{M}, v \Vdash A$ for all $v \in W$ with $(w, v) \in R_{\mathsf{C}}$, we prove that $\mathcal{M}, v \Vdash A$ for all $v \in W$ with $(w, v) \in (R_{\mathsf{E}})^n$ by induction on the positive integer $n$.
The **base case $n = 1$** immediately follows from $\mathcal{M}, w \Vdash {\left[s\right]\!}_{\mathsf{E}}A$.
**Induction step.** Let $(w, v^\prime) \in (R_{\mathsf{E}})^n$ and $(v^\prime, v) \in R_{\mathsf{E}}$ for some $v, v^\prime \in W$. By induction hypothesis, $\mathcal{M}, v^\prime \Vdash A$. Since $\mathcal{M}, w \Vdash {\left[t\right]\!}_{\mathsf{C}}(A \rightarrow {\left[s\right]\!}_{\mathsf{E}}A)$, we get $\mathcal{M}, v^\prime \Vdash A \rightarrow {\left[s\right]\!}_{\mathsf{E}}A$. Thus, $\mathcal{M}, v^\prime \Vdash {\left[s\right]\!}_{\mathsf{E}}A$, which yields $\mathcal{M}, v \Vdash A$.
Finally, we conclude that $\mathcal{M}, w \Vdash {\left[{\mathsf{ind}(t,s)}\right]\!}_{\mathsf{C}}A$.
(axiom necessitation)
: Let $A$ be an axiom and $c$ be a proof constant such that ${\left[c\right]\!}_{\circledast}A \in {\mathcal{CS}}$. Since $A$ is an axiom, $\mathcal{M}, w \Vdash A$ for all $w \in W$, as shown above. Since $\mathcal{M}$ is an AF-model meeting ${\mathcal{CS}}$, we also have $A \in {\mathcal{E}}_{\circledast}(w, c)$ for all $w \in W$ by the constant specification closure condition. Thus, $\mathcal{M}, w \Vdash {\left[c\right]\!}_{\circledast}A$ for all $w \in W$.
Let ${\mathcal{CS}}$ be a constant specification. A set $\Phi$ of formulae is called *${\mathcal{CS}}$-consistent* if $\Phi \nvdash_{\mathcal{CS}}\phi$ for some formula $\phi$. A set $\Phi$ is called *maximal ${\mathcal{CS}}$-consistent* if it is ${\mathcal{CS}}$-consistent and has no ${\mathcal{CS}}$-consistent proper extensions.
Whenever safe, we do not mention the constant specification and only talk about consistent and maximal consistent sets. It can be easily shown that maximal consistent sets contain all axioms of ${{{\mathsf{LP}}_{h}}^{{\mathsf{C}}}}$ and are closed under modus ponens.
For a set $\Phi$ of formulae, we define $$\Phi / {\circledast}\colonequals \{ A \; : \; \text{there is a } t \in {\textnormal{Tm}}_{\circledast}\text{ such that } {\left[t\right]\!}_{\circledast}A \in \Phi \}
\rlap{\enspace.}$$
Let ${\mathcal{CS}}$ be a constant specification. The *canonical AF-model $\mathcal{M} = (W, R, {\mathcal{E}}, {\nu})$ meeting ${\mathcal{CS}}$* is defined as follows:
1. $W \colonequals \{ w \subseteq {{\textnormal{Fm}}_{{{{\mathsf{LP}}_{h}}^{{\mathsf{C}}}}}}\; : \; w \text{ is a maximal ${\mathcal{CS}}$-consistent set} \}$;
2. $R_i \colonequals \{ (w,v) \in W \times W \; : \; w / i \subseteq v \}$;
3. ${\mathcal{E}}_{\circledast}(w, t) \colonequals \{ A \in {{\textnormal{Fm}}_{{{{\mathsf{LP}}_{h}}^{{\mathsf{C}}}}}}\; : \; {\left[t\right]\!}_{\circledast}A \in w \}$;
4. ${\nu}(P_n) \colonequals \{ w \in W \; : \; P_n \in w \}$.
\[canonicalmodelisfittingmodel\] Let ${\mathcal{CS}}$ be a constant specification. The canonical AF-model meeting ${\mathcal{CS}}$ is an AF-model meeting ${\mathcal{CS}}$.
The proof of reflexivity and transitivity of each $R_i$, as well as the argument for the constant specification, application, sum, and inspection closure conditions, is the same as in the single-agent case (see [@Fit05APAL]). We show the remaining five closure conditions:
(tupling)
: Assume $A \in {\mathcal{E}}_{i}(w, t_{i})$ for all $1 \leq {i}\leq {h}$. By definition of ${\mathcal{E}}_i$, we have ${\left[t_{i}\right]\!}_{i}A \in w$ for all $1 \leq {i}\leq {h}$. Therefore, by the tupling axiom and maximal consistency, ${\left[{\left\langle t_1, \dots, t_{h}\right\rangle}\right]\!}_{\mathsf{E}}A \in w$. Thus, $A \in {\mathcal{E}}_{\mathsf{E}}(w, {\left\langle t_1, \dots, t_{h}\right\rangle})$.
(projection)
: Assume $A \in {\mathcal{E}}_{\mathsf{E}}(w, t)$. Thus, we have ${\left[t\right]\!}_{\mathsf{E}}A \in w$. Then, by the projection axiom and maximal consistency, ${\left[{\pi}_{i}t\right]\!}_{i}A \in w$, and thus $A \in {\mathcal{E}}_{i}(w, {\pi}_{i}t)$.
(co-closure)
: Assume $A \in {\mathcal{E}}_{\mathsf{C}}(w, t)$. Thus, ${\left[t\right]\!}_{\mathsf{C}}A \in w$, and, by the co-closure axioms and maximal consistency, ${\left[{{\mathsf{ccl}}_1(t)}\right]\!}_{\mathsf{E}}A \in w$ and ${\left[{{\mathsf{ccl}}_2(t)}\right]\!}_{\mathsf{E}}{\left[t\right]\!}_{\mathsf{C}}A \in w$. Hence, $A \in {\mathcal{E}}_{\mathsf{E}}(w, {{\mathsf{ccl}}_1(t)})$ and ${\left[t\right]\!}_{\mathsf{C}}A \in {\mathcal{E}}_{\mathsf{E}}(w, {{\mathsf{ccl}}_2(t)})$.
(induction)
: Assume $A \in {\mathcal{E}}_{\mathsf{E}}(w, s)$ and $(A \rightarrow {\left[s\right]\!}_{\mathsf{E}}A) \in {\mathcal{E}}_{\mathsf{C}}(w, t)$. Then we have ${\left[s\right]\!}_{\mathsf{E}}A \in w$ and ${\left[t\right]\!}_{\mathsf{C}}(A \rightarrow {\left[s\right]\!}_{\mathsf{E}}A) \in w$. From ${\vdash}_{\mathcal{CS}}{\left[s\right]\!}_{\mathsf{E}}A \rightarrow A$ (Lemma \[lem:basicproperties1\].\[lem:basicproperties1:Erefl\]) and the induction axiom, it follows by maximal consistency that $A \in w$ and ${\left[{\mathsf{ind}(t,s)}\right]\!}_{\mathsf{C}}A \in w$. Therefore, $A \in {\mathcal{E}}_{\mathsf{C}}(w, {\mathsf{ind}(t,s)})$.
(monotonicity)
: We show only the case of ${*}= {\mathsf{C}}$ since the other cases are the same as in [@Fit05APAL]. It is sufficient to prove by induction on the positive integer $n$ that $$\label{ts:eq:mon:1}
\text{if } {\left[t\right]\!}_{\mathsf{C}}A \in w \text{ and } (w, v) \in (R_{\mathsf{E}})^n, \text{ then } {\left[t\right]\!}_C A \in v
\rlap{\enspace.}$$
**Base case $n = 1$.** Assume $(w, v) \in R_{\mathsf{E}}$, i.e., $w / {i}\subseteq v$ for some ${i}$. As ${\left[t\right]\!}_{\mathsf{C}}A \in w$, ${\left[{\pi}_{i}{{\mathsf{ccl}}_2(t)}\right]\!}_{i}{\left[t\right]\!}_{\mathsf{C}}A \in w$ by maximal consistency, and hence ${\left[t\right]\!}_{\mathsf{C}}A \in w / {i}\subseteq v$. The argument for the **induction step** is similar.
Now assume $(w, v) \in R_{\mathsf{C}}= \bigcup_{n=1}^{\infty} (R_{\mathsf{E}})^n$ and $A \in {\mathcal{E}}_{\mathsf{C}}(w, t)$, i.e., ${\left[t\right]\!}_{\mathsf{C}}A \in w$. As shown above, ${\left[t\right]\!}_{\mathsf{C}}A \in v$. Thus, $A \in {\mathcal{E}}_{\mathsf{C}}(v, t)$.
\[nonstandardbehaviour\] Let $R_{\mathsf{C}}^\prime$ denote the binary relation on $W$ given by $$(w, v) \in R_{\mathsf{C}}^\prime \quad \text{if and only if} \quad w / {\mathsf{C}}\subseteq v
\rlap{\enspace.}$$ An argument similar to the one just used for monotonicity shows that $R_{\mathsf{C}}\subseteq R_{\mathsf{C}}^\prime$. However, the converse does not hold for any pure ${\mathsf{C}}$-axiomatically appropriate constant specification ${\mathcal{CS}}$, which we demonstrate by adapting an example from [@HM95]. Let $$\Phi \colonequals \{ {\left[s_n\right]\!}_{\mathsf{E}}\dots {\left[s_1\right]\!}_{\mathsf{E}}P \; : \; n \geq 1, \, s_1, \dots, s_n \in {\textnormal{Tm}}_{\mathsf{E}}\} \cup \{ \neg {\left[t\right]\!}_{\mathsf{C}}P \; : \; t \in {\textnormal{Tm}}_{\mathsf{C}}\}
\rlap{\enspace.}$$ This set is ${\mathcal{CS}}$-consistent for any $P \in {\textnormal{Prop}}$.
To see this, let $\Phi^\prime \subseteq \Phi$ be finite and let $m$ denote the maximal number of terms such that ${\left[s_m\right]\!}_{\mathsf{E}}\dots {\left[s_1\right]\!}_{\mathsf{E}}P \in \Phi^\prime$. Define the model $\mathcal{N} \colonequals ({\mathbb{N}}, R^\mathcal{N}, {\mathcal{E}}^\mathcal{N}, {\nu}^\mathcal{N})$ by
- $R^\mathcal{N}_i \colonequals \{ (n, n+1) \in {\mathbb{N}}^2 \; : \; n \mod h = i \} \cup \{ (n, n) \in {\mathbb{N}}^2 \; : \; n \in {\mathbb{N}}\}$;
- ${\mathcal{E}}^\mathcal{N} (n, s) \colonequals {{\textnormal{Fm}}_{{{{\mathsf{LP}}_{h}}^{{\mathsf{C}}}}}}$ for all $n \in {\mathbb{N}}$ and terms $s \in {\textnormal{Tm}}$;
- ${\nu}^\mathcal{N}(P) \colonequals \{ 1, 2, \dots, m+1 \} \subseteq {\mathbb{N}}$.
Clearly, $\mathcal{N}$ meets any constant specification; in particular, it meets ${\mathcal{CS}}$. It can also be easily verified that $\mathcal{N}, 1 \Vdash \Phi^\prime$; therefore, $\Phi^\prime$ is ${\mathcal{CS}}$-consistent.
Since $\Phi$ is ${\mathcal{CS}}$-consistent, there exists a maximal ${\mathcal{CS}}$-consistent set $w \supseteq \Phi$. Let us show that the set $\Psi \colonequals \{ \neg P \} \cup (w / {\mathsf{C}})$ is also ${\mathcal{CS}}$-consistent. Indeed, if it were not the case, there would exist formulae $B_1, \dots, B_n \in w / {\mathsf{C}}$ such that $${\vdash}_{\mathcal{CS}}B_1 \rightarrow (B_2 \rightarrow \dots \rightarrow (B_n \rightarrow P) \dots )
\rlap{\enspace.}$$ Then, by Corollary \[constructivenecessitation\], there would exist a term $s \in {\textnormal{Tm}}_{\mathsf{C}}$ such that $${\vdash}_{\mathcal{CS}}{\left[s\right]\!}_{\mathsf{C}}{(B_1 \rightarrow (B_2 \rightarrow \dots \rightarrow (B_n \rightarrow P) \dots ) )}
\rlap{\enspace.}$$ But this would imply ${\left[( \dots (s \cdot t_1) \cdots t_{n-1}) \cdot t_n\right]\!}_{\mathsf{C}}{P} \in w$ for ${\left[t_j\right]\!}_{\mathsf{C}}{B_j} \in w$, $1 \leq j \leq n$, a contradiction with the consistency of $w$.
Let $v$ be a maximal ${\mathcal{CS}}$-consistent set that contains $\Psi$, i.e., $v \supseteq \Psi$. Clearly, $w / {\mathsf{C}}\subseteq v$, i.e., $(w, v) \in R_{\mathsf{C}}^\prime$, but $(w, v) \notin R_{\mathsf{C}}$ because this would imply $P \in v$, which cannot happen. It follows that $R_{\mathsf{C}}\subsetneq R_{\mathsf{C}}^\prime$.
Similarly, we can define $R_{\mathsf{E}}^\prime$ by $(w, v) \in R_{\mathsf{E}}^\prime$ if and only if $w / {\mathsf{E}}\subseteq v$. However, $ R_{\mathsf{E}}^\prime = R_{\mathsf{E}}$ for any ${\mathsf{C}}$-axiomatically appropriate constant specification ${\mathcal{CS}}$. Indeed, is easy to show that $R_{\mathsf{E}}\subseteq R_{\mathsf{E}}^\prime$. For the converse, assume $(w, v) \notin R_{\mathsf{E}}$, then $(w, v) \notin R_i$ for all $1 \leq i \leq h$. So there are formulae $A_1, \dots, A_{h}$ such that ${\left[t_i\right]\!}_i A_i \in w$ for some $t_i \in {\textnormal{Tm}}_i$, but $A_i \notin v$. Now let ${\left[c_i\right]\!}_{\mathsf{C}}(A_i \rightarrow A_1 \vee \dots \vee A_{h}) \in {\mathcal{CS}}$ for constants $c_1, \dots, c_{h}$. Then ${\left[{\downarrow\!}_i c_i \cdot t_i\right]\!}_i (A_1 \vee \dots \vee A_{h}) \in w$ for all $1 \leq i \leq h$, so ${\left[{\left\langle {\downarrow\!}_1 c_1 \cdot t_1, \dots, {\downarrow\!}_{h}c_{h}\cdot t_{h}\right\rangle}\right]\!}_{\mathsf{E}}(A_1 \vee \dots \vee A_{h}) \in w$. However, $A_i \notin v$ for any $1 \leq i \leq h$; therefore, by the maximal consistency of $v$, $A_1 \vee \dots \vee A_{h}\notin v$ either. Hence, $w / {\mathsf{E}}\nsubseteq v$, so $(w, v) \notin R_{\mathsf{E}}^\prime$.
\[truthlemma\] Let ${\mathcal{CS}}$ be a constant specification and $\mathcal{M}$ be the canonical AF-model meeting ${\mathcal{CS}}$. For all formulae $A$ and all worlds $w \in W$, $$A \in w \text{ if and only if } \mathcal{M}, w \Vdash A
\rlap{\enspace.}$$
The proof is by induction on the structure of $A$. The cases for propositional variables and propositional connectives are immediate by the definition of $\Vdash$ and by the maximal consistency of $w$. We check the remaining cases:
**Case** $A$ is ${\left[t\right]\!}_i B$. Assume $A \in w$. Then $B \in w / i$ and $B \in {\mathcal{E}}_i(w,t)$. Consider any $v$ such that $(w, v) \in R_i$. Since $w / i \subseteq v$, it follows that $B \in v$, and thus, by induction hypothesis, $\mathcal{M}, v \Vdash B$. And $\mathcal{M}, w \Vdash A$ immediately follows from this.
For the converse, assume $\mathcal{M}, w \Vdash {\left[t\right]\!}_i B$. By definition of $\Vdash$ we get $B \in {\mathcal{E}}_i (w, t)$, from which ${\left[t\right]\!}_i B \in w$ immediately follows by definition of ${\mathcal{E}}_i$.
**Case** $A$ is ${\left[t\right]\!}_{\mathsf{E}}B$. Assume $A \in w$ and consider any $v$ such that $(w, v) \in R_{\mathsf{E}}$. Then $(w, v) \in R_i$ for some $1 \leq i\leq h$, i.e., $w / i \subseteq v$. By definition of ${\mathcal{E}}_{\mathsf{E}}$, we get $B \in {\mathcal{E}}_{\mathsf{E}}(w, t)$. By maximal consistency of $w$, it follows that ${\left[{\pi}_i t\right]\!}_i B \in w$, and thus $B \in w / i \subseteq v$. Since, by induction hypothesis, $\mathcal{M}, v \Vdash B$, we conclude that $\mathcal{M}, w \Vdash A$. The argument for the converse repeats the one from the previous case.
**Case** $A$ is ${\left[t\right]\!}_{\mathsf{C}}B$. Assume $A \in w$ and consider any $v$ such that $(w, v)\in R_{\mathsf{C}}$, i.e., $(w, v) \in (R_{\mathsf{E}})^n$ for some $n \geq 1$. As in the previous cases, $B \in {\mathcal{E}}_{\mathsf{C}}(w, t)$ by definition of ${\mathcal{E}}_{\mathsf{C}}$. By we find $A \in v$, and thus, by ${\mathsf{C}}$-reflexivity and maximal consistency, also $B \in v$. Hence, by the induction hypothesis $\mathcal{M}, v \Vdash B$. Now $\mathcal{M}, w \Vdash A$ immediately follows. The argument for the converse repeats the one from the previous cases.
Note that the converse directions in the proof above are far from trivial in the modal case, see e.g. [@HM95]. The last case, in particular, usually requires more sophisticated methods that guarantee the finiteness of the model.
${{{\mathsf{LP}}_{h}}^{{\mathsf{C}}}}({\mathcal{CS}})$ is sound and complete with respect to the class of AF-models meeting ${\mathcal{CS}}$, i.e., for all formulae $A \in {{\textnormal{Fm}}_{{{{\mathsf{LP}}_{h}}^{{\mathsf{C}}}}}}$, $${\vdash}_{\mathcal{CS}}A \text{ if and only if\/ } \Vdash_{\mathcal{CS}}A
\rlap{\enspace.}$$
Soundness has already been shown in Lemma \[soundness\]. For completeness, let $\mathcal{M}$ be the canonical AF-model meeting ${\mathcal{CS}}$ and assume $\nvdash_{\mathcal{CS}}A$. Then $\{ \neg A \}$ is ${\mathcal{CS}}$-consistent and hence is contained in some maximal ${\mathcal{CS}}$-consistent set $w \in W$. So, by Lemma \[truthlemma\], $\mathcal{M}, w \Vdash \neg A$, and hence, by Lemma \[canonicalmodelisfittingmodel\], $\nVdash_{\mathcal{CS}}A$.
M-models were introduced as semantics for ${\mathsf{LP}}$ by Mkrtychev [@Mkr97LFCS]. They form a subclass of F-models (see [@Fit05APAL]).
An *M-model* is a singleton AF-model.
${{{\mathsf{LP}}_{h}}^{{\mathsf{C}}}}({\mathcal{CS}})$ is also sound and complete with respect to the class of M-models meeting ${\mathcal{CS}}$.
Soundness follows immediately from Lemma \[soundness\]. Now assume that $\nvdash_{\mathcal{CS}}A$, then $\{ \neg A \}$ is ${\mathcal{CS}}$-consistent, and hence $\mathcal{M}, w \Vdash \neg A$ for some world $w_0 \in W$ in the canonical AF-model $\mathcal{M} = (W, R, {\mathcal{E}}, {\nu})$ meeting ${\mathcal{CS}}$.
Let $\mathcal{M}^\prime = (W^\prime, R^\prime, {\mathcal{E}}^\prime, {\nu}^\prime)$ be the restriction of $\mathcal{M}$ to $\{ w_0 \}$, i.e., $W^\prime \colonequals \{ w_0 \}$, $R^\prime_{\circledast}\colonequals \{ (w_0, w_0) \}$ for any ${\circledast}$, ${\mathcal{E}}^\prime \colonequals {\mathcal{E}}\upharpoonright (W^\prime \times {\textnormal{Tm}})$, and ${\nu}^\prime(P_n) \colonequals {\nu}(P_n) \cap W^\prime$.
Since $\mathcal{M}^\prime$ is clearly an M-model meeting ${\mathcal{CS}}$, it remains to demonstrate that $\mathcal{M}^\prime, w_0 \Vdash B$ if and only if $\mathcal{M}, w_0 \Vdash B$ for all formulae $B$. We proceed by induction on the structure of $B$. The cases where either $B$ is a propositional variable or its primary connective is propositional are trivial. Therefore, we only show the case of $B = {\left[t\right]\!}_{\circledast}C$. First, observe that $$\label{eq:mmodel:1}
\mathcal{M}, w_0 \Vdash {\left[t\right]\!}_{\circledast}C \text{ if and only if } C \in {\mathcal{E}}^\prime_{\circledast}(w_0, t)
\rlap{\enspace.}$$ Indeed, by Lemma \[truthlemma\], $\mathcal{M}, w_0 \Vdash {\left[t\right]\!}_{\circledast}C$ if and only if ${\left[t\right]\!}_{\circledast}C \in w_0$, which, by definition of the canonical AF-model, is equivalent to $C \in {\mathcal{E}}_{\circledast}(w_0, t) = {\mathcal{E}}^\prime_{\circledast}(w_0, t)$.
If $\mathcal{M}, w_0 \Vdash {\left[t\right]\!}_{\circledast}C$, then $\mathcal{M}, w_0 \Vdash C$ since $R_{\circledast}$ is reflexive. By induction hypothesis, $\mathcal{M}^\prime, w_0 \Vdash C$. By we have $C \in {\mathcal{E}}^\prime_{\circledast}(w_0, t)$, and thus $\mathcal{M}^\prime, w_0 \Vdash {\left[t\right]\!}_{\circledast}C$.
If $\mathcal{M}, w_0 \nVdash {\left[t\right]\!}_{\circledast}C$, then by we have $C \notin {\mathcal{E}}^\prime_{\circledast}(w, t)$, so $\mathcal{M}^\prime, w_0 \nVdash {\left[t\right]\!}_{\circledast}C$.
${{{\mathsf{LP}}_{h}}^{{\mathsf{C}}}}({\mathcal{CS}})$ enjoys the finite model property with respect to AF-models.
Conservativity {#sect:conserv}
==============
Yavorskaya in [@TYav08TOCSnonote] introduced a two-agent version of ${\mathsf{LP}}$, which we extend to an arbitrary $h$ in the natural way:
The language of ${{\mathsf{LP}}_{h}}$ is obtained from that of ${{{\mathsf{LP}}_{h}}^{{\mathsf{C}}}}$ by restricting the set of operations to $\cdot_i$, $+_i$, and $!_i$ and by dropping all terms from ${\textnormal{Tm}}_{\mathsf{E}}$ and ${\textnormal{Tm}}_{\mathsf{C}}$. The axioms are restricted to application, sum, reflexivity, and inspection for each $i$. The definition of constant specification is changed accordingly.
We show that ${{{\mathsf{LP}}_{h}}^{{\mathsf{C}}}}$ is conservative over ${{\mathsf{LP}}_{h}}$ by adapting a technique from [@Fit08AMAI].
The mapping $\times : {{\textnormal{Fm}}_{{{{\mathsf{LP}}_{h}}^{{\mathsf{C}}}}}}\to {{\textnormal{Fm}}_{{{\mathsf{LP}}_{h}}}}$ is defined as follows:
1. $P^\times \colonequals P$ for propositional variables $P \in {\textnormal{Prop}}$;
2. $\times$ commutes with propositional connectives;
3. $({\left[t\right]\!}_{\circledast}A)^\times \colonequals
\begin{cases}
A^\times & \text{if $t$~contains a subterm } s \in {\textnormal{Tm}}_{\mathsf{E}}\cup {\textnormal{Tm}}_{\mathsf{C}}, \\
{\left[t\right]\!}_{\circledast}A^\times & \text{otherwise.}
\end{cases}$
Let ${\mathcal{CS}}$ be a constant specification for ${{{\mathsf{LP}}_{h}}^{{\mathsf{C}}}}$. For an arbitrary formula $A \in {{\textnormal{Fm}}_{{{\mathsf{LP}}_{h}}}}$, if ${{{\mathsf{LP}}_{h}}^{{\mathsf{C}}}}({\mathcal{CS}}) {\vdash}A$, then ${{\mathsf{LP}}_{h}}({\mathcal{CS}}^\times) {\vdash}A$.
Since $A^\times = A$ for any $A \in {{\textnormal{Fm}}_{{{\mathsf{LP}}_{h}}}}$, it suffices to demonstrate that for any formula $D \in {{\textnormal{Fm}}_{{{{\mathsf{LP}}_{h}}^{{\mathsf{C}}}}}}$, if ${{{\mathsf{LP}}_{h}}^{{\mathsf{C}}}}({\mathcal{CS}}) {\vdash}D$, then ${{\mathsf{LP}}_{h}}({\mathcal{CS}}^\times) {\vdash}D^\times$, which can be done by induction on the derivation of $D$.
[**Case** ]{}when $D$ is a propositional tautology, then so is $D^\times$.
[**Case** ]{}when $D = {\left[t\right]\!}_{i}B \rightarrow B$ is an instance of the reflexivity axiom. Then $D^\times$ is either ${\left[t\right]\!}_{i}B^\times \rightarrow B^\times$ or $B^\times \rightarrow B^\times$, i.e., an instance of the reflexivity axiom of ${{\mathsf{LP}}_{h}}$ or a propositional tautology respectively.
[**Case** ]{}when $D = {\left[t\right]\!}_{*}(B \rightarrow C) \rightarrow ({\left[s\right]\!}_{*}B \rightarrow {\left[t \cdot s\right]\!}_{*}C)$ is an instance of the application axiom. We distinguish the following possibilities:
1. Both $t$ and $s$ contain a subterm from ${\textnormal{Tm}}_{\mathsf{E}}\cup {\textnormal{Tm}}_{\mathsf{C}}$. Then $D^\times$ has the form $(B^\times \rightarrow C^\times) \rightarrow (B^\times \rightarrow C^\times)$, which is a propositional tautology and, thus, an axiom of ${{\mathsf{LP}}_{h}}$.
2. Neither $t$ nor $s$ contains a subterm from ${\textnormal{Tm}}_{\mathsf{E}}\cup {\textnormal{Tm}}_{\mathsf{C}}$. Then $D^\times$ is an instance of the application axiom of ${{\mathsf{LP}}_{h}}$.
3. Term $t$ contains a subterm from ${\textnormal{Tm}}_{\mathsf{E}}\cup {\textnormal{Tm}}_{\mathsf{C}}$ while $s$ does not. Then $D^\times$ is $(B^\times \rightarrow C^\times) \rightarrow ({\left[s\right]\!}_{i}B^\times \rightarrow C^\times)$, which can be derived in ${{\mathsf{LP}}_{h}}({\mathcal{CS}}^\times)$ from the reflexivity axiom ${\left[s\right]\!}_{i}B^\times \rightarrow B^\times$ by propositional reasoning. In this case, translation $\times$ does not map an axiom of ${{{\mathsf{LP}}_{h}}^{{\mathsf{C}}}}$ to an axiom of ${{\mathsf{LP}}_{h}}$.
4. Term $s$ contains a subterm from ${\textnormal{Tm}}_{\mathsf{E}}\cup {\textnormal{Tm}}_{\mathsf{C}}$ while $t$ does not. Then $D^\times$ is ${\left[t\right]\!}_{i}(B^\times \rightarrow C^\times) \rightarrow (B^\times \rightarrow C^\times)$, an instance of the reflexivity axiom of ${{\mathsf{LP}}_{h}}$.
[**Case** ]{}when $D = {\left[t\right]\!}_{*}B \rightarrow {\left[t + s\right]\!}_{*}B$ is an instance of the sum axiom. Then $D^\times$ becomes $B^\times \rightarrow B^\times$, ${\left[t\right]\!}_{i}B^\times \rightarrow B^\times$, or ${\left[t\right]\!}_{i}B^\times \rightarrow {\left[t + s\right]\!}_{i}B^\times$, i.e., a propositional tautology, an instance of the reflexivity axiom of ${{\mathsf{LP}}_{h}}$, or an instance of the sum axiom of ${{\mathsf{LP}}_{h}}$ respectively. The sum axiom ${\left[s\right]\!}_{*}B \rightarrow {\left[t + s\right]\!}_{*}B$ is treated in the same manner.
[**Case** ]{}when $D = {\left[t\right]\!}_{i}B \rightarrow {\left[! t\right]\!}_{i}{\left[t\right]\!}_{i}B$ is an instance of the inspection axiom. Then $D^\times$ is either the propositional tautology $B^\times \rightarrow B^\times$ or ${\left[t\right]\!}_{i}B^\times \rightarrow {\left[! t\right]\!}_{i}{\left[t\right]\!}_{i}B^\times$, an instance of the inspection axiom of ${{\mathsf{LP}}_{h}}$.
[**Case** ]{}when $D = {\left[t_1\right]\!}_1 B \wedge \dots \wedge {\left[t_{h}\right]\!}_{h}B \rightarrow {\left[{\left\langle t_1, \dots, t_{h}\right\rangle}\right]\!}_{\mathsf{E}}B$ is an instance of the tupling axiom. We distinguish the following possibilities:
1. At least one of the $t_i$’s contains a subterm from ${\textnormal{Tm}}_{\mathsf{E}}\cup {\textnormal{Tm}}_{\mathsf{C}}$. Then $D^\times$ has the form $C_1 \wedge \dots \wedge C_{h}\rightarrow B^\times$ with at least one $C_i = B^\times$ and is, therefore, a propositional tautology.
2. None of the $t_i$’s contains a subterm from ${\textnormal{Tm}}_{\mathsf{E}}\cup {\textnormal{Tm}}_{\mathsf{C}}$. Then $D^\times$ has the form ${\left[t_1\right]\!}_1 B^\times \wedge \dots \wedge {\left[t_{h}\right]\!}_{h}B^\times \rightarrow B^\times$, which can be derived in ${{\mathsf{LP}}_{h}}({\mathcal{CS}}^\times)$ from the reflexivity axiom. This is another case when translation $\times$ does not map an axiom of ${{{\mathsf{LP}}_{h}}^{{\mathsf{C}}}}$ to an axiom of ${{\mathsf{LP}}_{h}}$.
[**Case** ]{}when $D$ is an instance of the projection axiom ${\left[t\right]\!}_{\mathsf{E}}B \rightarrow {\left[{\pi}_i t\right]\!}_i B$ or of the co-closure axiom, i.e., ${\left[t\right]\!}_{\mathsf{C}}B \rightarrow {\left[{{\mathsf{ccl}}_1(t)}\right]\!}_{\mathsf{E}}B$ or ${\left[t\right]\!}_{\mathsf{C}}B \rightarrow {\left[{{\mathsf{ccl}}_2(t)}\right]\!}_{\mathsf{E}}{\left[t\right]\!}_{\mathsf{C}}B$. Then $D^\times$ is the propositional tautology $B^\times \rightarrow B^\times$.
[**Case** ]{}when $D = B \wedge {\left[t\right]\!}_{\mathsf{C}}(B \rightarrow {\left[s\right]\!}_{\mathsf{E}}B) \rightarrow {\left[{\mathsf{ind}(t,s)}\right]\!}_{\mathsf{C}}B$ is an instance of the induction axiom. Then $D^\times$ is $B^\times \wedge (B^\times \rightarrow B^\times) \rightarrow B^\times$, a propositional tautology.
[**Case** ]{}when $D$ is derived by modus ponens is trivial.
[**Case** ]{}when $D$ is ${\left[c\right]\!}_{\circledast}B \in {\mathcal{CS}}$. Then $D^\times$ is either $B^\times$ or ${\left[c\right]\!}_{i}B^\times$. In the former case, $B$ is an axiom of ${{{\mathsf{LP}}_{h}}^{{\mathsf{C}}}}$, and hence $B^\times$ is derivable in ${{\mathsf{LP}}_{h}}({\mathcal{CS}}^\times)$, as shown above; in the latter case, ${\left[c\right]\!}_{i}B^\times \in {\mathcal{CS}}^\times$.
Note that ${\mathcal{CS}}^\times$ need not, in general, be a constant specification for ${{\mathsf{LP}}_{h}}$ because, as noted above, for an axiom $D$ of ${{{\mathsf{LP}}_{h}}^{{\mathsf{C}}}}$, its image $D^\times$ is not always an axiom of ${{\mathsf{LP}}_{h}}$. To ensure that ${\mathcal{CS}}^\times$ is a proper constant specification, $(A \rightarrow B) \rightarrow ({\left[s\right]\!}_{i}A \rightarrow B)$ and ${\left[t_1\right]\!}_1 A \wedge \dots \wedge {\left[t_{h}\right]\!}_{h}A \rightarrow A$ have to be made axioms of ${{\mathsf{LP}}_{h}}$. Another option is to use Fitting’s concept of *embedding* one justification logic into another, which involves replacing constants in $D$ with more complicated terms in $D^\times$ (see [@Fit08AMAI] for details).
Forgetful Projection and a Word on Realization {#sect:real}
==============================================
Most justification logics are introduced as explicit counterparts to particular modal logics in the strict sense described in Sect. \[sec:intro\]. Although the realization theorem for ${{{\mathsf{LP}}_{h}}^{{\mathsf{C}}}}$ remains an open problem, in this section we prove that each theorem of our logic ${{{\mathsf{LP}}_{h}}^{{\mathsf{C}}}}$ states a valid modal fact if all terms are replaced with the corresponding modalities, which is one direction of the realization theorem. We also discuss approaches to the harder opposite direction.
We start with recalling the modal language of common knowledge. Modal formulae are defined by the following grammar $$A \coloncolonequals P_j {\mathrel{\mid}}\neg A {\mathrel{\mid}}(A \wedge A) {\mathrel{\mid}}(A \vee A) {\mathrel{\mid}}(A \rightarrow A) {\mathrel{\mid}}\Box_i A {\mathrel{\mid}}{\mathsf{E}}A {\mathrel{\mid}}{\mathsf{C}}A
\rlap{\enspace,}$$ where $P_j \in {\textnormal{Prop}}$. The set of all modal formulae is denoted by ${{\textnormal{Fm}}_{{{{\mathsf{S4}}_{h}}^{{\mathsf{C}}}}}}$.
The Hilbert system ${{{\mathsf{S4}}_{h}}^{{\mathsf{C}}}}$ [@HM95] is given by the modal axioms of ${\mathsf{S4}}$ for individual agents, by the necessitation rule for $\Box_1, \dots, \Box_{h}$, and ${\mathsf{C}}$, by modus ponens, and by the axioms $$\begin{gathered}
{\mathsf{C}}(A \rightarrow B) \rightarrow ({\mathsf{C}}A \rightarrow {\mathsf{C}}B),
\qquad
{\mathsf{C}}A \rightarrow A,
\qquad
{\mathsf{E}}A \leftrightarrow \Box_1 A \wedge \dots \wedge \Box_{h}A, \\
A \wedge {\mathsf{C}}(A \rightarrow {\mathsf{E}}A) \rightarrow {\mathsf{C}}A,
\qquad\qquad
{\mathsf{C}}A \rightarrow {\mathsf{E}}(A \wedge {\mathsf{C}}A).\end{gathered}$$
\[def:forgetfulprojection\] The mapping ${\circ}\colon {{\textnormal{Fm}}_{{{{\mathsf{LP}}_{h}}^{{\mathsf{C}}}}}}\rightarrow {{\textnormal{Fm}}_{{{{\mathsf{S4}}_{h}}^{{\mathsf{C}}}}}}$ is defined as follows:
1. $P^{\circ}\colonequals P$ for propositional variables $P \in {\textnormal{Prop}}$;
2. ${\circ}$ commutes with propositional connectives;
3. $({\left[t\right]\!}_i A)^{\circ}\colonequals \Box_i A^{\circ}$;
4. $({\left[t\right]\!}_{\mathsf{E}}A)^{\circ}\colonequals {\mathsf{E}}A^{\circ}$;
5. $({\left[t\right]\!}_{\mathsf{C}}A)^{\circ}\colonequals {\mathsf{C}}A^{\circ}$.
\[l:fp:1\] Let ${\mathcal{CS}}$ be any constant specification. For any formula $A \in {{\textnormal{Fm}}_{{{{\mathsf{LP}}_{h}}^{{\mathsf{C}}}}}}$, if ${{{\mathsf{LP}}_{h}}^{{\mathsf{C}}}}({\mathcal{CS}}) {\vdash}A$, then ${{{\mathsf{S4}}_{h}}^{{\mathsf{C}}}}{\vdash}A^{\circ}$.
The proof is by easy induction on the derivation of $A$.
A realization is a mapping $r \colon {{\textnormal{Fm}}_{{{{\mathsf{S4}}_{h}}^{{\mathsf{C}}}}}}\to {{\textnormal{Fm}}_{{{{\mathsf{LP}}_{h}}^{{\mathsf{C}}}}}}$ such that $(r(A))^{\circ}= A$. We usually write $A^r$ instead of $r(A)$.
We can think of a realization as a function that replaces occurrences of modal operators (including ${\mathsf{E}}$ and ${\mathsf{C}}$) with evidence terms of the corresponding type. The problem of realization for a given pure ${\mathsf{C}}$-axiomatically appropriate constant specification ${\mathcal{CS}}$ can be stated as follows: $$\text{Is there a realization~$r$ such that
${{{\mathsf{LP}}_{h}}^{{\mathsf{C}}}}({\mathcal{CS}}) {\vdash}A^r$
for any theorem~$A$ of~${{{\mathsf{S4}}_{h}}^{{\mathsf{C}}}}$?}$$ A positive answer to this question would constitute the harder direction of the realization theorem, which is often demonstrated using induction on a cut-free sequent proof of the modal formula.
Cut-free systems for ${{{\mathsf{S4}}_{h}}^{{\mathsf{C}}}}$ are presented in [@aj05] and [@bs09]. They are based on an infinitary $\omega$-rule of the form $${\displaystyle{\frac{{\mathsf{E}}^m A, \Gamma \quad \text{for all } m \geq 1}{{\mathsf{C}}A, \Gamma}}} \qquad (\omega).$$ However, realization of such a rule meets with serious difficulties in reaching uniformity among the realizations of the approximants ${\mathsf{E}}^m A$.
A finitary cut-free system is obtained in [@jks07] by finitizing this $\omega$-rule via the finite model property. Unfortunately, the “somewhat unusual” structural properties of the resulting system (see discussion in [@jks07]) make it hard to use it for realization.
The non-constructive, semantic realization method from [@Fit05APAL] cannot be applied directly because of the non-standard behavior of the canonical model (see Remark \[nonstandardbehaviour\]).
Perhaps the infinitary system presented in [@BucKuzStu09M4M], which is finitely branching but admits infinite branches, can help in proving the realization theorem for ${{{\mathsf{LP}}_{h}}^{{\mathsf{C}}}}$. For now this remains work in progress.
Coordinated attack {#s:attack:1}
==================
To illustrate our logic, we will now analyze the coordinated attack problem along the lines of [@FHMV95], where additional references can be found. Let us briefly recall this classical problem. Suppose two divisions of an army, located in different places, are about to attack an enemy. They have some means of communication, but these may be unreliable, and the only way to secure a victory is to attack simultaneously. How should generals $G$ and $H$ who command the two divisions coordinate their attacks? Of course, general $G$ could send a message $m_1^G$ with the time of attack to general $H$. Let us use the proposition ${{\text{\emph{del}}}}$ to denote the fact that the message with the time of attack has been delivered. If the generals trust the authenticity of the message, say because of a signature, the message itself can be taken as evidence that it has been delivered. So general $H$, upon receiving the message, knows the time of attack, i.e., ${\left[m_1^G\right]\!}_H {{\text{\emph{del}}}}$. However, since communication is unreliable, $G$ considers it possible that his message has not been delivered. But if general $H$ sends an acknowledgment $m_2^H$, he in turn cannot be sure whether the acknowledgment has reached $G$, which prompts yet another acknowledgment $m_3^G$ by general $G$, and so on.
In fact, common knowledge of ${{\text{\emph{del}}}}$ is a necessary condition for the attack. Indeed, it is reasonable to assume it to be common knowledge between the generals that they should only attack simultaneously or not attack at all, i.e., that they attack only if both know that they attack: ${\left[t\right]\!}_{\mathsf{C}}({\text{\emph{att}}}\rightarrow {\left[s\right]\!}_{\mathsf{E}}{\text{\emph{att}}})$ for some terms $s$ and $t$. Thus, by the induction axiom, we get ${\text{\emph{att}}}\rightarrow {\left[{\mathsf{ind}(t,s)}\right]\!}_{\mathsf{C}}{\text{\emph{att}}}$. Another reasonable assumption is that it is common knowledge that neither general attacks unless the message with the time of attack has been delivered: ${\left[r\right]\!}_{\mathsf{C}}({\text{\emph{att}}}\rightarrow {\text{\emph{del}}})$ for some term $r$. Using the application axiom, we obtain ${\text{\emph{att}}}\rightarrow {\left[r \cdot {\mathsf{ind}(t,s)}\right]\!}_{\mathsf{C}}{\text{\emph{del}}}$.
We now show that common knowledge of ${{\text{\emph{del}}}}$ cannot be achieved and that, therefore, no attack will take place, no matter how many messages and acknowledgments $m_1^G$, $m_2^H$, $m_3^G$, …are sent by the generals even if all the messages are successfully delivered.
In the classical modeling without evidence, the reason is that the sender of the last message always considers the possibility that his last message, say $m_{2k}^H$, has not been delivered. To give a flavor of the argument carried out in detail in [@FHMV95], we provide a countermodel where $m_2^H$ is the last message, it has been delivered, but $H$ is unsure of that, i.e., ${\left[m_1^G\right]\!}_H {{{\text{\emph{del}}}}}$, ${\left[m_2^H\right]\!}_G {{\left[m_1^G\right]\!}_H {{{\text{\emph{del}}}}}}$, but $\lnot {\left[s\right]\!}_H {\left[m_2^H\right]\!}_G {{\left[m_1^G\right]\!}_H {{{\text{\emph{del}}}}}}$ for all terms $s$. Indeed, consider the model ${\mathcal{M}}$ with $W \colonequals \{ 0, 1, 2, 3 \}$, ${{\nu}}({{\text{\emph{del}}}}) \colonequals \{ 0, 1, 2 \}$, $R_G$ being the reflexive closure of $\{ (1, 2) \}$, $R_H$ being the reflexive closure of $\{ (0, 1), (2, 3) \}$, and any evidence function ${\mathcal{E}}$ such that ${{\text{\emph{del}}}}\in {\mathcal{E}}_H (0, m_{1}^G)$ and ${\left[m_1^G\right]\!}_H {{{\text{\emph{del}}}}} \in {\mathcal{E}}_G (0, m_{2}^H)$. Then, whatever ${\mathcal{E}}_{\mathsf{C}}$ is, we have $\mathcal{M}, 0 \nVdash {\left[s\right]\!}_H {\left[m_2^H\right]\!}_G {{\left[m_1^G\right]\!}_H {{{\text{\emph{del}}}}}}$ and $\mathcal{M}, 0 \nVdash {\left[t\right]\!}_{\mathsf{C}}{{\text{\emph{del}}}}$ for any $s$ and $t$ because $\mathcal{M}, 3 \nVdash {{\text{\emph{del}}}}$.
In our models with explicit evidence, there is an alternative possibility for the lack of knowledge: the absence of evidence. For example, $G$ may receive the acknowledgment $m_2^H$ but not consider it to be evidence for ${\left[m_1^G\right]\!}_H {{\text{\emph{del}}}}$ because the signature of $H$ is missing.
We now demonstrate that common knowledge of the time of attack cannot emerge, basing the argument solely on the lack of common knowledge evidence. A corresponding M-model ${\mathcal{M}}= (W, R, {\mathcal{E}}, {\nu})$ is obtained as follows: $W \colonequals \{ w \}$, $R_i \colonequals \{ (w, w) \}$, ${{\nu}}({{\text{\emph{del}}}}) \colonequals \{ w \}$, and ${{\mathcal{E}}}$ is the minimal evidence function such that ${{\text{\emph{del}}}}\in {\mathcal{E}}_H (w, m_{1}^G)$ and ${\left[m_1^G\right]\!}_H{{{\text{\emph{del}}}}} \in {\mathcal{E}}_G (w, m_{2}^H)$. In this model $M, w \nVdash {\left[t\right]\!}_{\mathsf{C}}{{\text{\emph{del}}}}$ for any evidence term $t$ because ${{\text{\emph{del}}}}\notin {\mathcal{E}}_{\mathsf{C}}(w, t)$ for any $t$. To show the latter statement, note that for any term $t$, by Lemma \[l:fp:1\], $$\label{eq:coordatt}
\nvdash {\left[m_1^G\right]\!}_H {{\text{\emph{del}}}}\wedge {\left[m_2^H\right]\!}_G {\left[m_1^G\right]\!}_H{{{\text{\emph{del}}}}} \rightarrow {\left[t\right]\!}_{\mathsf{C}}{{\text{\emph{del}}}}$$ because $${{{\mathsf{S4}}_{h}}^{{\mathsf{C}}}}\nvdash \Box_H {{\text{\emph{del}}}}\wedge \Box_G \Box_H {{\text{\emph{del}}}}\rightarrow {\mathsf{C}}{{\text{\emph{del}}}}\rlap{\enspace,}$$ which is easy to demonstrate. Thus, the negation of the formula from is satisfiable, and for each $t$ there is a world $w_t$ in the canonical AF-model with evidence function ${{\mathcal{E}}^{\text{can}}}$ such that ${{\text{\emph{del}}}}\in {{\mathcal{E}}^{\text{can}}}_H (w_t, m_1^G)$ and ${\left[m_1^G\right]\!}_H {{\text{\emph{del}}}}\in {{\mathcal{E}}^{\text{can}}}_G (w_t, m_2^H)$, but by the Truth Lemma \[truthlemma\], ${{\text{\emph{del}}}}\notin {{\mathcal{E}}^{\text{can}}}_{\mathsf{C}}(w_t, t)$. Since ${{\mathcal{E}}^{\text{can}}}\upharpoonright (\{ w_t \} \times {\textnormal{Tm}})$ satisfies all the closure conditions, minimality of ${{\mathcal{E}}}$ implies that ${{\mathcal{E}}}_{\mathsf{C}}(w, s) \subseteq {{\mathcal{E}}^{\text{can}}}_{\mathsf{C}}(w_t, s)$ for any term $s$. In particular, ${{\text{\emph{del}}}}\notin {\mathcal{E}}_{\mathsf{C}}(w, t)$ for any term $t$.
Conclusions {#sect:concl}
===========
We have presented an explicit evidence system ${{{\mathsf{LP}}_{h}}^{{\mathsf{C}}}}$ with common knowledge, which is a conservative extension of the multi-agent explicit evidence logic ${{\mathsf{LP}}_{h}}$. The major open problem at the moment remains proving the realization theorem, one direction of which we have demonstrated.
Our analysis of the coordinated attack problem in the language of ${{{\mathsf{LP}}_{h}}^{{\mathsf{C}}}}$ shows that access to explicit evidence creates more alternatives than the classical modal approach. In particular, the lack of knowledge can occur either because messages are not delivered or because evidence of authenticity is missing.
We have mostly concentrated on the study of ${\mathsf{C}}$-axiomatically appropriate constant specifications. For modeling distributed systems with different reasoning capabilities of agents, it is also interesting to consider ${i}$-axiomatic appropriate, ${\mathsf{E}}$-axiomatic appropriate, and mixed constant specifications, where only certain aspects of reasoning are common knowledge.
We established soundness and completeness with respect to AF-models and singleton M-models. Can other semantics for justification logics such as (arithmetical) provability semantics [@Art95TR; @Art01BSL] and game semantics [@Ren09ICnonote] be adapted to ${{{\mathsf{LP}}_{h}}^{{\mathsf{C}}}}$?
There are further interesting questions: Is ${{{\mathsf{LP}}_{h}}^{{\mathsf{C}}}}$ decidable and, if yes, what is its complexity compared to that of ${{{\mathsf{S4}}_{h}}^{{\mathsf{C}}}}$? How robust is our treatment of common knowledge if the individual modalities are taken to be of type $\textsf{K}$, $\textsf{K5}$, etc.?
[^1]: The first and second authors are supported by Swiss National Science Foundation grant 200021–117699.
[^2]: Minimality here is understood in the sense of the minimal transfer of evidence.
|
---
abstract: 'Experiments have been performed using microscopic beads to probe the small scale mechanics of actin solutions. We show that that there are a number of regimes possible as a function of the size of the probing particle. In certain cases we argue that the quasi-static response resembles a smectic crystal rather than an isotropic solid, implying an anomalous scaling of the mechanical response of actin solutions as a function of the size of the probing particles.'
address: 'PCT, ESPCI, 10 rue Vauquelin, 75231 Paris Cedex 05, France. '
author:
- 'A. C. Maggs'
title: 'On micro-bead mechanics with actin filaments'
---
The mechanics and rheology of actin filaments are a beautiful model system for the study of the dynamics and mechanics of semi-dilute polymers [@kas; @kas2]. They are characterized by length scales which are easily accessible with optical techniques allowing the detailed study of phenomena such as tube dynamics. However, the macroscopic rheology of these systems has been hard to master from the experimental point of view. Difficulties of purification and sample preparation lead to orders of magnitude variations in such fundamental objects such as the value of the plateau modulus [@pollard; @zaner; @sackmanntube; @sackmanntube2; @janmey], the standard measure of the response of an entangled polymer solution to external perturbations.
To get around these problems of macroscopic sample preparation and also to probe the local viscoelastic behaviour of these materials a number of experimental groups have started using small, colloidal beads to study the local mechanics of these materials [@zaner2; @sackmannbead; @amblard; @schmidt1; @fred; @weitz]. One either pulls on the particles using super-paramagnetic beads in a magnetic field, or one simply observes the fluctuations of the particles undergoing Brownian motion. In this letter I shall try to attack the problem as to what exactly one measures in these experiments. In particular how large do these particles have to be in order to measure a macroscopic elastic modulus and when do we expect to be sensitive to the individual filament properties?
In contrast with flexible polymers solutions there are two principal length scales present in a semi-dilute solution of actin: the mesh size and the persistence length. Naive application of scaling ideas thus becomes a highly ambiguous exercise because an arbitrarily large number of intermediated lengths can be created by considering $\xi^{1-\alpha} l_p^{\alpha}$ with $\xi$ the mesh size and $l_p$ the persistence length. This ambiguity in lengths also translates into an ambiguity in the plateau modulus which can be expressed as $k_B T$ per characteristic volume. As an example of this difficulty we might quote two attempts to calculate the modulus in actin solutions with scaling approaches [@fredmod; @kroy] where two completely different results are found due in part due to this problem. Indeed this proliferation of lengths is already known for the tube geometry where one finds both $\alpha =+1/5$ and $\alpha = -1/5$, [@semenov; @semenovbis; @odjik]. We shall show in this article that a new intermediate scale with $\alpha=3/5$ becomes crucial in the understanding of the elasticity of actin solutions at length scales probed with micrometer sized beads. At these scales we show that the elastic response is highly anisotropic and resembles that of a smectic, with anomalous penetration of the response into the sample and unusual scaling of the response with the size of the probing particle.
Note that in this letter I am interested in the low frequency mechanics and thus I exclude from the discussion high frequency fluctuation measurements (up to $20KHz$) which have been recently performed [@fred] and am interested in a quasi-static regime between $10^{-3}Hz$ and $10Hz$. The reason for considering this range scale will become clear during the discussion.
A coherent picture of the large scale mechanics of non-crosslinked actin solutions is now available. The actin system is usually polymerized [@schmidt1] in conditions such that the mean distance between filaments, $\xi$ is between $0.3\mu$ and $1\mu$. $\xi$ can be linked with the concentration of monomers $c$ by noting that $\xi\sim1/\sqrt{c d}$ with $d$ the size of actin monomers. A useful geometric quantity is the length of filament per unit volume $\rho
\sim 1/\xi^2$. The filament is characterized by its persistence length $l_p$ which is close to $15\mu$ [@gittes]. For a single weakly bent filaments the energy of a configuration is given by [@landau] $$E = k_B T l_p/2 \int (\partial_s^2 {\bf r_{\perp}}(s))^2
ds \label{landau}$$ where $ {\bf r_{\perp}}(s)$ is the transverse fluctuation of the filament about its equilibrium shape.
In a manner which is familiar from flexible polymers the individual filaments are confined to a tube whose diameter scales as $\xi^{6/5} /
l_p^{1/5}$ and the filament is confined to the tube by collisions between the filament and its neighbors every $l_e \sim \xi^{4/5} l_p
^{1/5}$ [@semenov; @semenovbis]. $l_e$ is is in some ways equivalent to the entanglement length in the Doi-Edwards tube model [@doi; @degennes; @morse]. The long time dynamics and mechanics are dominated by the reptation of filaments along their tubes [@sackmanntube; @morse; @me]. This process has a characteristic time, the reptation time, which defines the time scale beyond which the sample behaves like a viscous fluid (rather than an elastic solid) and can be as long as several hours [@sackmanntube]. Under macroscopic shear the longitudinal stresses in a filament relax relatively rapidly [@me] leaving a residual contribution to the free energy which comes from the modification of the free energy of confinement of the filament in its tube. A simple argument for this free energy is to count $k_B T$ per collision of the tube with the filament. Thus the macroscopic modulus varies as $G\sim\rho k_BT/l_e \sim
c^{1.4}/l_p^{1/5}$ as confirmed by an explicit calculation [@morse; @herve].
This picture of filaments confined to a tube is only true on time scales that are long enough for the filament to dynamically sample fluctuations on the scale of $l_e$. This time, which is determined by the bending elasticity of the filaments varies as $\tau_e \sim \eta
l_e^4/l_p k_B T \sim 10Hz$ [@herve]. This is our reason for restricting our treatment to lower frequencies, at higher frequencies one is presumably sensitive to individual filament dynamics (coupled by hydrodynamics) rather than the collective, entangled, modes that interest us in this letter. For frequencies lower than the inverse reptation time (ie frequencies comparable to $10^{-3}Hz$) the sample behaves as a fluid and the bead moves freely as filaments slide out of the way of the particles.
Before passing to the problem of the behavior of actin solutions we shall revise a Peierls like argument from which we can deduce the basic scaling behaviour of a normal elastic solid. We shall then adapt this argument to the case of semiflexible filaments: Consider a bead of radius $R$ embedded in an elastic medium in $d$ dimensions. If we pull on the particle with a force $f$ we can make the following variational ansatz in order to find the minimum energy configuration. Let us assume that the material is disturbed over a distance $l$ from the bead then the elastic energy, will scale in the following manner $$E_{var} \sim G \int (\nabla a)^2 dV
\label{evar}$$ where $a$ is an amplitude of displacement, $G$ an elastic constant and the integral is over the variational volume $V \sim l^d$. This scales as $$\label{evar2}
E_{var} \sim G (a/l)^2 l^d$$ We see that in less than two dimension an arbitrarily small force is able to displace the bead large distances because $E_{var}$ can be made small by increasing the variational parameter $l$. In three dimensions, however, the energy diverges with $l$ and has a lower bound for small $l$ due to the short wavelength cutoff coming from the finite size of the bead. Thus the minimum energy is found for $l \sim R$ and we deduce that $E_{var} \approx G a^2 R$. At constant force the displacement scales inversely with the bead size, $$\label{inverse}
a \approx f/ G R$$ A full calculation of the response of an isotropic viscoelastic material has recently been performed and confirms this simple scaling argument [@fred].
We see that there is an anisotropy in the problem coming from the direction in which we apply the force $f$, and we should worry that the volume excited is not spherical as has been assumed in the argument. Let us perform a slightly more elaborate variational treatment where we assume that the volume $V$ is characterized by an disk of dimensions $l \times l \times D$ where the particle excites modes of wavelength $l$ which penetrate $D$ into the sample in the direction of $f$. In this case our estimate for $E_{var}$ is $$\label{evar3}
E_{var} \sim (l^2 D) G ( ({{a}/ {l}} )^2 + ({{a}/ {D}})^2)$$ Where $ a/l$ and $a/D$ are the estimates of the components of the strain tensor in the material. Taking $D$ as a variational parameter one sees that $D \sim l$ and the problem reduces to that considered above. This is in fact a crude statement of the principle of St. Vernet that a force on a body with a wavelength $l$ decays into the body over the same length scale, which is a elementary property of periodic harmonic functions in three dimensions.
How must this argument be modified in the actin system? Experiments are performed with bead which vary in size from $.1 \mu $ and $10
\mu$. The smallest beads pass between the filaments and diffuse almost freely [@schmidt1]; they will not concern us any further. Are we able to use continuum elastic arguments (like that above) to deduce the experimental stress stain relationships? We now argue that in actin solutions there are now two contributions to the variational energy $E_{var}$. For large beads the normal continuum elasticity (summarized above) dominates, for smaller beads however a new, and novel elastic response is found: Consider a volume $V$ distorted by a force on a particle of size $R$. Again we take this volume as anisotropic with dimensions $ l \times l
\times D$. In this volume the filaments which traverse the volume bend with a wavelength $l$ and there is a bending contribution to the total energy, coming from eq. (\[landau\]) which varies as $$\label{e1}
E_1 \sim (l^2 D) (a^2 k_B T l_p/l^4) \rho$$ The three multiplicative factors are respectively the volume excited, the bending energy per unit length of filament and the filament density within the volume. $a$ is again the typical amplitude of the excitation in the volume. To this bending contribution one must add the equivalent of $E_{var}$. When we impose the bending on the volume $V$ there is also a variation in the geometry of the confining tubes. For instance in the direction of $f$ the tubes are compressed by a factor comparable to $a/D$. Thus there is thus a contribution to the energy coming from the macroscopic bulk modulus of the form $$\label{e2}
E_2 \sim (l^2 D) ( (a/l)^2 + (a/D)^2) (\rho k_B T/l_e)$$ where we have again respectively the volume, the square elastic stress and the macroscopic elastic modulus. We can now optimize $E_1 + E_2$ by minimizing over $D$. However, we first notice that there are two term linear in $D$ and that depending on the value of $l$ one or the other will dominate. If we look at wavelengths $$\label{lc}
l< l_c = \sqrt{l_e l_p} \sim \xi^{2/5} l_p^{3/5}$$ the contribution from $E_1$ dominates over that from $E_2$. When $l >
l_c$ the second contribution dominates. We conclude that there is an important new length scale in the problem. When we look at excitations with wavelengths greater than $l_c$ the two contributions in $E_2$ are going to dominate the elasticity and we are back to the case of normal continuum elastic theory. However in the short wavelength limit $l < \sqrt{l_e l_p}$ we find that the elastic energy is given by $$\label{energy}
E_{eff} \sim a^2 \rho (l^2 D) (k_B T/D^2 l_e + l_p/l^4)$$ Minimizing the energy over $D$ we find several surprising results. Firstly $$\label{aniso}
l^2 \sim D \sqrt{l_e l_p}$$ the theorem of St. Vernet does not apply. Secondly substituting eq. (\[aniso\]) in (\[energy\]) gives $$\label{efinal}
E_{eff} \sim \rho a^2k_B T \sqrt{ {{l_p} \over {l_e}}}$$ which should be compared with the corresponding result for a normal solid eq. (\[evar2\]). From eq. (\[aniso\]) we see that the volume excited scales in an anisotropic fashion with the wavelength, quite unlike normal elastic solids. One is reminded of the penetration of excitation into a smectic liquid crystal with an effective energy for fluctuations of the form $ E \sim K \int[
(\partial_x^2 u)^2 + (\partial_y^2 u)^2 + \beta (\partial_z u)^2 ] dV $ where the $z$ axis is defined by the direction of application of the force.
The length scale $l$ is absent in the energy (\[efinal\]) which has not been minimized over the wavelength; this is analogous to the case of a normal elastic material eq. (\[evar2\]) in two rather than three dimensions. It suggests that we are in the lower critical dimension for the problem (at least in a certain range of wavelengths) and that a fuller treatment will bring out logarithmic corrections to our picture. We also deduce, since eq. (\[efinal\]) is independent of $l$, that the amplitude of displacement of a particle should be independent of its size, in contrast to the dependence discussed above eq. (\[inverse\]): We are no longer dominated by the short wavelength cutoff in the energy integral eq. (\[evar\]) despite being in three dimensions
Finally we note that for coherency in this picture we require that the depth of penetration of the excitation into the sample $D$ be greater than the distance between the filaments $\xi$, otherwise a continuum description as used here must break down. This implies that $l>
l_l=\sqrt{\xi\sqrt{ l_e l_p}} = \xi^{7/10} l_p^{3/10}$. On wavelengths shorter than this, one is presumably sensitive to filaments directly in contact with the probing particle and do not feel the three dimensional nature of the sample. Substituting typical values for material constants, $\xi\sim.5\mu$, $l_p \sim 15\mu$ we find that $l_e \sim 1 \mu$. The crossover length scale $ l_c \sim 4 \mu$. The short wavelength cutoff $l_l \sim 1.5
\mu$. We thus expect the following series of crossovers as a function of probing wavelength. (a) For $\xi< l < l_l$ one probes the bending of individual filaments. (b) For $l_l < l < l_c$ collective excitations of the solution become important with anomalous penetration of the excitation into the sample. (c) For $l> l_c$ the elasticity becomes isotropic. These crossovers are too closely spaced to be experimentally studied in great detail, however we conclude that to measure a valid macroscopic response function particle sizes should be substantially greater than $l_c=4\mu$.
Until now we have only considered the low frequency response of a sample, that is for times long enough that all longitudinal stresses have relaxed along the tube. It has been shown [@me] that one expects two plateau moduli as a function of frequency. The low frequency plateau used in the above discussion comes from variation in tube geometry under sample shear. The second much larger contribution which dominates at higher frequencies comes from coupling of the shear to the longitudinal density fluctuations of the filament in its tube. Can we see the crossover between the low frequency and high frequency behaviour with micro-bead techniques? This question is difficult to answer, the static approach used above is not adapted to answer this dynamic question however we can certainly expect that the frequency of crossover between the two regimes will vary with the bead size.
The regime of the high plateau in macroscopic rheology is delimited by the two times $\tau_e \sim 0.1s$ and $ \tau_e (l_p/l_e)^2 \sim 10s$. This second time is the time needed for excitations to diffuse a distance $l_p$ along the tube. It is important because macroscopic shear produces density fluctuations along the tube which are coherent over a distance $l_p$. When we excite a sample with a wavelength $l$, which is smaller the $l_p$, we expect that the window of times for the observation of this high plateau is reduced to the interval between $\tau_e$ and $\tau_e (l/l_e)^2$. For the smallest beads this high second plateau should almost completely disappear. Even with larger beads the elastic modulus should be substantially underestimated over certain frequency ranges. More detailed discussion of this regime seems to be difficult without a detailed [*dynamic*]{} theory of the coupling of the bend and longitudinal degrees of freedom.
To conclude actin mechanics shows a quite rich series of crossover in the response function $G(q,\omega)$. We have simple arguments for the wavevector dependence of this function at frequencies between $10^{-3}Hz$ and $10Hz$. Further work requires a full dynamic theory of the coupling between bending and density fluctuations. $G(q,\omega)$ is more complicated than might be expected; recent experiments which interpret elastic stress propagation in terms of an isotropic elastic theory characterized by an elastic constant and Poisson ratio [@sackmannpoisson] may be missing some interesting physics on length scales smaller than $4\mu$.
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ibvs2.sty
Introduction
============
The bright globular cluster M5 (NGC 5904) has been the subject of many variable star searches for more than a hundred years. The first variables were discovered by Bailey (1902). The catalogue of variable stars in globular clusters (CVSGC; Clement et al. 2001) lists 169 variables, mostly of the RR Lyrae type, with 5 SX Phe stars, one W Virginis star (CW), one RV Tau, one (possibly two) eclipsing binaries, and one U Gem type star. However, there are also a number of uncertain classifications and some variables have an unknown type, or it is not even clear if they are truly variable. A new study of the variable stars in M5 is therefore pertinent.
As part of our program of CCD time-series observations of variable star populations in globular clusters (GC), we performed CCD $V$ and $I$ photometry of the globular cluster M5. Difference image analysis (DIA) has proven to be very efficient in identifying variable stars even in the crowded central regions of GCs (e.g. Arellano Ferro et al. 2013 and references therein). Exploration of our collection of light curves of all stars in the field of our images down to $V \sim$ 18.5 mag allowed us to identify twelve variables not previously detected; one SX Phe and eleven semi-regular variables (SR). In the present note, we report on their identifications, equatorial coordinates, ephemerides, and light curves. We argue that the known variable V155, previously classified as RRc, is in fact a contact eclipsing binary or EW. Furthermore, we have explored the light curves of a group of stars whose variability has not been confirmed and that are marked as probable non-variables in the CVSGC. Finally, we offer detailed identifications for some of the known variables in crowded regions that were misidentified in previous studies. We shall also address the cases of the cataclysmic variable or U Gem type V101 and of the variable blue straggler V159.
Observations and reductions
===========================
The observations were acquired on 11 nights between 29th February 2012 and 9th April 2014 with the 2.0m telescope of the Hanle Observatory, India. A total of 385 and 384 images in $V$ and $I$, respectively, were obtained. Image data were calibrated using bias and flat-field correction procedures. We used DIA to extract high-precision time-series photometry employing the [DanDIA]{} pipeline for the data reduction process (Bramich et al. 2013), which includes an algorithm that models the convolution kernel matching the PSF of a pair of images of the same field as a discrete pixel array (Bramich 2008). We have also applied a post-calibration method developed by Bramich & Freudling (2012) which determines appropriate per-image magnitude offsets to correct for errors in the fitted value of the photometric scale factor $p$. We derived offsets of the order of $\sim0.02$ and $\sim0.03$ mag in $V$ and $I$, respectively. The instrumental magnitudes are calculated via the difference flux, the reference flux and the photometric scale factor by the equation;
$$m_{\mbox{\scriptsize ins}}(t) = 25.0 - 2.5 \log \left[f_{\mbox{\scriptsize ref}} +
\frac{f_{\mbox{\scriptsize diff}}(t)}{p(t)}
\right].
\label{eqn:mag}$$
The difference fluxes $f_{\mbox{\scriptsize diff}}$ are measured by scaling the known PSF to the difference images at the position of each star. Since the constant stars have been fully subtracted in the difference images, the difference fluxes for the variables are very precise. The reference fluxes $f_{\mbox{\scriptsize ref}}$ are, however, measured on the reference image by PSF fitting and they have the potential to suffer from the usual problems caused by blending. For the variables in the most crowded parts of the reference image, where the probability of blending is high, the brightness of a variable star may be overestimated, and its amplitude underestimated (see Section 2.3 of Bramich et al. 2011 for a more in-depth discussion of the caveats of DIA).
The instrumental magnitudes were transformed to the standard Johnson-Kron-Cousins magnitudes using secondary photometric standards in the field of view (FoV) from Stetson (2000) covering the full range of colours.
All of our $VI$ photometry for the stars discussed in this paper is provided in Table \[tab:vi\_phot\]. Just a small portion of this table is given in the printed version of this paper, while the full table is only available in electronic form.
Exploration of suspected non-variables in the CVSGC
===================================================
In the CVSGC there are 23 stars classified as (probably) non-variables or “constant” (CST, CST? or ?); these are V22, V23, V46, V48, V49, V51, V124, V136, V138, V140, V141, V143-V154. Except for V22 and V141, which are outside of the FoV of our images, we have $VI$ light curves for all of them. We have carried out a quick exploration of the light curves to comment on their possible variability or otherwise.
Var ID $<V>$ rms $V$ local rms Var ID $<V>$ rms $V$ local rms
-------- ------- --------- ----------- -------- ------- --------- -----------
V23 14.40 0.017 0.023 V145 15.26 0.017 0.033
V46 17.93 0.150 0.184 V146 15.65 0.019 0.040
V48 14.24 0.012 0.022 V147 15.12 0.015 0.031
V49 15.78 0.013 0.043 V148 15.62 0.018 0.040
V51 14.06 0.013 0.020 V149 17.75 0.062 0.160
V124 14.88 0.015 0.028 V150 18.08 0.056 0.208
V136 14.93 0.018 0.028 V151 18.02 0.054 0.198
V138 13.17 0.011 0.016 V152 17.72 0.043 0.160
V140 14.76 0.072 0.028 V153 18.05 0.049 0.203
V143 15.42 0.029 0.036 V154 17.93 0.046 0.184
V144 15.41 0.020 0.036
: Mean magnitudes and rms for stars whose variability is not confirmed. Local rms refers to the upper limit of the main rms distribution for a given value of $<V>$, represented by the continuous line in Fig. 2.
Firstly we have identified the 21 stars in our FoV on the colour-magnitude diagram (CMD) of Fig. 1 and the RMS diagram of Fig. 2. All of them are marked with red squares. In the RMS diagram we draw an arbitrarily set line above which all variables seem to fall and hence it can serve as a guide of detectability when judging the variabilty of a given candidate. While true variables are expected to have significantly larger rms values than this upper limit, we note however that non-true variables may lie above that limit if they are near a true variable due to flux contamination (e.g. V140, see below), or that true variables may be found below that limit, particularly those of very small amplitude (e.g. the SX Phe star V164). Thus, individual explorations of the light curves of specific cases is required. In Table 2 we list the mean magnitudes and rms values in the $V$ light curves. In the last column we list the value of the upper limit rms corresponding to the mean magntiude. Except for V140, all the stars listed above fall below the threshold. From the individual explorations we found that V140 does in fact show some variations. However we argue that these are the due to flux contamination by a nearby variable (V175) as it will be discussed in $\S$ \[newvar\]. For the rest, we found no signs of variability confirming their classification as non-variables (or CST) in the CVSGC.
Comments, identification and classification correction, for some previously known variables
===========================================================================================
During the process of variable star identification we noticed that stars V25, V36, V53, V74, V102 and V108 are all very close to a neighbour of similar brightness or much brighter. While checking the finding charts of the discovery papers, we found that their identifications are dubious or definitely wrong, mainly due to the fact that the stars are not resolved in old plates and/or that they are close to the cluster central regions. Here we offer a precise identification and a few comments on each variable. We have confirmed the variable nature of these stars by phasing the light curves of the two candidates and by blinking the difference images. It was also noted that the equatorial coordinates in the CVSGC of the variable V140 point to a non-variable star. We address the cases of V50 whose variability has not been clearly established and of V155 that needs a reclassification. To avoid confusion in future work we include here in Table 3 the correct equatorial coordinates of all these stars. Below we offer a brief comment on individual stars including the U Gem type star V101, and binary V159.
V25 is a very close pair that in our images is heavily blended. In the finding chart of the discovery paper (Bailey 1902), the star looks like a single one. Careful blinking of the difference images makes it clear that the true variable is the western star of the pair, as identified in Fig. 4.
V36 is also called V135 (see CVSGC for M5, 2014 update). The star is incorrectly identified in the chart of Caputo et al. (1999), labelled as V135, as the south-western star of the pair. The RRab variable is actually the north-eastern and brighter star of the pair.
V50 sits on the tip of the RGB. Bailey (1917) suggested a period of 106d that was not confirmed by Oosterhoff (1941) who described the variation as irregular. Our data suggest a period of 107.6d, in good agreement with Bailey’s result. Therefore we classify the star as a semi-regular late-type variable (SRA). Its light curve, phased with the above period, is shown in Fig. 3.
V53 is not resolved in the finding chart of Bailey (1902) and not identified afterwards. The correct variable is the eastern star of the pair.
V74 is not resolved in the finding chart of Bailey (1902) and not identified afterwards. The correct variable is the western star of the pair.
V101 is a cataclysmic variable of the U Gem type. It was discovered by Oosterhoff (1941) who classified it as SS Cyg (or dwarf nova). Two outbursts of amplitude 2.7 mag within 100 days in the $V$ light curve were detected by Kaluzny et al. (1999) who argue in favour of a short duty cycle with a characteristic time of about 3.4 hours. Our $VI$ light curves, displayed in the mosaic of Fig. 3, span 770 days and two outbursts are clearly seen at HJD 2456029.4 and 2456312.5 d reaching 18.5 and 18.0 mag in $V$, and 18.0 and 17.0 mag in $I$ respectively.
V102 The identification chart in Oosterhoff (1941) shows a strong blend close to the saturated central region that prevents an accurate identification. The authentic variable is the SE star of the pair.
V108 The variability of this star was announced by Kadla et al. (1987) (see also Gerachencko 1987). It was identified by Drissen & Shara (1998) in their [*Hubble Space Telescope (HST)*]{} image but mistakenly labelled as V22. The identification of the star by Caputo et al. (1999), now labelled as V108, points to the wrong star to the east of the real variable. We confirm that the variable star is the western star of the pair, in agreement with Drissen & Shara’s (1998) identification.
Var ID Variable type RA(J2000.0) Dec.(J2000.0)
-------- --------------- ------------- ---------------
V25 RRab 15 18 30.98 +02 02 42.5
V36 RRab 15 18 32.66 +02 03 58.9
V50 SRA 15 18 36.04 +02 06 37.8
V53 RRc 15 18 37.92 +02 05 06.8
V74 RRab 15 18 47.19 +02 07 25.7
V101 U Gem 15 18 14.51 +02 05 35.7
V102 RRab 15 18 34.37 +02 04 34.4
V108 RRc 15 18 33.79 +02 04 47.0
V140 CST 15 18 36.18 +02 05 13.2
V155 EW 15 18 33.40 +02 05 12.2
V159 E 15 18 32.88 +02 04 36.5
: Known variables with corrected classifications, equatorial coordinates and identifications in Fig. 4.
V140 is identified by Caputo et al. (1999) but the equatorial coordinates given in the CVSGC have a typo error producing some confusion in the identification and in the variable nature of the star which is classified as a probable non-variable. The star pointed to by the CVSGC at 15:18:36.18; +02:03:13.1 is not variable. The correct coordinates of the star identified by Caputo et al. (1999) are given in Table 3. However, this V140 is very close to a much brighter star and a careful blinking of the difference images clearly reveals that the authentic variable is the brighter star, which we have identified as the new SRA variable V175 (see $\S
\ref{newvar}$). Both the light curves for V140 and V175 are shown in the mosaic of Fig. 3. The light curve of V140 has been contaminated by the real variations in the much brighter star V175. We conclude that V140 is not a variable star while V175 is an SRA variable.
V155 was discovered by Drissen & Shara (1998) and they classified it as RRc. However, the star lies near the RGB on the CMD (see Fig. 1) and the light curve in Fig. 3 is similar to that of an eclipsing binary of the EW type, or contact binary, when phased with the ephemerides P= 0.664865 days and the epoch 245 6504.2067 d. Note that two different depths of the minima, particularly visible in the $V$ light curve, are implied.
V159 is classified in the CVSGC as a probable eclipsing binary. The star was identified as variable by Drissen & Shara (1998) on their HST images and labelled as V28. The V159 name was given by Caputo et al. (1999) on their finding chart. We find that this star is highly blended in our images, which affects the star’s position on the CMD. We detected two clear eclipses in the $V$ band at 2455989.52 and 2456750.44 d, of about 0.15 mag depth (see Fig. 3). However, we are not able to determine the periodicity although we confirm the star as an eclipsing binary.
New variable stars in M5 {#newvar}
========================
To search for new variables in the field of M5 we have used several approaches. We isolated all stars in regions of the CMD where most variable stars in a GC tend to be found. This includes the horizontal branch, the blue stragglers region and the RGB. We identified all previously known variables in the field of our images and studied in detail the light curves of the rest of the stars. This procedure allowed us to identify a new large amplitude ($A_V \sim 0.6$ mag) SX Phe star V170, for which we identify only one period. Then we explored the difference images for clear variations; this approach allowed us to discover the variability of six SRA (V171, V172, and V174-V177). A third approach was via the rms diagram of Fig. 2. It can be seen from this diagram that our photometry achieves uncertainties between 7 and 20 mmag at the bright end. High values of rms are generally produced by variable stars. For instance the group of stars with rms above 0.1 mag and with $V \sim$ 15 are all RR Lyrae stars which are not discussed in the present paper. Through this procedure we identified five additional SRA stars V173 and V178-V181. For some of these new SRA stars we have been able to estimate a periodicity.
The new variables, their equatorial coordinates, periods and epochs are listed in Table 4 and their position on the CMD and the rms diagram are displayed in Figs. 1 and 2 respectively. Their light curves in the $V$ and $I$ bands, phased whenever possible or as a function of HJD otherwise, are displayed in Fig. 3.
In the identification chart of Fig 4, we have marked a detailed identification for all variable stars discussed in this paper. No effort has been made to identify the numerous known variables listed in the CVSGC since that will be the subject of a future paper.
[**Table4.**]{} New variable stars found in M5, V170-V181.
------ ---------- ------------- ------------- ---------- -------------- ------- ------- ------- -------
Name Variable RA Dec. Period Epoch $<V>$ $<I>$ $A_V$ $A_I$
type (J2000.0) (J2000.0) days (-245 0000.) mag mag mag mag
V170 SX Phe 15 18 32.14 +02 04 20.4 0.089467 6063.3361 15.95 15.63 0.57 0.41
V171 SRA 15 18 34.26 +02 04 24.2 28.8 6312.5083 12.17 10.50 0.25 0.14
V172 SRA 15 18 31.59 +02 04 41.4 – – 12.15 10.47 0.23 0.13
V173 SRA 15 18 28.42 +02 04 29.8 43.1 6504.1686 12.28 10.86 0.13 0.13
V174 SRA 15 18 34.18 +02 06 25.5 80.6 6063.4183 12.03 10.33 0.33 0.15
V175 SRA 15 18 36.22 +02 05 11.3 – – 12.40 10.94 0.18 0.13
V176 SRA 15 18 37.38 +02 06 08.2 133.3 5989.3064 12.46 11.13 0.22 0.20
V177 SRA 15 18 41.40 +02 06 00.9 – – 12.51 11.19 0.13 0.10
V178 SRA 15 18 33.10 +02 04 58.0 141.6 5987.4759 12.39 11.03 0.12 0.10
V179 SRA 15 18 33.42 +02 04 59.6 – – 12.18 10.61 0.12 0.11
V180 SRA 15 18 35.82 +02 03 42.4 – – 12.27 10.86 0.24 0.24
V181 SRA 15 18 45.40 +02 04 30.9 – – 12.64 11.36 0.07 0.08
------ ---------- ------------- ------------- ---------- -------------- ------- ------- ------- -------
ACKNOWLEDGMENTS: We are thankful to the referee Dr. Ádám Sódor for his valuable comments and suggestions. We acknowledge the support from DGAPA-UNAM, Mexico grant through projects IN104612 and IN106615 and from NPRP grant \# X-019-1-006 from the Qatar National Research Fund (a member of Qatar Foundation).
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|
---
abstract: 'In this work we present the ultra-relativistic $\mathcal{N}$-extended AdS Chern-Simons supergravity theories in three spacetime dimensions invariant under $\mathcal{N}$-extended AdS Carroll superalgebras. We first consider the $(2,0)$ and $(1,1)$ cases; subsequently, we generalize our analysis to $\mathcal{N}=(\mathcal{N},0)$, with $\mathcal{N}$ even, and to $\mathcal{N}=(p,q)$, with $p=q$, such that, in particular, $\mathcal{N}=p+q$ is even. The $\mathcal{N}$-extended AdS Carroll superalgebras are obtained through the Carrollian (i.e., ultra-relativistic) contraction applied to an $so(2)$ extension of $\mathfrak{osp}(2|2)\otimes \mathfrak{sp}(2)$, to $\mathfrak{osp}(2|1)\otimes \mathfrak{osp}(2,1)$, to an $\mathfrak{so}(\mathcal{N})$ extension of $\mathfrak{osp}(2|\mathcal{N})\otimes \mathfrak{sp}(2)$, and to the direct sum of an $\mathfrak{so}(p) \oplus \mathfrak{so}(q)$ algebra and $\mathfrak{osp}(2|p)\otimes \mathfrak{osp}(2,q)$, respectively. We also analyze the flat limit ($\ell \rightarrow \infty$, being $\ell$ the length parameter) of the aforementioned $\mathcal{N}$-extended Chern-Simons AdS Carroll supergravities, in which we recover the ultra-relativistic $\mathcal{N}$-extended (flat) Chern-Simons supergravity theories invariant under $\mathcal{N}$-extended super-Carroll algebras. The flat limit is applied at the level of the superalgebras, Chern-Simons actions, supersymmetry transformation laws, and field equations.'
---
****
**Farhad Ali**$^{\ast}$, **Lucrezia Ravera**$^{\star}$\
$^{\ast}$*Department of Mathematics, Kohat University of Science and Technology,*\
*26000 Kohat KPK, Pakistan.*\
$^{\star}$*INFN, Sezione di Milano,*\
*Via Celoria 16, I-20133 Milano, Italy.*\
`farhadali@kust.edu.pk`, `lucrezia.ravera@mi.infn.it`
Introduction
============
In the study and understanding of several physical models, such as Newtonian gravity, Maxwell’s electromagnetism, special and general relativity, string and supergravity theory, spacetime symmetries have played a key role. Most of these theories are based on relativistic symmetries. On the other hand, during the years models with non-relativistic symmetries have also been developed and analyzed, and and are still the subject of in-depth studies.
In this context, Carroll symmetries [@LL; @Bacry:1968zf], which arise when the velocity of light is sent to zero (i.e., in the ultra-relativistic limit, $c \rightarrow 0$), have attracted some interest over recent years. Models with Carroll symmetries occurred in the literature in the study of tachyon condensation [@Gibbons:2002tv], warped conformal field theories [@Hofman:2014loa], and in the context of tensionless strings [@Bagchi:2013bga; @Bagchi:2015nca; @Bagchi:2016yyf; @Bagchi:2017cte; @Bagchi:2018wsn].
Concerning gravity theories, models of Carrollian (i.e., ultra-relativistic) gravity have been developed and analyzed in [@Hartong:2015xda; @Bergshoeff:2016soe; @Bergshoeff:2017btm]. In particular, in [@Bergshoeff:2016soe] the authors focused on the construction of non- and ultra-relativistic Chern-Simons (CS) type actions in $2+1$ dimensions including a spin-$3$ field coupled to gravity. Moreover, in [@Bergshoeff:2015wma], the geometry of flat and curved (Anti-de Sitter, AdS for short) Carroll space and the symmetries of a particle moving in such a space, both in the bosonic as well as in the supersymmetric case, were investigated. Afterwards, in the work [@Matulich:2019cdo], which concerns the classification of gravitational theories in $2+1$ dimensions and limits of their actions, the AdS Carroll CS gravity theory was discussed for the first time.[^1]
It has further been shown that non-relativistic symmetry groups play a remarkable role in holography [@Bagchi:2009my; @Christensen:2013lma; @Christensen:2013rfa; @Hartong:2014oma; @Bergshoeff:2014uea; @Hartong:2015wxa; @Bagchi:2010eg; @Bagchi:2012cy; @Bagchi:2016bcd; @Lodato:2016alv; @Bagchi:2019xfx; @Duval:2014uva; @Duval:2014lpa; @Ciambelli:2018xat; @Ciambelli:2018wre; @Ciambelli:2018ojf; @Campoleoni:2018ltl]. In particular, in [@Bagchi:2010eg], connections among the Bondi-Metzner-Sachs (BMS) algebra, Carrollian physics, and holography of flat space were noticed and followed up in [@Bagchi:2012cy] (see also [@Bagchi:2016bcd; @Lodato:2016alv], the latter developed in the context of supergravity, and [@Bagchi:2019xfx]). Besides, in [@Duval:2014uva; @Duval:2014lpa] conformal extensions of the Carroll group were explored and related to the BMS group, and in [@Ciambelli:2018xat; @Ciambelli:2018wre; @Ciambelli:2018ojf; @Campoleoni:2018ltl] the authors shewed how Carrollian structures and geometry emerge in the framework of flat holography and fluid/gravity correspondence.
Recently, in [@Ravera:2019ize] the construction of the three-dimensional $\mathcal{N}=1$ CS supergravity theory invariant under the so-called AdS Carroll superalgebra (ultra-relativistic contraction of the $\mathcal{N}=1$ AdS superalgebra, see [@Bergshoeff:2015wma]), together with the study of its flat limit, has been presented for the first time. In the study done in [@Ravera:2019ize], the method introduced in [@Concha:2016zdb] was adopted. Specifically, in [@Concha:2016zdb] the authors presented a generalization of the standard Inönü-Wigner (IW) contraction [@IW; @WW] by rescaling not only the generators of a Lie (super)algebra but also the arbitrary constants appearing in the components of the invariant tensor of the same Lie (super)algebra, the latter being the key ingredient for working out a CS action, invariant under the IW contracted (super)algebra by construction. This procedure was further improved in [@Ravera:2019ize] by considering dimensionful generators from the very beginning, on the same lines of [@Concha:2018jxx]. As shown in the same paper, this allows to obtain, for instance, the Poincaré limit from the $\mathfrak{osp}(2|2) \otimes \mathfrak{sp}(2)$ CS supergravity action directly considering the flat limit $\ell \rightarrow \infty$, being $\ell$ the length parameter related to the cosmological constant.
Moving to higher $\mathcal{N}$, we have that the $\mathcal{N} = 2$ supersymmetric extensions of the Poincaré and AdS algebras are not unique, and can be subdivided into two inequivalent classes: The $(2, 0)$ and the $(1,1)$ cases. Here we mention that, in the extension to the $(p,q)$ case, when either $p$ or $q$ is greater than one some subtleties arise. Indeed, even though the $(p, q)$ Poincaré superalgebra can be derived as an IW contraction of the $(p, q)$ AdS superalgebra, the Poincaré limit applied at the level of the (CS) action requires to enlarge the AdS superalgebra, considering, in particular, a direct sum of an $\mathfrak{so}(p)\oplus \mathfrak{so}(q)$ algebra and the $(p,q)$ AdS superalgebra $\mathfrak{osp}(2|p)\otimes \mathfrak{osp}(2,q)$ [@Howe:1995zm; @Giacomini:2006dr; @deAzcarraga:2011pa]; this is related to the fact that, as it was proven in [@Howe:1995zm], the semi-direct extension of the $\mathfrak{so}(p)\oplus \mathfrak{so}(q)$ automorphism algebra by the $(p,q)$ Poincaré superalgebra allows to produce a non-degenerate invariant tensor which is used to construct a well-defined three-dimensional CS $(p,q)$ Poincaré supergravity theory (in particular, when either $p$ or $q$ is greater than one, it is not possible to obtain a non-degenerate invariant tensor without considering this extension).
It is well assumed that a three-dimensional (super)gravity theory can be described by a CS action as a gauge theory, offering an interesting toy model to approach higher-dimensional theories [@DK; @Deser; @PvN; @AT1; @RPvN; @Witten; @AT2; @NG; @Banados:1996hi]. In the last decades, diverse three-dimensional supergravity models have been studied, and, in this context, there has also been a growing interest to extend AdS and Poincaré supergravity theories to other symmetries (see [@Concha:2018jxx; @Concha:2019icz] and references therein).
In the present work, we apply the method of [@Concha:2016zdb] with the improvements of [@Ravera:2019ize] to develop in a systematic way the ultra-relativistic $\mathcal{N}$-extended AdS CS supergravity theories in three (that is $2+1$) spacetime dimensions invariant under $\mathcal{N}$-extended AdS Carroll superalgebras. In particular, we will distinguish between the two $\mathcal{N}$-extended cases $\mathcal{N}=(\mathcal{N},0)$ and $\mathcal{N}=(p,q)$, generalizing the results presented at the algebraic level in [@Bergshoeff:2015wma] and also developing the associated CS supergravity theories in three dimensions. More specifically, we start by considering the $(2,0)$ and $(1,1)$ cases, and then generalize our analysis to $\mathcal{N}=(\mathcal{N},0)$, with $\mathcal{N}$ even, and to $\mathcal{N}=(p,q)$, that is $\mathcal{N}=p+q$, with $p=q$.[^2] The $\mathcal{N}$-extended AdS Carroll superalgebras are obtained through the Carrollian (i.e., ultra-relativistic) contraction applied to an $so(2)$ extension of $\mathfrak{osp}(2|2)\otimes \mathfrak{sp}(2)$, to $\mathfrak{osp}(2|1)\otimes \mathfrak{osp}(2,1)$, to an $\mathfrak{so}(\mathcal{N})$ extension of $\mathfrak{osp}(2|\mathcal{N})\otimes \mathfrak{sp}(2)$ (with $\mathcal{N}$ even), and to the direct sum of an $\mathfrak{so}(p) \oplus \mathfrak{so}(q)$ algebra and $\mathfrak{osp}(2|p)\otimes \mathfrak{osp}(2,q)$ (with $p=q$), respectively. Let us mention that the $\mathcal{N}=(\mathcal{N},0)$ case (and thus also the $\mathcal{N}=(2,0)$ one) will be more subtle, since it will require the definitions of new supersymmetry generators in order to properly study the Carroll limit, on the same lines of what was done in [@Bergshoeff:2015wma] (see also [@Lukierski:2006tr], which deals with non-relativistic superalgebras, and references therein). The ultra-relativistic $\mathcal{N}$-extended AdS Carroll supergravity actions are constructed à la CS, by exploiting the non-vanishing components of the corresponding invariant tensor. The aforementioned actions are all based on a non-degenerate, invariant bilinear form (i.e., an invariant metric). Our result was also an open problem suggested in Ref. [@Bergshoeff:2015wma], and it represents the $\mathcal{N}$-extended generalization of [@Ravera:2019ize].
Subsequently, we study the flat limit ($\ell \rightarrow \infty$, being $\ell$ the length parameter) of the aforesaid $\mathcal{N}$-extended CS AdS Carroll supergravities, in which we recover the ultra-relativistic $\mathcal{N}$-extended (flat) CS supergravity theories invariant under $\mathcal{N}$-extended super-Carroll algebras. The flat limit is applied at the level of the superalgebras, CS actions, supersymmetry transformation laws, and field equations.
The remain of the paper is organized as follows: In Section \[20case\], we first introduce a new $\mathcal{N}=(2,0)$ AdS Carroll superalgebra, which is obtained as the ultra-relativistic contraction of an $\mathfrak{so}(2)$ extension of $\mathfrak{osp}(2|2)\otimes \mathfrak{sp}(2)$. Here, the $\mathfrak{so}(2)$ extension is necessary in order to end up with an invariant non-degenerate inner product in the ultra-relativistic limit, providing a well-defined CS action. In fact, this allows us to subsequently develop the three-dimensional CS supergravity action invariant under the $(2,0)$ AdS Carroll superalgebra, which we call $(2,0)$ CS AdS Carroll supergravity in $2+1$ dimensions. In Section \[11case\], we repeat the same analysis for the $(1,1)$ case, ending up with the CS supergravity action invariant under the $(1,1)$ AdS Carroll superalgebra. Subsequently, we generalize our study to the cases of $\mathcal{N}=(\mathcal{N},0)$, with $\mathcal{N}$ even, and $\mathcal{N}=(p,q)$, with $p=q$, respectively in Section \[ncase\] and \[pqcase\]. In Section \[flatlimit\], we discuss the flat limit $\ell \rightarrow \infty$ of the $\mathcal{N}$-extended CS AdS Carroll supergravities introduced in the previous part of the work. Section \[conclusions\] contains some final comments and remarks.
$(2,0)$ AdS Carroll supergravity in $2+1$ dimensions {#20case}
====================================================
In this section, we first introduce a new $\mathcal{N}=(2,0)$ AdS Carroll superalgebra, which is obtained as the ultra-relativistic contraction of an $\mathfrak{so}(2)$ extension of $\mathfrak{osp}(2|2)\otimes \mathfrak{sp}(2)$. The $\mathfrak{so}(2)$ extension of $\mathfrak{osp}(2|2)\otimes \mathfrak{sp}(2)$ is needed in order to end up with an invariant non-degenerate inner product in the ultra-relativistic limit, namely with a well-defined invariant tensor (this is reminescent of what was done in [@Howe:1995zm] in the case of relativistic theories), in such a way to be able to construct a well-defined CS action. Indeed, this allows us to subsequently develop the three-dimensional CS supergravity action invariant under this $\mathcal{N}=(2,0)$ AdS Carroll superalgebra, which we call $(2,0)$ CS AdS Carroll supergravity.
Let us mention, here, that a $\mathcal{N}=(2,0)$ AdS Carroll superalgebra has been first introduced in [@Bergshoeff:2015wma]. Nevertheless, due to the degeneracy of the invariant tensor for that superalgebra, one could not construct a well-defined CS action in that case. On the other hand, as we will see in the following, our $\mathcal{N}=(2,0)$ AdS Carroll superalgebra will be different from the one presented in [@Bergshoeff:2015wma], allowing, in particular, the formulation of a three-dimensional CS action in the supergravity context.
$\mathcal{N}=(2,0)$ AdS Carroll superalgebra
--------------------------------------------
An $\mathcal{N}=(2,0)$ supersymmetric extension of the AdS Carroll algebra was obtained in [@Bergshoeff:2015wma] as the ultra-relativistic contraction of $\mathfrak{osp}(2|2)\otimes \mathfrak{sp}(2)$, the latter being generated by the set $\lbrace \tilde{J}_{AB}, \tilde{P}_A , \tilde{Z}^{ij}, \tilde{Q}^i_\alpha \rbrace$, with $A, B , \ldots=0,1,2$, $\alpha=1,2$, and $i=1,2$, where $\tilde{J}_{AB}$ are the Lorentz generators, $\tilde{P}_A$ represent the spacetime translations, $\tilde{Z}^{ij} = \epsilon^{ij} \tilde{Z}$ are internal symmetry generators, and $\tilde{Q}^i_\alpha$ are the supersymmetry generators ($2$-components Majorana spinor charges). The (anti)commutation relations of $\mathfrak{osp}(2|2)\otimes \mathfrak{sp}(2)$ read as follows: $$\label{osp22nomod}
\begin{split}
& \left[ \tilde{J}_{AB}, \tilde{J}_{CD}\right] =\eta _{BC}\tilde{J}_{AD}-\eta _{AC}\tilde{J}_{BD}-\eta
_{BD}\tilde{J}_{AC}+\eta _{AD}\tilde{J}_{BC} \, , \\
& \left[ \tilde{J}_{AB},\tilde{P}_{C}\right] =\eta _{BC}\tilde{P}_{A}-\eta _{AC}\tilde{P}_{B} \, , \\
& \left[ \tilde{P}_A , \tilde{P}_B \right] = \frac{1}{\ell^2} \tilde{J}_{AB} \, , \\
& \left[ \tilde{J}_{AB},\tilde{Q}^i_{\alpha }\right] =-\frac{1}{2}\left( \Gamma _{AB}\tilde{Q}^i\right)_{\alpha } \, , \quad \quad \left[ \tilde{P}_{A}, \tilde{Q}^i_{\alpha }\right] =-\frac{1}{2 \ell} \left( \Gamma _{A}\tilde{Q}^i\right) _{\alpha } \, , \\
& \left[ \tilde{Z} , \tilde{Q}^i_{\alpha } \right] = - \epsilon^{ij} \tilde{Q}^j_\alpha \, , \\
& \left\{ \tilde{Q}^i_{\alpha }, \tilde{Q}^j_{\beta }\right\} = \delta^{ij} \left[ - \frac{1}{2 \ell} \left(\Gamma^{AB} C \right)_{\alpha \beta} \tilde{J}_{AB} + \left( \Gamma ^{A}C\right) _{\alpha \beta }\tilde{P}_{A} \right] + \frac{1}{\ell} \epsilon^{ij} C_{\alpha \beta} \tilde{Z} \, ,
\end{split}$$ where $\ell$ is a length parameter, $C$ denotes the charge conjugation matrix, and $\Gamma_A$ and $\Gamma_{AB}$ represent the Dirac matrices in three dimensions. The generators $\tilde{J}_{AB}$, $\tilde{P}_A $, $\tilde{Z}^{ij}$, and $\tilde{Q}^i_\alpha$ have a dual description in terms of $1$-form fields, $\tilde{\omega}^{AB}$ (spin connection), $\tilde{V}^A$ (vielbein), $\tilde{z}_{ij}$ ($1$-form field dual to the generator $\tilde{Z}^{ij}$), and $\tilde{\psi}_i^\alpha$ (gravitinos), respectively.
Here, we consider the ultra-relativistic contraction of an $\mathfrak{so}(2)$ extension of $\mathfrak{osp}(2|2)\otimes \mathfrak{sp}(2)$, involving an extra generator $\tilde{S}^{ij} = \epsilon^{ij} \tilde{S}$. This will allow the formulation of a well-defined ultra-relativistic CS action, based on a non-degenerate invariant tensor, which would not be possible by considering the $\mathcal{N}=(2,0)$ superalgebra of [@Bergshoeff:2015wma]. In particular, we extend by adding the extra $\tilde{S}$ generator and we perform, on the same lines of [@Howe:1995zm], the redefinition $$\tilde{T} \equiv \tilde{Z} - \ell \tilde{S} \, ,$$ to eliminate $\tilde{Z}$ in favour of $\tilde{T}$ (this redefinition is particularly convenient for discussing the flat limit, see also [@Howe:1995zm]). Consequently, we rewrite the (anti)commutation relations as follows (we adopt dimensionful generators from the very beginning, on the same lines of [@Concha:2018jxx]):[^3] $$\label{osp22}
\begin{split}
& \left[ \tilde{J}_{AB}, \tilde{J}_{CD}\right] =\eta _{BC}\tilde{J}_{AD}-\eta _{AC}\tilde{J}_{BD}-\eta
_{BD}\tilde{J}_{AC}+\eta _{AD}\tilde{J}_{BC} \, , \\
& \left[ \tilde{J}_{AB},\tilde{P}_{C}\right] =\eta _{BC}\tilde{P}_{A}-\eta _{AC}\tilde{P}_{B} \, , \\
& \left[ \tilde{P}_A , \tilde{P}_B \right] = \frac{1}{\ell^2} \tilde{J}_{AB} \, , \\
& \left[ \tilde{J}_{AB},\tilde{Q}^i_{\alpha }\right] =-\frac{1}{2}\left( \Gamma _{AB}\tilde{Q}^i\right)_{\alpha } \, , \quad \quad \left[ \tilde{P}_{A}, \tilde{Q}^i_{\alpha }\right] =-\frac{1}{2 \ell} \left( \Gamma _{A}\tilde{Q}^i\right) _{\alpha } \, , \\
& \left[ \tilde{T} , \tilde{Q}^i_{\alpha } \right] = - \epsilon^{ij} \tilde{Q}^j_\alpha \, , \\
& \left\{ \tilde{Q}^i_{\alpha }, \tilde{Q}^j_{\beta }\right\} = \delta^{ij} \left[ - \frac{1}{2 \ell} \left(\Gamma^{AB} C \right)_{\alpha \beta} \tilde{J}_{AB} + \left( \Gamma ^{A}C\right) _{\alpha \beta }\tilde{P}_{A} \right] + \epsilon^{ij} C_{\alpha \beta} \left( \frac{1}{\ell} \tilde{T} + \tilde{S} \right) \, .
\end{split}$$ Note that, in the flat limit $\ell \rightarrow \infty$, $\tilde{S}$ becomes the central element of the $\mathcal{N}=(2,0)$ Poincaré superalgebra extended with the extra $so(2)$ generator $\tilde{T}$ (see [@Howe:1995zm]).
The non-vanishing components of an invariant tensor for the superalgebra , which will be useful in the sequel, are given by $$\label{invtosp22}
\begin{split}
& \langle \tilde{J}_{AB} \tilde{J}_{CD} \rangle = \alpha_0 \left( \eta_{AD} \eta_{BC} - \eta_{AC} \eta_{BD} \right) \, , \\
& \langle \tilde{J}_{AB} \tilde{P}_{C} \rangle = \alpha_1 \epsilon_{ABC} \, , \\
& \langle \tilde{P}_{A} \tilde{P}_{B} \rangle = \frac{\alpha_0}{\ell^2} \eta_{AB} \, , \\
& \langle \tilde{T} \tilde{T} \rangle = - 2 \alpha_0 \, , \\
& \langle \tilde{T} \tilde{S} \rangle = 2 \alpha_1 \, , \\
& \langle \tilde{S} \tilde{S} \rangle = - \frac{2 \alpha_1}{\ell} \, , \\
& \langle \tilde{Q}^i_\alpha \tilde{Q}^j_\beta \rangle = 2 \left(\alpha_1 -\frac{\alpha_0}{\ell} \right) C_{\alpha \beta} \delta^{ij} \, ,
\end{split}$$ where $\alpha_0$ and $\alpha_1$ are arbitrary constants and $\epsilon_{ABC}$ is the Levi-Civita symbol in three dimensions.
To take the Carrollian (i.e., ultra-relativistic) contraction of the superalgebra , we decompose the indices as $$\label{indexdec}
A \rightarrow (0,a) \, , \quad a=1,2 \, .$$ This induces the following decomposition of the generators: $$\label{gendec}
\tilde{J}_{AB} \rightarrow \lbrace \tilde{J}_{ab}, \tilde{J}_{a0} \equiv \tilde{K}_a \rbrace \, , \quad
\tilde{P}_A \rightarrow \lbrace \tilde{P}_a , \tilde{P}_0 \equiv \tilde{H} \rbrace \, .$$ We also have $$\label{gammadec}
\Gamma_{AB} \rightarrow \lbrace \Gamma_{ab} , \Gamma_{a0} \rbrace \, , \quad \Gamma_A \rightarrow \lbrace \Gamma_a , \Gamma_0 \rbrace \, .$$ Furthermore, we define new supersymmetry charges by $$\label{newsusy22}
\tilde{Q}^{\pm}_\alpha = \frac{1}{\sqrt{2}} \left( \tilde{Q}^1_\alpha \pm \left(\Gamma_0\right)_{\alpha \beta} \tilde{Q}^2_\beta \right) \, .$$ Then, we rescale the generators with a parameter $\sigma$ as follows: $$\label{resc}
\tilde{H} \rightarrow \sigma H \, , \quad \tilde{K}_a \rightarrow \sigma K_a \, , \quad \tilde{S} \rightarrow \sigma S \, , \quad \tilde{Q}^\pm_\alpha \rightarrow \sqrt{\sigma} Q^\pm_\alpha \, .$$ Taking the limit $\sigma \rightarrow \infty$[^4] and removing the tilde symbol also on the generators that we have not rescaled, we end up with a new $\mathcal{N}=(2,0)$ AdS Carroll superalgebra (differing from the one of [@Bergshoeff:2015wma], due to the presence of the generator $S$), which fulfills the following non-trivial (anti)commutation relations: $$\label{adscarrollsuper22}
\begin{split}
& \left[ K_a , J_{bc} \right] = \delta_{ab} K_c - \delta_{ac} K_b \, , \\
& \left[ J_{ab}, P_{c}\right] =\delta_{bc}P_{a}-\delta _{ac}P_{b} \, , \\
& \left[ K_{a}, P_{b}\right] = - \delta_{ab} H \, , \\
& \left[ P_a , P_b \right] = \frac{1}{\ell^2} J_{ab} \, , \\
& \left[ P_a , H \right] = \frac{1}{\ell^2} K_{a} \, , \\
& \left[ J_{ab},Q^\pm_{\alpha }\right] =-\frac{1}{2}\left( \Gamma _{ab} Q^\pm \right)_{\alpha } \, , \\
& \left[ P_{a}, Q^\pm_{\alpha }\right] =-\frac{1}{2 \ell} \left( \Gamma _{a}Q^\mp\right) _{\alpha } \, , \\
& \left[ T, Q^+_{\alpha }\right] = \left( \Gamma _{0} \right)_{\alpha \beta} Q^+_\beta \, , \\
& \left[ T, Q^-_{\alpha }\right] = - \left( \Gamma _{0} \right)_{\alpha \beta} Q^-_\beta \, , \\
& \left\{ Q^+_{\alpha }, Q^+_{\beta }\right\} = \left( \Gamma ^{0}C\right) _{\alpha \beta } \left( H + S \right) \, , \\
& \left\{ Q^+_{\alpha }, Q^-_{\beta }\right\} = - \frac{1}{\ell} \left(\Gamma^{a0} C \right)_{\alpha \beta} K_{a} \, , \\
& \left\{ Q^-_{\alpha }, Q^-_{\beta }\right\} = \left( \Gamma ^{0}C\right) _{\alpha \beta } \left( H - S \right) \, .
\end{split}$$ We will now construct a CS supergravity action in three dimensions invariant under the $\mathcal{N}=(2,0)$ AdS Carroll superalgebra : The $(2,0)$ AdS Carroll CS supergravity action.
$(2,0)$ AdS Carroll supergravity action
---------------------------------------
The general form of a three-dimensional CS action is given by $$\label{genCS}
I_{CS}=\frac{k}{4 \pi} \int_\mathcal{M} \Big \langle A dA + \frac{2}{3} A^3 \Big \rangle = \frac{k}{4 \pi} \int_\mathcal{M} \Big \langle A dA + \frac{1}{3} A \left[A,A \right] \Big \rangle \, ,$$ where $k=1/(4G)$ is the CS level of the theory (and for gravitational theories it is related to the gravitational constant $G$), $A$ corresponds to the gauge connection $1$-form, $\langle \ldots \rangle$ denotes the invariant tensor, and the integral is over a three-dimensional manifold $\mathcal{M}$.[^5] The CS action can also be rewritten as $$I_{CS}=\frac{k}{4 \pi} \int_\mathcal{M} \Big \langle A F - \frac{1}{3} A^3 \Big \rangle \, ,$$ in terms of the curvature $2$-form $F= dA + A^2 = dA + \frac{1}{2}\left[A,A \right]$.
In the case of the $\mathcal{N}=(2,0)$ AdS Carroll superalgebra , the connection $1$-form reads[^6] $$\label{connadscarrollsuper22}
A = \frac{1}{2} \omega^{ab} J_{ab} + k^a K_a + V^a P_a + h H + t T + s S + \psi^+ Q^+ + \psi^- Q^- \, ,$$ where $\omega^{ab}$, $k^a$, $V^a$, $h$, $t$, $s$, $\psi^+$, and $\psi^-$ are the $1$-form fields dual to the generators $J_{ab}$, $K_a$, $P_a$, $H$, $T$, $S$, $Q^+$, and $Q^-$, respectively.
The corresponding curvature $2$-form $F$ is $$\label{curv2f22}
F = \frac{1}{2} \mathcal{R}^{ab} J_{ab} + \mathcal{K}^a K_a + R^a P_a + \mathcal{H} H + \mathcal{T} T + \mathcal{S} S + \nabla \psi^+ Q^+ + \nabla \psi^- Q^- \, ,$$ with $$\label{curvadscarrollsuper22}
\begin{split}
\mathcal{R}^{ab} & = d \omega^{ab} + \frac{1}{\ell^2} V^a V^b = R^{ab} + \frac{1}{\ell^2} V^a V^b \, , \\
\mathcal{K}^a & = d k^a + \omega^a_{\phantom{a} b} k^b + \frac{1}{\ell^2} V^a h + \frac{1}{\ell} \bar{\psi}^+ \Gamma^{a0} \psi^- = \mathfrak{K}^a + \frac{1}{\ell^2} V^a h + \frac{1}{\ell} \bar{\psi}^+ \Gamma^{a0} \psi^- \, , \\
R^a & = d V^a + \omega^a_{\phantom{a}b} V^b \, , \\
\mathcal{H} & = d h + V^a k_a - \frac{1}{2} \bar{\psi}^+ \Gamma^0 \psi^+ - \frac{1}{2} \bar{\psi}^- \Gamma^0 \psi^- = \mathfrak{H} - \frac{1}{2} \bar{\psi}^+ \Gamma^0 \psi^+ - \frac{1}{2} \bar{\psi}^- \Gamma^0 \psi^- \, , \\
\mathcal{T} & = d t \, , \\
\mathcal{S} & = d s - \frac{1}{2} \bar{\psi}^+ \Gamma^0 \psi^+ + \frac{1}{2} \bar{\psi}^- \Gamma^0 \psi^- \, , \\
\nabla \psi^+ & = d \psi^+ + \frac{1}{4} \omega^{ab} \Gamma_{ab} \psi^+ + \frac{1}{2 \ell} V^a \Gamma_a \psi^- - t \Gamma_0 \psi^+ \, , \\
\nabla \psi^- & = d \psi^- + \frac{1}{4} \omega^{ab} \Gamma_{ab} \psi^- + \frac{1}{2 \ell} V^a \Gamma_a \psi^+ + t \Gamma_0 \psi^- \, .
\end{split}$$
Now, in order to construct a CS action (that is an action of the form ) invariant under the $\mathcal{N}=(2,0)$ super-AdS Carroll group, we require the connection $1$-form given in and the corresponding non-vanishing components of the invariant tensor.
Concerning the invariant tensor, that is the fundamental ingredient for the construction of a CS action, we now apply the method of [@Concha:2016zdb], which consists in rescaling not only the generators but also the coefficients appearing in the invariant tensor before applying a contraction, in order to end up with a non-trivial invariant tensor for the contracted (super)algebra on which the desired CS theory will be based. Specifically, we consider the non-vanishing components of the invariant tensor for the $\mathfrak{so}(2)$ extension of $\mathfrak{osp}(2|2)\otimes \mathfrak{sp}(2)$ (see ) given in , we decompose the indices as in and consider the new supersymmetry charges , and then we rescale not only the generators in compliance with but also the coefficients appearing in as follows: $$\label{alsc}
\alpha_0 \rightarrow \alpha_0 \, , \quad \alpha_1 \rightarrow \sigma \alpha_1 \, .$$ In this way, taking the limit $\sigma \rightarrow \infty$, we end up with the following non-vanishing components of an invariant tensor for the $\mathcal{N}=(2,0)$ AdS Carroll superalgebra: $$\label{invadscarrollsuper22}
\begin{split}
& \langle J_{ab} J_{cd} \rangle = \alpha_0 \left( \delta_{ad} \delta_{bc} - \delta_{ac} \delta_{bd} \right) \, , \\
& \langle J_{ab} H \rangle = \alpha_1 \epsilon_{ab} \, , \\
& \langle K_a P_b \rangle = - \alpha_1 \epsilon_{ab} \, , \\
& \langle P_{a} P_{b} \rangle = \frac{\alpha_0}{\ell^2} \delta_{ab} \, , \\
& \langle T T \rangle = - 2 \alpha_0\, , \\
& \langle T S \rangle = 2 \alpha_1 \, , \\
& \langle Q^+_\alpha Q^+_\beta \rangle = \langle Q^-_\alpha Q^-_\beta \rangle = 2 \alpha_1 C_{\alpha \beta} \, .
\end{split}$$ This bilinear form is non-degenerate if $\alpha_1 \neq 0$.
Thus, using the connection $1$-form in and the non-vanishing components of the invariant tensor given in in the general expression for a three-dimensional CS action, we can finally write the $(2,0)$ AdS Carroll CS supergravity action in three spacetime dimensions invariant under , which reads as follows: $$\label{CSAC22}
\begin{split}
I^{(2,0)}_{CS} & = \frac{k}{4 \pi} \int_\mathcal{M} \Bigg \lbrace \frac{\alpha_0}{2} \left( \omega^a_{\phantom{a} b} R^b_{\phantom{b} a} + \frac{2}{\ell^2} V^a R_a - 4 t dt \right) + \alpha_1 \bigg( \epsilon_{ab} R^{ab} h - 2 \epsilon_{ab} \mathfrak{K}^a V^b + \frac{1}{\ell^2} \epsilon_{ab} V^a V^b h \\
& + 4 t ds + 2 \bar{\psi}^+ \nabla \psi^+ + 2 \bar{\psi}^- \nabla \psi^- \bigg) - d \left( \frac{\alpha_1}{2} \epsilon_{ab} \omega^{ab} h - \alpha_1 \epsilon_{ab} k^a V^b - 2 \alpha_1 t s \right) \Bigg \rbrace \, ,
\end{split}$$ written in terms of the curvatures appearing in . We can see that in we have two different sectors, one proportional to $\alpha_0$ and the other proportional to $\alpha_1$. Observe that the term proportional to $\alpha_0$ corresponds to the exotic Lagrangian involving the Lorentz contribution, a torsional piece, and a contribution from the $1$-form field $t$, while it does not contain any contribution from the $1$-form fields $\psi^+$ and $\psi^-$. Let us mention that the CS action can also be rewritten up to boundary terms as $$\label{CSAC22uptobdy}
\begin{split}
I^{(2,0)}_{CS} & = \frac{k}{4 \pi} \int_\mathcal{M} \Bigg \lbrace \frac{\alpha_0}{2} \left( \omega^a_{\phantom{a} b} R^b_{\phantom{b} a} + \frac{2}{\ell^2} V^a R_a - 4 t dt \right) + \alpha_1 \bigg( \epsilon_{ab} R^{ab} h - 2 \epsilon_{ab} \mathfrak{K}^a V^b + \frac{1}{\ell^2} \epsilon_{ab} V^a V^b h \\
& + 4 t ds + 2 \bar{\psi}^+ \nabla \psi^+ + 2 \bar{\psi}^- \nabla \psi^- \bigg) \Bigg \rbrace \, .
\end{split}$$
The CS action , characterized by two coupling constants $\alpha_0$ and $\alpha_1$, is invariant by construction under the $\mathcal{N}=(2,0)$ AdS Carroll superalgebra . In particular, the local gauge transformations $\delta_\lambda A = d \lambda + \left[A, \lambda \right]$ with gauge parameter $$\label{gpar22}
\lambda = \frac{1}{2} \lambda^{ab} J_{ab} + \kappa^a K_a + \lambda^a P_a + \tau H + \varphi T + \varsigma S + \varepsilon^+ Q^+ + \varepsilon^- Q^-$$ are given by $$\label{gaugetr22}
\begin{split}
\delta \omega^{ab} & = d \lambda^{ab} + \frac{2}{\ell^2} V^{[a} \lambda^{b]} \, , \\
\delta k^a & = d \kappa^a - \lambda^a_{\phantom{a} b} k^b + \omega^a_{\phantom{a} b} \kappa^b - \frac{1}{\ell^2} \lambda^a h + \frac{1}{\ell^2} V^a \tau - \frac{1}{\ell} \bar{\varepsilon}^+ \Gamma^{a0} \psi^- - \frac{1}{\ell} \bar{\varepsilon}^- \Gamma^{a0} \psi^+ \, , \\
\delta V^a & = d \lambda^a - \lambda^a_{\phantom{a}b} V^b + \omega^a_{\phantom{a}b} \lambda^b \, , \\
\delta h & = d \tau - \lambda^a k_a + V^a \kappa_a + \bar{\varepsilon}^+ \Gamma^0 \psi^+ + \bar{\varepsilon}^- \Gamma^0 \psi^- \, , \\
\delta t & = d \varphi \, , \\
\delta s & = d \varsigma + \bar{\varepsilon}^+ \Gamma^0 \psi^+ - \bar{\varepsilon}^- \Gamma^0 \psi^- \, , \\
\delta \psi^+ & = d \varepsilon^+ - \frac{1}{4} \lambda^{ab} \Gamma_{ab} \psi^+ + \frac{1}{4} \omega^{ab} \Gamma_{ab} \varepsilon^+ - \frac{1}{2 \ell} \lambda^a \Gamma_a \psi^- + \frac{1}{2 \ell} V^a \Gamma_a \varepsilon^- + \varphi \Gamma_0 \psi^+ - t \Gamma_0 \varepsilon^+ \, , \\
\delta \psi^- & = d \varepsilon^- - \frac{1}{4} \lambda^{ab} \Gamma_{ab} \psi^- + \frac{1}{4} \omega^{ab} \Gamma_{ab} \varepsilon^- - \frac{1}{2 \ell} \lambda^a \Gamma_a \psi^+ + \frac{1}{2 \ell} V^a \Gamma_a \varepsilon^+ - \varphi \Gamma_0 \psi^- + t \Gamma_0 \varepsilon^- \, .
\end{split}$$ Restricting ourselves to supersymmetry, we have $$\label{susytr22}
\begin{split}
\delta \omega^{ab} & = 0 \, , \\
\delta k^a & = - \frac{1}{\ell} \bar{\varepsilon}^+ \Gamma^{a0} \psi^- - \frac{1}{\ell} \bar{\varepsilon}^- \Gamma^{a0} \psi^+ \, , \\
\delta V^a & = 0 \, , \\
\delta h & = \bar{\varepsilon}^+ \Gamma^0 \psi^+ + \bar{\varepsilon}^- \Gamma^0 \psi^- \, , \\
\delta t & = 0 \, , \\
\delta s & = \bar{\varepsilon}^+ \Gamma^0 \psi^+ - \bar{\varepsilon}^- \Gamma^0 \psi^- \, , \\
\delta \psi^+ & = d \varepsilon^+ + \frac{1}{4} \omega^{ab} \Gamma_{ab} \varepsilon^+ + \frac{1}{2 \ell} V^a \Gamma_a \varepsilon^- - t \Gamma_0 \varepsilon^+ \, , \\
\delta \psi^- & = d \varepsilon^- + \frac{1}{4} \omega^{ab} \Gamma_{ab} \varepsilon^- + \frac{1}{2 \ell} V^a \Gamma_a \varepsilon^+ + t \Gamma_0 \varepsilon^- \, .
\end{split}$$
The equations of motion obtained from the variation of the action with respect to the fields $\omega^{ab}$, $k^a$, $V^a$, $h$, $t$, $s$, $\psi^+$, and $\psi^-$ are, respectively, $$\label{eom22}
\begin{split}
\delta \omega^{ab} & : \quad \alpha_0 \mathcal{R}^{ab} + \alpha_1 \epsilon^{ab} \mathcal{H}= 0 \, , \\
\delta k^a & : \quad \alpha_1 R^a = 0 \, , \\
\delta V^a & : \quad \frac{\alpha_0}{\ell^2} R^a + 2 \alpha_1 \epsilon_{ab} \mathcal{K}^b = 0 \, , \\
\delta h & : \quad \alpha_1 \mathcal{R}^{ab} = 0 \, , \\
\delta t & : \quad - \alpha_0 \mathcal{T} + \alpha_1 \mathcal{S} = 0 \, , \\
\delta s & : \quad \alpha_1 \mathcal{T} = 0 \, , \\
\delta \psi^+ & : \quad \alpha_1 \nabla \psi^+ = 0 \, , \\
\delta \psi^- & : \quad \alpha_1 \nabla \psi^- = 0 \, ,
\end{split}$$ up to boundary terms, and we can see that when $\alpha_1 \neq 0$ they reduce to the vanishing of the $(2,0)$ super-AdS Carroll curvature $2$-forms, namely $$\label{eomvac}
\mathcal{R}^{ab} = 0 \, , \quad \mathcal{K}^a = 0 \, , \quad R^a = 0 \, , \quad \mathcal{H}=0 \, , \quad \mathcal{T} =0 \, , \quad \mathcal{S} = 0 \, , \quad \nabla \psi^+ =0 \, , \quad \nabla \psi^- =0 \, .$$ Here, we can also observe that $\alpha_1 \neq 0$ is a sufficient condition to recover , meaning that one could consistently set $\alpha_0=0$, which corresponds to the vanishing of the exotic term in the CS action .
$(1,1)$ AdS Carroll supergravity in $2+1$ dimensions {#11case}
====================================================
In this section, we repeat the analysis done in Section \[20case\] in the $(1,1)$ case. To this aim, we first review the derivation of the $\mathcal{N}=(1,1)$ AdS Carroll superalgebra introduced in [@Bergshoeff:2015wma], which is obtained as the ultra-relativistic contraction of $\mathfrak{osp}(2|1)\otimes \mathfrak{osp}(2,1)$. Then, we write the non-vanishing components of the invariant tensor of the $\mathcal{N}=(1,1)$ AdS Carroll superalgebra (obtained as the Carrollian contraction of the non-vanishing components of the invariant tensor for $\mathfrak{osp}(2|1)\otimes \mathfrak{osp}(2,1)$). This allows us to construct a three-dimensional CS supergravity action invariant under the $\mathcal{N}=(1,1)$ AdS Carroll superalgebra, which we call the $(1,1)$ AdS Carroll CS supergravity action.
Review of the $\mathcal{N}=(1,1)$ AdS Carroll superalgebra
----------------------------------------------------------
Let us briefly review the derivation of the $\mathcal{N}=(1,1)$ AdS Carroll superalgebra of [@Bergshoeff:2015wma] as the Carrollian contraction of $\mathfrak{osp}(2|1)\otimes \mathfrak{osp}(2,1)$.
The superalgebra $\mathfrak{osp}(2|1)\otimes \mathfrak{osp}(2,1)$ is generated by the set $\lbrace \tilde{J}_{AB}, \tilde{P}_A , \tilde{Q}^+_\alpha , \tilde{Q}^-_\alpha \rbrace$ obeying the following (anti)commutation relations: $$\label{osp11}
\begin{split}
& \left[ \tilde{J}_{AB}, \tilde{J}_{CD}\right] =\eta _{BC}\tilde{J}_{AD}-\eta _{AC}\tilde{J}_{BD}-\eta
_{BD}\tilde{J}_{AC}+\eta _{AD}\tilde{J}_{BC} \, , \\
& \left[ \tilde{J}_{AB},\tilde{P}_{C}\right] =\eta _{BC}\tilde{P}_{A}-\eta _{AC}\tilde{P}_{B} \, , \\
& \left[ \tilde{P}_A , \tilde{P}_B \right] = \frac{1}{\ell^2} \tilde{J}_{AB} \, , \\
& \left[ \tilde{J}_{AB},\tilde{Q}^\pm_{\alpha }\right] =-\frac{1}{2}\left( \Gamma _{AB}\tilde{Q}^\pm\right)_{\alpha } \, , \quad \quad \left[ \tilde{P}_{A}, \tilde{Q}^\pm_{\alpha }\right] =\mp \frac{1}{2 \ell} \left( \Gamma _{A}\tilde{Q}^\pm \right) _{\alpha } \, , \\
& \left\{ \tilde{Q}^+_{\alpha }, \tilde{Q}^+_{\beta }\right\} = - \frac{1}{2 \ell} \left(\Gamma^{AB} C \right)_{\alpha \beta} \tilde{J}_{AB} + \left( \Gamma ^{A}C\right) _{\alpha \beta }\tilde{P}_{A} \, , \\
& \left\{ \tilde{Q}^-_{\alpha }, \tilde{Q}^-_{\beta }\right\} = \frac{1}{2 \ell} \left(\Gamma^{AB} C \right)_{\alpha \beta} \tilde{J}_{AB} + \left( \Gamma ^{A}C\right) _{\alpha \beta }\tilde{P}_{A} \, .
\end{split}$$ Note that by taking the flat limit $\ell \rightarrow \infty$ of one recovers the $\mathcal{N}=(1,1)$ Poincaré superalgebra.
The non-vanishing components of an invariant tensor for the superalgebra are $$\label{invtosp11}
\begin{split}
& \langle \tilde{J}_{AB} \tilde{J}_{CD} \rangle = \alpha_0 \left( \eta_{AD} \eta_{BC} - \eta_{AC} \eta_{BD} \right) \, , \\
& \langle \tilde{J}_{AB} \tilde{P}_{C} \rangle = \alpha_1 \epsilon_{ABC} \, , \\
& \langle \tilde{P}_{A} \tilde{P}_{B} \rangle = \frac{\alpha_0}{\ell^2} \eta_{AB} \, , \\
& \langle \tilde{Q}^+_\alpha \tilde{Q}^+_\beta \rangle = 2 \left(\alpha_1 - \frac{\alpha_0}{\ell} \right) C_{\alpha \beta} \, , \\
& \langle \tilde{Q}^-_\alpha \tilde{Q}^-_\beta \rangle = 2 \left(\alpha_1 + \frac{\alpha_0}{\ell} \right) C_{\alpha \beta} \, ,
\end{split}$$ being $\alpha_0$ and $\alpha_1$ arbitrary independent constants.
Now, to take the Carrollian contraction of the superalgebra , we decompose the indices $A,B,\ldots = 0,1,2$ as in , which induces the decomposition , together with . Then, we rescale the generators with a parameter $\sigma$ as $$\label{resc11}
\tilde{H} \rightarrow \sigma H \, , \quad \tilde{K}_a \rightarrow \sigma K_a \, , \quad \tilde{Q}^\pm_\alpha \rightarrow \sqrt{\sigma} Q^\pm_\alpha \, .$$ Subsequently, taking the limit $\sigma \rightarrow \infty$ (and removing the tilde symbol also on the generators that we have not rescaled), we end up with the $\mathcal{N}=(1,1)$ AdS Carroll superalgebra introduced in Ref. [@Bergshoeff:2015wma], whose (anti)commutation relations read $$\label{adscarrollsuper11}
\begin{split}
& \left[ K_a , J_{bc} \right] = \delta_{ab} K_c - \delta_{ac} K_b \, , \\
& \left[ J_{ab}, P_{c}\right] =\delta_{bc}P_{a}-\delta _{ac}P_{b} \, , \\
& \left[ K_{a}, P_{b}\right] = - \delta_{ab} H \, , \\
& \left[ P_a , P_b \right] = \frac{1}{\ell^2} J_{ab} \, , \\
& \left[ P_a , H \right] = \frac{1}{\ell^2} K_{a} \, , \\
& \left[ J_{ab},Q^\pm_{\alpha }\right] =-\frac{1}{2}\left( \Gamma _{ab} Q^\pm \right)_{\alpha } \, , \quad \quad \left[ P_{a}, Q^\pm_{\alpha }\right] =\mp \frac{1}{2 \ell} \left( \Gamma _{a}Q^\pm \right) _{\alpha } \, , \\
& \left\{ Q^+_{\alpha }, Q^+_{\beta }\right\} = - \frac{1}{\ell} \left(\Gamma^{a0} C \right)_{\alpha \beta} K_{a} + \left( \Gamma ^{0}C\right) _{\alpha \beta } H \, , \\
& \left\{ Q^-_{\alpha }, Q^-_{\beta }\right\} = \frac{1}{\ell} \left(\Gamma^{a0} C \right)_{\alpha \beta} K_{a} + \left( \Gamma ^{0}C\right) _{\alpha \beta } H \, .
\end{split}$$ In the sequel we will construct a CS action in three-dimension invariant under the $\mathcal{N}=(1,1)$ AdS Carroll superalgebra .
$(1,1)$ AdS Carroll supergravity action
---------------------------------------
We will now construct a three-dimensional CS supergravity action invariant under the $\mathcal{N}=(1,1)$ AdS Carroll superalgebra , which we call the $(1,1)$ AdS Carroll CS supergravity action.
To this aim, we introduce the connection $1$-form $A$ associated with , that is $$\label{connadscarrollsuper11}
A = \frac{1}{2} \omega^{ab} J_{ab} + k^a K_a + V^a P_a + h H + \psi^+ Q^+ + \psi^- Q^- \, ,$$ being $\omega^{ab}$, $k^a$, $V^a$, $h$, $\psi^+$, and $\psi^-$ the $1$-form fields respectively dual to the generators $J_{ab}$, $K_a$, $P_a$, $H$, $Q^+$, and $Q^-$ obeying the (anti)commutation relations given in , and the related curvature $2$-form $F$, which reads $$\label{curv2f11}
F = \frac{1}{2} \mathcal{R}^{ab} J_{ab} + \mathcal{K}^a K_a + R^a P_a + \mathcal{H} H + \nabla \psi^+ Q^+ + \nabla \psi^- Q^- \, ,$$ with $$\label{curvadscarrollsuper11}
\begin{split}
\mathcal{R}^{ab} & = d \omega^{ab} + \frac{1}{\ell^2} V^a V^b = R^{ab} + \frac{1}{\ell^2} V^a V^b \, , \\
\mathcal{K}^a & = d k^a + \omega^a_{\phantom{a} b} k^b + \frac{1}{\ell^2} V^a h + \frac{1}{2\ell} \bar{\psi}^+ \Gamma^{a0} \psi^+ - \frac{1}{2\ell} \bar{\psi}^- \Gamma^{a0} \psi^- = \mathfrak{K}^a + \frac{1}{2\ell} \bar{\psi}^+ \Gamma^{a0} \psi^+ - \frac{1}{2\ell} \bar{\psi}^- \Gamma^{a0} \psi^- \, , \\
R^a & = d V^a + \omega^a_{\phantom{a}b} V^b \, , \\
\mathcal{H} & = d h + V^a k_a - \frac{1}{2} \bar{\psi}^+ \Gamma^0 \psi^+ - \frac{1}{2} \bar{\psi}^- \Gamma^0 \psi^- = \mathfrak{H} - \frac{1}{2} \bar{\psi}^+ \Gamma^0 \psi^+ - \frac{1}{2} \bar{\psi}^- \Gamma^0 \psi^- \, , \\
\nabla \psi^+ & = d \psi^+ + \frac{1}{4} \omega^{ab} \Gamma_{ab} \psi^+ + \frac{1}{2 \ell} V^a \Gamma_a \psi^+ \, , \\
\nabla \psi^- & = d \psi^- + \frac{1}{4} \omega^{ab} \Gamma_{ab} \psi^- - \frac{1}{2 \ell} V^a \Gamma_a \psi^- \, .
\end{split}$$
Now, we move to the explicit construction of a CS action invariant under the $\mathcal{N}=(1,1)$ super-AdS Carroll group, on the same lines of what we have previously done in Section \[20case\] for $\mathcal{N}=(2,0)$. Thus, we consider the non-vanishing components of the (relativistic) invariant tensor given in , we decompose the indices as in , and we rescale not only the generators in compliance with but also the coefficients appearing in as in . Then, taking the ultra-relativistic limit $\sigma \rightarrow \infty$, we get the following non-vanishing components of an invariant tensor for the $\mathcal{N}=(1,1)$ AdS Carroll superalgebra: $$\label{invadscarrollsuper11}
\begin{split}
& \langle J_{ab} J_{cd} \rangle = \alpha_0 \left( \delta_{ad} \delta_{bc} - \delta_{ac} \delta_{bd} \right) \, , \\
& \langle J_{ab} H \rangle = \alpha_1 \epsilon_{ab} \, , \\
& \langle K_a P_b \rangle = - \alpha_1 \epsilon_{ab} \, , \\
& \langle P_{a} P_{b} \rangle = \frac{\alpha_0}{\ell^2} \delta_{ab} \, , \\
& \langle Q^+_\alpha Q^+_\beta \rangle = \langle Q^-_\alpha Q^-_\beta \rangle = 2 \alpha_1 C_{\alpha \beta} \, .
\end{split}$$ This invariant tensor is non-degenerate if $\alpha_1 \neq 0$.
Substituting the connection $1$-form in and the non-vanishing components of the invariant tensor in the general expression for a three-dimensional CS action, we end up with the $(1,1)$ AdS Carroll CS supergravity action in $2+1$ spacetime dimensions, that is $$\label{CSAC11}
\begin{split}
I^{(1,1)}_{CS} & = \frac{k}{4 \pi} \int_\mathcal{M} \Bigg \lbrace \frac{\alpha_0}{2} \left( \omega^a_{\phantom{a} b} R^b_{\phantom{b} a} + \frac{2}{\ell^2} V^a R_a \right) + \alpha_1 \bigg( \epsilon_{ab} R^{ab} h - 2 \epsilon_{ab} \mathfrak{K}^a V^b + \frac{1}{\ell^2} \epsilon_{ab} V^a V^b h \\
& + 2 \bar{\psi}^+ \nabla \psi^+ + 2 \bar{\psi}^- \nabla \psi^- \bigg) - d \left( \frac{\alpha_1}{2} \epsilon_{ab} \omega^{ab} h - \alpha_1 \epsilon_{ab} k^a V^b \right) \Bigg \rbrace \, ,
\end{split}$$ which can also be rewritten omitting boundary terms as follows: $$\begin{split}
I^{(1,1)}_{CS} & = \frac{k}{4 \pi} \int_\mathcal{M} \Bigg \lbrace \frac{\alpha_0}{2} \left( \omega^a_{\phantom{a} b} R^b_{\phantom{b} a} + \frac{2}{\ell^2} V^a R_a \right) + \alpha_1 \bigg( \epsilon_{ab} R^{ab} h - 2 \epsilon_{ab} \mathfrak{K}^a V^b + \frac{1}{\ell^2} \epsilon_{ab} V^a V^b h \\
& + 2 \bar{\psi}^+ \nabla \psi^+ + 2 \bar{\psi}^- \nabla \psi^- \bigg) \Bigg \rbrace \, .
\end{split}$$ The action has been written in terms of the curvatures appearing in , it involves two different sectors, respectively proportional to $\alpha_0$ (which corresponds to the exotic Lagrangian) and to $\alpha_1$, and it is invariant by construction under the $\mathcal{N}=(1,1)$ AdS Carroll superalgebra . The local gauge transformations $\delta_\lambda A = d \lambda + \left[A, \lambda \right]$ with gauge parameter $$\label{gpar11}
\lambda = \frac{1}{2} \lambda^{ab} J_{ab} + \kappa^a K_a + \lambda^a P_a + \tau H + \varepsilon^+ Q^+ + \varepsilon^- Q^-$$ are $$\label{gaugetr11}
\begin{split}
\delta \omega^{ab} & = d \lambda^{ab} + \frac{2}{\ell^2} V^{[a} \lambda^{b]} \, , \\
\delta k^a & = d \kappa^a - \lambda^a_{\phantom{a} b} k^b + \omega^a_{\phantom{a} b} \kappa^b - \frac{1}{\ell^2} \lambda^a h + \frac{1}{\ell^2} V^a \tau - \frac{1}{\ell} \bar{\varepsilon}^+ \Gamma^{a0} \psi^+ + \frac{1}{\ell} \bar{\varepsilon}^- \Gamma^{a0} \psi^- \, , \\
\delta V^a & = d \lambda^a - \lambda^a_{\phantom{a}b} V^b + \omega^a_{\phantom{a}b} \lambda^b \, , \\
\delta h & = d \tau - \lambda^a k_a + V^a \kappa_a + \bar{\varepsilon}^+ \Gamma^0 \psi^+ + \bar{\varepsilon}^- \Gamma^0 \psi^- \, , \\
\delta \psi^+ & = d \varepsilon^+ - \frac{1}{4} \lambda^{ab} \Gamma_{ab} \psi^+ + \frac{1}{4} \omega^{ab} \Gamma_{ab} \varepsilon^+ - \frac{1}{2 \ell} \lambda^a \Gamma_a \psi^+ + \frac{1}{2 \ell} V^a \Gamma_a \varepsilon^+ \, , \\
\delta \psi^- & = d \varepsilon^- - \frac{1}{4} \lambda^{ab} \Gamma_{ab} \psi^- + \frac{1}{4} \omega^{ab} \Gamma_{ab} \varepsilon^- + \frac{1}{2 \ell} \lambda^a \Gamma_a \psi^- - \frac{1}{2 \ell} V^a \Gamma_a \varepsilon^- \, ,
\end{split}$$ and, restricting ourselves to supersymmetry, we are left with the following transformation rules: $$\label{susytr11}
\begin{split}
\delta \omega^{ab} & = 0 \, , \\
\delta k^a & = - \frac{1}{\ell} \bar{\varepsilon}^+ \Gamma^{a0} \psi^+ + \frac{1}{\ell} \bar{\varepsilon}^- \Gamma^{a0} \psi^- \, , \\
\delta V^a & = 0 \, , \\
\delta h & = \bar{\varepsilon}^+ \Gamma^0 \psi^+ + \bar{\varepsilon}^- \Gamma^0 \psi^- \, , \\
\delta \psi^+ & = d \varepsilon^+ + \frac{1}{4} \omega^{ab} \Gamma_{ab} \varepsilon^+ + \frac{1}{2 \ell} V^a \Gamma_a \varepsilon^+ \, , \\
\delta \psi^- & = d \varepsilon^- + \frac{1}{4} \omega^{ab} \Gamma_{ab} \varepsilon^- - \frac{1}{2 \ell} V^a \Gamma_a \varepsilon^- \, .
\end{split}$$
The equations of motion obtained from the variation of the action with respect to the $1$-form fields $\omega^{ab}$, $k^a$, $V^a$, $h$, $\psi^+$, and $\psi^-$ are $$\label{eom11}
\begin{split}
\delta \omega^{ab} & : \quad \alpha_0 \mathcal{R}^{ab} + \alpha_1 \epsilon^{ab} \mathcal{H}= 0 \, , \\
\delta k^a & : \quad \alpha_1 R^a = 0 \, , \\
\delta V^a & : \quad \frac{\alpha_0}{\ell^2} R^a + 2 \alpha_1 \epsilon_{ab} \mathcal{K}^b = 0 \, , \\
\delta h & : \quad \alpha_1 \mathcal{R}^{ab} = 0 \, , \\
\delta \psi^+ & : \quad \alpha_1 \nabla \psi^+ = 0 \, , \\
\delta \psi^- & : \quad \alpha_1 \nabla \psi^- = 0 \, ,
\end{split}$$ respectively; for $\alpha_1 \neq 0$, they reduce to the vanishing of the $(1,0)$ super-AdS Carroll curvature $2$-forms, namely $$\label{eomvac11}
\mathcal{R}^{ab} = 0 \, , \quad \mathcal{K}^a = 0 \, , \quad R^a = 0 \, , \quad \mathcal{H}=0 \, , \quad \nabla \psi^+ =0 \, , \quad \nabla \psi^- =0 \, .$$ Analogously to what happened in the $(2,0)$ case discussed in Section \[20case\], we can see that $\alpha_1 \neq 0$ is a sufficient condition to recover , which means that one could consistently set $\alpha_0=0$, making the exotic term in the CS action disappear.
$(\mathcal{N},0)$ AdS Carroll supergravity theories in $2+1$ dimensions {#ncase}
=======================================================================
Now, we generalize our analysis to the $(\mathcal{N},0)$ case, with $\mathcal{N}$ even. First, we present the derivation of the $\mathcal{N}=(\mathcal{N},0)$ AdS Carroll superalgebra as the Carrollian contraction of an $\mathfrak{so}(\mathcal{N})$ extension of $\mathfrak{osp}(2|\mathcal{N})\otimes \mathfrak{sp}(2)$. This also provides us with a non-degenerate invariant tensor in the ultra-relativistic limit. Then, we can subsequently formulate a well-defined three-dimensional CS supergravity action invariant under the aforesaid $\mathcal{N}=(\mathcal{N},0)$ AdS Carroll superalgebra.
$\mathcal{N}=(\mathcal{N},0)$ AdS Carroll superalgebra
------------------------------------------------------
Let us first take the direct sum of $\mathfrak{osp}(2|\mathcal{N})\otimes \mathfrak{sp}(2)$ and an $\mathfrak{so}(\mathcal{N})$ algebra (we consider $\mathcal{N}$ even), that is reminiscent of what was done in Ref. [@Howe:1995zm]. In this case, the non-trivial (anti)commutation relations are $$\label{ospnnonmod}
\begin{split}
& \left[ \tilde{J}_{AB}, \tilde{J}_{CD}\right] =\eta _{BC}\tilde{J}_{AD}-\eta _{AC}\tilde{J}_{BD}-\eta
_{BD}\tilde{J}_{AC}+\eta _{AD}\tilde{J}_{BC} \, , \\
& \left[ \tilde{J}_{AB},\tilde{P}_{C}\right] =\eta _{BC}\tilde{P}_{A}-\eta _{AC}\tilde{P}_{B} \, , \quad \quad \left[ \tilde{P}_A , \tilde{P}_B \right] = \frac{1}{\ell^2} \tilde{J}_{AB} \, , \\
& \left[ \tilde{Z}^{ij} , \tilde{Z}^{kl} \right] = \delta^{jk} \tilde{Z}^{il} - \delta^{ik} \tilde{Z}^{jl}- \delta^{jl} \tilde{Z}^{ik} + \delta^{il} \tilde{Z}^{jk} \, , \\
& \left[ \tilde{S}^{ij} , \tilde{S}^{kl} \right] = - \frac{1}{\ell} \left( \delta^{jk} \tilde{S}^{il} - \delta^{ik} \tilde{S}^{jl}- \delta^{jl} \tilde{S}^{ik} + \delta^{il} \tilde{S}^{jk} \right) \, , \\
& \left[ \tilde{J}_{AB},\tilde{Q}^i_{\alpha }\right] =-\frac{1}{2}\left( \Gamma _{AB}\tilde{Q}^i\right)_{\alpha } \, , \quad \quad \left[ \tilde{P}_{A}, \tilde{Q}^i_{\alpha }\right] =- \frac{1}{2 \ell} \left( \Gamma _{A}\tilde{Q}^i \right) _{\alpha } \, , \\
& \left[ \tilde{Z}^{ij}, \tilde{Q}^k_{\alpha }\right] = \delta^{jk} \tilde{Q}^i_\alpha - \delta^{ik} \tilde{Q}^j_\alpha \, , \\
& \left\{ \tilde{Q}^i_{\alpha }, \tilde{Q}^j_{\beta }\right\} = \delta^{ij} \left[ - \frac{1}{2 \ell} \left(\Gamma^{AB} C \right)_{\alpha \beta} \tilde{J}_{AB} + \left( \Gamma ^{A}C\right) _{\alpha \beta }\tilde{P}_{A} \right] + \frac{1}{\ell} C_{\alpha \beta} \tilde{Z}^{ij} \, ,
\end{split}$$ with $A, B, \ldots=0,1,2$, $i,j,\ldots=1,\ldots,\mathcal{N}$ (where we have considered $\mathcal{N}=2x$, $x=1,\ldots,\frac{\mathcal{N}}{2}$), and where $\tilde{Z}^{ij}=-\tilde{Z}^{ji}$, $\tilde{S}^{ij}=-\tilde{S}^{ij}$. Then, we do the following redefinition (on the same lines of [@Howe:1995zm]): $$\tilde{T}^{ij} \equiv \tilde{Z}^{ij} - \ell \tilde{S}^{ij} \, ,$$ which is a generalization of the one performed in Section \[20case\]. Thus, we can now rewrite the (anti)commutation relations as $$\label{ospn}
\begin{split}
& \left[ \tilde{J}_{AB}, \tilde{J}_{CD}\right] =\eta _{BC}\tilde{J}_{AD}-\eta _{AC}\tilde{J}_{BD}-\eta
_{BD}\tilde{J}_{AC}+\eta _{AD}\tilde{J}_{BC} \, , \\
& \left[ \tilde{J}_{AB},\tilde{P}_{C}\right] =\eta _{BC}\tilde{P}_{A}-\eta _{AC}\tilde{P}_{B} \, , \\
& \left[ \tilde{P}_A , \tilde{P}_B \right] = \frac{1}{\ell^2} \tilde{J}_{AB} \, , \\
& \left[ \tilde{T}^{ij} , \tilde{T}^{kl} \right] = \delta^{jk} \tilde{T}^{il} - \delta^{ik} \tilde{T}^{jl}- \delta^{jl} \tilde{T}^{ik} + \delta^{il} \tilde{T}^{jk} \, , \\
& \left[ \tilde{T}^{ij} , \tilde{S}^{kl} \right] = \delta^{jk} \tilde{S}^{il} - \delta^{ik} \tilde{S}^{jl}- \delta^{jl} \tilde{S}^{ik} + \delta^{il} \tilde{S}^{jk} \, , \\
& \left[ \tilde{S}^{ij} , \tilde{S}^{kl} \right] = - \frac{1}{\ell} \left( \delta^{jk} \tilde{S}^{il} - \delta^{ik} \tilde{S}^{jl}- \delta^{jl} \tilde{S}^{ik} + \delta^{il} \tilde{S}^{jk} \right) \, , \\
& \left[ \tilde{J}_{AB},\tilde{Q}^i_{\alpha }\right] =-\frac{1}{2}\left( \Gamma _{AB}\tilde{Q}^i\right)_{\alpha } \, , \\
& \left[ \tilde{P}_{A}, \tilde{Q}^i_{\alpha }\right] =- \frac{1}{2 \ell} \left( \Gamma _{A}\tilde{Q}^i \right) _{\alpha } \, , \\
& \left[ \tilde{T}^{ij}, \tilde{Q}^k_{\alpha }\right] = \delta^{jk} \tilde{Q}^i_\alpha - \delta^{ik} \tilde{Q}^j_\alpha \, , \\
& \left\{ \tilde{Q}^i_{\alpha }, \tilde{Q}^j_{\beta }\right\} = \delta^{ij} \left[ - \frac{1}{2 \ell} \left(\Gamma^{AB} C \right)_{\alpha \beta} \tilde{J}_{AB} + \left( \Gamma ^{A}C\right) _{\alpha \beta }\tilde{P}_{A} \right] + C_{\alpha \beta} \left(\frac{1}{\ell} \tilde{T}^{ij} + \tilde{S}^{ij} \right) \, .
\end{split}$$ Observe that, taking the limit $\ell \rightarrow \infty$ of , we get the $\mathcal{N}=(\mathcal{N},0)$ Poincaré superalgebra involving a semi-direct $\mathfrak{so}(\mathcal{N})$ extension (with $\mathcal{N}$ even).
The non-vanishing components of an invariant tensor for , which will be useful in the sequel, are given by $$\label{invtospn}
\begin{split}
& \langle \tilde{J}_{AB} \tilde{J}_{CD} \rangle = \alpha_0 \left( \eta_{AD} \eta_{BC} - \eta_{AC} \eta_{BD} \right) \, , \\
& \langle \tilde{J}_{AB} \tilde{P}_{C} \rangle = \alpha_1 \epsilon_{ABC} \, , \\
& \langle \tilde{P}_{A} \tilde{P}_{B} \rangle = \frac{\alpha_0}{\ell^2} \eta_{AB} \, , \\
& \langle \tilde{T}^{ij} \tilde{T}^{kl} \rangle = 2 \alpha_0 \left( \delta^{il} \delta^{kj} - \delta^{ik} \delta^{lj} \right) \, , \\
& \langle \tilde{T}^{ij} \tilde{S}^{kl} \rangle = - 2 \alpha_1 \left( \delta^{il} \delta^{kj} - \delta^{ik} \delta^{lj} \right) \, , \\
& \langle \tilde{S}^{ij} \tilde{S}^{kl} \rangle = \frac{2 \alpha_1}{\ell} \left( \delta^{il} \delta^{kj} - \delta^{ik} \delta^{lj} \right) \, , \\
& \langle \tilde{Q}^i_\alpha \tilde{Q}^j_\beta \rangle = 2 \left(\alpha_1 - \frac{\alpha_0}{\ell} \right) C_{\alpha \beta} \delta^{ij} \, ,
\end{split}$$ being $\alpha_0$ and $\alpha_1$ arbitrary independent constants.
In order to take the ultra-relativistic contraction of , we decompose the indices $A,B,\ldots = 0,1,2$ as in , which induces the decomposition , together with the gamma matrices decomposition .
Moreover, we define, on the same lines of what was done in [@Lukierski:2006tr] (see also references therein) in the case of non-relativistic theories, new supersymmetry charges by $$\label{newsusyn}
\tilde{Q}^{\pm \, \lambda} _\alpha = \frac{1}{\sqrt{2}} \left( \tilde{Q}^\lambda_\alpha \pm \left(\Gamma_0\right)_{\alpha \beta} \tilde{Q}^{x + \lambda}_\beta \right) \, ,$$ where in we consider $\lambda, \mu , \ldots = 1, \ldots, x$ (these new indices must not be confused with the spinor ones $\alpha, \beta, \ldots = 1,2$), generalizing to the $\mathcal{N}=(\mathcal{N},0)$ case, with $\mathcal{N}$ even, what we have previously done in Section \[20case\] (see, in particular, ). This also reflects on the generators $\tilde{T}^{ij}$ and $\tilde{S}^{ij}$, which are now respectively described by $$\label{dectijsij}
\begin{split}
& \tilde{T}^{\lambda \mu} \, , \quad \tilde{T}'^{\lambda \mu} \equiv \tilde{T}^{\lambda + x \ \mu + x} \, , \quad \tilde{U}^{\lambda \mu} \equiv \tilde{T}^{x+\lambda \ \mu} \, , \quad \tilde{U}'^{\lambda \mu} \equiv \tilde{T}^{\lambda \ x + \mu} \, , \\
& \tilde{S}^{\lambda \mu} \, , \quad \tilde{S}'^{\lambda \mu} \equiv \tilde{S}^{\lambda + x \ \mu + x} \, , \quad \tilde{V}^{\lambda \mu} \equiv \tilde{S}^{x+\lambda \ \mu} \, , \quad \tilde{V}'^{\lambda \mu} \equiv \tilde{S}^{\lambda \ x + \mu} \, ,
\end{split}$$ which satisfy the symmetry properties $$\label{symmprop}
\begin{split}
& \tilde{T}^{\lambda \mu} = - \tilde{T}^{\mu \lambda} \, , \quad \tilde{T}'^{\lambda \mu} = - \tilde{T}'^{\mu \lambda} \, , \quad \tilde{U}^{\lambda \mu} = - \tilde{U}'^{\mu \lambda} \, , \\
& \tilde{S}^{\lambda \mu} = - \tilde{S}^{\mu \lambda} \, , \quad \tilde{S}'^{\lambda \mu} = - \tilde{S}'^{\mu \lambda} \, , \quad \tilde{V}^{\lambda \mu} = - \tilde{V}'^{\mu \lambda} \, .
\end{split}$$
In particular, using , , and , together with the decomposition and , and defining $$\label{xydef}
\begin{split}
& \tilde{X}^{[\lambda \mu]} \equiv \frac{1}{2} \left( \tilde{T}^{\lambda \mu} + \tilde{T}'^{\lambda \mu} \right) \, , \quad \tilde{X}'^{[\lambda \mu]} \equiv \frac{1}{2} \left( \tilde{T}^{\lambda \mu} - \tilde{T}'^{\lambda \mu} \right) \, , \\
& \tilde{Y}^{[\lambda \mu]} \equiv \frac{1}{2} \left( \tilde{S}^{\lambda \mu} + \tilde{S}'^{\lambda \mu} \right) \, , \quad \tilde{Y}'^{[\lambda \mu]} \equiv \frac{1}{2} \left( \tilde{S}^{\lambda \mu} - \tilde{S}'^{\lambda \mu} \right) \, ,
\end{split}$$ the (anti)commutation relations in become $$\label{antinew}
\begin{split}
\lbrace \tilde{Q}^{\pm \, \lambda} _\alpha , \tilde{Q}^{\pm \, \mu} _\beta \rbrace & = \delta^{\lambda \mu} \left[ - \frac{1}{2 \ell} \left( \Gamma^{ab} C \right)_{\alpha \beta} \tilde{J}_{ab} + \left( \Gamma^0 C \right)_{\alpha \beta} \tilde{H} \right] + C_{\alpha \beta} \left( \frac{1}{\ell} \tilde{X}^{[\lambda \mu]} + \tilde{Y}^{[\lambda \mu]} \right) \\
& \mp \left( \Gamma^0 C \right)_{\alpha \beta} \left( \frac{1}{\ell} \tilde{U}^{(\lambda \mu)} + \tilde{V}^{(\lambda \mu)} \right) \, , \\
\lbrace \tilde{Q}^{+ \, \lambda} _\alpha , \tilde{Q}^{- \, \mu} _\beta \rbrace & = \delta^{\lambda \mu} \left[ - \frac{1}{\ell} \left( \Gamma^{a0} C \right)_{\alpha \beta} \tilde{K}_{a} + \left( \Gamma^a C \right)_{\alpha \beta} \tilde{P}_a \right] + C_{\alpha \beta} \left( \frac{1}{\ell} \tilde{X}'^{[\lambda \mu]} + \tilde{Y}'^{[\lambda \mu]} \right) \\
& - \left( \Gamma^0 C \right)_{\alpha \beta} \left( \frac{1}{\ell} \tilde{U}^{[\lambda \mu]} + \tilde{V}^{[\lambda \mu]} \right) \, ,
\end{split}$$ where $$\label{uvsas}
\begin{split}
& \tilde{U}^{(\lambda \mu)} = \frac{1}{2} \left( \tilde{U}^{\lambda \mu} + \tilde{U}^{\mu \lambda} \right) \, , \quad \tilde{U}^{[\lambda \mu]} = \frac{1}{2} \left( \tilde{U}^{\lambda \mu} - \tilde{U}^{\mu \lambda} \right) \, , \\
& \tilde{V}^{(\lambda \mu)} = \frac{1}{2} \left( \tilde{V}^{\lambda \mu} + \tilde{V}^{\mu \lambda} \right) \, , \quad \tilde{V}^{[\lambda \mu]} = \frac{1}{2} \left( \tilde{V}^{\lambda \mu} - \tilde{V}^{\mu \lambda} \right) \, .
\end{split}$$ The generators $\tilde{X}^{[\lambda \mu]}$ in amount to $\frac{k(k-1)}{2}$ generators, and the same holds for $\tilde{X}'^{[\lambda \mu]}$, $\tilde{U}^{[\lambda \mu]}$, $\tilde{Y}^{[\lambda \mu]}$, $\tilde{Y}'^{[\lambda\mu]}$, and $\tilde{V}^{[\lambda \mu]}$, while the generators $\tilde{U}^{(\lambda \mu)}$ are $\frac{k(k+1)}{2}$ generators, and the same holds for the generators $\tilde{V}^{(\lambda \mu)}$.
Furthermore, from the commutation relations involving the generators $T^{ij}$ and $S^{ij}$, we get the following non-vanishing ones (recall that we have $\tilde{U}'^{\lambda \mu} =- \tilde{U}^{\mu \lambda}$ and $\tilde{V}'^{\lambda \mu} =- \tilde{V}^{\mu \lambda}$): $$\label{newtttsss}
\begin{split}
& \left[ \tilde{T}^{\lambda \mu} , \tilde{T}^{\nu \rho} \right] = \delta^{\mu \nu} \tilde{T}^{\lambda \rho} - \delta^{\lambda \nu} \tilde{T}^{\mu \rho}- \delta^{\mu \rho} \tilde{T}^{\lambda \nu} + \delta^{\lambda \rho} \tilde{T}^{\mu \nu} \, , \\
& \left[ \tilde{T}^{\lambda \mu} , \tilde{U}^{\nu \rho} \right] = \delta^{\mu \rho} \tilde{U}^{\nu \lambda} - \delta^{\lambda \rho} \tilde{U}^{\nu \mu} \, , \\
& \left[ \tilde{T}'^{\lambda \mu} , \tilde{T}'^{\nu \rho} \right] = \delta^{\mu \nu} \tilde{T}'^{\lambda \rho} - \delta^{\lambda \nu} \tilde{T}'^{\mu \rho}- \delta^{\mu \rho} \tilde{T}'^{\lambda \nu} + \delta^{\lambda \rho} \tilde{T}'^{\mu \nu} \, , \\
& \left[ \tilde{T}'^{\lambda \mu} , \tilde{U}^{\nu \rho} \right] = \delta^{\mu \nu} \tilde{U}^{\lambda \rho}- \delta^{\lambda \nu} \tilde{U}^{\mu \rho} \, , \\
& \left[ \tilde{U}^{\lambda \mu} , \tilde{U}^{\nu \rho} \right] = - \delta^{\lambda \nu} \tilde{T}^{\mu \rho} - \delta^{\mu \rho} \tilde{T}'^{\lambda \nu} \, , \\
& \left[ \tilde{T}^{\lambda \mu} , \tilde{S}^{\nu \rho} \right] = \delta^{\mu \nu} \tilde{S}^{\lambda \rho} - \delta^{\lambda \nu} \tilde{S}^{\mu \rho}- \delta^{\mu \rho} \tilde{S}^{\lambda \nu} + \delta^{\lambda \rho} \tilde{S}^{\mu \nu} \, , \\
& \left[ \tilde{T}^{\lambda \mu} , \tilde{V}^{\nu \rho} \right] = \delta^{\mu \rho} \tilde{V}^{\nu \lambda} - \delta^{\lambda \rho} \tilde{V}^{\nu \mu} \, , \\
& \left[ \tilde{T}'^{\lambda \mu} , \tilde{S}'^{\nu \rho} \right] = \delta^{\mu \nu} \tilde{S}'^{\lambda \rho} - \delta^{\lambda \nu} \tilde{S}'^{\mu \rho}- \delta^{\mu \rho} \tilde{S}'^{\lambda \nu} + \delta^{\lambda \rho} \tilde{S}'^{\mu \nu} \, , \\
& \left[ \tilde{T}'^{\lambda \mu} , \tilde{V}^{\nu \rho} \right] = \delta^{\mu \nu} \tilde{V}^{\lambda \rho}- \delta^{\lambda \nu} \tilde{V}^{\mu \rho} \, , \\
& \left[ \tilde{U}^{\lambda \mu} , \tilde{S}^{\nu \rho} \right] = - \delta^{\mu \rho} \tilde{V}^{\lambda \nu} + \delta^{\mu \nu} \tilde{V}^{\lambda \rho} \, , \\
& \left[ \tilde{U}^{\lambda \mu} , \tilde{S}'^{\nu \rho} \right] = - \delta^{\lambda \rho} \tilde{V}^{\nu \mu} + \delta^{\lambda \nu} \tilde{V}^{\rho \mu} \, , \\
& \left[ \tilde{U}^{\lambda \mu} , \tilde{V}^{\nu \rho} \right] = - \delta^{\lambda \nu} \tilde{S}^{\mu \rho} - \delta^{\mu \rho} \tilde{S}'^{\lambda \nu} \, , \\
& \left[ \tilde{S}^{\lambda \mu} , \tilde{S}^{\nu \rho} \right] = - \frac{1}{\ell} \left( \delta^{\mu \nu} \tilde{S}^{\lambda \rho} - \delta^{\lambda \nu} \tilde{S}^{\mu \rho}- \delta^{\mu \rho} \tilde{S}^{\lambda \nu} + \delta^{\lambda \rho} \tilde{S}^{\mu \nu} \right) \, , \\
& \left[ \tilde{S}^{\lambda \mu} , \tilde{V}^{\nu \rho} \right] = - \frac{1}{\ell} \left( \delta^{\mu \rho} \tilde{V}^{\nu \lambda} - \delta^{\lambda \rho} \tilde{V}^{\nu \mu} \right) \, , \\
& \left[ \tilde{S}'^{\lambda \mu} , \tilde{S}'^{\nu \rho} \right] = - \frac{1}{\ell} \left( \delta^{\mu \nu} \tilde{S}'^{\lambda \rho} - \delta^{\lambda \nu} \tilde{S}'^{\mu \rho}- \delta^{\mu \rho} \tilde{S}'^{\lambda \nu} + \delta^{\lambda \rho} \tilde{S}'^{\mu \nu} \right) \, , \\
& \left[ \tilde{S}'^{\lambda \mu} , \tilde{V}^{\nu \rho} \right] = - \frac{1}{\ell} \left( \delta^{\mu \nu} \tilde{V}^{\lambda \rho}- \delta^{\lambda \nu} \tilde{V}^{\mu \rho} \right) \, , \\
& \left[ \tilde{V}^{\lambda \mu} , \tilde{V}^{\nu \rho} \right] = \frac{1}{\ell} \left(\delta^{\lambda \nu} \tilde{S}^{\mu \rho} + \delta^{\mu \rho} \tilde{S}'^{\lambda \nu} \right) \, ,
\end{split}$$ and $$\label{newtq}
\begin{split}
& \left[ \tilde{T}^{\lambda \mu}, \tilde{Q}^{\pm \, \nu } _\alpha \right] = \frac{1}{2} \left[\delta^{\mu \nu} \left( \tilde{Q}^{+ \, \lambda } _\alpha + \tilde{Q}^{- \, \lambda } _\alpha \right) - \delta^{\lambda \nu} \left( \tilde{Q}^{+ \, \mu } _\alpha + \tilde{Q}^{- \, \mu } _\alpha \right) \right] \, , \\
& \left[ \tilde{T}'^{\lambda \mu}, \tilde{Q}^{\pm \, \nu } _\alpha \right] = \pm \frac{1}{2} \left[\delta^{\mu \nu} \left( \tilde{Q}^{+ \, \lambda } _\alpha - \tilde{Q}^{- \, \lambda } _\alpha \right) - \delta^{\lambda \nu} \left( \tilde{Q}^{+ \, \mu } _\alpha - \tilde{Q}^{- \, \mu } _\alpha \right) \right] \, , \\
& \left[ \tilde{U}^{\lambda \mu}, \tilde{Q}^{\pm \, \nu } _\alpha \right] = \mp \frac{1}{2} \left( \Gamma_0 \right)_{\alpha \beta} \left[ \delta^{\lambda \nu} \left( \tilde{Q}^{+ \, \mu } _\beta + \tilde{Q}^{- \, \ \mu } _\beta \right) \pm \delta^{\mu \nu} \left( \tilde{Q}^{+ \, \lambda } _\beta - \tilde{Q}^{- \, \lambda } _\beta \right) \right] \, .
\end{split}$$
Now, let us rescale the generators with a parameter $\sigma$ as $$\label{rescn}
\tilde{H} \rightarrow \sigma H \, , \quad \tilde{K}_a \rightarrow \sigma K_a \, , \quad \tilde{S}^{\lambda \mu} \rightarrow \sigma S^{\lambda \mu} \, , \quad \tilde{S}'^{\lambda \mu} \rightarrow \sigma S'^{\lambda \mu} \, , \quad \tilde{V}^{\lambda \mu} \rightarrow \sigma V^{\lambda \mu} \, , \quad \tilde{Q}^{\pm \, \lambda}_\alpha \rightarrow \sqrt{\sigma} Q^{\pm \, \lambda}_\alpha \, ,$$ where we have also removed the tilde symbol on the generators. Taking the limit $\sigma \rightarrow \infty$ (and removing the tilde symbol also on the generators that we have not rescaled), we end up with the $\mathcal{N}=(\mathcal{N},0)$ AdS Carroll superalgebra (with $\mathcal{N}$ even), whose non-trivial (anti)commutation relations read as follows (recall the definitions , , since we have expressed the anticommutation relations in terms of the combinations given in that expressions, together with the fact that $\tilde{U}'^{\lambda \mu} =- \tilde{U}^{\mu \lambda}$ and $\tilde{V}'^{\lambda \mu} =- \tilde{V}^{\mu \lambda}$, and the (anti)commutation relations , , and ): $$\label{adscarrollsupern}
\begin{split}
& \left[ K_a , J_{bc} \right] = \delta_{ab} K_c - \delta_{ac} K_b \, , \\
& \left[ J_{ab}, P_{c}\right] =\delta_{bc}P_{a}-\delta _{ac}P_{b} \, , \\
& \left[ K_{a}, P_{b}\right] = - \delta_{ab} H \, , \\
& \left[ P_a , P_b \right] = \frac{1}{\ell^2} J_{ab} \, , \quad \quad \left[ P_a , H \right] = \frac{1}{\ell^2} K_{a} \, , \\
& \left[ {T}^{\lambda \mu} , {T}^{\nu \rho} \right] = \delta^{\mu \nu} {T}^{\lambda \rho} - \delta^{\lambda \nu} {T}^{\mu \rho}- \delta^{\mu \rho} {T}^{\lambda \nu} + \delta^{\lambda \rho} {T}^{\mu \nu} \, , \\
& \left[ {T}^{\lambda \mu} , {U}^{\nu \rho} \right] = \delta^{\mu \rho} {U}^{\nu \lambda} - \delta^{\lambda \rho} {U}^{\nu \mu} \, , \\
& \left[ {T}'^{\lambda \mu} , {T}'^{\nu \rho} \right] = \delta^{\mu \nu} {T}'^{\lambda \rho} - \delta^{\lambda \nu} {T}'^{\mu \rho}- \delta^{\mu \rho} {T}'^{\lambda \nu} + \delta^{\lambda \rho} {T}'^{\mu \nu} \, , \\
& \left[ {T}'^{\lambda \mu} , {U}^{\nu \rho} \right] = \delta^{\mu \nu} {U}^{\lambda \rho}- \delta^{\lambda \nu} {U}^{\mu \rho} \, , \\
& \left[ {U}^{\lambda \mu} , {U}^{\nu \rho} \right] = - \delta^{\lambda \nu} {T}^{\mu \rho} - \delta^{\mu \rho} {T}'^{\lambda \nu} \, , \\
& \left[ {T}^{\lambda \mu} , {S}^{\nu \rho} \right] = \delta^{\mu \nu} {S}^{\lambda \rho} - \delta^{\lambda \nu} {S}^{\mu \rho}- \delta^{\mu \rho} {S}^{\lambda \nu} + \delta^{\lambda \rho} {S}^{\mu \nu} \, , \\
& \left[ {T}^{\lambda \mu} , {V}^{\nu \rho} \right] = \delta^{\mu \rho} {V}^{\nu \lambda} - \delta^{\lambda \rho} {V}^{\nu \mu} \, , \\
& \left[ {T}'^{\lambda \mu} , {S}'^{\nu \rho} \right] = \delta^{\mu \nu} {S}'^{\lambda \rho} - \delta^{\lambda \nu} {S}'^{\mu \rho}- \delta^{\mu \rho} {S}'^{\lambda \nu} + \delta^{\lambda \rho} {S}'^{\mu \nu} \, , \\
& \left[ {T}'^{\lambda \mu} , {V}^{\nu \rho} \right] = \delta^{\mu \nu} {V}^{\lambda \rho}- \delta^{\lambda \nu} {V}^{\mu \rho} \, , \\
& \left[ {U}^{\lambda \mu} , {S}^{\nu \rho} \right] = - \delta^{\mu \rho} {V}^{\lambda \nu} + \delta^{\mu \nu} {V}^{\lambda \rho} \, , \\
& \left[ {U}^{\lambda \mu} , {S}'^{\nu \rho} \right] = - \delta^{\lambda \rho} {V}^{\nu \mu} + \delta^{\lambda \nu} {V}^{\rho \mu} \, , \\
& \left[ {U}^{\lambda \mu} , {V}^{\nu \rho} \right] = - \delta^{\lambda \nu} {S}^{\mu \rho} - \delta^{\mu \rho} {S}'^{\lambda \nu} \, , \\
& \left[ J_{ab}, Q^{\pm \, \lambda} _\alpha \right] =-\frac{1}{2}\left( \Gamma _{ab} Q^{\pm \, \lambda} \right)_{\alpha } \, , \quad \quad \left[ P_{a}, Q^{\pm \, \lambda} _\alpha \right] =- \frac{1}{2 \ell} \left( \Gamma _{a} Q^{\mp \, \lambda} \right) _\alpha \, , \\
& \left[ {T}^{\lambda \mu}, {Q}^{\pm \, \nu } _\alpha \right] = \frac{1}{2} \left[\delta^{\mu \nu} \left( {Q}^{+ \, \lambda } _\alpha + {Q}^{- \, \lambda } _\alpha \right) - \delta^{\lambda \nu} \left( {Q}^{+ \, \mu } _\alpha + {Q}^{- \, \mu } _\alpha \right) \right] \, , \\
& \left[ {T}'^{\lambda \mu}, {Q}^{\pm \, \nu } _\alpha \right] = \pm \frac{1}{2} \left[\delta^{\mu \nu} \left( {Q}^{+ \, \lambda } _\alpha - {Q}^{- \, \lambda } _\alpha \right) - \delta^{\lambda \nu} \left( {Q}^{+ \, \mu } _\alpha - {Q}^{- \, \mu } _\alpha \right) \right] \, , \\
& \left[ {U}^{\lambda \mu}, {Q}^{\pm \, \nu } _\alpha \right] = \mp \frac{1}{2} \left( \Gamma_0 \right)_{\alpha \beta} \left[ \delta^{\lambda \nu} \left( {Q}^{+ \, \mu } _\beta + {Q}^{- \, \ \mu } _\beta \right) \pm \delta^{\mu \nu} \left( {Q}^{+ \, \lambda } _\beta - {Q}^{- \, \lambda } _\beta \right) \right] \, ,
\end{split}$$ $$\nonumber
\begin{split}
& \lbrace {Q}^{+ \, \lambda} _\alpha , {Q}^{+ \, \mu} _\beta \rbrace = \left( \Gamma^0 C \right)_{\alpha \beta} \left( \delta^{\lambda \mu} {H} - {V}^{(\lambda \mu)} \right) + C_{\alpha \beta} {Y}^{[\lambda \mu]} \, , \\
& \lbrace {Q}^{+ \, \lambda} _\alpha , {Q}^{- \, \mu} _\beta \rbrace = - \frac{1}{\ell} \delta^{\lambda \mu} \left( \Gamma^{a0} C \right)_{\alpha \beta} {K}_{a} + C_{\alpha \beta} {Y}'^{[\lambda \mu]} - \left( \Gamma^0 C \right)_{\alpha \beta} {V}^{[\lambda \mu]} \, , \\
& \lbrace {Q}^{- \, \lambda} _\alpha , {Q}^{- \, \mu} _\beta \rbrace = \left( \Gamma^0 C \right)_{\alpha \beta} \left( \delta^{\lambda \mu} {H} + {V}^{(\lambda \mu)} \right) + C_{\alpha \beta} {Y}^{[\lambda \mu]} \, .
\end{split}$$ Notice that if we restrict ourselves to the special case $\mathcal{N}=(2,0)$, that is $x=1$, after some algebraic calculations, exploiting the definitions , , , and the symmetry properties , we exactly reproduce the $\mathcal{N}=(2,0)$ AdS Carroll superalgebra obtained in Section \[20case\], given by .[^7]
In the sequel, we will construct a CS action in $2+1$ dimensions invariant under .
$(\mathcal{N},0)$ AdS Carroll supergravity
------------------------------------------
We can now move to the formulation of a three-dimensional CS supergravity action invariant under the superalgebra . We call this action $(\mathcal{N},0)$ AdS Carroll CS supergravity action (where, in our analysis, $\mathcal{N}$ is even).
To this aim, let us start by introducing the connection $1$-form $A$ associated with the superalgebra , that is $$\label{connadscarrollsupern}
\begin{split}
A & = \frac{1}{2} \omega^{ab} J_{ab} + k^a K_a + V^a P_a + h H \\
& + \frac{1}{2} t^{\lambda \mu} T_{\lambda \mu} + \frac{1}{2} t'^{\lambda \mu} T'_{\lambda \mu} + u^{\lambda \mu} U_{\lambda \mu} + \frac{1}{2} s^{\lambda \mu} S_{\lambda \mu} + \frac{1}{2} s'^{\lambda \mu} S'_{\lambda \mu} + v^{\lambda \mu} V_{\lambda \mu} \\
& + \psi^+_\lambda Q^{+ \, \lambda} + \psi^-_\lambda Q^{- \, \lambda} \, ,
\end{split}$$ where $\omega^{ab}$, $k^a$, $V^a$, $h$, $t^{\lambda \mu}$, $t'^{\lambda \mu}$, $u^{\lambda \mu}$, $s^{\lambda \mu}$, $s'^{\lambda \mu}$, $v^{\lambda \mu}$, $\psi^+_\lambda$, and $\psi^-_\lambda$ are the $1$-form fields dual to the generators $J_{ab}$, $K_a$, $P_a$, $H$, $T_{\lambda \mu}$, $T'_{\lambda \mu}$, $U_{\lambda \mu}$, $S_{\lambda \mu}$, $S'_{\lambda \mu}$, $V_{\lambda \mu}$, $Q^{+ \, \lambda}$, and $Q^{- \, \lambda}$, respectively.
The corresponding curvature $2$-form $F$ is $$\label{curv2fn}
\begin{split}
F & = \frac{1}{2} \mathcal{R}^{ab} J_{ab} + \mathcal{K}^a K_a + R^a P_a + \mathcal{H} H \\
& + \frac{1}{2} \mathcal{T}^{\lambda \mu} T_{\lambda \mu} + \frac{1}{2} \mathcal{T}'^{\lambda \mu} T'_{\lambda \mu} + \mathcal{U}^{\lambda \mu} U_{\lambda \mu} + \frac{1}{2} \mathcal{S}^{\lambda \mu} S_{\lambda \mu} + \frac{1}{2} \mathcal{S}'^{\lambda \mu} S'_{\lambda \mu} + \mathcal{V}^{\lambda \mu} V_{\lambda \mu} \\
& + \nabla \psi^+_\lambda Q^{+ \, \lambda} + \nabla \psi^-_\lambda Q^{- \, \lambda} \, ,
\end{split}$$ with $$\label{curvadscarrollsupern}
\begin{split}
\mathcal{R}^{ab} & = d \omega^{ab} + \frac{1}{\ell^2} V^a V^b = R^{ab} + \frac{1}{\ell^2} V^a V^b \, , \\
\mathcal{K}^a & = d k^a + \omega^a_{\phantom{a} b} k^b + \frac{1}{\ell^2} V^a h + \frac{1}{\ell} \bar{\psi}^{+ \, \lambda} \Gamma^{a0} \psi^{- \, \lambda} = \mathfrak{K}^a + \frac{1}{\ell^2} V^a h + \frac{1}{\ell} \bar{\psi}^{+ \, \lambda} \Gamma^{a0} \psi^{- \, \lambda} \, , \\
R^a & = d V^a + \omega^a_{\phantom{a}b} V^b \, , \\
\mathcal{H} & = d h + V^a k_a - \frac{1}{2} \bar{\psi}^{+ \, \lambda} \Gamma^0 \psi^{+ \, \lambda} - \frac{1}{2} \bar{\psi}^{- \, \lambda} \Gamma^0 \psi^{- \, \lambda} = \mathfrak{H} - \frac{1}{2} \bar{\psi}^{+ \, \lambda} \Gamma^0 \psi^{+ \, \lambda} - \frac{1}{2} \bar{\psi}^{- \, \lambda} \Gamma^0 \psi^{- \, \lambda} \, , \\
\mathcal{T}^{\lambda \mu} & = d t^{\lambda \mu} + {t^\lambda}_\nu t^{\nu \mu} + {u'^{[\lambda}}_\nu u^{\nu \vert \mu]} \, , \\
\mathcal{T}'^{\lambda \mu} & = d t'^{\lambda \mu} + {t'^\lambda}_\nu t'^{\nu \mu} + {u^{[\lambda}}_{ \nu} u'^{\nu \vert \mu]} \, , \\
\mathcal{U}^{\lambda \mu} & = d u^{\lambda \mu} + {u^\lambda}_\nu t^{\nu \mu} + {t'^\lambda}_\nu u^{\nu \mu} \, , \\
\mathcal{S}^{\lambda \mu} & = d s^{\lambda \mu} + 2 {t^\lambda}_\nu s^{\nu \mu} + 2 {u'^{[\lambda}}_\nu v^{\nu \vert \mu]} - \frac{1}{2} \bar{\psi}^{+ \, \lambda} \psi^{+ \, \mu} - \frac{1}{2} \bar{\psi}^{- \, \lambda} \psi^{- \, \mu} - \bar{\psi}^{+ \, [ \lambda} \psi^{- \, \mu ]} \, , \\
\mathcal{S}'^{\lambda \mu} & = d s'^{\lambda \mu} + 2 {t'^\lambda}_\nu s'^{\nu \mu} + 2 {u^{[\lambda}}_\nu v'^{\nu \vert \mu]} - \frac{1}{2} \bar{\psi}^{+ \, \lambda} \psi^{+ \, \mu} - \frac{1}{2} \bar{\psi}^{- \, \lambda} \psi^{- \, \mu} + \bar{\psi}^{+ \, [ \lambda} \psi^{- \, \mu ]} \, , \\
\mathcal{V}^{\lambda \mu} & = d v^{\lambda \mu} + {v^\lambda}_\nu t^{\nu \mu} + {t'^\lambda}_\nu v^{\nu \mu} + {u^\lambda}_\nu s^{\nu \mu} + {s'^\lambda}_\nu u^{\nu \mu} + \frac{1}{2} \bar{\psi}^{+ \, \lambda} \Gamma^0 \psi^{+ \, \mu} - \frac{1}{2} \bar{\psi}^{- \, \lambda} \Gamma^0 \psi^{- \, \mu} \\
& + \bar{\psi}^{+ \, [ \lambda} \Gamma^0 \psi^{- \, \mu ]} \, , \\
\nabla \psi^{+ \, \lambda} & = d \psi^{+ \, \lambda} + \frac{1}{4} \omega^{ab} \Gamma_{ab} \psi^{+ \, \lambda} + \frac{1}{2 \ell} V^a \Gamma_a \psi^{- \, \lambda} + \frac{1}{2} t^{\lambda \mu} \psi^+_\mu + \frac{1}{2} t^{\lambda \mu} \psi^-_\mu + \frac{1}{2} t'^{\lambda \mu} \psi^+_\mu - \frac{1}{2} t'^{\lambda \mu} \psi^-_\mu \\
& + u^{(\lambda \mu)} \Gamma_0 \psi^+_\mu + u^{[\lambda \mu]} \Gamma_0 \psi^-_\mu \, , \\
\nabla \psi^{- \, \lambda} & = d \psi^{- \, \lambda} + \frac{1}{4} \omega^{ab} \Gamma_{ab} \psi^{- \, \lambda} + \frac{1}{2 \ell} V^a \Gamma_a \psi^{+ \, \lambda} + \frac{1}{2} t^{\lambda \mu} \psi^+_\mu + \frac{1}{2} t^{\lambda \mu} \psi^-_\mu - \frac{1}{2} t'^{\lambda \mu} \psi^+_\mu + \frac{1}{2} t'^{\lambda \mu} \psi^-_\mu \\
& - u^{(\lambda \mu)} \Gamma_0 \psi^-_\mu - u^{[\lambda \mu]} \Gamma_0 \psi^+_\mu \, ,
\end{split}$$ where we have used $$\label{vu}
\begin{split}
& u^{\lambda \mu} = t^{\lambda + x \ \mu} = - t^{\mu \ \lambda + x} = - u'^{\mu \lambda} \, , \\
& v^{\lambda \mu} = s^{\lambda + x \ \mu} = - s^{\mu \ \lambda + x} = - v'^{\mu \lambda} \, ,
\end{split}$$ and where we have $$\label{uvsymmform}
\begin{split}
& u^{(\lambda \mu)} = \frac{1}{2} \left( u^{\lambda \mu} + u^{\mu \lambda} \right) \, , \\
& u^{[\lambda \mu]} = \frac{1}{2} \left( u^{\lambda \mu} - u^{\mu \lambda} \right) \, .
\end{split}$$
In order to develop a CS action invariant under , we have to consider the non-vanishing components of the invariant tensor in , decompose the indices as in , exploit , , , and rescale not only the generators in compliance with but also the coefficients appearing in as in . Consequently, in the ultra-relativistic limit $\sigma \rightarrow \infty$ we get the following non-vanishing components of an invariant tensor for : $$\label{invadscarrollsupern}
\begin{split}
& \langle J_{ab} J_{cd} \rangle = \alpha_0 \left( \delta_{ad} \delta_{bc} - \delta_{ac} \delta_{bd} \right) \, , \\
& \langle J_{ab} H \rangle = \alpha_1 \epsilon_{ab} \, , \\
& \langle K_a P_b \rangle = - \alpha_1 \epsilon_{ab} \, , \\
& \langle P_{a} P_{b} \rangle = \frac{\alpha_0}{\ell^2} \delta_{ab} \, , \\
& \langle {T}^{\lambda \mu} {T}^{\nu \rho} \rangle = \langle {T}'^{\lambda \mu} {T}'^{\nu \rho} \rangle = 2 \alpha_0 \left( \delta^{\lambda \rho} \delta^{\nu \mu} - \delta^{\lambda \nu} \delta^{\rho \mu} \right) \, , \\
& \langle {U}^{\lambda \mu} {U}^{\nu \rho} \rangle = - 2 \alpha_0 \delta^{\lambda \nu} \delta^{\rho \mu} \, , \\
& \langle {T}^{\lambda \mu} {S}^{\nu \rho} \rangle = \langle {T}'^{\lambda \mu} {S}'^{\nu \rho} \rangle = - 2 \alpha_1 \left( \delta^{\lambda \rho} \delta^{\nu \mu} - \delta^{\lambda \nu} \delta^{\rho \mu} \right) \, , \\
& \langle {U}^{\lambda \mu} {V}^{\nu \rho} \rangle = 2 \alpha_1 \delta^{\lambda \nu} \delta^{\rho \mu} \, , \\
& \langle Q^{+ \, \lambda}_\alpha Q^{+ \, \mu}_\beta \rangle = \langle Q^{- \, \lambda}_\alpha Q^{- \, \mu}_\beta \rangle = 2 \alpha_1 C_{\alpha \beta} \delta^{\lambda \mu} \, .
\end{split}$$ The invariant tensor for above is non-degenerate if $\alpha_1 \neq 0$.
Then, substituting the connection $1$-form in and the non-zero components of the invariant tensor into , we end up with the three-dimensional $(\mathcal{N},0)$ AdS Carroll CS supergravity action, which reads $$\label{CSACn}
\begin{split}
I^{(\mathcal{N},0)}_{CS} & = \frac{k}{4 \pi} \int_\mathcal{M} \Bigg \lbrace \frac{\alpha_0}{2} \bigg( \omega^a_{\phantom{a} b} R^b_{\phantom{b} a} + \frac{2}{\ell^2} V^a R_a + 2 {t^\lambda}_\mu {d t^\mu}_\lambda + \frac{4}{3} {t^\lambda}_\mu {t^\mu}_\nu {t^\nu}_\lambda + 2 {t'^\lambda}_\mu {d t'^\mu}_\lambda + \frac{4}{3} {t'^\lambda}_\mu {t'^\mu}_\nu {t'^\nu}_\lambda \\
& + 4 {u^\lambda}_\mu {d u'^\mu}_\lambda - 4 t_{\lambda \mu} {u'^{\lambda}}_{\nu} u^{\nu \mu} - 4 t'_{\lambda \mu} {u^{\lambda}}_{\nu} u'^{\nu \mu} \bigg) + \alpha_1 \bigg[ \epsilon_{ab} R^{ab} h - 2 \epsilon_{ab} \mathfrak{K}^a V^b + \frac{1}{\ell^2} \epsilon_{ab} V^a V^b \\
& - 2 {t^\lambda}_\mu \left( {ds^\mu}_\lambda + {t^\mu}_\nu {s^\nu}_\lambda \right) - 2 {t'^\lambda}_\mu \left( {ds'^\mu}_\lambda + {t'^\mu}_\nu {s'^\nu}_\lambda \right) - 4 {u^\lambda}_\mu {d v'^\mu}_\lambda - 2 {u'^\lambda}_\mu {u^\mu}_\nu {s^\nu}_\lambda - 2 {u^\lambda}_\mu {u'^\mu}_\nu {s'^\nu}_\lambda \\
& - 4 {u'^\lambda}_\mu {v^\mu}_\nu {t^\nu}_\lambda - 4 {u^\lambda}_\mu {v'^\mu}_\nu {t'^\nu}_\lambda + 2 \bar{\psi}^{+ \, \lambda} \nabla \psi^{+ \, \lambda} + 2 \bar{\psi}^{- \, \lambda} \nabla \psi^{- \, \lambda} \bigg] \\
& - d \left( \frac{\alpha_1}{2} \epsilon_{ab} \omega^{ab} h - \alpha_1 \epsilon_{ab} k^a V^b + \alpha_1 {t^\lambda}_\mu {s^\mu}_\lambda + \alpha_1 {t'^\lambda}_\mu {s'^\mu}_\lambda + 2 \alpha_1 {u^\lambda}_\mu {v'^\mu}_\lambda \right) \Bigg \rbrace \, ,
\end{split}$$ where we have also exploited . The action has been written in terms of the curvatures appearing in and it involves two coupling constants, that are $\alpha_0$ and $\alpha_1$. Up to boundary terms, can be rewritten as $$\begin{split}
I^{(\mathcal{N},0)}_{CS} & = \frac{k}{4 \pi} \int_\mathcal{M} \Bigg \lbrace \frac{\alpha_0}{2} \bigg( \omega^a_{\phantom{a} b} R^b_{\phantom{b} a} + \frac{2}{\ell^2} V^a R_a + 2 {t^\lambda}_\mu {d t^\mu}_\lambda + \frac{4}{3} {t^\lambda}_\mu {t^\mu}_\nu {t^\nu}_\lambda + 2 {t'^\lambda}_\mu {d t'^\mu}_\lambda + \frac{4}{3} {t'^\lambda}_\mu {t'^\mu}_\nu {t'^\nu}_\lambda \\
& + 4 {u^\lambda}_\mu {d u'^\mu}_\lambda - 4 t_{\lambda \mu} {u'^{\lambda}}_{\nu} u^{\nu \mu} - 4 t'_{\lambda \mu} {u^{\lambda}}_{\nu} u'^{\nu \mu} \bigg) + \alpha_1 \bigg[ \epsilon_{ab} R^{ab} h - 2 \epsilon_{ab} \mathfrak{K}^a V^b + \frac{1}{\ell^2} \epsilon_{ab} V^a V^b \\
& - 2 {t^\lambda}_\mu \left( {ds^\mu}_\lambda + {t^\mu}_\nu {s^\nu}_\lambda \right) - 2 {t'^\lambda}_\mu \left( {ds'^\mu}_\lambda + {t'^\mu}_\nu {s'^\nu}_\lambda \right) - 4 {u^\lambda}_\mu {d v'^\mu}_\lambda - 2 {u'^\lambda}_\mu {u^\mu}_\nu {s^\nu}_\lambda - 2 {u^\lambda}_\mu {u'^\mu}_\nu {s'^\nu}_\lambda \\
& - 4 {u'^\lambda}_\mu {v^\mu}_\nu {t^\nu}_\lambda - 4 {u^\lambda}_\mu {v'^\mu}_\nu {t'^\nu}_\lambda + 2 \bar{\psi}^{+ \, \lambda} \nabla \psi^{+ \, \lambda} + 2 \bar{\psi}^{- \, \lambda} \nabla \psi^{- \, \lambda} \bigg] \Bigg \rbrace \, .
\end{split}$$ The contribution proportional to $\alpha_0$ corresponds to the exotic Lagrangian, and, in the present case, it involves, besides the Lorentz and torsional terms, also pieces including the $1$-form fields $t^{\lambda \mu}$, $t'^{\lambda \mu}$, and $u^{\lambda \mu}$. On the other hand, the contribution proportional to $\alpha_1$ also includes terms involving the $1$-form fields $s^{\lambda \mu}$, $s'^{\lambda \mu}$, and $v^{\lambda \mu}$, plus the spinor $1$-form fields $\psi^{+ \, \lambda}$ and $\psi^{- \, \lambda}$.
The CS action is invariant by construction under , and the local gauge transformations $\delta_\lambda A = d \lambda + \left[A, \lambda \right]$ with gauge parameter $$\label{gparn}
\begin{split}
\lambda & = \frac{1}{2} \lambda^{ab} J_{ab} + \kappa^a K_a + \lambda^a P_a + \tau H + \frac{1}{2} \varrho^{\lambda \mu} t_{\lambda \mu} + \frac{1}{2} \varrho'^{\lambda \mu} t'_{\lambda \mu} + \varphi^{\lambda \mu} u_{\lambda \mu} \\
& + \frac{1}{2} \vartheta^{\lambda \mu} s_{\lambda \mu} + \frac{1}{2} \vartheta'^{\lambda \mu} s_{\lambda \mu} + \varsigma^{\lambda \mu} v_{\lambda \mu} + \varepsilon^{+ \, \lambda} Q^+_\lambda + \varepsilon^{- \, \lambda} Q^-_\lambda
\end{split}$$ are given by $$\label{gaugetrn}
\begin{split}
\delta \omega^{ab} & = d \lambda^{ab} + \frac{2}{\ell^2} V^{[a} \lambda^{b]} \, , \\
\delta k^a & = d \kappa^a - \lambda^a_{\phantom{a} b} k^b + \omega^a_{\phantom{a} b} \kappa^b - \frac{1}{\ell^2} \lambda^a h + \frac{1}{\ell^2} V^a \tau - \frac{1}{\ell} \bar{\varepsilon}^{+ \, \lambda} \Gamma^{a0} \psi^{- \, \lambda} - \frac{1}{\ell} \bar{\varepsilon}^{- \, \lambda} \Gamma^{a0} \psi^{+ \, \lambda} \, , \\
\delta V^a & = d \lambda^a - \lambda^a_{\phantom{a}b} V^b + \omega^a_{\phantom{a}b} \lambda^b \, , \\
\delta h & = d \tau - \lambda^a k_a + V^a \kappa_a + \bar{\varepsilon}^{+ \, \lambda} \Gamma^0 \psi^{+ \, \lambda} + \bar{\varepsilon}^{- \, \lambda} \Gamma^0 \psi^{- \, \lambda} \, , \\
\delta t^{\lambda \mu} & = d \varrho^{\lambda \mu} - 2 {\varrho^{[\lambda}}_\nu t^{\nu \vert \mu]} - 2 {\varphi'^{[\lambda}}_\nu u^{\nu \vert \mu]} \, , \\
\delta t'^{\lambda \mu} & = d \varrho'^{\lambda \mu} - 2 {\varrho'^{[\lambda}}_\nu t'^{\nu \vert \mu]} - 2 {\varphi^{[\lambda}}_{ \nu} u'^{\nu \vert \mu]} \, , \\
\delta u^{\lambda \mu} & = d \varphi^{\lambda \mu} - {\varphi^\lambda}_\nu t^{\nu \mu} + {u^\lambda}_\nu \varrho^{\nu \mu} - {\varrho'^\lambda}_\nu u^{\nu \mu} + {t'^\lambda}_\nu \varphi^{\nu \mu} \, , \\
\delta s^{\lambda \mu} & = d \vartheta^{\lambda \mu} - 2 {\varrho^{[\lambda}}_\nu s^{\nu \vert \mu]} + 2 {t^{[\lambda}}_\nu \vartheta^{\nu \vert \mu]} - 2 {\varphi'^{[\lambda}}_\nu v^{\nu \vert \mu]} + 2 {u'^{[\lambda}}_\nu \varsigma^{\nu \vert \mu]} + \bar{\varepsilon}^{+ \, [\lambda} \psi^{+ \, \mu]} + \bar{\varepsilon}^{- \, [\lambda} \psi^{- \, \mu]} \\
& + \bar{\varepsilon}^{+ \, [ \lambda} \psi^{- \, \mu ]} + \bar{\varepsilon}^{- \, [ \lambda} \psi^{+ \, \mu ]} \, , \\
\delta s'^{\lambda \mu} & = d \vartheta'^{\lambda \mu} - 2 {\varrho'^{[\lambda}}_\nu s'^{\nu \vert \mu]} + 2 {t'^{[\lambda}}_\nu \vartheta'^{\nu \vert \mu]} - 2 {\varphi^{[\lambda}}_\nu v'^{\nu \vert \mu]} + 2 {u^{[\lambda}}_\nu \varsigma'^{\nu \vert \mu]} + \bar{\varepsilon}^{+ \, [\lambda} \psi^{+ \, \mu]} + \bar{\varepsilon}^{- \, [\lambda} \psi^{- \, \mu]} \\
& - \bar{\varepsilon}^{+ \, [ \lambda} \psi^{- \, \mu ]} - \bar{\varepsilon}^{- \, [ \lambda} \psi^{+ \, \mu ]} \, , \\
\delta v^{\lambda \mu} & = d \varsigma^{\lambda \mu} - {\varsigma^\lambda}_\nu t^{\nu \mu} + {v^\lambda}_\nu \varrho^{\nu \mu} - {\varrho'^\lambda}_\nu v^{\nu \mu} + {t'^\lambda}_\nu \varsigma^{\nu \mu} - {\varphi^\lambda}_\nu s^{\nu \mu} + {u^\lambda}_\nu \vartheta^{\nu \mu} - {\vartheta'^\lambda}_\nu u^{\nu \mu} + {s'^\lambda}_\nu \varphi^{\nu \mu} \\
& - \bar{\varepsilon}^{+ \, \lambda} \Gamma^0 \psi^{+ \, \mu} + \bar{\varepsilon}^{- \, \lambda} \Gamma^0 \psi^{- \, \mu} - \bar{\varepsilon}^{+ \, [ \lambda} \Gamma^0 \psi^{- \, \mu ]} + \bar{\varepsilon}^{- \, [ \lambda} \Gamma^0 \psi^{+ \, \mu ]} \, , \\
\delta \psi^{+ \, \lambda} & = d \varepsilon^{+ \, \lambda} - \frac{1}{4} \lambda^{ab} \Gamma_{ab} \psi^{+ \, \lambda} + \frac{1}{4} \omega^{ab} \Gamma_{ab} \varepsilon^{+ \, \lambda} - \frac{1}{2 \ell} \lambda^a \Gamma_a \psi^{- \, \lambda} + \frac{1}{2 \ell} V^a \Gamma_a \varepsilon^{- \, \lambda} \\
& - \frac{1}{2} \varrho^{\lambda \mu} \psi^+_\mu + \frac{1}{2} t^{\lambda \mu} \varepsilon^+_\mu - \frac{1}{2} \varrho^{\lambda \mu} \psi^-_\mu + \frac{1}{2} t^{\lambda \mu} \varepsilon^-_\mu - \frac{1}{2} \varrho'^{\lambda \mu} \psi^+_\mu + \frac{1}{2} t'^{\lambda \mu} \varepsilon^+_\mu + \frac{1}{2} \varrho'^{\lambda \mu} \psi^-_\mu - \frac{1}{2} t'^{\lambda \mu} \varepsilon^-_\mu \\
& - \varphi^{(\lambda \mu)} \Gamma_0 \psi^+_\mu + u^{(\lambda \mu)} \Gamma_0 \varepsilon^+_\mu - \varphi^{[\lambda \mu]} \Gamma_0 \psi^-_\mu + u^{[\lambda \mu]} \Gamma_0 \varepsilon^-_\mu \, , \\
\delta \psi^{- \, \lambda} & = d \varepsilon^{- \, \lambda} - \frac{1}{4} \lambda^{ab} \Gamma_{ab} \psi^{- \, \lambda} + \frac{1}{4} \omega^{ab} \Gamma_{ab} \varepsilon^{- \, \lambda} - \frac{1}{2 \ell} \lambda^a \Gamma_a \psi^{+ \, \lambda} + \frac{1}{2 \ell} V^a \Gamma_a \varepsilon^{+ \, \lambda} \\
& - \frac{1}{2} \varrho^{\lambda \mu} \psi^+_\mu + \frac{1}{2} t^{\lambda \mu} \varepsilon^+_\mu - \frac{1}{2} \varrho^{\lambda \mu} \psi^-_\mu + \frac{1}{2} t^{\lambda \mu} \varepsilon^-_\mu + \frac{1}{2} \varrho'^{\lambda \mu} \psi^+_\mu - \frac{1}{2} t'^{\lambda \mu} \varepsilon^+_\mu - \frac{1}{2} \varrho'^{\lambda \mu} \psi^-_\mu + \frac{1}{2} t'^{\lambda \mu} \varepsilon^-_\mu \\
& + \varphi^{(\lambda \mu)} \Gamma_0 \psi^-_\mu - u^{(\lambda \mu)} \Gamma_0 \varepsilon^-_\mu + \varphi^{[\lambda \mu]} \Gamma_0 \psi^+_\mu - u^{[\lambda \mu]} \Gamma_0 \varepsilon^+_\mu \, ,
\end{split}$$ where we have also used the properties and definitions $$\label{parsymm}
\begin{split}
& \varphi^{\lambda \mu} = - \varphi'^{\mu \lambda} \, , \\
& \varphi^{(\lambda \mu)} \equiv \frac{1}{2} \left( \varphi^{\lambda \mu} + \varphi^{\mu \lambda} \right) \, , \\
& \varphi^{[\lambda \mu]} = \frac{1}{2} \left( \varphi^{\lambda \mu} - \varphi^{\mu \lambda} \right) \, .
\end{split}$$ Restricting ourselves to supersymmetry, we get the following supersymmetry transformation laws: $$\label{susytrn}
\begin{split}
\delta \omega^{ab} & = 0 \, , \\
\delta k^a & = - \frac{1}{\ell} \bar{\varepsilon}^{+ \, \lambda} \Gamma^{a0} \psi^{- \, \lambda} - \frac{1}{\ell} \bar{\varepsilon}^{- \, \lambda} \Gamma^{a0} \psi^{+ \, \lambda} \, , \\
\delta V^a & = 0 \, , \\
\delta h & = \bar{\varepsilon}^{+ \, \lambda} \Gamma^0 \psi^{+ \, \lambda} + \bar{\varepsilon}^{- \, \lambda} \Gamma^0 \psi^{- \, \lambda} \, , \\
\delta t^{\lambda \mu} & = 0 \, , \quad \quad \delta t'^{\lambda \mu} = 0 \, , \quad \quad \delta u^{\lambda \mu} = 0 \, , \\
\delta s^{\lambda \mu} & = \bar{\varepsilon}^{+ \, [\lambda} \psi^{+ \, \mu]} + \bar{\varepsilon}^{- \, [\lambda} \psi^{- \, \mu]} + \bar{\varepsilon}^{+ \, [ \lambda} \psi^{- \, \mu ]} + \bar{\varepsilon}^{- \, [ \lambda} \psi^{+ \, \mu ]} \, , \\
\delta s'^{\lambda \mu} & = \bar{\varepsilon}^{+ \, [\lambda} \psi^{+ \, \mu]} + \bar{\varepsilon}^{- \, [\lambda} \psi^{- \, \mu]} - \bar{\varepsilon}^{+ \, [ \lambda} \psi^{- \, \mu ]} - \bar{\varepsilon}^{- \, [ \lambda} \psi^{+ \, \mu ]} \, , \\
\delta v^{\lambda \mu} & = - \bar{\varepsilon}^{+ \, \lambda} \Gamma^0 \psi^{+ \, \mu} + \bar{\varepsilon}^{- \, \lambda} \Gamma^0 \psi^{- \, \mu} - \bar{\varepsilon}^{+ \, [ \lambda} \Gamma^0 \psi^{- \, \mu ]} + \bar{\varepsilon}^{- \, [ \lambda} \Gamma^0 \psi^{+ \, \mu ]} \, , \\
\delta \psi^{+ \, \lambda} & = d \varepsilon^{+ \, \lambda} + \frac{1}{4} \omega^{ab} \Gamma_{ab} \varepsilon^{+ \, \lambda} + \frac{1}{2 \ell} V^a \Gamma_a \varepsilon^{- \, \lambda} + \frac{1}{2} t^{\lambda \mu} \varepsilon^+_\mu + \frac{1}{2} t^{\lambda \mu} \varepsilon^-_\mu + \frac{1}{2} t'^{\lambda \mu} \varepsilon^+_\mu - \frac{1}{2} t'^{\lambda \mu} \varepsilon^-_\mu \\
& + u^{(\lambda \mu)} \Gamma_0 \varepsilon^+_\mu + u^{[\lambda \mu]} \Gamma_0 \varepsilon^-_\mu \, , \\
\delta \psi^{- \, \lambda} & = d \varepsilon^{- \, \lambda} + \frac{1}{4} \omega^{ab} \Gamma_{ab} \varepsilon^{- \, \lambda} + \frac{1}{2 \ell} V^a \Gamma_a \varepsilon^{+ \, \lambda} + \frac{1}{2} t^{\lambda \mu} \varepsilon^+_\mu + \frac{1}{2} t^{\lambda \mu} \varepsilon^-_\mu - \frac{1}{2} t'^{\lambda \mu} \varepsilon^+_\mu + \frac{1}{2} t'^{\lambda \mu} \varepsilon^-_\mu \\
& - u^{(\lambda \mu)} \Gamma_0 \varepsilon^-_\mu - u^{[\lambda \mu]} \Gamma_0 \varepsilon^+_\mu \, .
\end{split}$$
Finally, one can prove that from the variation of the action with respect to the $1$-form fields $\omega^{ab}$, $k^a$, $V^a$, $h$, $t^{\lambda \mu}$, $t'^{\lambda \mu}$, $u^{\lambda \mu}$, $s^{\lambda \mu}$, $s'^{\lambda \mu}$, $v^{\lambda \mu}$, $\psi^{+ \, \lambda}$, and $\psi^{- \, \lambda}$, we get, respectively, the equations of motion $$\label{eomn}
\begin{split}
\delta \omega^{ab} & : \quad \alpha_0 \mathcal{R}^{ab} + \alpha_1 \epsilon^{ab} \mathcal{H}= 0 \, , \\
\delta k^a & : \quad \alpha_1 R^a = 0 \, , \\
\delta V^a & : \quad \frac{\alpha_0}{\ell^2} R^a + 2 \alpha_1 \epsilon_{ab} \mathcal{K}^b = 0 \, , \\
\delta h & : \quad \alpha_1 \mathcal{R}^{ab} = 0 \, , \\
\delta t^{\lambda \mu} & : \quad \alpha_0 \mathcal{T}^{\lambda \mu} + \alpha_1 \mathcal{S}^{\lambda \mu} = 0 \, , \\
\delta t'^{\lambda \mu} & : \quad \alpha_0 \mathcal{T}'^{\lambda \mu} + \alpha_1 \mathcal{S}'^{\lambda \mu} = 0 \, , \\
\delta u^{\lambda \mu} & : \quad - \alpha_0 \mathcal{U}^{\lambda \mu} + \alpha_1 \mathcal{V}^{\lambda \mu} = 0 \, , \\
\delta s^{\lambda \mu} & : \quad \alpha_1 \mathcal{T}^{\lambda \mu} = 0 \, , \\
\delta s'^{\lambda \mu} & : \quad \alpha_1 \mathcal{T}'^{\lambda \mu} = 0 \, , \\
\delta v^{\lambda \mu} & : \quad \alpha_1 \mathcal{U}^{\lambda \mu} = 0 \, , \\
\delta \psi^{+ \, \lambda} & : \quad \alpha_1 \nabla \psi^{+ \, \lambda} = 0 \, , \\
\delta \psi^{- \, \lambda} & : \quad \alpha_1 \nabla \psi^{- \, \lambda} = 0 \, ,
\end{split}$$ written up to boundary contributions. We observe that, for $\alpha_1 \neq 0$, the equations reduce precisely to the vanishing of the $(\mathcal{N},0)$ super-AdS Carroll curvature $2$-forms given in , that is to say $$\label{eomvacn}
\begin{split}
& \mathcal{R}^{ab} = 0 \, , \quad \mathcal{K}^a = 0 \, , \quad R^a = 0 \, , \quad \mathcal{H}=0 \, , \\
& \mathcal{T}^{\lambda \mu} = 0 \, , \quad \mathcal{T}'^{\lambda \mu} = 0 \, , \quad \mathcal{U}^{\lambda \mu} = 0 \, , \\
& \mathcal{S}^{\lambda \mu} = 0 \, , \quad \mathcal{S}'^{\lambda \mu} = 0 \, , \quad \mathcal{V}^{\lambda \mu} = 0 \, , \\
& \nabla \psi^{+ \, \lambda} =0 \, , \quad \nabla \psi^{- \, \lambda} = 0 \, .
\end{split}$$ Also in this case, we notice that $\alpha_1 \neq 0$ is a sufficient condition to recover , which means that we can consistently impose $\alpha_0=0$ in the CS action , making the exotic term disappear.
Let us also mention that, restricting ourselves to the special case $\mathcal{N}=(2,0)$, that is $x=1$, after some algebraic calculations, we exactly reproduce the results of Section \[20case\], that is to say, in fact, the $(2,0)$ AdS Carroll supergravity theory.
$(p,q)$ AdS Carroll supergravity theories in $2+1$ dimensions {#pqcase}
=============================================================
We finally extend our analysis to the $(p,q)$ case, with $p=q$. We first derive the $\mathcal{N}=(p,q)$ AdS Carroll superalgebra as the Carrollian contraction of the direct sum of an $\mathfrak{so}(p) \oplus \mathfrak{so}(q)$ algebra and $\mathfrak{osp}(2|p)\otimes \mathfrak{osp}(2,q)$. This allows us to end up with a non-degenerate invariant tensor in the ultra-relativistic limit, and to consequently construct the three-dimensional CS supergravity theory invariant under the aforesaid $\mathcal{N}=(p,q)$ AdS Carroll superalgebra.
$\mathcal{N}=(p,q)$ AdS Carroll superalgebra
--------------------------------------------
Let us begin by considering the direct sum of the $\mathfrak{osp}(2|p)\otimes \mathfrak{osp}(2,q)$ superalgebra and an $\mathfrak{so}(p) \oplus \mathfrak{so}(q)$ algebra (we consider $p=q$). The non-zero (anti)commutation relations are the following ones (see [@Howe:1995zm]): $$\label{osppqnonmod}
\begin{split}
& \left[ \tilde{J}_{AB}, \tilde{J}_{CD}\right] =\eta _{BC}\tilde{J}_{AD}-\eta _{AC}\tilde{J}_{BD}-\eta
_{BD}\tilde{J}_{AC}+\eta _{AD}\tilde{J}_{BC} \, , \\
& \left[ \tilde{J}_{AB},\tilde{P}_{C}\right] =\eta _{BC}\tilde{P}_{A}-\eta _{AC}\tilde{P}_{B} \, , \quad \quad \left[ \tilde{P}_A , \tilde{P}_B \right] = \frac{1}{\ell^2} \tilde{J}_{AB} \, , \\
& \left[ \tilde{Z}^{ij} , \tilde{Z}^{kl} \right] = \delta^{jk} \tilde{Z}^{il} - \delta^{ik} \tilde{Z}^{jl}- \delta^{jl} \tilde{Z}^{ik} + \delta^{il} \tilde{Z}^{jk} \, , \\
& \left[ \tilde{Z}^{IJ} , \tilde{Z}^{KL} \right] = \delta^{JK} \tilde{Z}^{IL} - \delta^{IK} \tilde{Z}^{JL}- \delta^{JL} \tilde{Z}^{IK} + \delta^{IL} \tilde{Z}^{JK} \, , \\
& \left[ \tilde{S}^{ij} , \tilde{S}^{kl} \right] = - \frac{1}{\ell} \left( \delta^{jk} \tilde{S}^{il} - \delta^{ik} \tilde{S}^{jl}- \delta^{jl} \tilde{S}^{ik} + \delta^{il} \tilde{S}^{jk} \right) \, , \\
& \left[ \tilde{S}^{IJ} , \tilde{S}^{KL} \right] = - \frac{1}{\ell} \left( \delta^{JK} \tilde{S}^{IL} - \delta^{IK} \tilde{S}^{JL}- \delta^{JL} \tilde{S}^{IK} + \delta^{IL} \tilde{S}^{JK} \right) \, , \\
& \left[ \tilde{J}_{AB},\tilde{Q}^i_{\alpha }\right] =-\frac{1}{2}\left( \Gamma _{AB}\tilde{Q}^i\right)_{\alpha } \, , \quad \quad \left[ \tilde{J}_{AB},\tilde{Q}^I_{\alpha }\right] =-\frac{1}{2}\left( \Gamma _{AB}\tilde{Q}^I\right)_{\alpha } \, , \\
& \left[ \tilde{P}_{A}, \tilde{Q}^i_{\alpha }\right] =- \frac{1}{2 \ell} \left( \Gamma _{A}\tilde{Q}^i \right) _{\alpha } \, , \quad \quad \left[ \tilde{P}_{A}, \tilde{Q}^I_{\alpha }\right] = \frac{1}{2 \ell} \left( \Gamma _{A}\tilde{Q}^I \right) _{\alpha } \, ,
\end{split}$$ $$\nonumber
\begin{split}
& \left[ \tilde{Z}^{ij}, \tilde{Q}^k_{\alpha }\right] = \delta^{jk} \tilde{Q}^i_\alpha - \delta^{ik} \tilde{Q}^j_\alpha \, , \quad \quad \left[ \tilde{Z}^{IJ}, \tilde{Q}^K_{\alpha }\right] = \delta^{JK} \tilde{Q}^I_\alpha - \delta^{IK} \tilde{Q}^J_\alpha \, , \\
& \left\{ \tilde{Q}^i_{\alpha }, \tilde{Q}^j_{\beta }\right\} = \delta^{ij} \left[ - \frac{1}{2 \ell} \left(\Gamma^{AB} C \right)_{\alpha \beta} \tilde{J}_{AB} + \left( \Gamma ^{A}C\right) _{\alpha \beta }\tilde{P}_{A} \right] + \frac{1}{\ell} C_{\alpha \beta} \tilde{Z}^{ij} \, , \\
& \left\{ \tilde{Q}^I_{\alpha }, \tilde{Q}^J_{\beta }\right\} = \delta^{IJ} \left[ \frac{1}{2 \ell} \left(\Gamma^{AB} C \right)_{\alpha \beta} \tilde{J}_{AB} + \left( \Gamma ^{A}C\right) _{\alpha \beta }\tilde{P}_{A} \right] - \frac{1}{\ell} C_{\alpha \beta} \tilde{Z}^{IJ} \, ,
\end{split}$$ with $A, B , \ldots=0,1,2$, $i,j,\ldots=1,\ldots,p$, $I,J,\ldots=1,\ldots, q$, and where $\tilde{Z}^{ij}=-\tilde{Z}^{ji}$, $\tilde{S}^{ij}=-\tilde{S}^{ij}$, and $\tilde{Z}^{IJ} = - \tilde{Z}^{JI}$, $\tilde{S}^{IJ}= - \tilde{S}^{JI}$.
We now perform, on the same lines of [@Howe:1995zm], the redefinition $$\tilde{T}^{ij} \equiv \tilde{Z}^{ij} - \ell \tilde{S}^{ij} \, , \quad \tilde{T}^{IJ} \equiv \tilde{Z}^{IJ} - \ell \tilde{S}^{IJ} \, ,$$ which is analogous to the ones we have done in Sections \[20case\] and \[ncase\]. Consequently, we can rewrite the (anti)commutation relations as $$\label{osppq}
\begin{split}
& \left[ \tilde{J}_{AB}, \tilde{J}_{CD}\right] =\eta _{BC}\tilde{J}_{AD}-\eta _{AC}\tilde{J}_{BD}-\eta
_{BD}\tilde{J}_{AC}+\eta _{AD}\tilde{J}_{BC} \, , \\
& \left[ \tilde{J}_{AB},\tilde{P}_{C}\right] =\eta _{BC}\tilde{P}_{A}-\eta _{AC}\tilde{P}_{B} \, , \\
& \left[ \tilde{P}_A , \tilde{P}_B \right] = \frac{1}{\ell^2} \tilde{J}_{AB} \, , \\
& \left[ \tilde{T}^{ij} , \tilde{T}^{kl} \right] = \delta^{jk} \tilde{T}^{il} - \delta^{ik} \tilde{T}^{jl}- \delta^{jl} \tilde{T}^{ik} + \delta^{il} \tilde{T}^{jk} \, , \\
& \left[ \tilde{T}^{IJ} , \tilde{T}^{KL} \right] = \delta^{JK} \tilde{T}^{IL} - \delta^{IK} \tilde{T}^{JL}- \delta^{JL} \tilde{T}^{IK} + \delta^{IL} \tilde{T}^{JK} \, , \\
& \left[ \tilde{T}^{ij} , \tilde{S}^{kl} \right] = \delta^{jk} \tilde{S}^{il} - \delta^{ik} \tilde{S}^{jl}- \delta^{jl} \tilde{S}^{ik} + \delta^{il} \tilde{S}^{jk} \, , \\
& \left[ \tilde{T}^{IJ} , \tilde{S}^{KL} \right] = \delta^{JK} \tilde{S}^{IL} - \delta^{IK} \tilde{S}^{JL}- \delta^{JL} \tilde{S}^{IK} + \delta^{IL} \tilde{S}^{JK} \, , \\
& \left[ \tilde{S}^{ij} , \tilde{S}^{kl} \right] = - \frac{1}{\ell} \left( \delta^{jk} \tilde{S}^{il} - \delta^{ik} \tilde{S}^{jl}- \delta^{jl} \tilde{S}^{ik} + \delta^{il} \tilde{S}^{jk} \right) \, , \\
& \left[ \tilde{S}^{IJ} , \tilde{S}^{KL} \right] = - \frac{1}{\ell} \left( \delta^{JK} \tilde{S}^{IL} - \delta^{IK} \tilde{S}^{JL}- \delta^{JL} \tilde{S}^{IK} + \delta^{IL} \tilde{S}^{JK} \right) \, , \\
& \left[ \tilde{J}_{AB},\tilde{Q}^i_{\alpha }\right] =-\frac{1}{2}\left( \Gamma _{AB}\tilde{Q}^i\right)_{\alpha } \, , \\
& \left[ \tilde{J}_{AB},\tilde{Q}^I_{\alpha }\right] =-\frac{1}{2}\left( \Gamma _{AB}\tilde{Q}^I\right)_{\alpha } \, , \\
& \left[ \tilde{P}_{A}, \tilde{Q}^i_{\alpha }\right] =- \frac{1}{2 \ell} \left( \Gamma _{A}\tilde{Q}^i \right) _{\alpha } \, , \\
& \left[ \tilde{P}_{A}, \tilde{Q}^I_{\alpha }\right] = \frac{1}{2 \ell} \left( \Gamma _{A}\tilde{Q}^I \right) _{\alpha } \, , \\
& \left[ \tilde{T}^{ij}, \tilde{Q}^k_{\alpha }\right] = \delta^{jk} \tilde{Q}^i_\alpha - \delta^{ik} \tilde{Q}^j_\alpha \, , \\
& \left[ \tilde{T}^{IJ}, \tilde{Q}^K_{\alpha }\right] = \delta^{JK} \tilde{Q}^I_\alpha - \delta^{IK} \tilde{Q}^J_\alpha \, , \\
& \left\{ \tilde{Q}^i_{\alpha }, \tilde{Q}^j_{\beta }\right\} = \delta^{ij} \left[ - \frac{1}{2 \ell} \left(\Gamma^{AB} C \right)_{\alpha \beta} \tilde{J}_{AB} + \left( \Gamma ^{A}C\right) _{\alpha \beta }\tilde{P}_{A} \right] + C_{\alpha \beta} \left(\frac{1}{\ell} \tilde{T}^{ij} + \tilde{S}^{ij} \right) \, , \\
& \left\{ \tilde{Q}^I_{\alpha }, \tilde{Q}^J_{\beta }\right\} = \delta^{IJ} \left[ \frac{1}{2 \ell} \left(\Gamma^{AB} C \right)_{\alpha \beta} \tilde{J}_{AB} + \left( \Gamma ^{A}C\right) _{\alpha \beta }\tilde{P}_{A} \right] - C_{\alpha \beta} \left(\frac{1}{\ell} \tilde{T}^{IJ} + \tilde{S}^{IJ} \right) \, .
\end{split}$$ Notice that taking the flat limit $\ell \rightarrow \infty$ of , one recovers the $\mathcal{N}=(p,q)$ Poincaré superalgebra of [@Howe:1995zm] (here we have restricted ourselves to the case $p=q$).
The non-vanishing components of an invariant tensor for the superalgebra , that will be useful in the following study, are given by $$\label{invtosppq}
\begin{split}
& \langle \tilde{J}_{AB} \tilde{J}_{CD} \rangle = \alpha_0 \left( \eta_{AD} \eta_{BC} - \eta_{AC} \eta_{BD} \right) \, , \\
& \langle \tilde{J}_{AB} \tilde{P}_{C} \rangle = \alpha_1 \epsilon_{ABC} \, , \\
& \langle \tilde{P}_{A} \tilde{P}_{B} \rangle = \frac{\alpha_0}{\ell^2} \eta_{AB} \, , \\
& \langle \tilde{T}^{ij} \tilde{T}^{kl} \rangle = 2 \alpha_0 \left( \delta^{il} \delta^{kj} - \delta^{ik} \delta^{lj} \right) \, , \\
& \langle \tilde{T}^{IJ} \tilde{T}^{KL} \rangle = 2 \alpha_0 \left( \delta^{IL} \delta^{KJ} - \delta^{IK} \delta^{LJ} \right) \, , \\
& \langle \tilde{T}^{ij} \tilde{S}^{kl} \rangle = - 2 \alpha_1 \left( \delta^{il} \delta^{kj} - \delta^{ik} \delta^{lj} \right) \, , \\
& \langle \tilde{T}^{IJ} \tilde{S}^{KL} \rangle = 2 \alpha_1 \left( \delta^{IL} \delta^{KJ} - \delta^{IK} \delta^{LJ} \right) \, , \\
& \langle \tilde{S}^{ij} \tilde{S}^{kl} \rangle = \frac{2 \alpha_1}{\ell} \left( \delta^{il} \delta^{kj} - \delta^{ik} \delta^{lj} \right) \, , \\
& \langle \tilde{S}^{IJ} \tilde{S}^{KL} \rangle = - \frac{2 \alpha_1}{\ell} \left( \delta^{IL} \delta^{KJ} - \delta^{IK} \delta^{LJ} \right) \, , \\
& \langle \tilde{Q}^i_\alpha \tilde{Q}^j_\beta \rangle = 2 \left(\alpha_1 - \frac{\alpha_0}{\ell} \right) C_{\alpha \beta} \delta^{ij} \, , \\
& \langle \tilde{Q}^I_\alpha \tilde{Q}^J_\beta \rangle = 2 \left(\alpha_1 + \frac{\alpha_0}{\ell} \right) C_{\alpha \beta} \delta^{IJ} \, ,
\end{split}$$ where $\alpha_0$ and $\alpha_1$ are arbitrary constants.
In order to take the ultra-relativistic contraction of the superalgebra , we decompose, as usual, the indices $A,B,\ldots = 0,1,2$ as in , which induces the decomposition , together with . After that, we rescale the generators with a parameter $\sigma$ as $$\label{rescpq}
\tilde{H} \rightarrow \sigma H \, , \quad \tilde{K}_a \rightarrow \sigma K_a \, , \quad \tilde{S}^{ij} \rightarrow \sigma S^{ij} \, , \quad \tilde{S}^{IJ} \rightarrow \sigma S^{IJ} \, , \quad \tilde{Q}^i_\alpha \rightarrow \sqrt{\sigma} Q^i_\alpha \, , \quad \tilde{Q}^I_\alpha \rightarrow \sqrt{\sigma} Q^I_\alpha \, .$$ Then, taking the limit $\sigma \rightarrow \infty$ (and removing the tilde symbol also on the generators that we have not rescaled), we end up with the $\mathcal{N}=(p,q)$ AdS Carroll superalgebra whose non-trivial (anti)commutation relations read $$\label{adscarrollsuperpq}
\begin{split}
& \left[ K_a , J_{bc} \right] = \delta_{ab} K_c - \delta_{ac} K_b \, , \quad \quad \left[ J_{ab}, P_{c}\right] =\delta_{bc}P_{a}-\delta _{ac}P_{b} \, , \\
& \left[ K_{a}, P_{b}\right] = - \delta_{ab} H \, , \quad \quad \left[ P_a , P_b \right] = \frac{1}{\ell^2} J_{ab} \, , \quad \quad \left[ P_a , H \right] = \frac{1}{\ell^2} K_{a} \, , \\
& \left[ {T}^{ij} , {T}^{kl} \right] = \delta^{jk} {T}^{il} - \delta^{ik} {T}^{jl}- \delta^{jl} {T}^{ik} + \delta^{il} {T}^{jk} \, , \\
& \left[ {T}^{IJ} , {T}^{KL} \right] = \delta^{JK} {T}^{IL} - \delta^{IK} {T}^{JL}- \delta^{JL} {T}^{IK} + \delta^{IL} {T}^{JK} \, , \\
& \left[ {T}^{ij} , {S}^{kl} \right] = \delta^{jk} {S}^{il} - \delta^{ik} {S}^{jl}- \delta^{jl} {S}^{ik} + \delta^{il} {S}^{jk} \, , \\
& \left[ {T}^{IJ} , {S}^{KL} \right] = \delta^{JK} {S}^{IL} - \delta^{IK} {S}^{JL}- \delta^{JL} {S}^{IK} + \delta^{IL} {S}^{JK} \, , \\
& \left[ J_{ab},Q^i_{\alpha }\right] =-\frac{1}{2}\left( \Gamma _{ab} Q^i \right)_{\alpha } \, , \quad \quad \left[ J_{ab},Q^I_{\alpha }\right] =-\frac{1}{2}\left( \Gamma _{ab} Q^I \right)_{\alpha } \, , \\
& \left[ P_{a}, Q^i_{\alpha }\right] =- \frac{1}{2 \ell} \left( \Gamma _{a}Q^i \right) _{\alpha } \, , \quad \quad \left[ P_{a}, Q^I_{\alpha }\right] = \frac{1}{2 \ell} \left( \Gamma _{a}Q^I \right) _{\alpha } \, ,
\end{split}$$ $$\nonumber
\begin{split}
& \left[ {T}^{ij}, {Q}^k_{\alpha }\right] = \delta^{jk} {Q}^i_\alpha - \delta^{ik} {Q}^j_\alpha \, , \quad \quad \left[ {T}^{IJ}, {Q}^K_{\alpha }\right] = \delta^{JK} {Q}^I_\alpha - \delta^{IK} {Q}^J_\alpha \, , \\
& \left\{ Q^i_{\alpha }, Q^j_{\beta }\right\} = \delta^{ij} \left[- \frac{1}{\ell} \left(\Gamma^{a0} C \right)_{\alpha \beta} K_{a} + \left( \Gamma ^{0}C\right) _{\alpha \beta } H \right] + C_{\alpha \beta} S^{ij} \, , \\
& \left\{ Q^I_{\alpha }, Q^J_{\beta }\right\} = \delta^{ij} \left[ \frac{1}{\ell} \left(\Gamma^{a0} C \right)_{\alpha \beta} K_{a} + \left( \Gamma ^{0}C\right) _{\alpha \beta } H \right] - C_{\alpha \beta} S^{IJ} \, .
\end{split}$$ Notice that if we restrict ourselves to the special case $\mathcal{N}=(1,1)$, that is $p=q=1$, we exactly reproduce the $\mathcal{N}=(1,1)$ AdS Carroll superalgebra obtained in Section \[11case\], namely .
In the following, we will construct a three-dimensional CS action invariant under .
$(p,q)$ AdS Carroll supergravity
--------------------------------
We will now construct a three-dimensional CS supergravity action invariant under the superalgebra just introduced. We call this action $(p,q)$ AdS Carroll CS supergravity action (recall that, in our case, $p=q$, such that $\mathcal{N}=p+q$ is even).
To this aim, let us first introduce the connection $1$-form $A$ associated with , namely $$\label{connadscarrollsuperpq}
A = \frac{1}{2} \omega^{ab} J_{ab} + k^a K_a + V^a P_a + h H + \frac{1}{2} t^{ij} T_{ij} + \frac{1}{2} t^{IJ} T_{IJ} + \frac{1}{2} s^{ij} S_{ij} + \frac{1}{2} s^{IJ} S_{IJ} + \psi_i Q^i + \psi_I Q^I \, ,$$ being $\omega^{ab}$, $k^a$, $V^a$, $h$, $t^{ij}$, $t^{IJ}$, $s^{ij}$, $s^{IJ}$, $\psi_i$, and $\psi_I$ the $1$-form fields respectively dual to the generators $J_{ab}$, $K_a$, $P_a$, $H$, $T_{ij}$, $T_{IJ}$, $S_{ij}$, $S_{IJ}$, $Q^i$, and $Q^I$ (obeying the (anti)commutation relations given in ), and the related curvature $2$-form $F$, that is $$\label{curv2fpq}
F = \frac{1}{2} \mathcal{R}^{ab} J_{ab} + \mathcal{K}^a K_a + R^a P_a + \mathcal{H} H + \frac{1}{2} \mathcal{T}^{ij} T_{ij} + \frac{1}{2} \mathcal{T}^{IJ} T_{IJ} + \frac{1}{2} \mathcal{S}^{ij} S_{ij} + \frac{1}{2} \mathcal{S}^{IJ} S_{IJ} + \nabla \psi_i Q^i + \nabla \psi_I Q^I \, ,$$ with $$\label{curvadscarrollsuperpq}
\begin{split}
\mathcal{R}^{ab} & = d \omega^{ab} + \frac{1}{\ell^2} V^a V^b = R^{ab} + \frac{1}{\ell^2} V^a V^b \, , \\
\mathcal{K}^a & = d k^a + \omega^a_{\phantom{a} b} k^b + \frac{1}{\ell^2} V^a h + \frac{1}{2\ell} \bar{\psi}^i \Gamma^{a0} \psi^i - \frac{1}{2\ell} \bar{\psi}^I \Gamma^{a0} \psi^I = \mathfrak{K}^a + \frac{1}{2\ell} \bar{\psi}^i \Gamma^{a0} \psi^i - \frac{1}{2\ell} \bar{\psi}^I \Gamma^{a0} \psi^I \, , \\
R^a & = d V^a + \omega^a_{\phantom{a}b} V^b \, , \\
\mathcal{H} & = d h + V^a k_a - \frac{1}{2} \bar{\psi}^i \Gamma^0 \psi^i - \frac{1}{2} \bar{\psi}^I \Gamma^0 \psi^I = \mathfrak{H} - \frac{1}{2} \bar{\psi}^i \Gamma^0 \psi^i - \frac{1}{2} \bar{\psi}^I \Gamma^0 \psi^I \, , \\
\mathcal{T}^{ij} & = d t^{ij} + {t^i}_k t^{kj} \, , \\
\mathcal{T}^{IJ} & = d t^{IJ} + {t^I}_K t^{KJ} \, ,\\
\mathcal{S}^{ij} & = d s^{ij} + 2 {t^i}_k s^{kj} - \bar{\psi}^i \psi^j \, , \\
\mathcal{S}^{IJ} & = d s^{IJ} + 2 {t^I}_K s^{KJ} + \bar{\psi}^I \psi^J \, ,\\
\nabla \psi^i & = d \psi^i + \frac{1}{4} \omega^{ab} \Gamma_{ab} \psi^i + \frac{1}{2 \ell} V^a \Gamma_a \psi^i + t^{ij} \psi_j \, , \\
\nabla \psi^I & = d \psi^I + \frac{1}{4} \omega^{ab} \Gamma_{ab} \psi^I - \frac{1}{2 \ell} V^a \Gamma_a \psi^I + t^{IJ} \psi_J \, .
\end{split}$$
We can now move to the explicit construction of a CS action invariant under . To this aim, consider the non-vanishing components of the invariant tensor given in , decompose the indices as in , and rescale not only the generators in compliance with but also the coefficients appearing in as in . Consequently, the Carroll limit $\sigma \rightarrow \infty$ leads to the following non-vanishing components of an invariant tensor for the superalgebra : $$\label{invadscarrollsuperpq}
\begin{split}
& \langle J_{ab} J_{cd} \rangle = \alpha_0 \left( \delta_{ad} \delta_{bc} - \delta_{ac} \delta_{bd} \right) \, , \\
& \langle J_{ab} H \rangle = \alpha_1 \epsilon_{ab} \, , \\
& \langle K_a P_b \rangle = - \alpha_1 \epsilon_{ab} \, , \\
& \langle P_{a} P_{b} \rangle = \frac{\alpha_0}{\ell^2} \delta_{ab} \, , \\
& \langle {T}^{ij} {T}^{kl} \rangle = 2 \alpha_0 \left( \delta^{il} \delta^{kj} - \delta^{ik} \delta^{lj} \right) \, , \\
& \langle {T}^{IJ} {T}^{KL} \rangle = 2 \alpha_0 \left( \delta^{IL} \delta^{KJ} - \delta^{IK} \delta^{LJ} \right) \, , \\
& \langle {T}^{ij} {S}^{kl} \rangle = - 2 \alpha_1 \left( \delta^{il} \delta^{kj} - \delta^{ik} \delta^{lj} \right) \, , \\
& \langle {T}^{IJ} {S}^{KL} \rangle = 2 \alpha_1 \left( \delta^{IL} \delta^{KJ} - \delta^{IK} \delta^{LJ} \right) \, , \\
& \langle Q^i_\alpha Q^j_\beta \rangle = 2 \alpha_1 C_{\alpha \beta} \delta^{ij} \, , \\
& \langle Q^I_\alpha Q^J_\beta \rangle = 2 \alpha_1 C_{\alpha \beta} \delta^{IJ} \, .
\end{split}$$ The invariant tensor whose components are given in is non-degenerate when $\alpha_1 \neq 0$.
Then, substituting the connection $1$-form in and the non-zero components of the invariant tensor into the general expression , we end up with the $(p,q)$ AdS Carroll CS supergravity action in three dimensions, that is $$\label{CSACpq}
\begin{split}
I^{(p,q)}_{CS} & = \frac{k}{4 \pi} \int_\mathcal{M} \Bigg \lbrace \frac{\alpha_0}{2} \left( \omega^a_{\phantom{a} b} R^b_{\phantom{b} a} + \frac{2}{\ell^2} V^a R_a + 2 {t^i}_j {d t^j}_i + \frac{4}{3} {t^i}_j {t^j}_k {t^k}_i + 2 {t^I}_J {d t^J}_I + \frac{4}{3} {t^I}_J {t^J}_K {t^K}_I \right) \\
& + \alpha_1 \bigg[ \epsilon_{ab} R^{ab} h - 2 \epsilon_{ab} \mathfrak{K}^a V^b + \frac{1}{\ell^2} \epsilon_{ab} V^a V^b h - 2 {t^i}_j \left( {ds^j}_i + {t^j}_k {s^k}_i \right) + 2 {t^I}_J \left( {ds^J}_I + {t^J}_K {s^K}_I \right) \\
& + 2 \bar{\psi}^i \nabla \psi^i + 2 \bar{\psi}^I \nabla \psi^I \bigg] - d \left( \frac{\alpha_1}{2} \epsilon_{ab} \omega^{ab} h - \alpha_1 \epsilon_{ab} k^a V^b + \alpha_1 {t^i}_j {s^j}_i - \alpha_1 {t^I}_J {s^J}_I \right) \Bigg \rbrace \, ,
\end{split}$$ which is written in terms of the curvatures appearing in and it involves two coupling constants, that are $\alpha_0$ and $\alpha_1$. Up to boundary terms, the action can be reworked as follows: $$\begin{split}
I^{(p,q)}_{CS} & = \frac{k}{4 \pi} \int_\mathcal{M} \Bigg \lbrace \frac{\alpha_0}{2} \left( \omega^a_{\phantom{a} b} R^b_{\phantom{b} a} + \frac{2}{\ell^2} V^a R_a + 2 {t^i}_j {d t^j}_i + \frac{4}{3} {t^i}_j {t^j}_k {t^k}_i + 2 {t^I}_J {d t^J}_I + \frac{4}{3} {t^I}_J {t^J}_K {t^K}_I \right) \\
& + \alpha_1 \bigg[ \epsilon_{ab} R^{ab} h - 2 \epsilon_{ab} \mathfrak{K}^a V^b + \frac{1}{\ell^2} \epsilon_{ab} V^a V^b h - 2 {t^i}_j \left( {ds^j}_i + {t^j}_k {s^k}_i \right) + 2 {t^I}_J \left( {ds^J}_I + {t^J}_K {s^K}_I \right) \\
& + 2 \bar{\psi}^i \nabla \psi^i + 2 \bar{\psi}^I \nabla \psi^I \bigg] \Bigg \rbrace \, .
\end{split}$$ As usual, the contribution proportional to $\alpha_0$ corresponds to the exotic Lagrangian, and we can see that it involves, besides the Lorentz and torsional terms, also pieces including the $1$-form fields $t^{ij}$ and $t^{IJ}$. However, it does not contain terms involving $\psi^i$ and $\psi^I$. On the other hand, the contribution proportional to $\alpha_1$ also includes pieces involving the $1$-form fields $s^{ij}$, $s^{IJ}$, $\psi^i$, and $\psi^I$.
The action is invariant by construction under , and the local gauge transformations $\delta_\lambda A = d \lambda + \left[A, \lambda \right]$ with gauge parameter $$\label{gparpq}
\lambda = \frac{1}{2} \lambda^{ab} J_{ab} + \kappa^a K_a + \lambda^a P_a + \tau H + \frac{1}{2} \varphi^{ij} t_{ij} + \frac{1}{2} \varphi^{IJ} t_{IJ} + \frac{1}{2} \varsigma^{ij} s_{ij} + \frac{1}{2} \varsigma^{IJ} s_{IJ} + \varepsilon^i Q_i + \varepsilon^I Q_I$$ are given by $$\label{gaugetrpq}
\begin{split}
\delta \omega^{ab} & = d \lambda^{ab} + \frac{2}{\ell^2} V^{[a} \lambda^{b]} \, , \\
\delta k^a & = d \kappa^a - \lambda^a_{\phantom{a} b} k^b + \omega^a_{\phantom{a} b} \kappa^b - \frac{1}{\ell^2} \lambda^a h + \frac{1}{\ell^2} V^a \tau - \frac{1}{\ell} \bar{\varepsilon}^i \Gamma^{a0} \psi^i + \frac{1}{\ell} \bar{\varepsilon}^I \Gamma^{a0} \psi^I \, , \\
\delta V^a & = d \lambda^a - \lambda^a_{\phantom{a}b} V^b + \omega^a_{\phantom{a}b} \lambda^b \, , \\
\delta h & = d \tau - \lambda^a k_a + V^a \kappa_a + \bar{\varepsilon}^i \Gamma^0 \psi^i + \bar{\varepsilon}^I \Gamma^0 \psi^I \, , \\
\delta t^{ij} & = d \varphi^{ij} - 2 {\varphi^{[i}}_k t^{k \vert j]} \, , \\
\delta t^{IJ} & = d \varphi^{IJ} - 2 {\varphi^{[I}}_K t^{K \vert J]} \, ,\\
\delta s^{ij} & = d \varsigma^{ij} - 2 {\varphi^{[i}}_k s^{k \vert j]} + 2 {t^{[i}}_k \varsigma^{k \vert j]} + 2 \bar{\varepsilon}^{[i} \psi^{j]} \, , \\
\delta s^{IJ} & = d \varsigma^{IJ} + 2 {\varphi^{[I}}_K s^{K \vert J]} + 2 {t^{[I}}_K \varsigma^{K \vert J]} - 2 \bar{\varepsilon}^{[I} \psi^{J]} \, , \\
\delta \psi^i & = d \varepsilon^+ - \frac{1}{4} \lambda^{ab} \Gamma_{ab} \psi^i + \frac{1}{4} \omega^{ab} \Gamma_{ab} \varepsilon^i - \frac{1}{2 \ell} \lambda^a \Gamma_a \psi^i + \frac{1}{2 \ell} V^a \Gamma_a \varepsilon^i - \varphi^{ij} \psi_j + t^{ij} \varepsilon_j \, , \\
\delta \psi^I & = d \varepsilon^- - \frac{1}{4} \lambda^{ab} \Gamma_{ab} \psi^I + \frac{1}{4} \omega^{ab} \Gamma_{ab} \varepsilon^I + \frac{1}{2 \ell} \lambda^a \Gamma_a \psi^I - \frac{1}{2 \ell} V^a \Gamma_a \varepsilon^I - \varphi^{IJ} \psi_J + t^{IJ} \varepsilon_J \, .
\end{split}$$ Thus, the restriction to supersymmetry transformations gives us $$\label{susytrpq}
\begin{split}
\delta \omega^{ab} & =0 \, , \\
\delta k^a & = - \frac{1}{\ell} \bar{\varepsilon}^i \Gamma^{a0} \psi^i + \frac{1}{\ell} \bar{\varepsilon}^I \Gamma^{a0} \psi^I \, , \\
\delta V^a & = 0 \, , \\
\delta h & = \bar{\varepsilon}^i \Gamma^0 \psi^i + \bar{\varepsilon}^I \Gamma^0 \psi^I \, , \\
\delta t^{ij} & = 0 \, , \\
\delta t^{IJ} & = 0 \, ,\\
\delta s^{ij} & = 2 \bar{\varepsilon}^{[i} \psi^{j]} \, , \\
\delta s^{IJ} & = - 2 \bar{\varepsilon}^{[I} \psi^{J]} \, ,\\
\delta \psi^i & = d \varepsilon^+ + \frac{1}{4} \omega^{ab} \Gamma_{ab} \varepsilon^i + \frac{1}{2 \ell} V^a \Gamma_a \varepsilon^i + t^{ij} \varepsilon_j \, , \\
\delta \psi^I & = d \varepsilon^- + \frac{1}{4} \omega^{ab} \Gamma_{ab} \varepsilon^I - \frac{1}{2 \ell} V^a \Gamma_a \varepsilon^I + t^{IJ} \varepsilon_J \, .
\end{split}$$
Finally, the equations of motion obtained from the variation of with respect to the $1$-form fields $\omega^{ab}$, $k^a$, $V^a$, $h$, $t^{ij}$, $t^{IJ}$, $s^{ij}$, $s^{IJ}$, $\psi^i$, and $\psi^I$ are, respectively (up to boundary contributions), $$\label{eompq}
\begin{split}
\delta \omega^{ab} & : \quad \alpha_0 \mathcal{R}^{ab} + \alpha_1 \epsilon^{ab} \mathcal{H}= 0 \, , \\
\delta k^a & : \quad \alpha_1 R^a = 0 \, , \\
\delta V^a & : \quad \frac{\alpha_0}{\ell^2} R^a + 2 \alpha_1 \epsilon_{ab} \mathcal{K}^b = 0 \, , \\
\delta h & : \quad \alpha_1 \mathcal{R}^{ab} = 0 \, , \\
\delta t^{ij} & : \quad - \alpha_0 \mathcal{T}^{ij} + \alpha_1 \mathcal{S}^{ij} = 0 \, , \\
\delta t^{IJ} & : \quad - \alpha_0 \mathcal{T}^{IJ} - \alpha_1 \mathcal{S}^{IJ} = 0 \, , \\
\delta s^{ij} & : \quad \alpha_1 \mathcal{T}^{ij} = 0 \, , \\
\delta s^{IJ} & : \quad \alpha_1 \mathcal{T}^{IJ} = 0 \, , \\
\delta \psi^i & : \quad \alpha_1 \nabla \psi^i = 0 \, , \\
\delta \psi^I & : \quad \alpha_1 \nabla \psi^I = 0 \, ,
\end{split}$$ and, for $\alpha_1 \neq 0$, they reduce precisely to the vanishing of the $(p,q)$ super-AdS Carroll curvature $2$-forms given in , that is to say $$\label{eomvacpq}
\begin{split}
& \mathcal{R}^{ab} = 0 \, , \quad \mathcal{K}^a = 0 \, , \quad R^a = 0 \, , \quad \mathcal{H}=0 \, , \quad \mathcal{T}^{ij} = 0 \, , \quad \mathcal{T}^{IJ} = 0 \, , \quad \mathcal{S}^{ij} = 0 \, , \quad \mathcal{S}^{IJ} = 0 \, , \\
& \nabla \psi^i =0 \, , \quad \nabla \psi^I =0 \, .
\end{split}$$ We observe that, as usual, $\alpha_1 \neq 0$ is a sufficient condition to recover , meaning that $\alpha_0$ can be consistently set to zero, making the exotic term in the CS action disappear.
Let us finally mention that, if we restrict ourselves to the case $p=q=1$, we exactly reproduce the results of Section \[11case\], that is to say, as properly expected, the $(1,1)$ AdS Carroll supergravity theory.
Study of the flat limit $\ell \rightarrow \infty$ {#flatlimit}
=================================================
In the sequel, we study the flat limit $\ell \rightarrow \infty$, which can be directly applied to the $\mathcal{N}$-extended AdS Carroll superalgebras , , , and , to the corresponding curvature $2$-forms, respectively given by , , , and , to the related CS actions , , , and , to the transformation laws , , , (and, in particular to the supersymmetry transformation laws , , , ), and to the field equations of the respective theories, namely , , , and .
$(2,0)$ Carroll supergravity from the $\ell \rightarrow \infty$ limit
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In the limit $\ell \rightarrow \infty$, the (anti)commutation relations of the $\mathcal{N}=(2,0)$ AdS Carroll superalgebra reduce to the following non-vanishing ones: $$\label{adscarrollsuperF22}
\begin{split}
& \left[ K_a , J_{bc} \right] = \delta_{ab} K_c - \delta_{ac} K_b \, , \\
& \left[ J_{ab}, P_{c}\right] =\delta_{bc}P_{a}-\delta _{ac}P_{b} \, , \\
& \left[ K_{a}, P_{b}\right] = - \delta_{ab} H \, , \\
& \left[ J_{ab},Q^\pm_{\alpha }\right] =-\frac{1}{2}\left( \Gamma _{ab} Q^\pm \right)_{\alpha } \, , \\
& \left[ T, Q^+_{\alpha }\right] = \left( \Gamma _{0} \right)_{\alpha \beta} Q^+_\beta \, , \\
& \left[ T, Q^-_{\alpha }\right] = - \left( \Gamma _{0} \right)_{\alpha \beta} Q^-_\beta \, , \\
& \left\{ Q^+_{\alpha }, Q^+_{\beta }\right\} = \left( \Gamma ^{0}C\right) _{\alpha \beta } \left( H + S \right) \, , \\
& \left\{ Q^-_{\alpha }, Q^-_{\beta }\right\} = \left( \Gamma ^{0}C\right) _{\alpha \beta } \left( H - S \right) \, .
\end{split}$$ These are the (anti)commutation relations of a new $\mathcal{N}=(2,0)$, $D=3$ super-Carroll algebra, involving an extra generator $S$, which could also have been derived by considering the ultra-relativistic contraction of the $\mathcal{N}=(2,0)$, $D=3$ Poincaré superalgebra supplemented with an $so(2)$ extension consisting in the extra generator $\tilde{S}$ introduced at the relativistic level.
As $\ell \rightarrow \infty$, the $2$-form curvatures become $$\label{curvadscarrollsuperF22}
\begin{split}
\mathcal{R}^{ab} & = d \omega^{ab} = R^{ab} \, , \\
\mathcal{K}^a & = d k^a + \omega^a_{\phantom{a} b} k^b = \mathfrak{K}^a \, , \\
R^a & = d V^a + \omega^a_{\phantom{a}b} V^b \, , \\
\mathcal{H} & = d h + V^a k_a - \frac{1}{2} \bar{\psi}^+ \Gamma^0 \psi^+ - \frac{1}{2} \bar{\psi}^- \Gamma^0 \psi^- = \mathfrak{H} - \frac{1}{2} \bar{\psi}^+ \Gamma^0 \psi^+ - \frac{1}{2} \bar{\psi}^- \Gamma^0 \psi^- \, , \\
\mathcal{T} & = d t \, , \\
\mathcal{S} & = d s - \frac{1}{2} \bar{\psi}^+ \Gamma^0 \psi^+ + \frac{1}{2} \bar{\psi}^- \Gamma^0 \psi^- \, , \\
\nabla \psi^+ & = d \psi^+ + \frac{1}{4} \omega^{ab} \Gamma_{ab} \psi^+ - t \Gamma_0 \psi^+ \, , \\
\nabla \psi^- & = d \psi^- + \frac{1}{4} \omega^{ab} \Gamma_{ab} \psi^- + t \Gamma_0 \psi^- \, .
\end{split}$$ On the other hand, by applying the $\ell \rightarrow \infty$ limit to the three-dimensional CS action , we end up with $$\label{CSC22}
\begin{split}
I^{(2,0)}_{CS} \vert_{\ell \rightarrow \infty} & = \frac{k}{4 \pi} \int_\mathcal{M} \Bigg \lbrace \frac{\alpha_0}{2} \left( \omega^a_{\phantom{a} b} R^b_{\phantom{b} a} - 4 t dt \right) + \alpha_1 \bigg( \epsilon_{ab} R^{ab} h - 2 \epsilon_{ab} \mathfrak{K}^a V^b + 4 t ds \\
& + 2 \bar{\psi}^+ \nabla \psi^+ + 2 \bar{\psi}^- \nabla \psi^- \bigg) - d \left( \frac{\alpha_1}{2} \epsilon_{ab} \omega^{ab} h - \alpha_1 \epsilon_{ab} k^a V^b - 2 \alpha_1 t s \right) \Bigg \rbrace \, ,
\end{split}$$ which is written in terms of the super-Carroll curvatures appearing in . The latter must not be confused with the super-AdS Carroll ones given in , since correspond to the flat limit of . Here we signal that we have done a little abuse of notation. The action can also be derived by using the following non-vanishing components of the invariant tensor: $$\begin{split}
& \langle J_{ab} J_{cd} \rangle = \alpha_0 \left( \delta_{ad} \delta_{bc} - \delta_{ac} \delta_{bd} \right) \, , \\
& \langle J_{ab} H \rangle = \alpha_1 \epsilon_{ab} \, , \\
& \langle K_a P_b \rangle = - \alpha_1 \epsilon_{ab} \, , \\
& \langle T T \rangle = - 2 \alpha_0\, , \\
& \langle T S \rangle = 2 \alpha_1 \, , \\
& \langle Q^+_\alpha Q^+_\beta \rangle = \langle Q^-_\alpha Q^-_\beta \rangle = 2 \alpha_1 C_{\alpha \beta} \, ,
\end{split}$$ which are obtained by taking the limit $\ell \rightarrow \infty$ of , and the connection $1$-form for the $\mathcal{N}=(2,0)$ (flat) Carroll superalgebra in the general expression . Notice that the exotic term, which is the one proportional to $\alpha_0$ in , now reduces purely to the so-called Lorentz Lagrangian.
The CS action is invariant by construction under the super-Carroll group associated with . In particular, concerning the flat limit of the gauge transformations , we get the local gauge transformations $$\label{gaugetrF22}
\begin{split}
\delta \omega^{ab} & = d \lambda^{ab} \, , \\
\delta k^a & = d \kappa^a - \lambda^a_{\phantom{a} b} k^b + \omega^a_{\phantom{a} b} \kappa^b \, , \\
\delta V^a & = d \lambda^a - \lambda^a_{\phantom{a}b} V^b + \omega^a_{\phantom{a}b} \lambda^b \, , \\
\delta h & = d \tau - \lambda^a k_a + V^a \kappa_a + \bar{\varepsilon}^+ \Gamma^0 \psi^+ + \bar{\varepsilon}^- \Gamma^0 \psi^- \, , \\
\delta t & = d \varphi \, , \\
\delta s & = d \varsigma + \bar{\varepsilon}^+ \Gamma^0 \psi^+ - \bar{\varepsilon}^- \Gamma^0 \psi^- \, , \\
\delta \psi^+ & = d \varepsilon^+ - \frac{1}{4} \lambda^{ab} \Gamma_{ab} \psi^+ + \frac{1}{4} \omega^{ab} \Gamma_{ab} \varepsilon^+ + \varphi \Gamma_0 \psi^+ - t \Gamma_0 \varepsilon^+ \, , \\
\delta \psi^- & = d \varepsilon^- - \frac{1}{4} \lambda^{ab} \Gamma_{ab} \psi^- + \frac{1}{4} \omega^{ab} \Gamma_{ab} \varepsilon^- - \varphi \Gamma_0 \psi^- + t \Gamma_0 \varepsilon^- \, .
\end{split}$$ The restriction to supersymmetry transformations reads $$\label{susytrF22}
\begin{split}
\delta \omega^{ab} & = 0 \, , \\
\delta k^a & = 0 \, , \\
\delta V^a & = 0 \, , \\
\delta h & = \bar{\varepsilon}^+ \Gamma^0 \psi^+ + \bar{\varepsilon}^- \Gamma^0 \psi^- \, , \\
\delta t & = 0 \, , \\
\delta s & = \bar{\varepsilon}^+ \Gamma^0 \psi^+ - \bar{\varepsilon}^- \Gamma^0 \psi^- \, , \\
\delta \psi^+ & = d \varepsilon^+ + \frac{1}{4} \omega^{ab} \Gamma_{ab} \varepsilon^+ - t \Gamma_0 \varepsilon^+ \, , \\
\delta \psi^- & = d \varepsilon^- + \frac{1}{4} \omega^{ab} \Gamma_{ab} \varepsilon^- + t \Gamma_0 \varepsilon^- \, .
\end{split}$$
Finally, the equations of motion for the action (flat limit of the equations of motion given in ) are $$\label{eomF22}
\begin{split}
\delta \omega^{ab} & : \quad \alpha_0 \mathcal{R}^{ab} + \alpha_1 \epsilon^{ab} \mathcal{H}= 0 \, , \\
\delta k^a & : \quad \alpha_1 R^a = 0 \, , \\
\delta V^a & : \quad \alpha_1 \epsilon_{ab} \mathcal{K}^b = 0 \, , \\
\delta h & : \quad \alpha_1 \mathcal{R}^{ab} = 0 \, , \\
\delta t & : \quad - \alpha_0 \mathcal{T} + \alpha_1 \mathcal{S} = 0 \, , \\
\delta s & : \quad \alpha_1 \mathcal{T} = 0 \, , \\
\delta \psi^+ & : \quad \alpha_1 \nabla \psi^+ = 0 \, , \\
\delta \psi^- & : \quad \alpha_1 \nabla \psi^- = 0 \, ,
\end{split}$$ and we can see that, when $\alpha_1 \neq 0$, they exactly reduce to the vanishing of the curvature $2$-forms given in . We can also observe that, in analogy with the AdS case of Section \[20case\], also in the flat limit $\alpha_1 \neq 0$ results to be a sufficient condition to recover the vanishing of the curvature $2$-forms obtained in the flat limit, which means that one could consistently set $\alpha_0=0$ and thus neglect the exotic term (i.e., the Lorentz Lagrangian) in the CS action .[^8]
$(1,1)$ Carroll supergravity from the $\ell \rightarrow \infty$ limit
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The limit $\ell \rightarrow \infty$ performed on the (anti)commutation relations of the $\mathcal{N}=(1,0)$ AdS Carroll superalgebra leads to the following non-vanishing ones: $$\label{adscarrollsuperF11}
\begin{split}
& \left[ K_a , J_{bc} \right] = \delta_{ab} K_c - \delta_{ac} K_b \, , \\
& \left[ J_{ab}, P_{c}\right] =\delta_{bc}P_{a}-\delta _{ac}P_{b} \, , \\
& \left[ K_{a}, P_{b}\right] = - \delta_{ab} H \, , \\
& \left[ J_{ab},Q^\pm_{\alpha }\right] =-\frac{1}{2}\left( \Gamma _{ab} Q^\pm \right)_{\alpha } \, , \\
& \left\{ Q^\pm_{\alpha }, Q^\pm_{\beta }\right\} = \left( \Gamma ^{0}C\right) _{\alpha \beta } H \, .
\end{split}$$ These are the (anti)commutation relations of the $\mathcal{N}=(1,1)$, $D=3$ super-Carroll algebra (see [@Bergshoeff:2015wma], where corresponds to the superalgebra obtained in the $R \rightarrow \infty$ limit of the $\mathcal{N}=(1,1)$ AdS-Carroll superalgebra of Section C.4 of the same paper). It could be also obtained by considering the ultra-relativistic contraction of the $\mathcal{N}=(1,1)$, $D=3$ Poincaré superalgebra.
Taking $\ell \rightarrow \infty$, the $2$-form curvatures reduce to $$\label{curvadscarrollsuperF11}
\begin{split}
\mathcal{R}^{ab} & = d \omega^{ab} = R^{ab} \, , \\
\mathcal{K}^a & = d k^a + \omega^a_{\phantom{a} b} k^b = \mathfrak{K}^a \, , \\
R^a & = d V^a + \omega^a_{\phantom{a}b} V^b \, , \\
\mathcal{H} & = d h + V^a k_a - \frac{1}{2} \bar{\psi}^+ \Gamma^0 \psi^+ - \frac{1}{2} \bar{\psi}^- \Gamma^0 \psi^- = \mathfrak{H} - \frac{1}{2} \bar{\psi}^+ \Gamma^0 \psi^+ - \frac{1}{2} \bar{\psi}^- \Gamma^0 \psi^- \, , \\
\nabla \psi^+ & = d \psi^+ + \frac{1}{4} \omega^{ab} \Gamma_{ab} \psi^+ \, , \\
\nabla \psi^- & = d \psi^- + \frac{1}{4} \omega^{ab} \Gamma_{ab} \psi^- \, ,
\end{split}$$ and the $\ell \rightarrow \infty$ limit of the CS action leads us to the following three-dimensional one: $$\label{CSC11}
\begin{split}
I^{(1,1)}_{CS} \vert_{\ell \rightarrow \infty} & = \frac{k}{4 \pi} \int_\mathcal{M} \Bigg \lbrace \frac{\alpha_0}{2} \left( \omega^a_{\phantom{a} b} R^b_{\phantom{b} a} \right) + \alpha_1 \left( \epsilon_{ab} R^{ab} h - 2 \epsilon_{ab} \mathfrak{K}^a V^b + 2 \bar{\psi}^+ \nabla \psi^+ + 2 \bar{\psi}^- \nabla \psi^- \right) \\
& - d \left( \frac{\alpha_1}{2} \epsilon_{ab} \omega^{ab} h - \alpha_1 \epsilon_{ab} k^a V^b \right) \Bigg \rbrace \, ,
\end{split}$$ written in terms of the curvatures appearing in (again, we are doing a little abuse of notation). The action can also be derived by using the connection $1$-form for the $\mathcal{N}=(1,1)$ (flat) Carroll superalgebra together with the non-vanishing components of the invariant tensor $$\begin{split}
& \langle J_{ab} J_{cd} \rangle = \alpha_0 \left( \delta_{ad} \delta_{bc} - \delta_{ac} \delta_{bd} \right) \, , \\
& \langle J_{ab} H \rangle = \alpha_1 \epsilon_{ab} \, , \\
& \langle K_a P_b \rangle = - \alpha_1 \epsilon_{ab} \, , \\
& \langle Q^+_\alpha Q^+_\beta \rangle = \langle Q^-_\alpha Q^-_\beta \rangle = 2 \alpha_1 C_{\alpha \beta} \,
\end{split}$$ in the general expression for a three-dimensional CS action . Analogously to what happened in the $(2,0)$ flat theory, also in the current case the exotic term, proportional to $\alpha_0$, now reduces purely to the Lorentz Lagrangian.
By construction, the CS action is invariant under the $(1,1)$ super-Carroll group, that is associated with the superalgebra given in . In particular, taking the flat limit of the gauge transformations , we get the following local gauge transformations: $$\label{gaugetrF11}
\begin{split}
\delta \omega^{ab} & = d \lambda^{ab} \, , \\
\delta k^a & = d \kappa^a - \lambda^a_{\phantom{a} b} k^b + \omega^a_{\phantom{a} b} \kappa^b \, , \\
\delta V^a & = d \lambda^a - \lambda^a_{\phantom{a}b} V^b + \omega^a_{\phantom{a}b} \lambda^b \, , \\
\delta h & = d \tau - \lambda^a k_a + V^a \kappa_a + \bar{\varepsilon}^+ \Gamma^0 \psi^+ + \bar{\varepsilon}^- \Gamma^0 \psi^- \, , \\
\delta \psi^+ & = d \varepsilon^+ - \frac{1}{4} \lambda^{ab} \Gamma_{ab} \psi^+ + \frac{1}{4} \omega^{ab} \Gamma_{ab} \varepsilon^+ \, , \\
\delta \psi^- & = d \varepsilon^- - \frac{1}{4} \lambda^{ab} \Gamma_{ab} \psi^- + \frac{1}{4} \omega^{ab} \Gamma_{ab} \varepsilon^- \, ,
\end{split}$$ and restricting ourselves to supersymmetry transformations, we are left with $$\label{susytrF11}
\begin{split}
\delta \omega^{ab} & = 0 \, , \\
\delta k^a & = 0 \, , \\
\delta V^a & = 0 \, , \\
\delta h & = \bar{\varepsilon}^+ \Gamma^0 \psi^+ + \bar{\varepsilon}^- \Gamma^0 \psi^- \, , \\
\delta \psi^+ & = d \varepsilon^+ + \frac{1}{4} \omega^{ab} \Gamma_{ab} \varepsilon^+ \, , \\
\delta \psi^- & = d \varepsilon^- + \frac{1}{4} \omega^{ab} \Gamma_{ab} \varepsilon^- \, .
\end{split}$$
Concluding, the equations of motion for the action (flat limit of the equations of motion given in ) read as follows: $$\label{eomF11}
\begin{split}
\delta \omega^{ab} & : \quad \alpha_0 \mathcal{R}^{ab} + \alpha_1 \epsilon^{ab} \mathcal{H}= 0 \, , \\
\delta k^a & : \quad \alpha_1 R^a = 0 \, , \\
\delta V^a & : \quad \alpha_1 \epsilon_{ab} \mathcal{K}^b = 0 \, , \\
\delta h & : \quad \alpha_1 \mathcal{R}^{ab} = 0 \, , \\
\delta \psi^+ & : \quad \alpha_1 \nabla \psi^+ = 0 \, , \\
\delta \psi^- & : \quad \alpha_1 \nabla \psi^- = 0 \, .
\end{split}$$ When $\alpha_1 \neq 0$, the eqs. exactly reduce to the vanishing of the curvature $2$-forms in ($\alpha_1 \neq 0$ is a sufficient condition to recover the vanishing of the curvature $2$-forms , meaning that one could consistently set $\alpha_0=0$, omitting the exotic term, that is the Lorentz Lagrangian, in the CS action ).
Notice that the restriction to the purely bosonic part of the action yields exactly the three-dimensional CS gravity action invariant under the $D=3$ Carroll algebra [@LL; @Bacry:1968zf]. The aforesaid CS action involving purely bosonic terms is equivalent, as argued in [@Bergshoeff:2017btm], to the action found in [@Bergshoeff:2017btm] if we take the $D=3$ case in the same paper.
$(\mathcal{N},0)$ Carroll supergravity theories from the $\ell \rightarrow \infty$ limit
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Taking the flat limit $\ell \rightarrow \infty$ of the (anti)commutation relations , we get the following non-trivial ones: $$\label{adscarrollsuperFn}
\begin{split}
& \left[ K_a , J_{bc} \right] = \delta_{ab} K_c - \delta_{ac} K_b \, , \\
& \left[ J_{ab}, P_{c}\right] =\delta_{bc}P_{a}-\delta _{ac}P_{b} \, , \\
& \left[ K_{a}, P_{b}\right] = - \delta_{ab} H \, , \\
& \left[ {T}^{\lambda \mu} , {T}^{\nu \rho} \right] = \delta^{\mu \nu} {T}^{\lambda \rho} - \delta^{\lambda \nu} {T}^{\mu \rho}- \delta^{\mu \rho} {T}^{\lambda \nu} + \delta^{\lambda \rho} {T}^{\mu \nu} \, , \\
& \left[ {T}^{\lambda \mu} , {U}^{\nu \rho} \right] = \delta^{\mu \rho} {U}^{\nu \lambda} - \delta^{\lambda \rho} {U}^{\nu \mu} \, , \\
& \left[ {T}'^{\lambda \mu} , {T}'^{\nu \rho} \right] = \delta^{\mu \nu} {T}'^{\lambda \rho} - \delta^{\lambda \nu} {T}'^{\mu \rho}- \delta^{\mu \rho} {T}'^{\lambda \nu} + \delta^{\lambda \rho} {T}'^{\mu \nu} \, , \\
& \left[ {T}'^{\lambda \mu} , {U}^{\nu \rho} \right] = \delta^{\mu \nu} {U}^{\lambda \rho}- \delta^{\lambda \nu} {U}^{\mu \rho} \, , \\
& \left[ {U}^{\lambda \mu} , {U}^{\nu \rho} \right] = - \delta^{\lambda \nu} {T}^{\mu \rho} - \delta^{\mu \rho} {T}'^{\lambda \nu} \, , \\
& \left[ {T}^{\lambda \mu} , {S}^{\nu \rho} \right] = \delta^{\mu \nu} {S}^{\lambda \rho} - \delta^{\lambda \nu} {S}^{\mu \rho}- \delta^{\mu \rho} {S}^{\lambda \nu} + \delta^{\lambda \rho} {S}^{\mu \nu} \, , \\
& \left[ {T}^{\lambda \mu} , {V}^{\nu \rho} \right] = \delta^{\mu \rho} {V}^{\nu \lambda} - \delta^{\lambda \rho} {V}^{\nu \mu} \, , \\
& \left[ {T}'^{\lambda \mu} , {S}'^{\nu \rho} \right] = \delta^{\mu \nu} {S}'^{\lambda \rho} - \delta^{\lambda \nu} {S}'^{\mu \rho}- \delta^{\mu \rho} {S}'^{\lambda \nu} + \delta^{\lambda \rho} {S}'^{\mu \nu} \, , \\
& \left[ {T}'^{\lambda \mu} , {V}^{\nu \rho} \right] = \delta^{\mu \nu} {V}^{\lambda \rho}- \delta^{\lambda \nu} {V}^{\mu \rho} \, , \\
& \left[ {U}^{\lambda \mu} , {S}^{\nu \rho} \right] = - \delta^{\mu \rho} {V}^{\lambda \nu} + \delta^{\mu \nu} {V}^{\lambda \rho} \, , \\
& \left[ {U}^{\lambda \mu} , {S}'^{\nu \rho} \right] = - \delta^{\lambda \rho} {V}^{\nu \mu} + \delta^{\lambda \nu} {V}^{\rho \mu} \, , \\
& \left[ {U}^{\lambda \mu} , {V}^{\nu \rho} \right] = - \delta^{\lambda \nu} {S}^{\mu \rho} - \delta^{\mu \rho} {S}'^{\lambda \nu} \, , \\
& \left[ J_{ab}, Q^{\pm \, \lambda} _\alpha \right] =-\frac{1}{2}\left( \Gamma _{ab} Q^{\pm \, \lambda} \right)_{\alpha } \, , \\
& \left[ {T}^{\lambda \mu}, {Q}^{\pm \, \nu } _\alpha \right] = \frac{1}{2} \left[\delta^{\mu \nu} \left( {Q}^{+ \, \lambda } _\alpha + {Q}^{- \, \lambda } _\alpha \right) - \delta^{\lambda \nu} \left( {Q}^{+ \, \mu } _\alpha + {Q}^{- \, \mu } _\alpha \right) \right] \, , \\
& \left[ {T}'^{\lambda \mu}, {Q}^{\pm \, \nu } _\alpha \right] = \pm \frac{1}{2} \left[\delta^{\mu \nu} \left( {Q}^{+ \, \lambda } _\alpha - {Q}^{- \, \lambda } _\alpha \right) - \delta^{\lambda \nu} \left( {Q}^{+ \, \mu } _\alpha - {Q}^{- \, \mu } _\alpha \right) \right] \, , \\
& \left[ {U}^{\lambda \mu}, {Q}^{\pm \, \nu } _\alpha \right] = \mp \frac{1}{2} \left( \Gamma_0 \right)_{\alpha \beta} \left[ \delta^{\lambda \nu} \left( {Q}^{+ \, \mu } _\beta + {Q}^{- \, \ \mu } _\beta \right) \pm \delta^{\mu \nu} \left( {Q}^{+ \, \lambda } _\beta - {Q}^{- \, \lambda } _\beta \right) \right] \, , \\
& \lbrace {Q}^{+ \, \lambda} _\alpha , {Q}^{+ \, \mu} _\alpha \rbrace = \left( \Gamma^0 C \right)_{\alpha \beta} \left( \delta^{\lambda \mu} {H} - {V}^{(\lambda \mu)} \right) + C_{\alpha \beta} {Y}^{[\lambda \mu]} \, , \\
& \lbrace {Q}^{+ \, \lambda} _\alpha , {Q}^{- \, \mu} _\beta \rbrace = C_{\alpha \beta} {Y}'^{[\lambda \mu]} - \left( \Gamma^0 C \right)_{\alpha \beta} {V}^{[\lambda \mu]} \, , \\
& \lbrace {Q}^{- \, \lambda} _\alpha , {Q}^{- \, \mu} _\beta \rbrace = \left( \Gamma^0 C \right)_{\alpha \beta} \left( \delta^{\lambda \mu} {H} + {V}^{(\lambda \mu)} \right) + C_{\alpha \beta} {Y}^{[\lambda \mu]} \, .
\end{split}$$ The (anti)commutation relations are those of the $\mathcal{N}=(\mathcal{N},0)$, $D=3$ super-Carroll algebra (with $\mathcal{N}$ even), and one could also obtain it by taking the ultra-relativistic limit of the $\mathfrak{so}(\mathcal{N})$ extension of the $\mathcal{N}=(\mathcal{N},0)$, $D=3$ Poincaré superalgebra.
Moreover, as $\ell \rightarrow \infty$, the $2$-form curvatures become $$\label{curvadscarrollsuperFn}
\begin{split}
\mathcal{R}^{ab} & = d \omega^{ab} = R^{ab} \, , \\
\mathcal{K}^a & = d k^a + \omega^a_{\phantom{a} b} k^b = \mathfrak{K}^a \, , \\
R^a & = d V^a + \omega^a_{\phantom{a}b} V^b \, , \\
\mathcal{H} & = d h + V^a k_a - \frac{1}{2} \bar{\psi}^{+ \, \lambda} \Gamma^0 \psi^{+ \, \lambda} - \frac{1}{2} \bar{\psi}^{- \, \lambda} \Gamma^0 \psi^{- \, \lambda} = \mathfrak{H} - \frac{1}{2} \bar{\psi}^{+ \, \lambda} \Gamma^0 \psi^{+ \, \lambda} - \frac{1}{2} \bar{\psi}^{- \, \lambda} \Gamma^0 \psi^{- \, \lambda} \, , \\
\mathcal{T}^{\lambda \mu} & = d t^{\lambda \mu} + {t^\lambda}_\nu t^{\nu \mu} + {u'^{[\lambda}}_\nu u^{\nu \vert \mu]} \, , \\
\mathcal{T}'^{\lambda \mu} & = d t'^{\lambda \mu} + {t'^\lambda}_\nu t'^{\nu \mu} + {u^{[\lambda}}_{ \nu} u'^{\nu \vert \mu]} \, , \\
\mathcal{U}^{\lambda \mu} & = d u^{\lambda \mu} + {u^\lambda}_\nu t^{\nu \mu} + {t'^\lambda}_\nu u^{\nu \mu} \, , \\
\mathcal{S}^{\lambda \mu} & = d s^{\lambda \mu} + 2 {t^\lambda}_\nu s^{\nu \mu} + 2 {u'^{[\lambda}}_\nu v^{\nu \vert \mu]} - \frac{1}{2} \bar{\psi}^{+ \, \lambda} \psi^{+ \, \mu} - \frac{1}{2} \bar{\psi}^{- \, \lambda} \psi^{- \, \mu} - \bar{\psi}^{+ \, [ \lambda} \psi^{- \, \mu ]} \, , \\
\mathcal{S}'^{\lambda \mu} & = d s'^{\lambda \mu} + 2 {t'^\lambda}_\nu s'^{\nu \mu} + 2 {u^{[\lambda}}_\nu v'^{\nu \vert \mu]} - \frac{1}{2} \bar{\psi}^{+ \, \lambda} \psi^{+ \, \mu} - \frac{1}{2} \bar{\psi}^{- \, \lambda} \psi^{- \, \mu} + \bar{\psi}^{+ \, [ \lambda} \psi^{- \, \mu ]} \, , \\
\mathcal{V}^{\lambda \mu} & = d v^{\lambda \mu} + {v^\lambda}_\nu t^{\nu \mu} + {t'^\lambda}_\nu v^{\nu \mu} + {u^\lambda}_\nu s^{\nu \mu} + {s'^\lambda}_\nu u^{\nu \mu} + \frac{1}{2} \bar{\psi}^{+ \, \lambda} \Gamma^0 \psi^{+ \, \mu} - \frac{1}{2} \bar{\psi}^{- \, \lambda} \Gamma^0 \psi^{- \, \mu} \\
& + \bar{\psi}^{+ \, [ \lambda} \Gamma^0 \psi^{- \, \mu ]} \, , \\
\nabla \psi^{+ \, \lambda} & = d \psi^{+ \, \lambda} + \frac{1}{4} \omega^{ab} \Gamma_{ab} \psi^{+ \, \lambda} + \frac{1}{2} t^{\lambda \mu} \psi^+_\mu + \frac{1}{2} t^{\lambda \mu} \psi^-_\mu + \frac{1}{2} t'^{\lambda \mu} \psi^+_\mu - \frac{1}{2} t'^{\lambda \mu} \psi^-_\mu \\
& + u^{(\lambda \mu)} \Gamma_0 \psi^+_\mu + u^{[\lambda \mu]} \Gamma_0 \psi^-_\mu \, , \\
\nabla \psi^{- \, \lambda} & = d \psi^{- \, \lambda} + \frac{1}{4} \omega^{ab} \Gamma_{ab} \psi^{- \, \lambda} + \frac{1}{2} t^{\lambda \mu} \psi^+_\mu + \frac{1}{2} t^{\lambda \mu} \psi^-_\mu - \frac{1}{2} t'^{\lambda \mu} \psi^+_\mu + \frac{1}{2} t'^{\lambda \mu} \psi^-_\mu \\
& - u^{(\lambda \mu)} \Gamma_0 \psi^-_\mu - u^{[\lambda \mu]} \Gamma_0 \psi^+_\mu \, .
\end{split}$$
Applying the $\ell \rightarrow \infty$ limit to the CS action , we obtain $$\label{CSCn}
\begin{split}
I^{(\mathcal{N},0)}_{CS} \vert_{\ell \rightarrow \infty} & = \frac{k}{4 \pi} \int_\mathcal{M} \Bigg \lbrace \frac{\alpha_0}{2} \bigg( \omega^a_{\phantom{a} b} R^b_{\phantom{b} a} + 2 {t^\lambda}_\mu {d t^\mu}_\lambda + \frac{4}{3} {t^\lambda}_\mu {t^\mu}_\nu {t^\nu}_\lambda + 2 {t'^\lambda}_\mu {d t'^\mu}_\lambda + \frac{4}{3} {t'^\lambda}_\mu {t'^\mu}_\nu {t'^\nu}_\lambda \\
& + 4 {u^\lambda}_\mu {d u'^\mu}_\lambda - 4 t_{\lambda \mu} {u'^{\lambda}}_{\nu} u^{\nu \mu} - 4 t'_{\lambda \mu} {u^{\lambda}}_{\nu} u'^{\nu \mu} \bigg) + \alpha_1 \bigg[ \epsilon_{ab} R^{ab} h - 2 \epsilon_{ab} \mathfrak{K}^a V^b \\
& - 2 {t^\lambda}_\mu \left( {ds^\mu}_\lambda + {t^\mu}_\nu {s^\nu}_\lambda \right) - 2 {t'^\lambda}_\mu \left( {ds'^\mu}_\lambda + {t'^\mu}_\nu {s'^\nu}_\lambda \right) - 4 {u^\lambda}_\mu {d v'^\mu}_\lambda - 2 {u'^\lambda}_\mu {u^\mu}_\nu {s^\nu}_\lambda \\
& - 2 {u^\lambda}_\mu {u'^\mu}_\nu {s'^\nu}_\lambda - 4 {u'^\lambda}_\mu {v^\mu}_\nu {t^\nu}_\lambda - 4 {u^\lambda}_\mu {v'^\mu}_\nu {t'^\nu}_\lambda + 2 \bar{\psi}^{+ \, \lambda} \nabla \psi^{+ \, \lambda} + 2 \bar{\psi}^{- \, \lambda} \nabla \psi^{- \, \lambda} \bigg] \\
& - d \left( \frac{\alpha_1}{2} \epsilon_{ab} \omega^{ab} h - \alpha_1 \epsilon_{ab} k^a V^b + \alpha_1 {t^\lambda}_\mu {s^\mu}_\lambda + \alpha_1 {t'^\lambda}_\mu {s'^\mu}_\lambda + 2 \alpha_1 {u^\lambda}_\mu {v'^\mu}_\lambda \right) \Bigg \rbrace \, ,
\end{split}$$ which is written in terms of the super-Carroll curvatures appearing in (we emphasize that the latter must not be confused with the super-AdS Carroll ones given in ). Note that the exotic term in , that is the one proportional to $\alpha_0$, is now given by the Lorentz Lagrangian plus additional terms involving the $1$-form fields $t^{\lambda \mu}$, $t'^{\lambda \mu}$, and $u^{\lambda \mu}$. Let us further mention that the action can also be derived by using the non-vanishing components of the invariant tensor obtained by taking the limit $\ell \rightarrow \infty$ of together with the connection $1$-form for the $\mathcal{N}=(\mathcal{N},0)$ (flat) Carroll superalgebra in the general expression . In particular, taking the $\ell \rightarrow \infty$ limit of we are left with the non-vanishing components $$\begin{split}
& \langle J_{ab} J_{cd} \rangle = \alpha_0 \left( \delta_{ad} \delta_{bc} - \delta_{ac} \delta_{bd} \right) \, , \\
& \langle J_{ab} H \rangle = \alpha_1 \epsilon_{ab} \, , \\
& \langle K_a P_b \rangle = - \alpha_1 \epsilon_{ab} \, , \\
& \langle {T}^{\lambda \mu} {T}^{\nu \rho} \rangle = \langle {T}'^{\lambda \mu} {T}'^{\nu \rho} \rangle = 2 \alpha_0 \left( \delta^{\lambda \rho} \delta^{\nu \mu} - \delta^{\lambda \nu} \delta^{\rho \mu} \right) \, , \\
& \langle {U}^{\lambda \mu} {U}^{\nu \rho} \rangle = - 2 \alpha_0 \delta^{\lambda \nu} \delta^{\rho \mu} \, , \\
& \langle {T}^{\lambda \mu} {S}^{\nu \rho} \rangle = \langle {T}'^{\lambda \mu} {S}'^{\nu \rho} \rangle = - 2 \alpha_1 \left( \delta^{\lambda \rho} \delta^{\nu \mu} - \delta^{\lambda \nu} \delta^{\rho \mu} \right) \, , \\
& \langle {U}^{\lambda \mu} {V}^{\nu \rho} \rangle = 2 \alpha_1 \delta^{\lambda \nu} \delta^{\rho \mu} \, , \\
& \langle Q^{+ \, \lambda}_\alpha Q^{+ \, \mu}_\beta \rangle = \langle Q^{- \, \lambda}_\alpha Q^{- \, \mu}_\beta \rangle = 2 \alpha_1 C_{\alpha \beta} \delta^{\lambda \mu} \, .
\end{split}$$
The action is invariant by construction under the super-Carroll group associated with ; in particular, concerning the $\ell \rightarrow \infty$ limit of the local gauge transformations , we obtain $$\label{gaugetrFn}
\begin{split}
\delta \omega^{ab} & = d \lambda^{ab} \, , \\
\delta k^a & = d \kappa^a - \lambda^a_{\phantom{a} b} k^b + \omega^a_{\phantom{a} b} \kappa^b \, , \\
\delta V^a & = d \lambda^a - \lambda^a_{\phantom{a}b} V^b + \omega^a_{\phantom{a}b} \lambda^b \, , \\
\delta h & = d \tau - \lambda^a k_a + V^a \kappa_a + \bar{\varepsilon}^{+ \, \lambda} \Gamma^0 \psi^{+ \, \lambda} + \bar{\varepsilon}^{- \, \lambda} \Gamma^0 \psi^{- \, \lambda} \, , \\
\delta t^{\lambda \mu} & = d \varrho^{\lambda \mu} - 2 {\varrho^{[\lambda}}_\nu t^{\nu \vert \mu]} - 2 {\varphi'^{[\lambda}}_\nu u^{\nu \vert \mu]} \, , \\
\delta t'^{\lambda \mu} & = d \varrho'^{\lambda \mu} - 2 {\varrho'^{[\lambda}}_\nu t'^{\nu \vert \mu]} - 2 {\varphi^{[\lambda}}_{ \nu} u'^{\nu \vert \mu]} \, , \\
\delta u^{\lambda \mu} & = d \varphi^{\lambda \mu} - {\varphi^\lambda}_\nu t^{\nu \mu} + {u^\lambda}_\nu \varrho^{\nu \mu} - {\varrho'^\lambda}_\nu u^{\nu \mu} + {t'^\lambda}_\nu \varphi^{\nu \mu} \, , \\
\delta s^{\lambda \mu} & = d \vartheta^{\lambda \mu} - 2 {\varrho^{[\lambda}}_\nu s^{\nu \vert \mu]} + 2 {t^{[\lambda}}_\nu \vartheta^{\nu \vert \mu]} - 2 {\varphi'^{[\lambda}}_\nu v^{\nu \vert \mu]} + 2 {u'^{[\lambda}}_\nu \varsigma^{\nu \vert \mu]} + \bar{\varepsilon}^{+ \, [\lambda} \psi^{+ \, \mu]} + \bar{\varepsilon}^{- \, [\lambda} \psi^{- \, \mu]} \\
& + \bar{\varepsilon}^{+ \, [ \lambda} \psi^{- \, \mu ]} + \bar{\varepsilon}^{- \, [ \lambda} \psi^{+ \, \mu ]} \, , \\
\delta s'^{\lambda \mu} & = d \vartheta'^{\lambda \mu} - 2 {\varrho'^{[\lambda}}_\nu s'^{\nu \vert \mu]} + 2 {t'^{[\lambda}}_\nu \vartheta'^{\nu \vert \mu]} - 2 {\varphi^{[\lambda}}_\nu v'^{\nu \vert \mu]} + 2 {u^{[\lambda}}_\nu \varsigma'^{\nu \vert \mu]} + \bar{\varepsilon}^{+ \, [\lambda} \psi^{+ \, \mu]} + \bar{\varepsilon}^{- \, [\lambda} \psi^{- \, \mu]} \\
& - \bar{\varepsilon}^{+ \, [ \lambda} \psi^{- \, \mu ]} - \bar{\varepsilon}^{- \, [ \lambda} \psi^{+ \, \mu ]} \, , \\
\delta v^{\lambda \mu} & = d \varsigma^{\lambda \mu} - {\varsigma^\lambda}_\nu t^{\nu \mu} + {v^\lambda}_\nu \varrho^{\nu \mu} - {\varrho'^\lambda}_\nu v^{\nu \mu} + {t'^\lambda}_\nu \varsigma^{\nu \mu} - {\varphi^\lambda}_\nu s^{\nu \mu} + {u^\lambda}_\nu \vartheta^{\nu \mu} - {\vartheta'^\lambda}_\nu u^{\nu \mu} + {s'^\lambda}_\nu \varphi^{\nu \mu} \\
& - \bar{\varepsilon}^{+ \, \lambda} \Gamma^0 \psi^{+ \, \mu} + \bar{\varepsilon}^{- \, \lambda} \Gamma^0 \psi^{- \, \mu} - \bar{\varepsilon}^{+ \, [ \lambda} \Gamma^0 \psi^{- \, \mu ]} + \bar{\varepsilon}^{- \, [ \lambda} \Gamma^0 \psi^{+ \, \mu ]} \, , \\
\delta \psi^{+ \, \lambda} & = d \varepsilon^{+ \, \lambda} - \frac{1}{4} \lambda^{ab} \Gamma_{ab} \psi^{+ \, \lambda} + \frac{1}{4} \omega^{ab} \Gamma_{ab} \varepsilon^{+ \, \lambda} - \frac{1}{2} \varrho^{\lambda \mu} \psi^+_\mu + \frac{1}{2} t^{\lambda \mu} \varepsilon^+_\mu - \frac{1}{2} \varrho^{\lambda \mu} \psi^-_\mu + \frac{1}{2} t^{\lambda \mu} \varepsilon^-_\mu - \frac{1}{2} \varrho'^{\lambda \mu} \psi^+_\mu \\
& + \frac{1}{2} t'^{\lambda \mu} \varepsilon^+_\mu + \frac{1}{2} \varrho'^{\lambda \mu} \psi^-_\mu - \frac{1}{2} t'^{\lambda \mu} \varepsilon^-_\mu - \varphi^{(\lambda \mu)} \Gamma_0 \psi^+_\mu + u^{(\lambda \mu)} \Gamma_0 \varepsilon^+_\mu - \varphi^{[\lambda \mu]} \Gamma_0 \psi^-_\mu + u^{[\lambda \mu]} \Gamma_0 \varepsilon^-_\mu \, , \\
\delta \psi^{- \, \lambda} & = d \varepsilon^{- \, \lambda} - \frac{1}{4} \lambda^{ab} \Gamma_{ab} \psi^{- \, \lambda} + \frac{1}{4} \omega^{ab} \Gamma_{ab} \varepsilon^{- \, \lambda} - \frac{1}{2} \varrho^{\lambda \mu} \psi^+_\mu + \frac{1}{2} t^{\lambda \mu} \varepsilon^+_\mu - \frac{1}{2} \varrho^{\lambda \mu} \psi^-_\mu + \frac{1}{2} t^{\lambda \mu} \varepsilon^-_\mu + \frac{1}{2} \varrho'^{\lambda \mu} \psi^+_\mu \\
& - \frac{1}{2} t'^{\lambda \mu} \varepsilon^+_\mu - \frac{1}{2} \varrho'^{\lambda \mu} \psi^-_\mu + \frac{1}{2} t'^{\lambda \mu} \varepsilon^-_\mu + \varphi^{(\lambda \mu)} \Gamma_0 \psi^-_\mu - u^{(\lambda \mu)} \Gamma_0 \varepsilon^-_\mu + \varphi^{[\lambda \mu]} \Gamma_0 \psi^+_\mu - u^{[\lambda \mu]} \Gamma_0 \varepsilon^+_\mu \, .
\end{split}$$ Then, restricting ourselves to the supersymmetry transformations in the limit $\ell \rightarrow \infty$, we find $$\label{susytrFn}
\begin{split}
\delta \omega^{ab} & = 0 \, , \\
\delta k^a & = 0 \, , \\
\delta V^a & = 0 \, , \\
\delta h & = \bar{\varepsilon}^{+ \, \lambda} \Gamma^0 \psi^{+ \, \lambda} + \bar{\varepsilon}^{- \, \lambda} \Gamma^0 \psi^{- \, \lambda} \, , \\
\delta t^{\lambda \mu} & = 0 \, , \\
\delta t'^{\lambda \mu} & = 0 \, , \\
\delta u^{\lambda \mu} & = 0 \, , \\
\delta s^{\lambda \mu} & = \bar{\varepsilon}^{+ \, [\lambda} \psi^{+ \, \mu]} + \bar{\varepsilon}^{- \, [\lambda} \psi^{- \, \mu]} + \bar{\varepsilon}^{+ \, [ \lambda} \psi^{- \, \mu ]} + \bar{\varepsilon}^{- \, [ \lambda} \psi^{+ \, \mu ]} \, , \\
\delta s'^{\lambda \mu} & = \bar{\varepsilon}^{+ \, [\lambda} \psi^{+ \, \mu]} + \bar{\varepsilon}^{- \, [\lambda} \psi^{- \, \mu]} - \bar{\varepsilon}^{+ \, [ \lambda} \psi^{- \, \mu ]} - \bar{\varepsilon}^{- \, [ \lambda} \psi^{+ \, \mu ]} \, , \\
\delta v^{\lambda \mu} & = - \bar{\varepsilon}^{+ \, \lambda} \Gamma^0 \psi^{+ \, \mu} + \bar{\varepsilon}^{- \, \lambda} \Gamma^0 \psi^{- \, \mu} - \bar{\varepsilon}^{+ \, [ \lambda} \Gamma^0 \psi^{- \, \mu ]} + \bar{\varepsilon}^{- \, [ \lambda} \Gamma^0 \psi^{+ \, \mu ]} \, , \\
\delta \psi^{+ \, \lambda} & = d \varepsilon^{+ \, \lambda} + \frac{1}{4} \omega^{ab} \Gamma_{ab} \varepsilon^{+ \, \lambda} + \frac{1}{2} t^{\lambda \mu} \varepsilon^+_\mu + \frac{1}{2} t^{\lambda \mu} \varepsilon^-_\mu + \frac{1}{2} t'^{\lambda \mu} \varepsilon^+_\mu - \frac{1}{2} t'^{\lambda \mu} \varepsilon^-_\mu + u^{(\lambda \mu)} \Gamma_0 \varepsilon^+_\mu + u^{[\lambda \mu]} \Gamma_0 \varepsilon^-_\mu \, , \\
\delta \psi^{- \, \lambda} & = d \varepsilon^{- \, \lambda} + \frac{1}{4} \omega^{ab} \Gamma_{ab} \varepsilon^{- \, \lambda} + \frac{1}{2} t^{\lambda \mu} \varepsilon^+_\mu + \frac{1}{2} t^{\lambda \mu} \varepsilon^-_\mu - \frac{1}{2} t'^{\lambda \mu} \varepsilon^+_\mu + \frac{1}{2} t'^{\lambda \mu} \varepsilon^-_\mu - u^{(\lambda \mu)} \Gamma_0 \varepsilon^-_\mu - u^{[\lambda \mu]} \Gamma_0 \varepsilon^+_\mu \, .
\end{split}$$
Finally, we find that the equations of motion for the action (flat limit of the equations of motion given in ) read $$\label{eomFn}
\begin{split}
\delta \omega^{ab} & : \quad \alpha_0 \mathcal{R}^{ab} + \alpha_1 \epsilon^{ab} \mathcal{H}= 0 \, , \\
\delta k^a & : \quad \alpha_1 R^a = 0 \, , \\
\delta V^a & : \quad \alpha_1 \epsilon_{ab} \mathcal{K}^b = 0 \, , \\
\delta h & : \quad \alpha_1 \mathcal{R}^{ab} = 0 \, , \\
\delta t^{\lambda \mu} & : \quad \alpha_0 \mathcal{T}^{\lambda \mu} + \alpha_1 \mathcal{S}^{\lambda \mu} = 0 \, , \\
\delta t'^{\lambda \mu} & : \quad \alpha_0 \mathcal{T}'^{\lambda \mu} + \alpha_1 \mathcal{S}'^{\lambda \mu} = 0 \, , \\
\delta u^{\lambda \mu} & : \quad - \alpha_0 \mathcal{U}^{\lambda \mu} + \alpha_1 \mathcal{V}^{\lambda \mu} = 0 \, , \\
\delta s^{\lambda \mu} & : \quad \alpha_1 \mathcal{T}^{\lambda \mu} = 0 \, , \\
\delta s'^{\lambda \mu} & : \quad \alpha_1 \mathcal{T}'^{\lambda \mu} = 0 \, , \\
\delta v^{\lambda \mu} & : \quad \alpha_1 \mathcal{U}^{\lambda \mu} = 0 \, , \\
\delta \psi^{+ \, \lambda} & : \quad \alpha_1 \nabla \psi^{+ \, \lambda} = 0 \, , \\
\delta \psi^{- \, \lambda} & : \quad \alpha_1 \nabla \psi^{- \, \lambda} = 0 \, .
\end{split}$$ We can see that when $\alpha_1 \neq 0$, the eqs. reduce to the vanishing of the curvature $2$-forms given in ($\alpha_1 \neq 0$ is a sufficient condition to recover the vanishing of the curvature $2$-forms in ; the coefficient $\alpha_0$ can be consistently set to zero, making the exotic term disappear from the action .
Let us observe that, restricting ourselves to the purely bosonic theory, we end up with the $\mathcal{N}=(\mathcal{N},0)$ Carroll gravity theories (with $\mathcal{N}$ even) in three dimensions, invariant under the $\mathcal{N}=(\mathcal{N},0)$ Carroll algebra. At the purely bosonic level, the fields $t^{\lambda \mu}$, $t'^{\lambda \mu}$, $u^{\lambda \mu}$, $s^{\lambda \mu}$, $s'^{\lambda \mu}$, and $v^{\lambda \mu}$, and the corresponding terms in the action, can also be consistently discarded by performing an IW contraction.
On the other hand, considering the special case $\mathcal{N}=(2,0)$, that is $x=1$, after some algebraic calculations, we can prove that the $(2,0)$ theory in the flat limit previously discussed in this section is exactly reproduced.
$(p,q)$ Carroll supergravity theories from the $\ell \rightarrow \infty$ limit
------------------------------------------------------------------------------
Applying the flat limit $\ell \rightarrow \infty$ to the (anti)commutation relations given in , we get the following non-vanishing ones: $$\label{adscarrollsuperFpq}
\begin{split}
& \left[ K_a , J_{bc} \right] = \delta_{ab} K_c - \delta_{ac} K_b \, , \\
& \left[ J_{ab}, P_{c}\right] =\delta_{bc}P_{a}-\delta _{ac}P_{b} \, , \\
& \left[ K_{a}, P_{b}\right] = - \delta_{ab} H \, , \\
& \left[ {T}^{ij} , {T}^{kl} \right] = \delta^{jk} {T}^{il} - \delta^{ik} {T}^{jl}- \delta^{jl} {T}^{ik} + \delta^{il} {T}^{jk} \, , \\
& \left[ {T}^{IJ} , {T}^{KL} \right] = \delta^{JK} {T}^{IL} - \delta^{IK} {T}^{JL}- \delta^{JL} {T}^{IK} + \delta^{IL} {T}^{JK} \, , \\
& \left[ {T}^{ij} , {S}^{kl} \right] = \delta^{jk} {S}^{il} - \delta^{ik} {S}^{jl}- \delta^{jl} {S}^{ik} + \delta^{il} {S}^{jk} \, , \\
& \left[ {T}^{IJ} , {S}^{KL} \right] = \delta^{JK} {S}^{IL} - \delta^{IK} {S}^{JL}- \delta^{JL} {S}^{IK} + \delta^{IL} {S}^{JK} \, , \\
& \left[ J_{ab},Q^i_{\alpha }\right] =-\frac{1}{2}\left( \Gamma _{ab} Q^i \right)_{\alpha } \, , \\
& \left[ J_{ab},Q^I_{\alpha }\right] =-\frac{1}{2}\left( \Gamma _{ab} Q^I \right)_{\alpha } \, , \\
& \left[ {T}^{ij}, {Q}^k_{\alpha }\right] = \delta^{jk} {Q}^i_\alpha - \delta^{ik} {Q}^j_\alpha \, , \\
& \left[ {T}^{IJ}, {Q}^K_{\alpha }\right] = \delta^{JK} {Q}^I_\alpha - \delta^{IK} {Q}^J_\alpha \, , \\
& \left\{ Q^i_{\alpha }, Q^j_{\beta }\right\} = \delta^{ij} \left( \Gamma ^{0}C\right) _{\alpha \beta } H + C_{\alpha \beta} S^{ij} \, , \\
& \left\{ Q^I_{\alpha }, Q^J_{\beta }\right\} = \delta^{ij} \left( \Gamma ^{0}C\right) _{\alpha \beta } H - C_{\alpha \beta} S^{IJ} \, .
\end{split}$$ These are the (anti)commutation relations of the $\mathcal{N}=(p,q)$, $D=3$ super-Carroll algebra (with $p=q$), and we could also have obtained it by applying the Carroll contraction to the semi-direct extension of the $\mathfrak{so}(p) \oplus \mathfrak{so}(q)$ automorphism algebra by the $\mathcal{N}=(p,q)$, $D=3$ Poincaré superalgebra (see Ref. [@Howe:1995zm]).
Then, as $\ell \rightarrow \infty$, the $2$-form curvatures become $$\label{curvadscarrollsuperFpq}
\begin{split}
\mathcal{R}^{ab} & = d \omega^{ab} = R^{ab} \, , \\
\mathcal{K}^a & = d k^a + \omega^a_{\phantom{a} b} k^b = \mathfrak{K}^a \, , \\
R^a & = d V^a + \omega^a_{\phantom{a}b} V^b \, , \\
\mathcal{H} & = d h + V^a k_a - \frac{1}{2} \bar{\psi}^i \Gamma^0 \psi^i - \frac{1}{2} \bar{\psi}^I \Gamma^0 \psi^I = \mathfrak{H} - \frac{1}{2} \bar{\psi}^i \Gamma^0 \psi^i - \frac{1}{2} \bar{\psi}^I \Gamma^0 \psi^I \, , \\
\mathcal{T}^{ij} & = d t^{ij} + {t^i}_k t^{kj} \, , \\
\mathcal{T}^{IJ} & = d t^{IJ} + {t^I}_K t^{KJ} \, ,\\
\mathcal{S}^{ij} & = d s^{ij} + 2 {t^i}_k s^{kj} - \bar{\psi}^i \psi^j \, , \\
\mathcal{S}^{IJ} & = d s^{IJ} + 2 {t^I}_K s^{KJ} + \bar{\psi}^I \psi^J \, ,\\
\nabla \psi^i & = d \psi^i + \frac{1}{4} \omega^{ab} \Gamma_{ab} \psi^i + t^{ij} \psi_j \, , \\
\nabla \psi^I & = d \psi^I + \frac{1}{4} \omega^{ab} \Gamma_{ab} \psi^I + t^{IJ} \psi_J \, ,
\end{split}$$ and by applying the $\ell \rightarrow \infty$ limit to the CS action , we get $$\label{CSCpq}
\begin{split}
I^{(p,q)}_{CS} \vert_{\ell \rightarrow \infty} & = \frac{k}{4 \pi} \int_\mathcal{M} \Bigg \lbrace \frac{\alpha_0}{2} \left( \omega^a_{\phantom{a} b} R^b_{\phantom{b} a} + 2 {t^i}_j {d t^j}_i + \frac{4}{3} {t^i}_j {t^j}_k {t^k}_i + 2 {t^I}_J {d t^J}_I + \frac{4}{3} {t^I}_J {t^J}_K {t^K}_I \right) \\
& + \alpha_1 \bigg[ \epsilon_{ab} R^{ab} h - 2 \epsilon_{ab} \mathfrak{K}^a V^b - 2 {t^i}_j \left( {ds^j}_i + {t^j}_k {s^k}_i \right) + 2 {t^I}_J \left( {ds^J}_I + {t^J}_K {s^K}_I \right) \\
& + 2 \bar{\psi}^i \nabla \psi^i + 2 \bar{\psi}^I \nabla \psi^I \bigg] - d \left( \frac{\alpha_1}{2} \epsilon_{ab} \omega^{ab} h - \alpha_1 \epsilon_{ab} k^a V^b + \alpha_1 {t^i}_j {s^j}_i - \alpha_1 {t^I}_J {s^J}_I \right) \Bigg \rbrace \, ,
\end{split}$$ which is written in terms of the super-Carroll curvatures appearing in (let us stress that the latter must not be confused with the super-AdS Carroll ones given in ). We can see that the exotic term appearing in , namely the one proportional to $\alpha_0$, now is given by the Lorentz Lagrangian plus additional terms involving the $1$-form fields $t^{ij}$ and $t^{IJ}$.
Notice that the action can also be derived by using the non-vanishing components of the invariant tensor obtained by taking the limit $\ell \rightarrow \infty$ of together with the connection $1$-form for the $\mathcal{N}=(p,q)$ (flat) Carroll superalgebra in the general expression . Specifically, the limit $\ell \rightarrow \infty$ of gives us the following non-vanishing components: $$\begin{split}
& \langle J_{ab} J_{cd} \rangle = \alpha_0 \left( \delta_{ad} \delta_{bc} - \delta_{ac} \delta_{bd} \right) \, , \\
& \langle J_{ab} H \rangle = \alpha_1 \epsilon_{ab} \, , \\
& \langle K_a P_b \rangle = - \alpha_1 \epsilon_{ab} \, , \\
& \langle {T}^{ij} {T}^{kl} \rangle = 2 \alpha_0 \left( \delta^{il} \delta^{kj} - \delta^{ik} \delta^{lj} \right) \, , \\
& \langle {T}^{IJ} {T}^{KL} \rangle = 2 \alpha_0 \left( \delta^{IL} \delta^{KJ} - \delta^{IK} \delta^{LJ} \right) \, , \\
& \langle {T}^{ij} {S}^{kl} \rangle = - 2 \alpha_1 \left( \delta^{il} \delta^{kj} - \delta^{ik} \delta^{lj} \right) \, , \\
& \langle {T}^{IJ} {S}^{KL} \rangle = 2 \alpha_1 \left( \delta^{IL} \delta^{KJ} - \delta^{IK} \delta^{LJ} \right) \, , \\
& \langle Q^i_\alpha Q^j_\beta \rangle = 2 \alpha_1 C_{\alpha \beta} \delta^{ij} \, , \\
& \langle Q^I_\alpha Q^J_\beta \rangle = 2 \alpha_1 C_{\alpha \beta} \delta^{IJ} \, .
\end{split}$$
The CS action is invariant by construction under the super-Carroll group associated with , and, in particular, concerning the $\ell \rightarrow \infty$ limit of the local gauge transformations , we get $$\label{gaugetrFpq}
\begin{split}
\delta \omega^{ab} & = d \lambda^{ab} \, , \\
\delta k^a & = d \kappa^a - \lambda^a_{\phantom{a} b} k^b + \omega^a_{\phantom{a} b} \kappa^b \, , \\
\delta V^a & = d \lambda^a - \lambda^a_{\phantom{a}b} V^b + \omega^a_{\phantom{a}b} \lambda^b \, , \\
\delta h & = d \tau - \lambda^a k_a + V^a \kappa_a + \bar{\varepsilon}^i \Gamma^0 \psi^i + \bar{\varepsilon}^I \Gamma^0 \psi^I \, , \\
\delta t^{ij} & = d \varphi^{ij} - 2 {\varphi^{[i}}_k t^{k \vert j]} \, , \\
\delta t^{IJ} & = d \varphi^{IJ} - 2 {\varphi^{[I}}_K t^{K \vert J]} \, ,\\
\delta s^{ij} & = d \varsigma^{ij} - 2 {\varphi^{[i}}_k s^{k \vert j]} + 2 {t^{[i}}_k \varsigma^{k \vert j]} + 2 \bar{\varepsilon}^{[i} \psi^{j]} \, , \\
\delta s^{IJ} & = d \varsigma^{IJ} + 2 {\varphi^{[I}}_K s^{K \vert J]} + 2 {t^{[I}}_K \varsigma^{K \vert J]} - 2 \bar{\varepsilon}^{[I} \psi^{J]} \, , \\
\delta \psi^i & = d \varepsilon^+ - \frac{1}{4} \lambda^{ab} \Gamma_{ab} \psi^i + \frac{1}{4} \omega^{ab} \Gamma_{ab} \varepsilon^i - \varphi^{ij} \psi_j + t^{ij} \varepsilon_j \, , \\
\delta \psi^I & = d \varepsilon^- - \frac{1}{4} \lambda^{ab} \Gamma_{ab} \psi^I + \frac{1}{4} \omega^{ab} \Gamma_{ab} \varepsilon^I - \varphi^{IJ} \psi_J + t^{IJ} \varepsilon_J \, .
\end{split}$$ Thus, restricting ourselves to the supersymmetry transformations in the limit $\ell \rightarrow \infty$, we are left with $$\label{susytrFpq}
\begin{split}
\delta \omega^{ab} & =0 \, , \\
\delta k^a & = 0 \, , \\
\delta V^a & = 0 \, , \\
\delta h & = \bar{\varepsilon}^i \Gamma^0 \psi^i + \bar{\varepsilon}^I \Gamma^0 \psi^I \, , \\
\delta t^{ij} & = 0 \, , \\
\delta t^{IJ} & = 0 \, ,\\
\delta s^{ij} & = 2 \bar{\varepsilon}^{[i} \psi^{j]} \, , \\
\delta s^{IJ} & = - 2 \bar{\varepsilon}^{[I} \psi^{J]} \, ,\\
\delta \psi^i & = d \varepsilon^+ + \frac{1}{4} \omega^{ab} \Gamma_{ab} \varepsilon^i + t^{ij} \varepsilon_j \, , \\
\delta \psi^I & = d \varepsilon^- + \frac{1}{4} \omega^{ab} \Gamma_{ab} \varepsilon^I + t^{IJ} \varepsilon_J \, .
\end{split}$$
Concluding, the equations of motion for the action (flat limit of the equations of motion given in ) read as follows: $$\label{eomFpq}
\begin{split}
\delta \omega^{ab} & : \quad \alpha_0 \mathcal{R}^{ab} + \alpha_1 \epsilon^{ab} \mathcal{H}= 0 \, , \\
\delta k^a & : \quad \alpha_1 R^a = 0 \, , \\
\delta V^a & : \quad 2 \alpha_1 \epsilon_{ab} \mathcal{K}^b = 0 \, , \\
\delta h & : \quad \alpha_1 \mathcal{R}^{ab} = 0 \, , \\
\delta t^{ij} & : \quad - \alpha_0 \mathcal{T}^{ij} + \alpha_1 \mathcal{S}^{ij} = 0 \, , \\
\delta t^{IJ} & : \quad - \alpha_0 \mathcal{T}^{IJ} - \alpha_1 \mathcal{S}^{IJ} = 0 \, , \\
\delta s^{ij} & : \quad \alpha_1 \mathcal{T}^{ij} = 0 \, , \\
\delta s^{IJ} & : \quad \alpha_1 \mathcal{T}^{IJ} = 0 \, , \\
\delta \psi^i & : \quad \alpha_1 \nabla \psi^i = 0 \, , \\
\delta \psi^I & : \quad \alpha_1 \nabla \psi^I = 0 \, .
\end{split}$$ When $\alpha_1 \neq 0$, the eqs. exactly reduce to the vanishing of the curvature $2$-forms given in ($\alpha_1 \neq 0$ is a sufficient condition to recover the vanishing of the curvature $2$-forms , and the coefficient $\alpha_0$ can also be consistently set to zero, making the exotic term disappear from the action ).
Restricting ourselves to the purely bosonic theory, we end up with the $\mathcal{N}=(p,q)$ Carroll gravity theories (with $p=q$) in three dimensions, invariant under the $\mathcal{N}=(p,q)$ Carroll algebra. At the purely bosonic level, the fields $t^{ij}$, $t^{IJ}$, $s^{ij}$, and $s^{IJ}$, and the corresponding terms in the action, can also be consistently discarded by performing an IW contraction.
On the other hand, let us finally mention that, if we now consider the particular case $p=q=1$, we exactly reproduce the results previously obtained in this section for the $(1,1)$ theory in the flat limit.
All the studies of the flat limit presented in this section represent a new development and generalization of the previous works concerning Carroll superalgebras in three dimensions, in particular in the context of three-dimensional CS supergravity theories.
Conclusions
===========
Motivated by the recent development of applications of Carroll symmetries (in particular, by their prominent role in the context of holography), and by the fact that, nevertheless, the study of their supersymmetric extensions in the context of supergravity models still remains poorly explored, in this paper we have presented, in a systematic fashion, the ultra-relativistic $\mathcal{N}$-extended AdS CS supergravity theories in three ($2+1$) spacetime dimensions, which are invariant under $\mathcal{N}$-extended AdS Carroll superalgebras, extending the results recently presented in [@Ravera:2019ize] (where the construction of the three-dimensional $\mathcal{N}=1$ CS supergravity theory invariant under the so-called AdS Carroll superalgebra, ultra-relativistic contraction of the $\mathcal{N}=1$ AdS superalgebra [@Bergshoeff:2015wma], together with the study of its flat limit, has been presented for the first time). In particular, we have applied the method introduced in [@Concha:2016zdb] with the improvements of [@Ravera:2019ize] to construct the aforesaid ultra-relativistic $\mathcal{N}$-extended AdS CS supergravity theories.
We have first considered the $(2,0)$ and $(1,1)$ cases, and subsequently generalized our analysis to $\mathcal{N}=(\mathcal{N},0)$, with $\mathcal{N}$ even integer, and to $\mathcal{N}=(p,q)$, that is $\mathcal{N}=p+q$, with $p=q$. The $\mathcal{N}$-extended AdS Carroll superalgebras have been obtained through the Carrollian (i.e., ultra-relativistic) contraction applied to an $so(2)$ extension of $\mathfrak{osp}(2|2)\otimes \mathfrak{sp}(2)$, to $\mathfrak{osp}(2|1)\otimes \mathfrak{osp}(2,1)$, to an $\mathfrak{so}(\mathcal{N})$ extension of $\mathfrak{osp}(2|\mathcal{N})\otimes \mathfrak{sp}(2)$, and to the direct sum of an $\mathfrak{so}(p) \oplus \mathfrak{so}(q)$ algebra and $\mathfrak{osp}(2|p)\otimes \mathfrak{osp}(2,q)$, respectively.
A $\mathcal{N}=(2,0)$ AdS Carroll superalgebra in three dimensions was previously introduced in [@Bergshoeff:2015wma]. Nevertheless, the latter does not allow for a non-degenerate invariant tensor, meaning that one cannot construct a well-defined CS action based on this superalgebra. To overcome this point, we have considered an $so(2)$ extension of $\mathfrak{osp}(2|2)\otimes \mathfrak{sp}(2)$ and performed the ultra-relativistic contraction on it, ending up with a new $\mathcal{N}=(2,0)$ AdS Carroll superalgebra endowed with a non-degenerate invariant tensor. This has allowed us to develop the three-dimensional CS supergravity action invariant under this $\mathcal{N}=(2,0)$ AdS Carroll superalgebra. We have called this action the $(2,0)$ AdS Carroll CS supergravity action. We have done an analogous analysis in the $(1,1)$ case, and subsequently generalized our study to $\mathcal{N}=(\mathcal{N},0)$ (with $\mathcal{N}$ even) and to $\mathcal{N}=(p,q)$ (with $p=q$). In particular, after having introduced the ultra-relativistic superalgebras, we have constructed the respective CS supergravity theories in three-dimensions by exploiting the non-vanishing components of the corresponding invariant tensor. The aforementioned actions are all based on a non-degenerate, invariant bilinear form (i.e., an invariant metric), and each of them is characterized by two coupling constants and involve an exotic contribution. The results presented in this paper were also open problems suggested in Ref. [@Bergshoeff:2015wma], and they represent the $\mathcal{N}$-extended generalization of [@Ravera:2019ize]. Interestingly, one can observe that the CS formulation in the $\mathcal{N}$-extended cases $\mathcal{N}=(\mathcal{N},0)$ and $\mathcal{N}=(p,q)$ requires the presence of $\mathfrak{so}(\mathcal{N})$ and $\mathfrak{so}(p) \oplus \mathfrak{sp}(q)$ generators, respectively, also at the ultra-relativistic level, that is in the Carroll limit; thus, what happens at the relativistic level for three-dimensional $\mathcal{N}$-extended CS Poincaré and AdS supergravity theories (see [@Howe:1995zm]), that is the need to introduce the aforementioned extra generators (together with their dual $1$-form fields) in the theory in order to obtain a non-degenerate invariant tensor, has repercussions also on (and still holds at) the ultra-relativistic level.
We have also analyzed the flat limit $\ell \rightarrow \infty$ of the aforementioned models, in which we have recovered the ultra-relativistic $\mathcal{N}$-extended (flat) CS supergravity theories invariant under $\mathcal{N}$-extended super-Carroll algebras. The flat limit has been applied at the level of the superalgebras, CS actions, supersymmetry transformation laws, and field equations. Also all the studies of the flat limit presented in Section \[flatlimit\] represent a new development and generalization of the previous works presented in the literature concerning Carroll (super)algebras in three dimensions, in particular in the context of three-dimensional CS (super)gravity theories.
The recently discovered relations among the Carrollian world and flat holography suggest that this work might represents a starting point to go further in the analysis of supersymmetry invariance of flat supergravity in the presence of a non-trivial boundary, along the lines of [@Concha:2018ywv]. Besides, now, having well-defined three-dimensional CS (super)gravity theories respectively invariant under the $\mathcal{N}$-extended AdS-Carroll and Carroll (super)algebras, it would be intriguing to go beyond and study the asymptotic symmetry of these models, following, for instance, the prescription given in Ref. [@Concha:2018zeb]. It would also be interesting to further extend our analysis to more general amount of supersymmetry, involving also odd $\mathcal{N}$ cases, and to higher-dimensional models (recently, a study exploring the Carroll limit corresponding to M2- as well as M3-branes propagating over $D=11$ supergravity backgrounds in M-theory has been presented [@Roychowdhury:2019aoi]), where Carrollian (super)gravity theories still remain poorly explored. Finally, all these ultra-relativistic theories constructed à la CS could have some applications in the context of Carrollian fluids (and their relations with flat holography, see Refs. [@Ciambelli:2018xat; @Ciambelli:2018wre; @Ciambelli:2018ojf; @Campoleoni:2018ltl]).
Acknowledgments
===============
L.R. is grateful to L. Andrianopoli and B.L. Cerchiai for useful discussions and support.
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[^1]: Here let us also mention that the geometric realization of the (A)dS Carroll algebra corresponds to a null surface of (A)dS space (see Refs. [@Figueroa-OFarrill:2018ilb; @Figueroa-OFarrill:2019sex]).
[^2]: We concentrate on the $\mathcal{N}=(\mathcal{N},0)$ case with $\mathcal{N}$ even and on the $\mathcal{N}=(p,q)$ case with $p=q$ (such that, in particular, $\mathcal{N}=p+q$ is even), which involve some “restrictions” in order to reproduce a well-defined ultra-relativistic limit at the supersymmetric level (see also [@Lukierski:2006tr] and references therein, which deals with similar situations but in the non-relativistic limit $c \rightarrow \infty$).
[^3]: We use the metric $\eta_{AB}$ with the signature $(-,+,+)$.
[^4]: The limit $\sigma \rightarrow \infty$ corresponds to $\frac{1}{c} \rightarrow \infty$, being $c$ the velocity of light, that is $c \rightarrow 0$ (ultra-relativistic limit).
[^5]: In the sequel, we will omit the wedge product “$\wedge$” between differential forms.
[^6]: Here and in the following, for simplicity, we will omit the spinor index $\alpha$.
[^7]: In particular, when restricting ourselves to $\mathcal{N}=(2,0)$, namely $x=1$, we have $\lambda, \mu , \ldots = 1$ and $T^{\lambda \mu}=T^{11}=0$, $T'^{\lambda \mu}=T'^{11}=0$, $S^{\lambda \mu}=S^{11}=0$, $S'^{\lambda \mu}=S'^{11}=0$, $U^{\lambda \mu}=U^{11}=- T^{12}=- \epsilon^{12} T=-T$, $V^{\lambda \mu}=V^{11}=- S^{12}=- \epsilon^{12} S=-S$, and restricts itself to (when performing the Carroll limit, we also remove the tilde symbol on the generators); then, one can show that the superalgebra reduces to the one given by .
[^8]: Let us also observe that if we now restrict ourselves to the purely bosonic part of the action , we get a three-dimensional CS Carroll gravity action that is different from the one obtained in [@Ravera:2019ize] (see also [@Bergshoeff:2017btm]) by considering the purely bosonic contributions, due to the presence of the bosonic $1$-form fields $t$ and $s$ dual to the generators $T$ and $S$, respectively. Nevertheless, if we consider the purely bosonic level and set $t=s=0$ through an IW contraction, we have that the aforementioned actions coincide.
|
---
abstract: 'In this paper we study Gaussian ring ${\mbox{$\mathbb{Z}$}}[i]$ with a focus on representing Gaussian Mersenne primes $G_p$ in the form $x^2+7y^2$. Interestingly when such a form exists, one can observe that, $x\equiv \pm 1\pmod{8}$ and $y\equiv 0\pmod{8}$. To prove this property of Gaussian Mersenne primes, we show that Gaussian Mersenne primes splits completely in the cyclic quartic unramified extension of ${\mbox{$\mathbb{Q}$}}(\sqrt{-14})$ and have a trivial Artin symbol in this extension. We generalize this result for $d\equiv 7\pmod{24}$. We also attempt to give an alternate proof using Artin’s reciprocity law, which was earlier given by H. W. Lenstra and P. Stevenhagen to prove a similar property on ordinary Mersenne Primes.'
address: |
Dept. of Computer Science & Automation\
Indian Institute of Science, Bangalore - 560012
author:
- Sushma Palimar and Ambedkar Dukkipati
title: Gaussian Mersenne Primes of the form $x^2+dy^2$
---
[^1]
Introduction
============
The study of prime numbers in the form $x^2+dy^2$ for a fixed integer $d>0$ is ancient and formal study of this problem began with Fermat, later continued by Euler, which led him to discover quadratic reciprocity and many conjecture on $x^2+dy^2$ for $d>3$. The book [@6] is a good reference for this. Study of Mersenne primes in the form $x^2+dy^2$, for $d\in
[2,48]$ is carried out in detail in [@Jansen:2002:mersenneprimes]. In [@henstev:2000:artinreci] Lenstra and Stevenhagen prove an interesting property of Mersenne primes which was first observed by Lemmermeyer. This observation is given in the form of Theorem below. Here authors make use of Artin’s reciprocity to demonstrate this property of Mersenne primes.
\[mpthm\] Let $M_p$ be a Mersenne prime with $p\equiv 1\pmod{3}$. Write $M_p=x^2+7y^2$ with $x,y\in {\mbox{$\mathbb{Z}$}}$. Then $x$ is divisible by $8$.
In [@pal:2012:jis], authors call $\alpha=(2+\sqrt{2})$ and the norm $N (\frac{\alpha^{p}-1}{\alpha-1})$ is found. In this case whenever the norm $N(\frac{\alpha^{p}-1}{\alpha-1})$ is a rational prime, then those primes are quadratic residues of $7$ hence can be written in the form $x^{2}+7y^{2}$ and it is proved using Artin’s reciprocity that, $x$ is divisible by $8$ and $y\equiv \pm 3\pmod{8}$.
In Section \[Gp\] we study Gaussian Mersenne primes in the form $x^2+dy^2$ with a special focus on the form $x^2+7y^2$. Here we prove that whenever $G_p=x^2+7y^2$ then $x\equiv \pm 1\pmod{8}$ and $y\equiv 0\pmod{8}$ and generalize this result for all $d\equiv 7\pmod{24}$ in the next section.
Gaussian Mersenne Primes
-------------------------
Gaussian Mersenne primes and their primality test was first studied by Berrizbeitia and Iskra in [@1]. A Gaussian Mersenne number is an element of ${\mbox{$\mathbb{Z}$}}[i]$ given by $\mu_p=(1\pm i)^{p}-1$, for some rational prime $p$. The Gaussian Mersenne norm $G_p=2^{p}-\left(\frac{2}{p}\right)2^{\frac{p+1}{2}}+1$ is the norm of $\mu_p$, $N(\mu_p)$. If $G_p$ is a rational prime then we call $G_p$ a Gaussian Mersenne prime.
Some results from Class Field Theory {#cft}
------------------------------------
We take a short and quick revision of class field theory to fix notations. Gal(L/K) is the Galois group of the field extension $K\subset L$. $\mathcal{O}_{K}$, the ring of algebraic integers in a finite extension $K$ of ${\mbox{$\mathbb{Q}$}}$. We denote $I_{K}(\Delta(L/K))$ to be the group of all fractional ideals of $\mathcal{O}_{K}$ prime to $\Delta(L/K)$. The *Artin symbol* is denoted by $((L/K)/\mathfrak{P})$, for $\mathfrak{P}$, a prime of $L$ containing a prime $\mathfrak{p}$ of $\mathcal{O}_{K}$. Then for any $\alpha$ in $\mathcal{O}_{L}$ $$\left( \frac{L/K}{\mathfrak{P}}\right)(\alpha)\equiv
\alpha^{N(\mathfrak{p})} \pmod{\mathfrak{P}},$$ When $K\subset L$ is an Abelean extension, the Artin symbol can be written as $((L/K)/\mathfrak{p})$.
Let $K$ be a number field and $\mathfrak{m}$ a modulus of $K$. We define $P_{K,1}(m)$ to be the subgroup of $I_K(m)$ generated by the principal ideals $\alpha.{\mbox{$\mathbb{Z}$}}_K$, with $\alpha\in {\mbox{$\mathbb{Z}$}}_{K}$, satisfying $\alpha\equiv 1\bmod{ \mathfrak{m}_0}$, where $\mathfrak{m}_0$ is a nonzero $\mathcal{O}_{K}$ ideal and $\sigma(\alpha)>0$ for all infinite primes $\sigma$ of $K$ dividing $\mathfrak{m}_\infty $, is a product of distinct real infinite primes of $K$. Thus $Cl_{m}$ is the group $I_{K}(m)/P_{K,1}(m)$.
Given an order $\mathcal{O}$ in a quadratic field $K$, the index $f=[\mathcal{O}:\mathcal{O}_{K}]$ is called conductor of the order. The discriminant of $\mathcal{O}$ is $D=f^2d_k$, $d_k$ is the discriminant of the maximal order $\mathcal{O}_{K}$.
Let $K$ be a number field and $R=R_{m}(K)$ the ray class field of $K$ with modulus $m$. The ray class field of conductor $\mathfrak{m}=(1)$ is the Hilbert class field $H=H_1$ of $K$.
\[isodef\] If $\mathcal{O}$ is the order of $\mathcal{O}_{K}$ of conductor $f$, we define ring class group to be $I_K(f\mathcal{O}_K)/P_{K,\mathbb{Z}}(f\mathcal{O}_K)$, which is naturally isomorphic to the group of ideals prime to $f$ modulo the principal ideals prime to $f$. The ring class field of the maximal order $\mathcal{O}_{K}$ is the Hilbert class field of $K$.
Here, the bottom group $P_{K,\mathbb{Z}}(f\mathcal{O}_K)$ consists of the principal ideals which admit a generator $\alpha$ congruent to some integer $a$, relatively prime to $f$, i.e., $$\alpha\equiv a\pmod{f\mathcal{O}_{K}}$$ We have $m=[\mathcal{O}_{K}:\mathcal{O}]=1,2,3,4,6$ if and only if $P_{K,1}(m)=P_{K,\mathbb{Z}}(m)$. This is because, $m=1,2,3,4,6$ is nothing but $\phi(m)\leq 2$. And these are the cases where the global units 1 and –1 ‘fill up’ $({\mbox{$\mathbb{Z}$}}/m{\mbox{$\mathbb{Z}$}})^*$, and the ray class field equals the ring class field.
Gaussian Mersenne primes as $x^2+dy^2$ {#Gp}
======================================
In this section we derive two properties of Gaussian Mersenne primes and prove our main result that, whenever $G_{p}=x^{2}+7y^{2}$ then $x\equiv \pm 1\pmod{8}$ and $y\equiv 0\pmod{8}$.
We define $F_d$ to be the quadratic form $x^2+dy^2\in {\mbox{$\mathbb{Z}$}}[x,y]$. Let $p$ be a prime then $p$ is represented by $F_d$ if there exists $x>0$ and $y>0$ in ${\mbox{$\mathbb{Z}$}}$ such that $p=x^2+dy^2$.
\[drem3\] Let $d\equiv 3\bmod{4}$ be a square-free integer. Suppose $L=R_{2}({\mbox{$\mathbb{Q}$}}(\sqrt{-d}))(\sqrt{2})=H({\mbox{$\mathbb{Q}$}}(\sqrt{-2d}))$. Let $G_{p}$ be a Gaussian Mersenne prime unramified in $L$ then $F_d$ represents $G_{p}$ if and only if $F_{2d}$ represents $G_{p}$.
This property of Gaussian Mersenne primes stated in Proposition \[drem3\] is obeyed by Mersenne primes too, and the proof is similar to the proof given in [@Jansen:2002:mersenneprimes].
We have $G_{p}=2^{p}-\left(\frac{2}{p}\right)2^{\frac{p+1}{2}}+1$. For $p>3$, $G_{p}\equiv1\bmod{8}$. Hence, $G_{p}$ splits completely in ${\mbox{$\mathbb{Q}$}}(\sqrt{2})$ and the prime $G_{p}$ is unramified in $L$, which implies $G_{p}\nmid 2d$. We have, $[\mathcal{O}_{K}:{\mbox{$\mathbb{Z}$}}[\sqrt{-d}]]=2$, since $d\equiv 3\pmod{4}$ is a positive square-free integer. Now, by using the fact $G_{p}$ splits in ${\mbox{$\mathbb{Q}$}}(\sqrt{2})$ we get $F_{d}$ represents $G_{p}$ if and only if $G_{p}$ splits completely in $R_{2}({\mbox{$\mathbb{Q}$}}(\sqrt{-d}))(\sqrt{2})$. For $K={\mbox{$\mathbb{Q}$}}(\sqrt{-2d})$ we have $[\mathcal{O}_{K}:{\mbox{$\mathbb{Z}$}}[\sqrt{-2d}]=1$. Thus, $F_{2d}$ represents $G_{p}$ if and only if $G_{p}$ splits completely in $H({\mbox{$\mathbb{Q}$}}(\sqrt{-2d}))$. By assumption we have $R_{2}({\mbox{$\mathbb{Q}$}}(\sqrt{-d}))(\sqrt{2})=H({\mbox{$\mathbb{Q}$}}(\sqrt{-2d}))$ hence the result.
The property of Gaussian Mersenne primes in Proposition \[drem1\] is unique only to Gaussian Mersenne primes.
\[drem1\] Let $d\equiv 1\bmod{4}$ be a square-free integer. Suppose $L=H({\mbox{$\mathbb{Q}$}}(\sqrt{-d}))=H({\mbox{$\mathbb{Q}$}}(\sqrt{-2d}))$. Let $G_{p}$ be a Gaussian Mersenne prime unramified in $L$ then $F_d$ represents $G_{p}$ if and only if $F_{2d}$ represents $G_{p}$, which is not the case for usual Mersenne primes.
Proof follows from the previous proposition, except for the fact, $[\mathcal{O}_{K}:{\mbox{$\mathbb{Z}$}}[\sqrt{-d}]]=2$, which is $1$ in the present case, as $d\equiv 1\bmod{4}$.
Gaussian Mersenne primes as $x^2+7y^2$ {#impth}
--------------------------------------
Here we give first few examples of Gaussian Mersenne primes and their representation as $x^2+7y^2$. $$G_{7}=113=1+7\cdot4^{2}$$ $$G_{47}=140737471578113= 5732351^{2}+7\cdot3925696^{2}$$ $$G_{73}= 9444732965601851473921=96890022433^{2}+7\cdot2854983576^{2}$$ $G_{113}=10384593717069655112945804582584321=$ $$79288509938147361^{2}+7\cdot24195412519312600^{2}$$ One can check that, Gaussian Mersenne prime $G_p$ is $1$ $\text{ mod }{8}$ for all $p>3$. Also , $$\text { if }p\equiv 1\pmod{6} \text{ then } Gp\equiv1\pmod{7}$$ and $$\text{ if }p\equiv 5\pmod{6} \text{ then }Gp\equiv4\pmod{7} \text{ for all }p.$$ For $p>7$, a simple computation shows that, in each case $x\equiv \pm 1 \bmod{8}$ and $y\equiv 0\bmod{8}$. Also, from the above table it is not difficult to see that $y$ is exactly divisible by $4$. These two observations are proved as a lemma below.
\[mainlemma\] If $G_{p}$ is represented in the form $x^2+7y^2$ then $x\equiv \pm1\pmod{8}$ and $4|y$.
From the above discussion it is clear that, $G_p$ can be written in the form $x^2+7y^2$ if and only if $p\equiv\pm 1\pmod{8}$. Also, $G_{p}=2^{p}-2^{\frac{p+1}{2}}+1\equiv 1\pmod{8}$ for all $p>3$. We first show that, $y$ is divisible by $4$. For this we show that, $x$ is an odd integer and $y$ is an even integer. For if $x$ is even and $y$ is odd, then $x^{2}\equiv 0 \text{ or } 4\pmod{8}$ and $y^{2}\equiv 1\pmod{8}$, which implies $x^{2}+7y^{2}\equiv\ 7\text{ or }3\pmod{8}$, a contradiction. Hence $x$ should be an odd integer and $y$ should be an even integer. Now, clearly $x^{2}\equiv 1\pmod{8}$ and $y^{2}\equiv 0 \text{ or } 4\pmod{8}$. If $y^{2}\equiv 4\pmod{8}$ then $x^{2}+7y^{2}\equiv 5\pmod{8}$, again a contradiction. Hence $y\equiv 0\pmod{4}$ and we have $4|y$. Now we show that $x\equiv\pm 1\pmod{8}$. Clearly $x^2+7y^2\equiv x^2\pmod{16}$. Since $x$ is an odd integer either $x^2\equiv 1\pmod{16}$ or $x^2\equiv 9\pmod{16}$. If $x^2\equiv 9\pmod{16}$ then $G_{p}\equiv 9\pmod{16}$, a contradiction as $G_{p}\equiv1\pmod{16}$, for all $p\equiv\pm1\pmod{8}$. Thus, $x^{2}\equiv1\pmod{16}$. Hence finally $x\equiv \pm 1\pmod{8}$ and $y\equiv0\pmod{4}$.
Now it remains to prove that $y$ is divisible by $8$ in the expression $G_p=x^2+7y^2$. As a first step we transform the base field ${\mbox{$\mathbb{Q}$}}(\sqrt{-7})$ into ${\mbox{$\mathbb{Q}$}}(\sqrt{2})$. Then show that $Gp$ splits completely in the cyclic quartic unramified extension of ${\mbox{$\mathbb{Q}$}}(\sqrt{-14})$. This is because, $G_{p}\equiv 1\pmod{8}$ for all $p>3$ which shows that, $G_p$ is a prime in ${\mbox{$\mathbb{Q}$}}$ which splits in ${\mbox{$\mathbb{Q}$}}(\sqrt{2})$.
\[mainprop\] Let $p>7$ and $G_p=2^{p}-(\frac{2}{p})2^{\frac{p+1}{2}}+1$ be a Gaussian Mersenne prime. Then for $p\equiv\pm 1\pmod{8}$ the form $G_p= x^2+7y^2$ exists. Suppose that there exists a cyclic extension $H_4$ of $S={\mbox{$\mathbb{Q}$}}(\sqrt{-14})$ with $[H_4:S]=4$, $H_4\subset H(S)$ and $\sqrt{2}\in H_4$. Then $G_p$ splits completely in $H_4$.
Let $K={\mbox{$\mathbb{Q}$}}(\sqrt{-7})$ and let $J={\mbox{$\mathbb{Q}$}}(\sqrt{2})$. As $\sqrt{2}\in H_4$, consider an extension $J={\mbox{$\mathbb{Q}$}}(\sqrt{2})\subset H_4$ and $Gal(H_4/{\mbox{$\mathbb{Q}$}})$ is isomorphic to the dihedral group with $8$ elements. Thus, there are two conjugate field extensions of $J$, say $J_1$ and $J_2$ contained in $H_4$, namely $J_1=J(\sqrt{-1+2\sqrt{2}})$ and $J_2=J(\sqrt{-1-2\sqrt{2}})$. Consider the following field diagram.
(h4) at (0,2) [$H_4$]{}; (jk) at (0,1) [$JK$]{}; (j1) at (1,1) [$J_1$]{}; (j2) at (2,1) [$J_2$]{}; (k) at (-1,0) [$K$]{}; (s) at (0,0) [$S$]{}; (j) at (1,0) [$J$]{}; (q) at (0,-1) [${\mbox{$\mathbb{Q}$}}$]{};
\(q) – (k) – (jk) – (s) – (q) – (j) – (jk)– (h4) – (j1) – (j)– (j2) – (h4);
;
The discriminant of $JK$ over $J$ is $-7$. The discriminants $\Delta(J_1/J)$ and $\Delta(J_2/J)$ must be relatively prime. If not there is a prime of $J$ which is ramified in $J_1$ and $J_2$. This prime would be ramified in $JK$, because $\Delta(H_4/JK)=1$. But then the inertia field of this prime ideals equals $J$, which cannot be the case since $\Delta(H_4/JK)=1$, using the result on discriminant of tower of fields, and $-7=\Delta(J_1/J)\Delta(J_2/J)$. Since $G_p$ splits in ${\mbox{$\mathbb{Q}$}}(\sqrt{2})$, write $G_p=v_{p}\cdot \bar {v_p}$, identifying $v_{p}=x_1+\sqrt{2}y_1$ and $\bar {v_p}=x_1-\sqrt{2}y_1\in {\mbox{$\mathbb{Z}$}}_{J}$ for some $x_1,y_1\in {\mbox{$\mathbb{Z}$}}$. Clearly $\sigma(v_p)>0$ and $\sigma(\bar {v_p})>0$ for all $p$ and for all $\sigma$ in $Gal(J/{\mbox{$\mathbb{Q}$}})$. $$\text{ Claim: } v_p\equiv\bar v_p\equiv 1\pmod{\Delta(J_i/J)}\text { for } i=1,2 \text{ and for all } p$$ Here we consider two cases $p\equiv 1\pmod{6}$ and $p\equiv 5\pmod{6}$. For $p\equiv 1\pmod{6}$, we have $G_p\equiv 1\pmod{7}$ hence, $v_p\equiv\bar v_p\equiv 1\pmod{\Delta(J_1/J)}$ [ and ]{} $v_p\equiv\bar v_p\equiv 1\pmod{\Delta(J_2/J)} $. Since $\Delta(J_1/J)$ and $\Delta(J_2/J)$ are coprime, we get $v_p\equiv\bar {v_p}\equiv 1\pmod{\Delta(J_i/J) }$ for $i\in {1,2}$. Applying Artin map, $$\left(\frac{J_i/J}{v_{p}}\right)=\left(\frac{J_i/J}{\bar {v_{p}}}\right)=1 \text{ for } i\in {1,2},$$ since $v_p$ and $\bar v_p$ are in the kernel. The Galois group of $H_4/J$ is isomorphic to $Gal(J_1/J)\times Gal(J_2/J)$. So $$\left(\frac{H_4/J}{v_{p}}\right)=\left(\frac{H_4/J}{\bar {v_{p}}}\right)=1.$$ So, $G_p$ splits completely in $H_4$ for $p\equiv 1\pmod{6}$. For $p\equiv 5\pmod{6}$ to show that Artin symbol corresponding to $J_1/J$ and $J_2/J$ is trivial, the strategy has to be some what different. We know that, for $p\equiv 5\pmod{6}$, $v_p\cdot\bar v_p\equiv 1\pmod{8}$, which is a prime in ${\mbox{$\mathbb{Q}$}}$ splits in $J={\mbox{$\mathbb{Q}$}}(\sqrt{2})$, and therefore $J$ completed at $v_p$ is nothing but ${\mbox{$\mathbb{Q}$}}$ completed at $G_p=v_p\cdot \bar v_p$. Since $G_p\equiv 1\pmod{8}$, we have, $$\left(\frac{-1}{G_p}\right)=1$$ Now it remains to compute the Legendre symbol $\left(\frac{-7}{G_p}\right).$ If $p\equiv 5\pmod{6}$ then $G_p\equiv 4\pmod{7}$, hence $\left(\frac{G_p}{7}\right)=1$. Therefore by quadratic reciprocity we have,
$$\left(\frac{-7}{G_p}\right)=1,$$ and the claim follows.
From Proposition \[mainprop\] and Lemma \[mainlemma\] it is clear that $G_{p}$ splits completely in the cyclic extension of ${\mbox{$\mathbb{Q}$}}(\sqrt{-14})$. Now we are ready to prove our main result that, for $p>7$ and $p\equiv\pm 1\pmod{8}$ in the representation of $G_p$ as $G_p=x^2+7y^2$ for some $x,y\in {\mbox{$\mathbb{Z}$}}$, $y$ is divisible by $8$.
\[mainth\] Let $p>7$ and $G_p=2^{p}-(\frac{2}{p})2^{\frac{p+1}{2}}+1$ be a Gaussian Mersenne prime. Then for $p\equiv\pm 1\pmod{8}$ the form $G_p=x^2+7y^2$ exists for some $x,y\in {\mbox{$\mathbb{Z}$}}$ and $y$ is divisible by $8$.
Here we make use of the fact that, $G_{p}$ splits completely in the cyclic extension of ${\mbox{$\mathbb{Q}$}}(\sqrt{-14})$, so we refer to Proposition \[mainprop\] for rest of the proof. Consider the following lattice of fields.
(h4) at (0,2) [$H_4$]{}; (k1) at (-2,1) [$K_1$]{}; (k2) at (-1,1) [$K_2$]{}; (jk) at (0,1) [$JK$]{}; (j1) at (1,1) [$J_1$]{}; (j2) at (2,1) [$J_2$]{}; (k) at (-1,0) [$K$]{}; (s) at (0,0) [$S$]{}; (j) at (1,0) [$J$]{}; (q) at (0,-1) [${\mbox{$\mathbb{Q}$}}$]{};
\(q) – (k) – (k1) – (h4) – (k2) – (k) – (jk) – (s) – (q) – (j) – (jk)– (h4) – (j1) – (j)– (j2) – (h4);
;
It is clear that the Galois group of $H_4/{\mbox{$\mathbb{Q}$}}$ is the dihedral group with $8$ elements. Here $K={\mbox{$\mathbb{Q}$}}(\sqrt{-7})$, $J={\mbox{$\mathbb{Q}$}}(\sqrt{2})$, $S={\mbox{$\mathbb{Q}$}}(\sqrt{-14})$ and $\omega=\frac{1+\sqrt{-7}}{2}$. Let $\bar \omega$ be the conjugate of $\omega$ then $\omega\bar\omega=2$ and $2$ splits completely in $K$ as $2=(2,\omega)(2,\bar\omega)$. Now, $K_1$ and $K_2$ are the two conjugate field extensions of $K$, and $J_1$ and $J_2$ are the two conjugate field extensions of $J$ respectively. Discriminant of the extension fields are, $\Delta(S/{\mbox{$\mathbb{Q}$}})=-56$, $\Delta(K/{\mbox{$\mathbb{Q}$}})=-7$, $\Delta(J/{\mbox{$\mathbb{Q}$}})=8 $, $\Delta(JK/S)=1$, $\Delta(H_4/JK)=1$, $\Delta(JK/J)=-7$ and $\Delta(J_1/J)\Delta(J_2/J)=-7$. Clearly $\Delta(K_1/K)$ and $\Delta(K_2/K)$ are coprime, and $\Delta(K_1/K) \cdot \Delta(K_2/K)=8$. Since the discriminants are coprime and $K_1$ and $K_2$ are conjugates, $\Delta(K_1/K)=(2,\omega)^{3}$ and $\Delta(K_2/K)=(2,\bar\omega)^{3}$. Now, $\Delta(H_4/{\mbox{$\mathbb{Q}$}})=8^2$ and $\Delta(K_i/K)|8^2$. Let $\Delta=\Delta(K_1/K)$. Then there is a surjective group homomorphism:$$Cl_{\Delta}(K)\rightarrow
Gal(K_2/K).$$ The group $I_{K}{(\Delta)}/P_{K,1}(\Delta)=P_{K}(\Delta)/P_{K,1}(\Delta)$ is a subgroup of $Cl_{\Delta}(K)$ and contains the principal prime ideal $\pi=(x+y\sqrt{-7})$ of $K$. Any prime of ${\mbox{$\mathbb{Q}$}}$, which is inert in both $J$ and $K$ splits in $S$. For example, since $7\equiv1\pmod{3}$, the prime $3$ of ${\mbox{$\mathbb{Q}$}}$ is inert in $J$ and $K$ and splits in $S$. Thus the decomposition group of a prime $\mathfrak{p}_3$ in $H_4$ above $3$ is $S$. The prime $3{\mbox{$\mathbb{Z}$}}_{K}$ of $K$ does not split in $K_1$. Hence, $P_{K}(\Delta)/P_{K,1}(\Delta)$ maps surjective on $Gal(K_1/K)$. The map from $({\mbox{$\mathbb{Z}$}}_{K}/\Delta)^{*}$ to $P_{K}(\Delta)/P_{K,1}(\Delta)$ is surjective and the group $({\mbox{$\mathbb{Z}$}}_{K}/\Delta)^{*}$ is isomorphic to the group $({\mbox{$\mathbb{Z}$}}/8{\mbox{$\mathbb{Z}$}})^{*}$. We have the following group homomorphism, $$({\mbox{$\mathbb{Z}$}}/8{\mbox{$\mathbb{Z}$}})^{*}\simeq({\mbox{$\mathbb{Z}$}}_{K}/\Delta)^{*}\rightarrow P_{K}(\Delta)/P_{K,1}(\Delta)\rightarrow Gal(K_1/K).$$ Now $\{\pm 1\}$ is contained in the kernel of the $({\mbox{$\mathbb{Z}$}}_{K}/\Delta)^{*}\rightarrow P_{K}(\Delta)/P_{K,1}(\Delta)$ and this map is surjective hence, the kernel of $({\mbox{$\mathbb{Z}$}}_{K}/\Delta)^{*}\rightarrow P_{K}(\Delta)/P_{K,1}(\Delta)$ is $\{\pm 1\}$. Thus $P_{K}(\Delta)/P_{K,1}(\Delta)$ has two elements. So,$$({\mbox{$\mathbb{Z}$}}/8{\mbox{$\mathbb{Z}$}})^{*}\simeq({\mbox{$\mathbb{Z}$}}_{K}/\Delta)^{*}\rightarrow P_{K}(\Delta)/P_{K,1}(\Delta)\simeq Gal(K_1/K).$$ From Proposition \[mainprop\] it is clear that $G_{p}$ splits completely in $H_4$ and $H_4$ contains $\sqrt{2}$. For any prime ideal $\pi$ of $K$, $\left(\frac{K_1/K}{\pi}\right)=1$. Hence, $\pi=x+y\sqrt{-7}$ is identity in $P_{K}(\Delta)/P_{K,1}(\Delta)$. Thus, $$\label{maineq}x+y\sqrt{-7}\equiv\pm 1\bmod{\Delta}.$$ From Lemma \[mainlemma\], we have $4|y$. We assume that, $p>7$, so $G_{p}\equiv 1\pmod{32}$. Since, $4|y$ and $x\equiv\pm 1\pmod{8}$, we have, $$x^2+7y^2\equiv 1\pmod{16}$$ and $$(y\cdot\sqrt{-7})^{2}\equiv 0\pmod{16}.$$ That is $y\cdot\sqrt{-7}\equiv 0 \text{ or }4\pmod{8}$. If $y\cdot\sqrt{-7}\equiv 4\bmod{8}$, then there is a contradiction to $\pi\equiv x+y\sqrt{-7}\equiv\pm 1\bmod{\Delta}$ in and to the splitting of $\pi$ in $H_4$. Hence, $$\label{eq1}
y\cdot\sqrt{-7}\equiv 0 \bmod{8}$$ Now, for $K={\mbox{$\mathbb{Q}$}}(\sqrt{-7})$, let $\mathcal{O}_K$ be its ring of integers. The element $\omega=\frac{1+\sqrt{-7}}{2}$ is an integer of $K$, which is a zero of the polynomial $X^2-X+2$. As $-2\in {\mbox{$\mathbb{Z}$}}/8{\mbox{$\mathbb{Z}$}}$ satisfies $X^2-X+2=0$, we obtain a ring homomorphism $$\mathcal{O}_{K}\rightarrow {\mbox{$\mathbb{Z}$}}/8{\mbox{$\mathbb{Z}$}}$$ $$a+b\omega\rightarrow (a-2b)\bmod{8}.$$ Now $\sqrt{-7}=2\omega-1$ maps to $3$, confirming $y\equiv 0\pmod{8}$ from equation \[eq1\]. This completes the proof.
Artin’s reciprocity law
------------------------
Now we give an alternate proof using the similar techinique used in [@henstev:2000:artinreci] to prove Theorem \[mainth\].
Primes in a Quadratic field
---------------------------
In this section we refer mainly to [@henstev:2000:artinreci]. To prove the results in the previous section, we need to consider the field ${\mbox{$\mathbb{Q}$}}(\sqrt{-7})$. We know that, for $K={\mbox{$\mathbb{Q}$}}(\sqrt{-7})$, we obtain a ring homomorphism $$\mathcal{O}_{K}\rightarrow {\mbox{$\mathbb{Z}$}}/8{\mbox{$\mathbb{Z}$}}$$ $$a+b\omega\rightarrow (a-2b)\bmod{8}.$$ kernel $\mathfrak{a}$ of the map is generated by $8$ and $\omega+2$. Thus $\mathfrak{a}=({\omega+2})\mathcal{O}_{K}$. Their kernels are prime ideals of index $2$ in $\mathcal{O}_{K}$ with generators $\omega$ and $\bar \omega$. Also, the identity $\omega\bar\omega=2$ and $\omega+2=-\omega^{3}$ show that the ideal $\mathfrak{a}$ factors as the cube of prime $\omega\mathcal{O}_{K}$. Now consider an Abelian extension $L$ of $K$ with Galois group $G$, say $L=K[\beta]$ where $\beta$ is a zero of $x^2-\omega x-1$ and $L$ has dimension $2$ over $K$. $ \text{ with } \beta^{2}-\omega\beta-1=0.$ Now consider $$K={\mbox{$\mathbb{Q}$}}[\sqrt{-7}]={\mbox{$\mathbb{Q}$}}[\omega] \text{ with } \omega^{2}-\omega+2=0 \text{ and }$$ $$L=K[\beta] ,\qquad \beta^2-\omega\beta -1$$ The discriminant of $L/K$ is $\omega^2+4=\omega+2$ is non-zero and $L$ has dimension $2$ over $K$, hence $L$ is Abelian over $K$ with Galois group $G$ of order $2$, say $G=\{1,\rho\}$. The non-identity element $\rho$ of $G$ satisfies $\rho(\beta)=\omega-\beta=-1/\beta$. The discriminant $\omega+2=-\omega^3$ of the polynomial defining $L$ is not a square in $K$, hence we have $L=K[\sqrt{-\omega}]$.
The inclusion map ${\mathcal{O}_{K}}\rightarrow{\mathcal{O}_{L}}$ induces a ring homomorphism $k(\mathfrak{p})\rightarrow {\mathcal{O}_{L}}/{\mathfrak{p}\mathcal{O}_{L}}$, where $k(\mathfrak{p})=\mathcal{O}_{K}/\mathfrak{p}$, for a prime $\mathfrak{p}$ of $K$.
An outline of the proof via Artin Symbol
------------------------------------------
If $\mathfrak{p}=\pi{\mathcal{O}}_{K}$ is a prime of $K={\mbox{$\mathbb{Q}$}}[\sqrt{-7}]={\mbox{$\mathbb{Q}$}}[\omega]$ different from $\omega{\mathcal{O}}_{K}$, then the Artin symbol $\left(\frac{L/K}{\mathfrak{p}}\right)=1$ or $\rho$ according as $\pi$ maps to $\pm 1$ or to $\pm 3$ under the map ${\mathcal{O}}_{K}\rightarrow {\mbox{$\mathbb{Z}$}}/8{\mbox{$\mathbb{Z}$}}$ that sends $\omega$ to $-2$. For example, $\sqrt{-7}=2\omega-1$ maps to $3\bmod{8}$ and the number $(8k\pm1 )\pm8l\sqrt{-7}$ maps to $\pm 1\pmod{8}$ for $l,k\in {\mbox{$\mathbb{Z}$}}$. This is because, $\sqrt{-7}=2\omega-1$ and $\omega$ maps to $-2$ under $ {\mbox{$\mathbb{Z}$}}/8{\mbox{$\mathbb{Z}$}}$. We see that, for $p=47$, $G_{47}=140737471578113= 5732351^{2}+7\cdot3925696^{2} $ is nothing but, $$5732351\pm 3925696\sqrt{-7}\equiv-1\pm 3(0)\equiv -1\bmod{8}.$$ Consider an element of the form $x+\sqrt{-7}y$ of norm $x^2+7y^2$ which is a rational prime. The Artin symbol equals $1$ if $x+3y$ is $\pm 1\bmod{8}$, and $\rho$ otherwise. From Lemma \[mainlemma\], it is clear that $x$ leaves the remainder $\pm 1\bmod{8}$ and $y$ is divisible by $4$. Hence the Artin symbol is $1$ if and only if $y$ is divisible by $8$. This observation is equivalent to the assertion that, *any prime of $K={\mbox{$\mathbb{Q}$}}(\sqrt{-7})$ of norm $G_{p}$ has trivial Artin symbol in the quadratic extension $L=K[\omega]$*. The proof works in the extension $N=K(\sqrt{-\omega},\sqrt{-\bar {\omega}})$, which is the union of two conjugate quadratic extensions $L=K(\sqrt{-\omega})$ and $L=K(\sqrt{-\bar {\omega}})$. Further details can be filled in by following Theorem \[mainth\] and the last theorem in [@henstev:2000:artinreci] and [@pal:2012:jis].
Generalization of Theorem \[mainth\]
====================================
In this section we generalize the result of Theorem \[mainth\] for $K={\mbox{$\mathbb{Q}$}}(\sqrt{-d})$ that has the same properties as $K={\mbox{$\mathbb{Q}$}}(\sqrt{-7})$. In this case there exists a cyclic extension $H_4$ of $S={\mbox{$\mathbb{Q}$}}(\sqrt{-2\cdot d})$ and $\sqrt{2}\in H_4$, so that $G_p$ splits completely in $H_4$. For this one requires $\left(\frac{2}{d}\right)=1$ and $\left(\frac{-d}{G_p}\right)=1$. Thus, with these conditions we state the general form of Proposition \[mainprop\].
Let $d\equiv 3\pmod{4}$ be a square free integer and $d>0$. Let $G_p$ be a Gaussian Mersenne prime and $G_p=x^{2}+dy^{2}$ for $d\equiv 3\pmod{4}$ exists and $\left(\frac{2}{d}\right)=1=\left(\frac{-d}{G_p}\right) $. Suppose that there exists a cyclic extension $H_4$ of $S={\mbox{$\mathbb{Q}$}}(\sqrt{-2\cdot d})$ with $[H_4:S]=4$, $H_4\subset H(S)$ and $\sqrt{2}\in H_4$. Then $G_p$ splits completely in $H_4$.
Proof of this proposition is similar to Proposition \[mainprop\].
Thus, the generalized form of Theorem \[mainth\] is stated below, proof of this theorem is similar to the proof of Theorem \[mainth\].
Let $d\equiv 7\pmod{24}$ be a square free integer and $d>0$. Suppose $G_{p}$ is a Gaussian Mersenne prime which splits completely in a cyclic extension $H_4$ of $S={\mbox{$\mathbb{Q}$}}(\sqrt{-2\cdot d})$. If $G_p=x^{2}+dy^{2}$ for $d\equiv 7\pmod{24}$ exists then $8|y$.
Conclusion
==========
In the case of Mersenne primes $2^{p}-1$, whenever $2^{p}-1$ is a quadratic residue of $7$, then $2^p-1$ takes the form $x^2+7y^2$. Franz Lemmermeyer made an observation that, in this representation $x$ is divisible by $8$ and $y$ leaves the remainder $\pm 3$ when divided by $8$. This was later proved by H. W. Lenstra and P. Stevenhagen using Artin reciprocity law and presented this result on the occasion of birth centenary of Emil Artin on 3 March, 1998 at Universiteit van Amsterdam[@henstev:2000:artinreci]. In [@pal:2012:jis], authors made similar observations for rational primes $N(\frac{\alpha^{p}-1}{\alpha-1})$, for $\alpha=2+\sqrt{2}$ and proved using Artin’s reciprocity law. In this paper, we have made an observation that, if a Gaussian Mersenne prime is a quadratic residue of $7$, then in the representation of $G_{p}$ as $x^2+7y^2$, we have $x\equiv \pm 1\pmod{8}$ and $y$ is divisible by $8$ and proved this result using Artin reciprocity law and generalized this result for $d\equiv 7\pmod{24}$.
Acknowledgement
===============
The authors would like to thank Professor P. Stevenhagen, Universiteit Leiden, for his valuable insights of class field theory concepts. The authors would also like to thank Professor U.K.Anandavardhanan, IIT Bombay, for helping to improve earlier versions of this manuscript. The first author would like to acknowledge National Board for Higher Mathematics, Government of India for funding this work through Post Doctoral Fellowship.
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[^1]: Research by the first author is supported by National Board for Higher Mathematics-Post Doctoral Fellowship, Government of India.
|
---
abstract: |
The structure of a complete lattice formed by closed linear subspaces of a Hilbert space (i.e., a Hilbert lattice) entails some unreasonable consequences from the physical point of view. Specifically, this structure seems to contradict to the localized variant of the Kochen-Specker theorem according to which the bivaluation of a proposition represented by a closed linear subspace that does not belong to a Boolean algebra shared by the state, in which a quantum-mechanical system is prepared, must be value indefinite. For this reason, the Hilbert lattice structure seems to be too strong and needs to be weakened. The question is, how should it be weakened so that to support the quantum uncertainty principle and the Kochen-Specker theorem? Which logic will a weakened structure identify? The present paper tries to answer these questions.\
**Keywords:** Quantum mechanics; Closed linear subspaces; Lattice structures; Truth-value assignment; Supervaluationism\
author:
- |
Arkady Bolotin[^1]\
*Ben-Gurion University of the Negev, Beersheba (Israel)*
bibliography:
- 'Admissibility.bib'
title: Admissibility of truth assignments for quantum propositions in supervaluational logic
---
Introduction and preliminaries
==============================
The assumption of the structure of a complete lattice imposed on the set $\mathcal{L}(\mathcal{H})$ of closed linear subspaces $\mathcal{H}_a$, $\mathcal{H}_b$, … of a Hilbert space $\mathcal{H}$ (called *Hilbert lattice* [@Redei]) entails some unreasonable consequences from the physical point of view.\
To see this, let us introduce the proposition $P_a( |\Psi\rangle )$ set forth as follows
$$\label{DEF} P_a( |\Psi\rangle )
\equiv
\mathrm{Prop}
\left(
|\Psi\rangle \in \mathcal{H}_{|\Psi\rangle} \wedge \mathcal{H}_a
\right)
\;\;\;\; ,$$
where $\mathrm{Prop}(\cdot)$ is a propositional function (i.e., a sentence that becomes a proposition when all variables in it are given definite values), $\mathcal{H}_{|\Psi\rangle} $ is one of the closed linear subspaces of $\mathcal{L}(\mathcal{H})$ containing the state vector $|\Psi\rangle$ in which a quantum-mechanical system is prepared, $\mathcal{H}_a$ is the subspace of $\mathcal{L}(\mathcal{H})$ that represents the proposition $P_a$, and the subspace $\mathcal{H}_{|\Psi\rangle} \wedge \mathcal{H}_a$ (if it exists) is the meet of $\mathcal{H}_{|\Psi\rangle}$ and $\mathcal{H}_a$ such that
$$ \mathcal{H}_{|\Psi\rangle}, \mathcal{H}_a \in \mathcal{L}(\mathcal{H})
\;
\implies
\;
\mathcal{H}_{|\Psi\rangle} \wedge \mathcal{H}_a
=
\mathcal{H}_{|\Psi\rangle} \!\cap \mathcal{H}_a
\in \mathcal{L}(\mathcal{H})
\;\;\;\; ,$$
where $\cap$ is the set-theoretical intersection. By way of illustration, consider the simplest case of *a qubit*, a two-state quantum-mechanical system such as a one-half spin particle (say, an electron) whose spin may assume only two possible values: either $+\frac{\hbar}{2}$ (denoted as “up” or “$+$” for short) or $-\frac{\hbar}{2}$ (denoted as “down” or “$-$”). Suppose, the qubit is prepared in the pure state coinciding with the normalized eigenvector $|\Psi^{(z)}_{+}\rangle = [\begin{smallmatrix} 1 \\ 0 \end{smallmatrix}]$ of the Pauli matrix $\sigma^{(z)}$. This eigenvector lies in the range of the projection operator $\hat{P}^{(z)}_{+} = |\Psi^{(z)}_{+}\rangle\langle\Psi^{(z)}_{+}|$, the closed linear subspace of the two-dimensional Hilbert space $\mathcal{H} = \mathbb{C}^2$, namely,
$$ \mathrm{ran}(\hat{P}^{(z)}_{+}\!)
=
\left\{
\!\left[
\begingroup\SmallColSep
\begin{array}{r}
a \\
0
\end{array}
\endgroup
\right]\!
:
\,
a \in \mathbb{R}
\right\}
\;\;\;\; .$$
According to the definition (\[DEF\]), in the state $|\Psi^{(z)}_{+}\rangle$ the proposition “Spin along the $z$-axis is up”, denoted as $P^{(z)}_{+}$ and represented by the range $\mathrm{ran}(\hat{P}^{(z)}_{+})$, is given by
$$ P^{(z)}_{+}( |\Psi^{(z)}_{+}\rangle )
=
\mathrm{Prop}
\!
\left(
|\Psi^{(z)}_{+}\rangle
\in
\mathrm{ran}(\hat{P}^{(z)}_{+}\!)
\wedge
\mathrm{ran}(\hat{P}^{(z)}_{+}\!)
\right)
=
\mathrm{Prop}
\!
\left(
|\Psi^{(z)}_{+}\rangle
\in
\mathrm{ran}(\hat{P}^{(z)}_{+}\!)
\right)
\;\;\;\; ,$$
which is *a tautology*. By contrast, in the same state, the proposition “Spin along the $z$-axis is down”, denoted as $P^{(z)}_{-}$ and represented by the range
$$ \mathrm{ran}(\hat{P}^{(z)}_{-}\!)
=
\left\{
\!\left[
\begingroup\SmallColSep
\begin{array}{r}
0 \\
a
\end{array}
\endgroup
\right]\!
:
\,
a \in \mathbb{R}
\right\}
\;\;\;\;$$
containing the second normalized eigenvector $|\Psi^{(z)}_{-}\rangle = [\begin{smallmatrix} 0 \\ 1 \end{smallmatrix}]$ of the Pauli matrix $\sigma^{(z)}$, is given by
$$ P^{(z)}_{-}( |\Psi^{(z)}_{+}\rangle )
=
\mathrm{Prop}
\!
\left(
|\Psi^{(z)}_{+}\rangle
\in
\left\{
\!\left[
\begingroup\SmallColSep
\begin{array}{r}
a \\
0
\end{array}
\endgroup
\right]\!
:
a \in \mathbb{R}
\right\}
\cap \left\{
\!\left[
\begingroup\SmallColSep
\begin{array}{r}
0 \\
a
\end{array}
\endgroup
\right]\!
:
a \in \mathbb{R}
\right\}
\right)
\!
=
\mathrm{Prop}
\!
\left(
|\Psi^{(z)}_{+}\rangle
\in
\{ 0 \}
\right)
\;\;\;\; ,$$
which is *a contradiction* due to $|\Psi\rangle \neq 0$, i.e., the statement that any physically meaningful state of the quantum system differs from vector $0$. In a bivalent semantics (defined by *the bivaluation relation*, i.e., the function $b$ from the set of propositions into the set $\{0,1\}$ of bivalent truth values) this can be expressed equivalently, using $0$ for *false* and $1$ for *true*, in the following way:
$$ b
\left(
P^{(z)}_{+}( |\Psi^{(z)}_{+}\rangle )
\right)
=
b
\left(
\mathrm{Prop}
\!
\left(
|\Psi^{(z)}_{+}\rangle
\in
\mathrm{ran}(\hat{P}^{(z)}_{+}\!)
\right)
\right)
\!
=
1
\;\;\;\; ,$$
$$ b
\left(
P^{(z)}_{-}( |\Psi^{(z)}_{+}\rangle )
\right)
=
b
\left(
\mathrm{Prop}
\!
\left(
|\Psi^{(z)}_{+}\rangle
\in
\{ 0 \}
\right)
\right)
\!
=
0
\;\;\;\; .$$
Now, recall that in the Hilbert lattice structure, every pair of the elements $\mathcal{H}_a$ and $\mathcal{H}_b$ from $\mathcal{L}(\mathcal{H})$ has the meet [@Davey]. As a result, in the state $|\Psi^{(z)}_{+}\rangle$, the propositions “Spin along the $x$-axis is up” and “Spin along the $x$-axis is down” (denoted as $P^{(x)}_{+}$ and $P^{(x)}_{-}$, correspondingly, and represented by the ranges $\mathrm{ran}(\hat{P}^{(x)}_{+})$ and $\mathrm{ran}(\hat{P}^{(x)}_{-})$ containing in that order the normalized eigenvectors $|\Psi^{(x)}_{+}\rangle = \frac{1}{\sqrt{2}} [\begin{smallmatrix} 1 \\ 1 \end{smallmatrix}]$ and $|\Psi^{(x)}_{-}\rangle = \frac{1}{\sqrt{2}} [\begin{smallmatrix} \;\;\:1 \\ -1 \end{smallmatrix}]$ of the Pauli matrix $\sigma^{(x)}$) are *determined*
$$ \mathrm{ran}(\hat{P}^{(z)}_{+}\!)
\wedge
\mathrm{ran}(\hat{P}^{(x)}_{+}\!)
=
\left\{
\!\left[
\begingroup\SmallColSep
\begin{array}{r}
a \\
0
\end{array}
\endgroup
\right]\!
:
a \in \mathbb{R}
\right\}
\cap \left\{
\!\left[
\begingroup\SmallColSep
\begin{array}{r}
a \\
a
\end{array}
\endgroup
\right]\!
:
a \in \mathbb{R}
\right\}
=
\{ 0 \}
\;\;\;\; ,$$
$$ \mathrm{ran}(\hat{P}^{(z)}_{+}\!)
\wedge
\mathrm{ran}(\hat{P}^{(x)}_{-}\!)
=
\left\{
\!\left[
\begingroup\SmallColSep
\begin{array}{r}
a \\
0
\end{array}
\endgroup
\right]\!
:
a \in \mathbb{R}
\right\}
\cap \left\{
\!\left[
\begingroup\SmallColSep
\begin{array}{r}
a \\
-a
\end{array}
\endgroup
\right]\!
:
a \in \mathbb{R}
\right\}
=
\{ 0 \}
\;\;\;\; ,$$
and so they have a bivalent truth value:
$$\label{X12} b
\left(
P^{(x)}_{\pm}( |\Psi^{(z)}_{+}\rangle )
\right)
=
b
\left(
\mathrm{Prop}
\!
\left(
|\Psi^{(z)}_{+}\rangle
\in
\mathrm{ran}(\hat{P}^{(z)}_{+}\!)
\wedge
\mathrm{ran}(\hat{P}^{(x)}_{\pm}\!)
\right)
\right)
\!
=
0
\;\;\;\; .$$
However, by the postulates of quantum mechanics, measurements of spin along the different axes are incompatible. Since this requires that both propositions $P^{(x)}_{+}$ and $P^{(x)}_{-}$ be *undetermined* in the state where the proposition $P^{(z)}_{+}$ is verified (i.e., has the value of the truth), quantum logic (i.e., the complete orthomodular lattice based on the closed subspaces of a Hilbert space [@Mackey; @Ptak]) seems to contradict to the quantum mechanical uncertainty principle.\
Furthermore, quantum logic seems to contradict to *the localized variant of the Kochen-Specker theorem* [@Abbott] stating that the bivaluation of the proposition $P_a(|\Psi\rangle)$, represented by the closed linear subspace $\mathcal{H}_a$ that does not belong to a Boolean algebra shared by the state $|\Psi\rangle$ in which the quantum-mechanical system is prepared, must be value indefinite (e.g., instead of (\[X12\]) it must be $b( P^{(x)}_{\pm}( |\Psi^{(z)}_{+}\rangle ) ) = \frac{0}{0}$, where $\frac{0}{0}$ denotes an indeterminate value).\
Therefore, the Hilbert lattice structure seems to be too strong and thus needs to be weakened. The question is, how should it be weakened so that to support the uncertainty principle and the Kochen-Specker theorem? Which logic does a weakened structure identify?\
The present paper tries to answer these questions.\
From supervaluationary logic to invariant-subspace lattices
===========================================================
It was already observed in [@Chiara] that the Hilbert lattice structure could be weakened by the requirement for the meet operation to exist only for the subspaces belonging to a common Boolean algebra. However, it was not clear what semantics would associate with such a requirement. Let us show that it is *supervaluationism*.\
Recall that supervaluationism is a semantics in which a proposition can have a bivalent truth value even when its components do not. Therefore, it allows one to apply the tautologies of propositional logic in cases where bivalent truth values cannot be defined [@Varzi; @Keefe].\
Take, for example, *the Heisenberg cut*, i.e., the borderline that seemingly separates the quantum realm from the classical [@Zeh; @Schlosshauer; @Duvenhage]. This cut ensures that sufficiently small systems will manifest superpositions and hence quantum behavior; on the other hand, for sufficiently large systems, superpositional (quantum) behavior will be replaced by classical behavior. It leads to the question of where exactly it should be, where a classical system begins and a quantum system ends. It relates to another question of when the collapse of the wave function takes place and how long it takes (since this is essentially the same Heisenberg cut but with space replaced by time) [@Hajicek].\
Since the concept of a classical system appears to lack sharp boundaries and because of the subsequent indeterminacy surrounding the extension of the predicate “*is a classical system*”, no quantum particle (such as electron, protons, or neutron) constituting a composite system can be identified as making the difference between *being a classical system* and *not being a classical system*. Given then that one quantum particle does not make a classical system, it would seem to follow that two do not, thus three do not, and so on. In the end, no number of the quantum particles can constitute a classical system. This is a paradox since apparently true premises through seemingly uncontroversial reasoning lead to an apparently false conclusion.\
To resolve this paradox (known in the literature as *the Sorites Paradox* [@Keefe; @Hyde]), one may deny that there are such things as classical systems, i.e., the world is quantum rather than classical.\
A different approach to this paradox is offered by supervaluationism. Since it is logically true for any number $N$ of the quantum particles that it either does or does not make a classical system, the disjunction of the propositions $P$ = “$N$ quantum particle(s) constitute(s) a classical system” and $\neg P$ = “$N$ quantum particle(s) do(es) not constitute a classical system” is an instance of the valid schema. That is, $P \vee \neg P$ should be true regardless of whether its disjuncts can be described as either true or false. Otherwise stated, while it is true that there is some borderline separating everything governed by the wave function from a classical description, there is no particular number of the quantum particles $N$ for which it is true that it is the borderline (i.e., the Heisenberg cut). This implies that a semantics for a logic of propositions relating to quantum systems should not be truth-functional (that is, some propositions in this semantics may have no bivalent truth values which is called *truth-value gaps* [@Beziau]).\
As supervaluationism admits truth-value gaps, to interpret propositions relating to the quantum system in terms of a supervaluationary logic, *a lattice structure allowing truth-value gaps* should be imposed on the closed linear subspaces of the Hilbert space associated with the quantum system. A natural candidate for such a structure is the collection of invariant-subspace lattices that have no mutual nontrivial members.\
Recall that a subspace $\mathcal{H}_a \subseteq \mathcal{H}$ is called *an invariant subspace under the projection operator $\hat{P}_a$ on a finite Hilbert space $\mathcal{H}$* if
$$ \hat{P}_a\!
:\,
\mathcal{H}_a
\mapsto
\mathcal{H}_a
\;\;\;\; .$$
Concretely, the image of every vector $|\Psi\rangle$ in $\mathcal{H}_a$ under $\hat{P}_a$ remains within $\mathcal{H}_a$ which can be denoted as
$$ \hat{P}_a \mathcal{H}_a
=
\left\{
|\Psi\rangle \in \mathcal{H}_a
:\;
\hat{P}_a |\Psi\rangle
\right\}
\subseteq
\mathcal{H}_a
\;\;\;\; .$$
For example, the range and kernel of the projection operator $\hat{P}_a$, i.e., $\mathrm{ran}(\hat{P}_a)$ and $\mathrm{ker}(\hat{P}_a) = \mathrm{ran}(\hat{1} - \hat{P}_a)$, as well as the trivial subspaces $\mathrm{ran}(\hat{0}) = \{0\}$ and $\mathrm{ran}(\hat{1}) = \mathcal{H}$ (where $\hat{0}$ and $\hat{1}$ stand for the trivial projection operators) are the invariant subspaces under $\hat{P}_a$.\
Let $\mathcal{L}(\hat{P}_a)$ denote the set of the subspaces invariant under $\hat{P}_a$, namely,
$$ \mathcal{L}(\hat{P}_a)
\equiv
\left\{
\mathcal{H}_a \subseteq \mathcal{H}
:\;
\hat{P}_a \mathcal{H}_a
\subseteq
\mathcal{H}_a
\right\}
\;\;\;\; .$$
Recall that the family $\Sigma^{(Q)}$ of two or more nontrivial projection operators $\hat{P}^{(Q)}_n$, namely,
$$ \Sigma^{(Q)}
\equiv
\left\{
\hat{P}^{(Q)}_n
\right\}
\;\;\;\; ,$$
is called *a context* if the next two conditions hold:
$$ \hat{P}^{(Q)}_n, \hat{P}^{(Q)}_{m \neq n} \in \Sigma^{(Q)}
\;
\implies
\;
\hat{P}^{(Q)}_n \hat{P}^{(Q)}_{m \neq n}
=
\hat{P}^{(Q)}_{m \neq n} \hat{P}^{(Q)}_n
=
\hat{0}
\;\;\;\; ,$$
$$ \sum_{\hat{P}^{(Q)}_n \in \Sigma^{(Q)}} \hat{P}^{(Q)}_n
=
\hat{1}
\;\;\;\; .$$
Consider the set of the invariant subspaces *invariant under each $\hat{P}^{(Q)}_n \in \Sigma^{(Q)}$*:\
$$ \mathcal{L}(\Sigma^{(Q)})
\equiv
\!\!\!
\bigcap_{\hat{P}^{(Q)}_n \in \Sigma^{(Q)}}
\!\!\!
\mathcal{L}(\hat{P}^{(Q)}_n)
\;\;\;\; .$$
For the qubit where $\hat{P}^{(Q)}_{+} \hat{P}^{(Q)}_{-} = \hat{P}^{(Q)}_{-} \hat{P}^{(Q)}_{+} = \hat{0}$ and $\hat{P}^{(Q)}_{+} + \hat{P}^{(Q)}_{-} = \hat{1}$, this set is
$$ \mathcal{L}(\Sigma^{(Q)})
=
\mathcal{L}(\hat{P}^{(Q)}_{+})
\cap
\mathcal{L}(\hat{P}^{(Q)}_{-})
=
\left\{
\mathrm{ran}(\hat{0})
,\,
\mathrm{ran}(\hat{P}^{(Q)}_{+})
,\,
\mathrm{ran}(\hat{P}^{(Q)}_{-})
,\,
\mathrm{ran}(\hat{1})
\right\}
\;\;\;\; .$$
The elements of every set $\mathcal{L}(\Sigma^{(Q)})$ form a complete lattice called *the invariant-subspace lattice of the context $\Sigma^{(Q)}$* [@Radjavi]. It is straightforward to see that each invariant-subspace lattice only contains the subspaces belonging to the mutually commutable projection operators, that is, each $\mathcal{L}(\Sigma^{(Q)})$ is a Boolean algebra.\
Recall that two contexts are called *intertwined* if they share one or more common elements [@Svozil]. As any intertwined context has at least one individual element (i.e., one that is not shared by other contexts), each lattice $\mathcal{L}(\Sigma^{(Q)})$ has a nonempty set of individual subspaces. Clearly, in the structure of the invariant-subspace lattices $\mathcal{L}(\Sigma^{(Q)})$, $\mathcal{L}(\Sigma^{(R)})$, … imposed on the closed linear subspaces of the Hilbert space, the individual subspaces of the different lattices cannot meet each other. In symbols,
$$ \mathrm{ran}(\hat{P}_n^{(Q)})
\in
\mathcal{L}(\Sigma^{(Q)})
,\,
\mathrm{ran}(\hat{P}_m^{(R)})
\in
\mathcal{L}(\Sigma^{(R)})
\,
\implies
\,
\mathrm{ran}(\hat{P}_n^{(Q)})
\;\cancel{\;\wedge\;}\;
\mathrm{ran}(\hat{P}_m^{(R)})
\;\;\;\; ,$$
where $\mathrm{ran}(\hat{P}_n^{(Q)})$ and $\mathrm{ran}(\hat{P}_m^{(R)})$ denote the individual subspaces from the different lattices (i.e., $Q \neq R$), and the cancelation of the meet operation $\wedge$ indicates that this operation cannot be defined for such subspaces (recall that the meet is defined as an operation on pairs of elements from the same partially ordered set $\mathcal{L}$ [@Davey]).\
The nonexistence of the meet operation for pairs of the ranges that do not lie in a common invariant-subspace lattice corresponds to truth-value gaps in supervaluational logic. To be sure, let the quantum-mechanical system be prepared in the state $|\Psi^{(Q)}_n\rangle$ residing in $\mathrm{ran}(\hat{P}_n^{(Q)})$, the individual subspace from the lattice $\mathcal{L}(\Sigma^{(Q)})$. In this case, the proposition $P^{(R)}_m$ represented by the individual subspace $\mathrm{ran}(\hat{P}_m^{(R)})$ from the lattice $\mathcal{L}(\Sigma^{(R)})$ is undetermined and thus cannot have a bivalent truth value:
$$ b
\left(
P^{(R)}_{m}( |\Psi^{(Q)}_{n}\rangle )
\right)
=
b
\left(
\mathrm{Prop}
\!
\left(
|\Psi^{(Q)}_{n}\rangle
\in
\mathrm{ran}(\hat{P}_n^{(Q)})
\;\cancel{\;\wedge\;}\;
\mathrm{ran}(\hat{P}_m^{(R)})
\right)
\right)
\!
=
\frac{0}{0}
\;\;\;\; .$$
For example, in the state where the proposition “Spin along the $z$-axis is up” can be described as either true or false, both propositions “Spin along the $x$-axis is up” and “Spin along the $x$-axis is down” are neither true nor false: $b( P^{(x)}_{\pm}( |\Psi^{(z)}_{\pm}\rangle ) ) = b( \mathrm{Prop}( |\Psi^{(z)}_{\pm}\rangle \in \mathrm{ran}(\hat{P}^{(z)}_{+}) \;\cancel{\;\wedge\;}\; \mathrm{ran}(\hat{P}^{(x)}_{\pm}) ) ) = \frac{0}{0}$.\
Then again, in any invariant-subspace lattice on a finite-dimensional Hilbert space one has
$$ \mathrm{ran}(\hat{P}^{(R)}_m)
\wedge
\mathrm{ker}(\hat{P}^{(R)}_m)
=
\mathrm{ran}(\hat{0})
\;\;\;\; ,$$
$$ \mathrm{ran}(\hat{P}^{(R)}_m)
\vee
\mathrm{ker}(\hat{P}^{(R)}_m)
=
\left( \mathrm{ran}(\hat{0}) \right)^{\perp}
=
\mathrm{ran}(\hat{1})
\;\;\;\; ,$$
where $(\cdot)^{\perp}$ stands for the orthogonal complement of $(\cdot)$. Given that the subspaces $\mathrm{ran}(\hat{P}^{(R)}_m)$ and $\mathrm{ker}(\hat{P}^{(R)}_m)$ represent the proposition $P^{(R)}_m$ and its negation $\neg P^{(R)}_m$, while the subspaces $\mathrm{ran}(\hat{0})$ and $\mathrm{ran}(\hat{1})$ represent the conjunction and disjunction of $P^{(R)}_m$ and $\neg P^{(R)}_m$, respectively, one finds
$$ P^{(R)}_m
\wedge
\neg P^{(R)}_m
(|\Psi\rangle)
=
\mathrm{Prop}
\!
\left(
|\Psi\rangle
\in
\mathcal{H}_{|\Psi\rangle}
\wedge
\{0\}
\right)
\;\;\;\; ,$$
$$ P^{(R)}_m
\vee
\neg P^{(R)}_m
(|\Psi\rangle)
=
\mathrm{Prop}
\!
\left(
|\Psi\rangle
\in
\mathcal{H}_{|\Psi\rangle}
\wedge
\mathcal{H}
\right)
\;\;\;\; .$$
Since $\mathrm{Prop}(|\Psi\rangle\!\in\!\mathcal{H}_{|\Psi\rangle} \wedge \{0\})$ is a contradiction and $\mathrm{Prop}(|\Psi\rangle\!\in\!\mathcal{H}_{|\Psi\rangle} \wedge \mathcal{H})$ is a tautology, the conjunction and disjunction of $P^{(R)}_m$ and $\neg P^{(R)}_m$ are always false and true, correspondingly,
$$ b\left(
P^{(R)}_m
\wedge
\neg P^{(R)}_m
(|\Psi\rangle)
\right)
=
0
\;\;\;\; ,$$
$$ b\left(
P^{(R)}_m
\vee
\neg P^{(R)}_m
(|\Psi\rangle)
\right)
=
1
\;\;\;\; ,$$
even with undetermined $P^{(R)}_m(|\Psi\rangle)$ and $\neg P^{(R)}_m(|\Psi\rangle)$.\
Diagrams of lattice structures
==============================
To portray different lattice structures of the closed linear subspaces $\mathcal{H}_a$, $\mathcal{H}_b$, … of a Hilbert space, a modified version of a Hasse diagram can be used.\
![The bivaluation relation in the Hilbert sublattice $\mathcal{L}(\Sigma)$ of the qubit\[fig1\]](Figure1.eps)
Recall that the Hasse diagram is a type of mathematical diagram where each subspace corresponds to a vertex in the plane connected with another vertex by a line segment which goes upward from $\mathcal{H}_a$ to $\mathcal{H}_b$ whenever $\mathcal{H}_a \subseteq \mathcal{H}_b$ and there is no $\mathcal{H}_c$ such that $\mathcal{H}_a \subseteq \mathcal{H}_c \subseteq \mathcal{H}_b$ [@Kiena]. Besides the information on the transitive reduction, the Hasse diagram can also show the truth values of the propositions relating to the quantum system in a specific state by picturing the vertices that represent these propositions in the following way: the vertex is drawn as *a black square* if the proposition is *true*, the vertex is drawn as *a black circle* if the proposition is *false*, and the vertex is drawn as *a hollow circle* if the proposition *cannot be described as either true or false*. Moreover, the ellipse-like curves enclosing the subspaces belonging to the common contexts are added to the standard Hasse diagram.\
![The bivaluation relation in the invariant-subspace lattices of the qubit\[fig2\]](Figure2.eps)
The Figures \[fig1\] and \[fig2\] present the diagrams of the structures formed by the closed subspaces of the qubit. Specifically, the Figure \[fig1\] shows the Hilbert sublattice $\mathcal{L}(\Sigma)$ which is a lattice with the same meet and join operations as the Hilbert lattice $\mathcal{L}(\mathbb{C}^2)$ (the nontrivial closed subspaces of the sublattice $\mathcal{L}(\Sigma)$ correspond to the operators to measure the qubit spin along the $x$, $y$ and $z$ axes), while the Figure \[fig2\] demonstrates the invariant-subspace lattices $\mathcal{L}(\Sigma^{(z)})$ and $\mathcal{L}(\Sigma^{(x)})$. In both Figures, the truth values are given in the pure state $|\Psi\rangle \in \mathrm{ran}(\hat{P}^{(z)}_{+})$.\
Consider three contexts $\Sigma^{(1)}$, $\Sigma^{(2)}$ and $\Sigma^{(6)}$ on the Hilbert space $\mathbb{C}^4$ used in the paper [@Cabello] by Cabello et al. to prove the Bell-Kochen-Specker theorem. The projection operators composing these contexts are
$$ \hat{P}^{(1)}_1
=
\!\left[
\begingroup\SmallColSep
\begin{array}{r r r r}
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 1
\end{array}
\endgroup
\right]
\,
,
\,\,
\hat{P}^{(1)}_2
=
\!\left[
\begingroup\SmallColSep
\begin{array}{r r r r}
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 0
\end{array}
\endgroup
\right]
\,
,
\,\,
\hat{P}^{(1)}_3
=
\!
\frac{1}{2}
\!\left[
\begingroup\SmallColSep
\begin{array}{r r r r}
1 & 1 & 0 & 0 \\
1 & 1 & 0 & 0 \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0
\end{array}
\endgroup
\right]
\,
,
\,\,
\hat{P}^{(1)}_4
=
\!
\frac{1}{2}
\!\left[
\begingroup\SmallColSep
\begin{array}{r r r r}
1 & \bar{1} & 0 & 0 \\
\bar{1} & 1 & 0 & 0 \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0
\end{array}
\endgroup
\right]
\;\;\;\; ,$$
$$ \hat{P}^{(2)}_1
=
\!\left[
\begingroup\SmallColSep
\begin{array}{r r r r}
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 1
\end{array}
\endgroup
\right]
\,
,
\,\,
\hat{P}^{(2)}_2
=
\!\left[
\begingroup\SmallColSep
\begin{array}{r r r r}
0 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0
\end{array}
\endgroup
\right]
\,
,
\,\,
\hat{P}^{(2)}_3
=
\!
\frac{1}{2}
\!\left[
\begingroup\SmallColSep
\begin{array}{r r r r}
1 & 0 & 1 & 0 \\
0 & 0 & 0 & 0 \\
1 & 0 & 1 & 0 \\
0 & 0 & 0 & 0
\end{array}
\endgroup
\right]
\,
,
\,\,
\hat{P}^{(2)}_4
=
\!
\frac{1}{2}
\!\left[
\begingroup\SmallColSep
\begin{array}{r r r r}
1 & 0 & \bar{1} & 0 \\
0 & 0 & 0 & 0 \\
\bar{1} & 0 & 1 & 0 \\
0 & 0 & 0 & 0
\end{array}
\endgroup
\right]
\;\;\;\; ,$$
$$ \hat{P}^{(6)}_1
=
\!
\frac{1}{4}
\!\left[
\begingroup\SmallColSep
\begin{array}{r r r r}
1 & \bar{1} & \bar{1} & 1 \\
\bar{1} & 1 & 1 & \bar{1} \\
\bar{1} & 1 & 1 & \bar{1} \\
1 & \bar{1} & \bar{1} & 1
\end{array}
\endgroup
\right]
\,
,
\,\,
\hat{P}^{(6)}_2
=
\!
\frac{1}{4}
\!\left[
\begingroup\SmallColSep
\begin{array}{r r r r}
1 & 1 & 1 & 1 \\
1 & 1 & 1 & 1 \\
1 & 1 & 1 & 1 \\
1 & 1 & 1 & 1
\end{array}
\endgroup
\right]
\,
,
\,\,
\hat{P}^{(6)}_3
=
\!
\frac{1}{2}
\!\left[
\begingroup\SmallColSep
\begin{array}{r r r r}
1 & 0 & 0 & \bar{1} \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 \\
\bar{1} & 0 & 0 & 1
\end{array}
\endgroup
\right]
\,
,
\,\,
\hat{P}^{(6)}_4
=
\!
\frac{1}{2}
\!\left[
\begingroup\SmallColSep
\begin{array}{r r r r}
0 & 0 & 0 & 0 \\
0 & 1 & \bar{1} & 0 \\
0 & \bar{1} & 1 & 0 \\
0 & 0 & 0 & 0
\end{array}
\endgroup
\right]
\;\;\;\; ,$$
where $\bar{1}$ stands for $-1$.\
![The bivaluation relation in the invariant-subspace lattice $\mathcal{L}(\Sigma^{(1)})$ of Cabello’s set\[fig3\]](Figure3.eps)
![The bivaluation relation in the invariant-subspace lattice $\mathcal{L}(\Sigma^{(2)})$ of Cabello’s set\[fig4\]](Figure4.eps)
![The bivaluation relation in the invariant-subspace lattice $\mathcal{L}(\Sigma^{(6)})$ of Cabello’s set\[fig5\]](Figure5.eps)
The Figures \[fig3\], \[fig4\] and \[fig5\] show the Hasse diagrams of the invariant-subspace lattices based on the closed subspaces generated by those projection operators in addition to the truth values of the relating propositions given in the pure state $|\Psi\rangle \in \mathrm{ran}(\hat{P}^{(1)}_1)$. In the Figures \[fig3\]-\[fig5\], the following notations are used: “1” denotes $\mathrm{ran}(\hat{P}^{(1)}_1)$ (or $\mathrm{ran}(\hat{P}^{(2)}_1)$ or $\mathrm{ran}(\hat{P}^{(6)}_1)$, depending on the lattice), “1+2” denotes $\mathrm{ran}(\hat{P}^{(1)}_1 + \hat{P}^{(1)}_2)$, “1+2+3” denotes $\mathrm{ran}(\neg \hat{P}^{(1)}_4) = \mathrm{ran}(\hat{P}^{(1)}_1 + \hat{P}^{(1)}_2 + \hat{P}^{(1)}_3)$, and so on.\
It is obvious that $\mathrm{ran}(\hat{P}^{(2)}_1)$ and $\mathrm{ker}(\hat{P}^{(2)}_1) = \mathrm{ran}(\neg\hat{P}^{(2)}_1) = \mathrm{ran}(\hat{P}^{(2)}_2 + \hat{P}^{(2)}_3 + \hat{P}^{(2)}_4)$ are the shared closed subspaces belonging to both invariant-subspace lattices $\mathcal{L}(\Sigma^{(2)})$ and $\mathcal{L}(\Sigma^{(1)})$.\
What is more, the kernel of the projection operator $\hat{P}^{(6)}_4$ is the invariant subspace under any projection operator from the context $\Sigma^{(1)}$. To be sure,
$$ |\Psi\rangle
=
\!
\!\left[
\!
\begingroup\SmallColSep
\begin{array}{r}
0 \\
0 \\
0 \\
1
\end{array}
\endgroup
\!
\right]
\in
\mathrm{ker}(\hat{P}^{(6)}_4)
=
\!
\!\left\{
\!
\left[
\!
\begingroup\SmallColSep
\begin{array}{r}
c \\
b \\
b \\
a
\end{array}
\endgroup
\!
\right]
\!
\right\}
\,
\implies \,
\left\{ \begin{array}{r}
\hat{P}^{(1)}_1 |\Psi\rangle
=
\!\left[
\!
\begingroup\SmallColSep
\begin{array}{r}
0 \\
0 \\
0 \\
1
\end{array}
\endgroup
\!
\right]
\in
\!
\left\{
\!
\left[
\!
\begingroup\SmallColSep
\begin{array}{r}
c \\
b \\
b \\
a
\end{array}
\endgroup
\!
\right]
\!
\right\}
\\ \hat{P}^{(1)}_{m \neq 1} |\Psi\rangle
=
\!\left[
\!
\begingroup\SmallColSep
\begin{array}{r}
0 \\
0 \\
0 \\
0
\end{array}
\endgroup
\!
\right]
\in
\!
\left\{
\!
\left[
\!
\begingroup\SmallColSep
\begin{array}{r}
c \\
b \\
b \\
a
\end{array}
\endgroup
\!
\right]
\!
\right\}
\end{array}
\right.
\;\;\;\; ,$$
where $a,b,c \in \mathbb{R}$. Hence, $\mathrm{ker}(\hat{P}^{(6)}_4) = \mathrm{ran}(\hat{P}^{(6)}_1 + \hat{P}^{(6)}_2 + \hat{P}^{(6)}_3)$ belongs to both invariant-subspace lattices $\mathcal{L}(\Sigma^{(6)})$ and $\mathcal{L}(\Sigma^{(1)})$. The ordering of $\mathrm{ker}(\hat{P}^{(6)}_4)$ in $\mathcal{L}(\Sigma^{(1)})$ is as follows:
$$ \mathrm{ran}(\hat{P}^{(1)}_1)
\cap
\mathrm{ker}(\hat{P}^{(6)}_4)
=
\!
\!\left\{
\!
\left[
\!
\begingroup\SmallColSep
\begin{array}{r}
0 \\
0 \\
0 \\
a
\end{array}
\endgroup
\!
\right]
\!
\right\}
\cap \!\left\{
\!
\left[
\!
\begingroup\SmallColSep
\begin{array}{r}
c \\
b \\
b \\
a
\end{array}
\endgroup
\!
\right]
\!
\right\}
\!
=
\!
\!\left\{
\!
\left[
\!
\begingroup\SmallColSep
\begin{array}{r}
0 \\
0 \\
0 \\
a
\end{array}
\endgroup
\!
\right]
\!
\right\}
\,
\implies \,
\mathrm{ran}(\hat{P}^{(1)}_1)
\subseteq
\mathrm{ker}(\hat{P}^{(6)}_4)
\;\;\;\; .$$
It can be also shown that $\mathrm{ker}(\hat{P}^{(6)}_4) \in \mathcal{L}(\Sigma^{(2)})$ with the similar ordering.\
The text boxes of the shared nontrivial subspaces are colored grey in the diagrams.\
Conclusion remarks
==================
As it can be seen from the above diagrams, all the propositions represented by the subspaces that do not belong to the invariant-subspace lattices allocated by the state, in which the quantum system is prepared, are undetermined.\
This implies that the admissibility rules do not hold true in the structure of the invariant-subspace lattices imposed on the closed linear subspaces in the Hilbert space.\
Recall that according to the admissibility rules [@Abbott; @Svozil], (1) if among all the subspaces belonging to a context there is one that represents a true proposition, then the others must represent false propositions; what is more, (2) if there is a subspace belonging to a context such that it represents a false proposition, then another subspace in this context must represent either false or true proposition (as long as only one proposition is true).\
One can easily check with the presented above diagrams that the second admissibility rule holds true within the structure of the Hilbert sublattice $\mathcal{L}(\Sigma)$ but it does not apply to the structure of the invariant-subspace lattices.\
Accordingly, the statement of the Kochen-Specker theorem (expressing that there is no way to assign a bivalent truth value to every quantum proposition relating to a given context such that the admissibility rules hold true [@Kochen; @Peres]) need not to be proved in supervaluational logic since it is immediately evident in that logic.\
[^1]: $Email: arkadyv@bgu.ac.il$
|
---
abstract: 'The first direct experimental evidence for the Fermi surface (FS) driving the helical antiferromagnetic ordering in a gadolinium-yttrium alloy is reported. The presence of a FS sheet capable of nesting is revealed, and the nesting vector associated with the sheet is found to be in excellent agreement with the periodicity of the helical ordering.'
address:
- '$^{1}$H.H. Wills Physics Laboratory, University of Bristol, Tyndall Avenue, Bristol BS8 1TL, United Kingdom'
- '$^{2}$Department of Physics, University of Illinois at Chicago, Chicago, IL 60607, USA'
- '$^{3}$Département de Physique de la Matière Condensée, Université de Genève, 24 quai Ernest Ansermet, CH-1211 Genève 4, Switzerland'
- '$^{4}$Department of Physics, University of Warwick, Coventry CV4 7AL, United Kingdom'
author:
- 'H.M. Fretwell,$^{1,2}$ S.B. Dugdale,$^{1,3}$ M.A. Alam,$^{1}$ D.C.R. Hedley,$^{1}$ A. Rodriguez-Gonzalez,$^{1}$ and S.B. Palmer$^{4}$'
title: 'Fermi Surface as the Driving Mechanism for Helical Antiferromagnetic Ordering in Gd-Y Alloys'
---
A considerable body of theoretical and indirect experimental evidence indicates that the geometry of the Fermi surface (FS) drives a variety of ordering phenomena ; these include exotic magnetic ordering in the rare earths and their alloys [@kasuya:66], compositional ordering concentration density waves in binary alloys [@gyorffy:83], and magneto-oscillatory coupling in magnetic multilayers separated by non-magnetic spacer layers [@parkin:91]. Current theoretical understanding suggests that the ordering is governed by the “nesting” of specific sheets of FS in the disordered state. Nesting describes the coincidence of two approximately parallel FS sheets when translated by some distance in ${\bf k}$-space (i.e. by the “nesting vector”, $\bf Q$). In the presence of nesting, the disordered phase becomes unstable to an ordering modulation whose period is inversely proportional to the relevant nesting vector. However, in many cases the relevant features of the FS have never been directly observed.
Despite intense current interest in the concept of FS nesting as the driving force for modulating magnetic structures, there is a distinct dearth of direct experimental information about FS topologies in the relevant materials. This scarcity has mainly been due to the lack of a suitable technique. With recent developments in positron annihilation Fermiology (see e.g. [@west:95]), such studies are now possible. In this Letter, we provide the first direct evidence that a Gd-Y alloy which orders antiferromagnetically does indeed contain a FS sheet that has nesting properties. We compare the nesting vector directly calipered from the FS topology with those inferred via neutron diffraction on specimens from the same batch. We also provide preliminary discussion of our results in terms of the temperature dependence of the nesting vector in the helical phase and effects of the transitions on this nested FS.
A classic example of such a FS-driven ordering is the helical antiferromagnetic ordering in many of the heavier rare earths (e.g. Tb, Dy, Ho, Er). Here, the magnetic moments align in the basal planes with a rotation of the moment vectors in successive planes with a periodicity that is predicted to arise from the FS topology. The ordering has its origin in the coupling of the localised 4$f$ moments via the Ruderman-Kittel-Kasuya-Yosida (RKKY) indirect exchange interaction involving the conduction electrons [@kasuya:66]. The connection between the conduction electrons’ ability to establish magnetic order and the FS of the disordered paramagnetic state is most easily understood in terms of the wave-vector ($\bf q$) dependent susceptibility, $$\begin{aligned}
\chi(\bbox{q}) \propto
\sum_{\bbox{k},j,j'} \frac{\vert M(\bbox{k},\bbox{k+q}) \vert^{2}
f_{kj}(1-f_{k+q+Gj'})}
{\epsilon_{j'}(\bbox{k}+\bbox{q}+\bbox{G})-
\epsilon_{j}(\bbox{k})}.
\label{suscept}\end{aligned}$$ Here $M$ are the matrix elements involving the conduction and $f$ electron wave functions, $f_{kj}$ are the Fermi–Dirac distribution functions for reduced wave-vector ${\bf k}$ and band $j$, $\epsilon_{j}(\bbox{k})$ are the single particle energies and ${\bf G}$ is a reciprocal lattice vector. Subject to other constraints, the maximum in $\chi(\bbox{q})$ determines the most stable magnetic structure. If the maximum in $\chi(\bbox{q})$ occurs at ${\bf q}=0$, the material orders ferromagnetically. If the maximum occurs at a non-zero ${\bf q} = {\bf Q}$, then a more complex arrangement of the magnetic moments, such as helical antiferromagnetic order, takes place. The latter may happen if there are large parallel sections of FS to guarantee a sufficient number of terms in the sum with vanishingly small denominators at the nesting vector ${\bf Q}$. A so-called “webbing” feature [@loucks:66; @mattocks:78; @dugdale:97] in the FS of most of the rare earths provides the required parallel surfaces for nesting which drives the magnetic ordering.
Helical antiferromagnetic ordering is also observed in Gd-Y alloys in a certain composition and temperature range. The Gd FS does not contain the webbing feature [@west:98] and Gd orders ferromagnetically below 293K. The transition metal Y, on the other hand, possesses a strong webbing sheet [@loucks:66; @mattocks:78; @dugdale:97] but owing to the lack of magnetic moments is a paramagnet at all temperatures. Addition of small amounts (of the order of 0.5 at.%) of Tb [@child:68; @wakabayashi:74] or Er [@caudron:90] leads to the appearance of long range magnetic structures with ${\bf Q}$ vectors close to the nesting vector in the webbing feature in Y. The helical phase observed in a range of Gd-rich Gd-Y alloys is assumed to arise from a combination of magnetic moments contributed by Gd and the Y-induced nesting character of the FS. Gd-Y provides an ideal alloy system to study the generic magnetic behaviour in the rare earths because of the availability of good quality single crystal samples of sufficient size. Further, the concentration versus temperature magnetic phase diagram in these alloys is well established and is rich in interesting features, many of which are common to other rare earths [@bates:85; @melville:92]. In the concentration range of 30—40 at.% Y, the alloy shows a helical phase where the helix periodicity, often quoted as the interlayer turn angle of the basal plane moment vector, shows a reversible decrease with temperature. This may imply a temperature-dependent nesting vector (i.e. $T$-dependent changes in the webbing FS sheet). In the concentration-temperature regime which supports the antiferromagnetism, the application of a modest magnetic field (a few times 10$^{-2}$ tesla) along the $c$-axis leads to a ferromagnetic alignment of the moments along this axis. The specimen reverts to its antiferromagnetic state once the magnetic field is switched off. In addition to the suggestion that the Fermi surface of the magnetically disordered paramagnetic phase drives the helical ordering, there is the issue of the effect of helical ordering on the electron energy bands and therefore the impact on the FS itself.
The measurements of the FS topology were conducted via the so-called 2-Dimensional Angular Correlation of electron-positron Annihilation Radiation (2D-ACAR). A 2D-ACAR measurement yields a 2D projection (integration over one dimension) of the underlying two-photon momentum density, $\rho(\bbox{p})$. Within the independent particle model, $$\begin{aligned}
\rho(\bbox{p}) &=& \sum_{\bbox{k},j} \vert \int
d\bbox{r}\psi_{k,j}(\bbox{r})\psi_{+}(\bbox{r}) \exp (-
i\bbox{p}.\bbox{r})\vert ^{2} \nonumber\\
&=& \sum_{j,\bbox{k},\bbox{G}}
n^{j}(\bbox{k}) \vert C_{\bbox{G},j}(\bbox{k}) \vert ^{2}
\delta(\bbox{p}-\bbox{k}-\bbox{G}),\end{aligned}$$ where $\psi_{k,j}(\bbox{r})$ and $\psi_{+}(\bbox{r})$ are the electron and positron wave functions, respectively, and $n^{j}(\bbox{k})$ is the electron occupation density in ${\bf k}$-space in the $j^{\mbox{th}}$ band. The $C_{\bbox{G},j}(\bbox{k})$ are the Fourier coefficients of the electron-positron wave function product and the delta function expresses the conservation of crystal momentum. $\rho(\bbox{p})$ contains information about the occupied electron states and their momentum $\bbox{p} = \hbar
(\bbox{k +G})$ and the FS is reflected in the discontinuity in this occupancy at the Fermi momentum $\bbox{p_{F}} = \hbar (\bbox{k_{F} +
G})$. As already pointed out, the 2D-ACAR spectra represent projections of $\rho(\bbox{p})$, and the full 3D density can be reconstructed using tomographic techniques from a small number of projections with integrations along different crystallographic directions [@cormack:63; @cormack:64; @sznajd:90]. Finally, if the effects of the positron wave function (Eq. 2) are small such that the $C_{\bbox{G},j}(\bbox{k})$ are almost independent of ${\bf k}$, the full 3D ${\bf k}$-space occupation density can be obtained by folding back the 3D $\rho(\bbox{p})$ into the first Brillouin zone (BZ) according to the so-called LCW prescription [@lock:73]. By these means one is able to directly ‘image’ the FS in its full 3D form.
The sample under investigation was a Gd$_{0.62}$Y$_{0.38}$ single crystal which undergoes a helical antiferromagnetic transition below $\sim$200K where the nature of the helical phase and its periodicity has been extensively studied by one of us [@melville:92]. As part of a comprehensive programme, we have also studied the FS topology of the two pure elements Gd [@west:98; @dugdale:96] and Y [@dugdale:97; @dugdale:96]. In each case, the 3D $\rho(\bbox{p})$ and subsequently the 3D ${\bf k}$-space occupancy were reconstructed from five projections measured at 7.5 degree intervals encompassing the 30 degrees between the directions and (in the irreducible wedge of the hexagonal BZ). Following the usual processing of the measured 2D-ACAR spectra [@west:95], they were deconvoluted using a ‘Maximum Entropy’-based (MaxEnt) procedure [@dugdale:94] to suppress the unwanted smearing due to experimental resolution. The 3D $\rho(\bbox{p})$ was reconstructed from the measured projections (both raw and deconvoluted, henceforth referred to as ‘raw’ and ‘MaxEnt’) using Cormack’s method [@cormack:63; @cormack:64; @sznajd:90] before finally being folded back into the first BZ. The reconstruction method exploits the crystallographic symmetry which allows the reconstruction from only a few projections and has been rigorously tested [@sznajd:90; @dugdale:96] to show that no artefacts are introduced into the data. A more detailed description of the procedures used here can be found in [@sznajd:90] and further references therein. Using a threshold criterion to differentiate between the empty and occupied states [@manuel:82; @dugdale:97] it is possible to image the FS alone.
In Figure \[FS3d\], we show FS images of: (a) the calculated FS for Y [@loucks:66], where the webbing feature and the associated nesting vector, ${\bf Q_{0}}$, is marked by the double arrow; (b) the measured Fermi surface of Y [@dugdale:97] clearly showing the webbing feature; (c) the measured FS of Gd [@west:98; @dugdale:96] with the distinct lack of the webbing feature as expected from its ferromagnetic ordering (d) the current measurement of the alloy Gd$_{0.62}$Y$_{0.38}$ (in the disordered paramagnetic phase) where the nesting feature is clearly visible. The presented experimental FS images are extracted from MaxEnt deconvoluted data. It is noteworthy that in the paramagnetic phase of the Gd-rich alloy, the webbing feature is remarkably similar to that observed in pure Y. The fact that the FS of pure Gd does not show the webbing feature while that of the alloy has a strong nesting character clearly indicates that its helical ordering is driven by the nesting of this sheet.
We now investigate the webbing in greater detail. Figure \[hlmk\_raw\] shows the section through the webbing in the alloy sample (Fig. \[FS3d\]d) in the plane (on the face of the BZ in Fig. 1a). Here, lows (holes) are shown as black and highs (electrons) as white and the webbing is represented by the central region of holes. If the raw and Maxent reconstructions are normalised to contain the same number of electrons within the BZ and the raw reconstruction is subtracted from the Maxent reconstruction, the distribution shown in Figure \[hlmk\_diff\] results. This procedure amounts to an edge-enhancement that highlights the Fermi edges [@dugdale:94; @dugdale:96]. It enhances the discontinuity at the FS because it is in the vicinity of these discontinuities that the resolution function has its greatest smearing effect. Thus, the difference spectrum amplifies the signature of the Fermi edge.
Dugdale [*et al.*]{} [@dugdale:94] proposed that the locus of points where such a difference spectrum passes through zero defines the Fermi surface, providing an accurate method for calipering it. The zero crossing contour of Figure \[hlmk\_diff\] is shown in Figure \[hlmk\_zero\] which again clearly shows the yttrium-like webbing feature and its nesting properties. We estimate the width of the webbing parallel to the $c$-axis as $0.53 \pm 0.02 \times \left( \pi \over c \right)$, a value remarkably close to the nesting feature in pure Y of $0.55 \pm 0.02 \times \left( \pi
\over c \right)$ [@dugdale:97]. This would give rise to a period of helicity which corresponds to an inter-plane turn angle of $47.7 \pm 1.6$ degrees between the orientations of the magnetic moments in successive basal planes. As mentioned earlier, Bates [*et al.*]{} [@bates:85] measured the turn angle in the helical phase in a sample derived from the same ingot via neutron diffraction by inspecting the distance between the magnetic satellite Bragg peaks on either side of the nuclear peak along the $c$-axis. These results showed that the measured turn angles were strongly $T$-dependent, increasing linearly with temperature. The turn angles cannot be measured in the paramagnetic phase through neutron diffraction owing to the absence of the magnetic ordering and therefore the lack of the satellite peaks. However, a linear interpolation of the $T$-dependence of the turn angle returns a value of 48 degrees at 295K (the temperature of our measurement) which is in excellent agreement with the value obtained in our experiment.
Finally, it is worth mentioning our recent preliminary results from the same alloy sample measured at $T=140$K, well within the thermodynamic helical phase. In our experimental set-up, we use a magnetic field of $\sim$0.8T to focus the positrons on to the specimen. The field is applied parallel to the $c$-direction of the sample. It was not possible to carry out the measurements without the magnetic field owing to the small dimensions of the sample, as a substantial fraction of the defocused positrons annihilated in the sample holder and associated goniometer. As noted above, the applied field would force the magnetic moments to align along the easy axis giving rise to a $c$-axis ferromagnetic state. Our preliminary analysis of the FS of the alloy sample in this ferromagnetic state reconstructed from 3 projections again shows (figure not shown owing to lack of space) distinct indication of the webbing feature being present.
If the periodic antiferromagnetic ordering introduced superzone boundaries in the lattice, these may have distorted the FS feature in the webbing sheet. However, in our low temperature data, such degeneracy would have been lifted by the forced ferromagnetic ordering. If this is the case and the relevant energy bands are not significantly affected, then one would expect the webbing feature in the FS to be retained. Although theoretical calculations are necessary to confirm this, it is a reasonable assumption.
In conclusion, we have shown for the first time the exisitence of a nesting FS in a Gd-Y alloy ; this confirms theoretical predictions that the FS topology in the paramagnetic state is responsible for the antiferromagnetic ordering of the alloy.
The authors would like the thank the EPSRC (UK) for financial support. One of us (SBD) is grateful to the Royal Society (UK) and the Swiss National Science Foundation for the provision of a Research Fellowship.
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---
bibliography:
- 'nl4.bib'
---
SPIN-07/47\
ITP-UU-07/61
[**A note on non-linear electrodynamics, regular black holes and the entropy function.**]{}\
**Kevin Goldstein and Hossein Yavartanoo**\
[$^{1}$*Institute for Theoretical Physics,\
Utrecht University, Utrecht, The Netherlands*]{}\
$^{2}$ [*Center for Theoretical Physics and BK-21 Frontier Physics Division,\
Seoul National University, Seoul 151-747 KOREA*]{}
**Abstract**
We examine four dimensional magnetically charged extremal black holes in certain non-linear $U(1)$ gauge theories coupled to two derivative gravity. For a given coupling, one can tune the magnetic charge (or vice versa) so that the curvature singularity at the centre of the space-time is cancelled. Since these solutions have a horizon but no singularity, they have been called regular black holes. Contrary to recent claims in the literature, we find that the entropy function formalism reproduces the near horizon geometry and gives the correct entropy for these objects.
Introduction {#intro}
============
The Penrose cosmic censorship hypothesis states that, if singularities predicted by General Relativity occur in nature, they must be dressed by event horizons [@Hawking:1973uf]. Behind the veil of an event horizon, there is no causal contact from the interior to the exterior of a black hole, so the pathologies occurring at the singular region can have no influence on an external observer. However, the converse of the hypothesis is apparently not true — a horizon does not necessarily hide a singularity. Solutions with a horizon but no singularity have been called regular black holes.
The holographic principle, [@gr-qc/9310026; @9409089], states that the number of degrees of freedom describing the black hole is bounded by the area of the horizon. A stronger statement is that degrees of freedom living on the horizon can describe the physics of the interior completely. While the holographic principle is essentially a proposed feature of quantum gravity, one might wonder whether having a classically regular or singular solution has any quantitative or qualitative effect on the entropy and the physics at the horizon.
Earlier work on regular black hole models can be found in [@bardeen1968pg; @Barrabes:1995nk; @gr-qc/9403049; @Mars:1996np; @Cabo:1997rm] These regular solutions are referred to as “Bardeen black holes” [@Borde:1996df]. In addition, regular black hole solutions to Einstein equations with various physical sources were reported in [@AyonBeato:1998ub] and [@Magli:1997mw]. Among known regular black hole solutions, are the solutions to the coupled equations of nonlinear electrodynamics and general relativity found by Ayón-Beato and Garciá [@hep-th/9911174] and by Bronnikov [@gr-qc/0006014]. The latter describes magnetically charged black hole, and provides an interesting example of the system that could be both regular and extremal. In this note we are specially interested in the near horizon geometry of an extremal magnetically charged black hole non-linearly coupled to a $U(1)$ gauge field. For a given magnetic charge, one can tune the non-linear coupling so that the solution is regular.
In [@gr-qc/0403109], Matyjasek found the near horizon, $AdS_2\times S^2$ geometry of a particular magnetically charged extremal black hole. The entropy function formalism of Sen [@hep-th/0506177; @hep-th/0508042], is particularity useful for discussing the entropy of extremal black holes especially when non-linear or high derivative terms make a full analysis difficult. Since, the formalism is equivalent to solving Einsteins equations for the near horizon region, a priori, and assuming the near horizon geometry decouples, one would expect to be able to reproduce the results of [@gr-qc/0403109] using Sen’s approach. This issue has been studied recently, [@0705.2478; @0707.1933], and authors reported that, even at the level of two derivative gravity, the entropy function approach does not lead to the correct Bekenstein-Hawking. To account for this discrepancy, they claim that the entropy function approach is sensitive to whether the nature of the central region of the black hole is regular (linear) or singular (nonlinear).
Contrary to the claims of, [@0705.2478; @0707.1933], in this note we find that a straight forward application of the entropy function formalism reproduces the results of [@gr-qc/0403109]. The equation of motion derived from extremizing the entropy function are exactly the same as equation of motion at horizon found by extremizing the action, since the entropy function (up to Legendre transformation) is the Lagrangian at the horizon. The fact that the entropy is the value of entropy function at its extremum is derived from the Wald entropy formula, using the near horizon symmetries. Both of these results, just coming from careful consideration of the near horizon symmetries and have nothing to do with the regularity of the solution inside the horizon. Further more, we find that by varying the non-linear coupling, the regular solution can be smoothly connected to the extremal Reisner-Nordstrom solution of Einstein-Maxwell theory.
The paper is organised as follows. In section \[rbh\] we review a particular regular black hole solution of interest. Then, in section \[sec:ent\], we review the entropy function formalism and apply it to Einstein gravity coupled to non-linear electrodynamics. In section \[sec:rbh:E\] we consider the special case of the formalism applied to a regular black hole solution. Finally we end with the conclusion in section \[sec:conc\], having relegated some technical details about the large charge, small coupling expansion of the entropy to appendix \[sec:a1\].
Regular Black holes {#rbh}
===================
In this section we review a magnetically charged regular black hole solution of Einstein gravity coupled to non-linear electrodynamics and its extremal limit [@hep-th/9911174; @gr-qc/0006014; @gr-qc/0403109; @hep-th/0606185], mainly following [@gr-qc/0403109] with slightly different notation.
We consider an action given by, $$\label{Lag}
S= \frac{1}{16\pi} \int d^4x \sqrt{-g}({R} - {\mathcal L}_{F}(F^2))\; ,$$ where $F^2= F_{\mu\nu}F^{\mu\nu}$ and the non-linear $U(1)$ gauge field Lagrangian, ${\mathcal L}_{F}$, is,[^1] $$\label{lm}
{\mathcal L}_{F} = F^2 \cosh^{-2}\left((\lambda^{2}F^2/2)^{1/4} \right) \;.$$
The equations of motion corresponding to the metric and gauge field and the Bianchi identity are, $$\begin{aligned}
&& R_{\mu\nu} -\frac{1}{2} g_{\mu\nu} R
= \frac{\partial {\mathcal L}_{F}}{\partial (F^{2})}2 F_{\mu\lambda}F_{\nu}^{\phantom{\nu}\lambda}
-\frac{1}{2} {\mathcal L}_{F} g_{\mu\nu} \; , \\
&& \partial_{\mu} \left(\sqrt{-g} \frac{\partial {\mathcal L}_{F}}{\partial (F^{2})} F^{\mu\nu}\right) =0 \; , \\
&& \partial_{[\mu}F_{\alpha\beta]} =0 \;.\end{aligned}$$ For a magnetically charged black hole, the equation of motion for the gauge field and the Bianchi identity can be solved by, $$\label{MF}
F_{\theta\phi} = P \sin\theta,$$ where $P$ is the magnetic charge of the black hole. A static, spherically symmetric ansatz for the metric: $$ds^2= -a^{2}(r) dt^2 + \frac{dr^2}{a^{2}(r)} + r^2 d\Omega_2^2\; ,$$ can solve Einstein equations with, $$a^{2}(r) = 1-\frac{2m(r)}{r} \; ,$$ where, $$m(r)= m_\infty- \frac{|P|}{2|\lambda/P|^{1/2}} \tanh \frac{ |\lambda/P|^{1/2}}{r/|P|}.$$ The parameter, $m_\infty$, is an integration constant which can be fixed by employing the boundary condition $m(\infty)=M$, where $M$ is the black hole mass. Moreover demanding of the regularity of the line element as $r\rightarrow 0 $, yields, $$\label{reg:cond}
M=\frac{|P|}{2|\lambda/P|^{1/2}},$$ and consequently, $m(r)$ reads, $$m(r)=M\left(1-\tanh \frac{P^2}{2Mr}\right)
= \frac{|P|}{2|\lambda/P|^{1/2}}\left(1- \tanh \frac{ |\lambda/P|^{1/2}}{r/|P|}\right) \;.$$ The location of the inner and outer horizons, $r_{\pm}$, which are given by equation $a(r)=0$, can be expressed in terms of the real branches of the Lambert function, $W_i(x)$, as follows, $$\frac{r_+}{M} = -\frac{p^2}{W_0(-e^{p^2/4} p^2/4 )-p^2/4} \; ,
\quad \frac{r_-}{M} = -\frac{p^2}{W_{-1}(-e^{p^2/4} p^2/4 )-p^2/4}\;,$$ where, $p = P/M$, is the magnetic charge-to-mass ratio. The Lambert function[^2], is defined by the formula, $$\label{lambf}
e^{W(x)}W(x)=x \;.$$ This function has two real branches, called $W_0$ and $W_{-1}$, with the branch point at $x=-1/e$. Since the value of the principal branch of the Lambert function, $W_{0}$, at $1/e$, plays an important role in our discussion, we define $w_0 = W_{0}(1/e)$.
When $p =p_{ext} = 2 w_0^{1/2}$, $r_{+}=r_{-}$, and the two horizons merge into a degenerate horizon giving an extremal solution. Since we will be considering the near horizon geometry, we will eliminate the mass from our formulae as it is defined asymptotically. Using (\[reg:cond\]) we can express the condition for extremality and regularity, $p_{ext} = 2 w_0^{1/2}$, as, $$\label{eq:reg:cond2}
\frac{\lambda}{P} = w_{0}.$$ In other words for an extremal black hole to be regular we must tune the charge to coupling ratio to a particular value. A generic extremal, but not necessarily regular, solution to (\[Lag\]) will still have a degenerate horizon but presumably with a different charge to coupling ratio.
One can write the near horizon limit, found by [@gr-qc/0403109] as: $$\label{nh1}
ds^2= v_1\left(-\rho^2 dt^2 + \frac{d\rho^2}{\rho^2}\right) + v_2 \left(d\theta^2 + \sin^2\theta d\phi^2\right)\;.$$ with, $$\begin{aligned}
\label{result:rbh}
v_{2}&= \frac{4w_{0}}{(1+w_{0})^{2}}P^{2}\approx 0.68 P^{2}\;,\\
v_{1}&= \frac{8w_{0}}{(1+w_{0})^{3}}P^{2}\approx 1.07 P^{2}\;,\\
\frac{v_{2}}{v_{1}}&=\frac{1}{2}(1+w_{0})\approx 0.64\;,\end{aligned}$$ and the Bekenstein-Hawking entropy is, $$S_{BH}=\tfrac{1}{4}A=\pi v_{2}=\frac{4\pi w_{0}}{(1+w_{0})^{2}}P^{2}\;.$$
Entropy function Analysis {#sec:ent}
=========================
In this section we briefly review the entropy function formalism of Sen [@hep-th/0506177; @hep-th/0508042] and subsequently apply it to magnetically charged extremal solutions of (\[Lag\]).
Assuming a gauge and diffeomorphism invariant Lagrangian and a near horizon $AdS_{2}\times S^{2}$ geometry, the entropy function is defined as the Legendre transform, with respect to the electric charges, of the reduced Lagrangian evaluated at the horizon: $$\begin{aligned}
\label{EF}
{\mathcal E}({\vec u},{\vec v},{\vec Q}, {\vec P})
= 2\pi\bigg(e^{i}Q_{i}-f({\vec u},{\vec v}, {\vec e},{\vec P})\bigg)
= 2\pi\bigg(e^{i}Q_{i}-\int_{H}
d\theta d\varphi \sqrt{-G}{\cal L}\bigg), \end{aligned}$$ where ${e^{i}}$ are the electric fields, $Q_{i}=\partial f/\partial e^{i}$, are the electric charges conjugate to the electric field, $\vec u$ are the values of the scalar moduli at the horizon, and $v_1$, $v_2$ are the sizes of the $AdS_2$ and $S^2$, respectively. The near horizon equations of motion for a black hole carrying electric charges $\vec Q$ and magnetic charges $\vec P$, are equivalent to the extremisation of ${\cal E}$ with respect to $\vec u,\vec v$ and $\vec e$: $$\begin{aligned}
\frac{\partial {\mathcal E}}{\partial \vec u}=0\,, \qquad
\frac{\partial {\mathcal E}}{\partial v_i}=0\,, \qquad
\frac{\partial {\mathcal E}}{\partial \vec e}=0\,.
\label{attractor}\end{aligned}$$
Furthermore, the Wald entropy associated with the black hole is given by ${\mathcal E}$ at the extremum (\[attractor\]). If ${\mathcal E}$ has no flat directions, then the extremization of ${\mathcal E}$ determines ${\vec u}$, ${v_i}$, and ${\vec e}$, in terms of ${Q}$ and ${P}$. The extremal value of the Wald entropy, $S={\mathcal E}(\vec Q,\vec P)|_{extr}$, is independent of the asymptotic values of the scalar fields. This neatly demonstrates the attractor mechanism, [@Ferrara:1995ih; @Strominger:1996kf; @Ferrara:1996dd], with out requiring supersymmetry [@9702103]. The formalism can even be extended to rotating black holes which have less near horizon symmetry [@0606244]. However, since it only involves the near horizon geometry, a weakness of the formalism is that one implicitly assumes that the full solution exists, which is not always the case [@0507096].
We now specialise our discussion to the case of interest. Since the regular black hole solution has an extremal limit, one can use the entropy function formalism to find the near horizon geometry and the entropy. We take the near horizon $AdS_{2}\times S^{2}$ metric to be give by (\[nh1\]). From the definition (\[EF\]), using the Lagrangian (\[Lag\]), the entropy function evaluates to, $$\label{e}
{\cal E} = \pi
\left(
v_{2}-v_{1}+\tfrac{1}{2}v_{1}v_{2}{\cal L}_{F}(2P^{2}/v_{2}^{2})
\right)\; .$$ By extremizing this entropy function with respect to $v_1$ and $v_2$, we find following equations, $$\begin{aligned}
\label{eom1:0}
0&=-1+\frac{1}{2}v_{2}{\cal L}_{F}(2P^{2}/v_{2}^{2})\;,\\
0&=1 +\frac{1}{2}v_{1}\frac{{\partial}}{{\partial}v_{2}}\left[v_{2}{\cal L}_{F}(2P^{2}/v_{2}^{2})\right]\;.
\label{eom2:0}\end{aligned}$$ Substituting (\[eom1:0\]) into (\[e\]) gives, $$\label{eq:bh}
{\cal E}=\pi v_{2}=\tfrac{1}{4}A\; ,$$ which is just the Bekenstein-Hawking entropy. This result, which is independent of the form of ${\cal L}_{F}$, is to be expected, since, in the absence of higher derivative terms, the Bekenstein-Hawking and Wald entropies coincide.
Now, the equations of motion allow us to determine $v_{1}$ and $v_{2}$ in terms of $P$ and the coupling $\lambda$. The first equation, (\[eom1:0\]), determines $v_{2}$, and consequently the entropy, in terms of $P$ (and $\lambda$). Having found $v_{2}$, (\[eom2:0\]) allows us to determine $v_{1}$ in terms of $v_{2}$. Consequently, we see that extremising the entropy function completely determines the entropy and near horizon geometry in terms of $P$ (and $\lambda$).
We now consider explicitly finding $v_{1}$ and $v_{2}$ for a particular Lagrangian. Using the Lagrangian (\[lm\]), (\[eom1:0\]) and (\[eom2:0\]) become, $$\begin{aligned}
\label{eom1}
0&=-1+(P^{2}/v_{2})\cosh^{-2}(\sqrt{\lambda P/v_{2}})\;,\\
0&=1 -v_{1}\left({P/v_{2}}\right)^{2}\cosh^{-2}(\sqrt{\lambda P/v_{2}})
\nonumber\\
&+ v_{1}\sqrt{\lambda}\left({P/v_{2}}\right)^{5/2}\cosh^{-3}(\sqrt{\lambda P/v_{2}})
\sinh(\sqrt{\lambda P/v_{2}})\;,
\label{eom2}\end{aligned}$$ which agrees with the near horizon equations of motion found directly in [@gr-qc/0403109].
To solve (\[eom1\]), it is convenient to rewrite it as, $$\label{eq:x}
\cosh\xi = \gamma\xi,$$ where, $$\label{eq:def:u:gamma}
\xi = \sqrt{\lambda P/v_{2}},\qquad \gamma = (\lambda/P)^{-1/2}\;.$$ One can then graphically solve (\[eom1\]) by finding the intersection of $\cosh\xi$ and $\gamma\xi$ for various values of $\gamma$. We illustrate this procedure in figure \[fig:1\]. It is not hard to see that as we increase $\gamma$, there are either zero, one or two solutions to (\[eq:x\]). One can also see from figure \[fig:1\], that, as $\lambda/P\rightarrow0$ (i.e. $\gamma\rightarrow\infty$) , the two possible values for $\xi$ are, $$\label{eq:branches}
{\xi}|_{\frac{\lambda}{P}\rightarrow 0}\rightarrow
\left\{
\begin{array}{c}
\infty \\ 0
\end{array}
\right..$$ Notice that, since $\cosh x \geq 1 $, (\[eom1\]) also implies, $$\label{eq:ineq}
v_{2}\leq P^{2}.$$
![This figure illustrates the graphical solution of (\[eq:x\]) which is given by the intersection of $\cosh\xi$ and $\gamma\xi$. As we increase the gradient, $\gamma=\sqrt{P/\lambda}$, one obtains either no solutions, a tangential point (denoted by a red square above) or two solutions. The first point of a double intersection, labelled with a brown dot, corresponds to a point on what we call, for reasons that will be clear later, the large branch and the second intersection, labelled with a green triangle, is on the small branch.[]{data-label="fig:1"}](fig1){width="70.00000%"}
Now, we can define a (multi-valued) function, ${\mathcal{F}}(x)$, by, $$\label{eq:def:F}
\frac{{\cal F}(x)}{\cosh{{\cal F}(x)}}=\sqrt{x}\;,$$ so that we can formally write down a solution to (\[eq:x\]) as, $$\label{eq:sol:u}
\xi={\cal F}(\gamma^{-2})={\mathcal{F}}(\lambda/P)\;.$$ Then letting, $$\begin{aligned}
{\cal G}(x)=\frac{x}{{\mathcal{F}}^{2}(x)} \label{eq:def:ga}
{\stackrel{(\ref{eq:def:F})}{=}}\frac{1}{\cosh^{2}({\mathcal{F}}(x))}\;,
\end{aligned}$$ and using and (\[eq:def:u:gamma\]), we can write, $$\label{eq:def:g}
v_{2}=\frac{\left(\frac{\lambda}{P}\right)}{\xi^{2}}P^{2}= {\cal G}\left(\frac{\lambda}{P}\right)P^{2}.$$ which is of the generic form expected by dimensional analysis. Since (\[eq:x\]) may have two solutions, ${\cal F}$ and ${\cal G}$ both have two branches. Substituting (\[eq:branches\]) into (\[eq:def:ga\]) we find that, in the limit that the non-linear coupling goes to zero (or the charge becomes very large), $$\label{eq:g:branches}
{\cal G}(0)=1/\cosh^{2}({\cal F}(0))=\left\{
\begin{array}{ll}
0 & \mbox{(small branch, ${\mathcal{G}}_{S}$)}\\ 1& \mbox{(large branch, ${\mathcal{G}}_{L}$)}
\end{array}
\right..$$ We call the two branches of ${\cal G}$, the small and large branch. While it seems challenging to find an analytical expression for ${\mathcal{G}}$, it is very easy to evaluate it numerically. We have plotted ${\mathcal{G}}$, or in other words $v_{2}/P^{2}$, as a function of $\lambda/P$ in figure \[fig:2\]. We note that ${\mathcal{G}}$ decreases monotonically on the large branch, so that, for a fixed charge, the $\lambda=0$ solution is the most entropic.
![This figure shows $v_{2}$ as a function of $\lambda$ and $P$ found by numerically solving (\[eq:x\]). Specifically we plot, $v_{2}P^{-2}=\xi^{-2}\gamma^{-2}={\mathcal G}(\lambda/P)$. The large branch, ${\cal G}_{L}$, is plotted in brown and the small branch, ${\cal G}_{S}$, is plotted in green. We should exclude the shaded region, defined by $\gamma{^2}\xi^{2}-\gamma^{2}<1$ , in which, using (\[eq:sol:v1b\]), $v_{1}$ is negative. Since it is entirely contained within the shaded region, the small branch is unphysical. The regular black hole, denoted by a blue dot, is found on the big branch at $\lambda/P=w_{0}$. At the place where the branches meet, denoted by a red square, $v_{1}\rightarrow\infty$ (or $-\infty$ if we approach from below). []{data-label="fig:2"}](fig2){width="70.00000%"}
Having determined $v_{2}$ (at least in principle), we can find $v_{1}$ by substituting (\[eom1\]) into (\[eom2\]) and using (\[eq:def:g\]), we get, $$\begin{aligned}
\label{eq:sol:v1}
v_{1} &= v_{2}(1-\gamma^{-1}\sqrt{\xi^{2}\gamma^{2}-1})^{-1}\\
\label{eq:sol:v1b}
&= v_{2}(1-[\lambda/P]^{1/2}\sqrt{{\cal G}^{-1}(\lambda/P)-1})^{-1}\;,\end{aligned}$$ with (\[eq:ineq\]) ensuring reality.
For $v_{1}$ to be positive and finite, on sees that from (\[eq:sol:v1\]), we require $\xi^{2}\gamma^{2}-1> \gamma^{2}$. Now, at the branch point, the function $f(\xi)=\cosh\xi - \gamma\xi$ has a single zero, so we require that $f'(\xi)=0$ when $f(\xi)=0$. In other words, in addition to (\[eq:x\]) the branch point is determined by, $$\label{eq:fdash}
\sinh\xi = \gamma.$$ Combining (\[eq:x\])and (\[eq:fdash\]) gives, $$\gamma^{2}\xi^{2}-1=\gamma^{2}\;,$$ which, using (\[eq:sol:v1\]), implies that as we approach the branch point, $v_{1}\rightarrow\infty$ or in other words the $AdS_{2}$ approaches flat space. From figure \[fig:2\], we see that the small branch lies entirely in the region $\gamma{^2}\xi^{2}-\gamma^{2}<1$, and consequently, $v_{1}$ is always negative on it, making it unphysical. Discarding the small branch, we have used (\[eq:sol:v1b\]) to plot $v_{2}/v_{1}$ as a function of $\lambda/P$ in figure \[fig:3\]. We see that $v_{1}/v_{2}$ increases monotonically, eventually diverging at the branch point.
![This figure shows $v_{2}/v_{1}$ as a function of $\lambda/P$ found by numerically solving (\[eq:x\]) and using (\[eq:sol:v1\]). For $\lambda/P$ small, we have $v_{1}\approx v_{2}$. As we approach the branch point, denoted by a red square, $v_{1}$ diverges and $v_{2}/v_{1}\rightarrow0$. The regular black hole, at $\lambda/P=w_{0}$, is denoted by a blue dot. []{data-label="fig:3"}](fig3){width="75.00000%"}
Finally, one can actually obtain a large charge, small coupling expansion for ${\mathcal{G}}_{L}$. Assuming ${\mathcal{G}}_{L}$, has a nice Taylor expansion about zero, using (\[eq:g:branches\]) as a starting point, and by taking successive derivatives of (\[eq:def:F\]) and (\[eq:def:ga\]) one can recursively expand ${\mathcal{G}}_{L}(x)$ about zero. As discussed in appendix \[sec:a1\], we find that, $$\begin{aligned}
{\mathcal{G}}_{L}(x)&= 1-x-\tfrac{1}{3}x^{2}
+{\cal O}\left(x^{3}\right)\; ,
\label{taylor:L}\end{aligned}$$ so that we can write a large charge, small coupling expansion for the entropy, $$\begin{aligned}
{\cal E}= \pi P^{2}\left( 1-\frac{\lambda}{P}-\frac{1}{3}\frac{\lambda^{2}}{P^{2}}
+{\cal O}\left(\frac{\lambda^{3}}{P^{3}}\right)\right) \;.
\label{largeP}\end{aligned}$$ For completeness, we mention that on the small branch, as discussed in appendix \[sec:a1\], for $x$ small, we get, $$\label{eq:gs:approx}
{\mathcal{G}}_{S}(x)\approx x{\left[ W_{-1}(-\tfrac{1}{2}\sqrt{x})\right]^{-2}}\;.$$ where $W_{-1}$ is the non-principal real branch of the Lambert function.
As a check we note that on the large branch, taking $\lambda\rightarrow0$, we recover the usual near horizon extremal Einstein-Maxwell Reisner-Nordstrom solution with, $$\begin{aligned}
\label{result}
v_{1}=v_{2}= P^{2}\;.\end{aligned}$$
Entropy function and the regular black hole {#sec:rbh:E}
===========================================
In this section we confirm that the entropy function analysis of the regular black hole reproduces the near horizon geometry of the known solution found in [@gr-qc/0403109]. This merely entails considering the results of the previous section with the appropriate value of $\lambda/P$.
As discussed in section \[rbh\], the regular black hole corresponds to the point $\lambda/P={w_{0}}$, so that (\[eq:x\]) becomes, $$\label{eq:rbh:eom1}
\cosh\xi = {w_{0}}^{-1/2}\xi.$$ One can analytically check, using the property $w_{0}e^{w_{0}}=e^{-1}$, that (\[eq:rbh:eom1\]) has a solution, $$\label{u:rbh}
\xi=\frac{w_{0}+1}{2},$$ or in other words ${\mathcal{F}}(w_{0})=\frac{1}{2}(w_{0}+1)$. We have plotted the position of the regular solution as a blue dot in figures \[fig:2\] and \[fig:3\], from which we observe that it is on the large branch.
Finally, one can check that substituting the solution, (\[u:rbh\]), into (\[eq:def:g\]) and (\[eq:sol:v1\]) reproduces (\[result:rbh\]) and we are done.
Conclusion {#sec:conc}
==========
In this paper we examine entropy function formalism for regular magnetically charged black hole solution in Einstein-Hilbert gravity coupled with a certain non-linear $U(1)$ gauge field. The mass and charge of the full solution can be tuned so that it has no curvature singularity at the centre. In the extremal limit this corresponds to a particular charge to non-linear coupling ratio with an $AdS_2\times S^2$ near horizon geometry. Unsurprisingly we find that the entropy function analysis match with the exact solution found by solving the full Einstein equations. This is in contrast with the claim in the recent papers [@0705.2478; @0707.1933].
Indeed in the entropy function formalism, the equation of motion, which follow from extremizing the entropy function, are exactly the same as equation of motion at horizon found by extremizing the action, simply because the entropy function (up to Legendre transformation) is the Lagrangian at the horizon. The fact that the entropy is the value of entropy function at its extremum is derived from Wald entropy formula, using the near horizon symmetries. Both of these results apparently have nothing to do with the regularity of the solution inside the horizon.
We would like to thank Ashoke Sen for helpful comments. The work of K.G. is, in part, supported by the EU-RTN network contract MRTN-CT-2004-005104 and INTAS contract 03-51-6346. The work of H.Y is supported by the Korea Research Foundation Leading Scientist Grant (R02-2004-000-10150-0) and Star Faculty Grant (KRF-2005-084-C00003).
Large charge/small coupling expansion of the entropy {#sec:a1}
====================================================
In this appendix we discuss the expansion of ${\mathcal{G}}(x)$ about zero.
Taking the derivative of (\[eq:def:F\]) with respect to $x$ and solving for ${\mathcal{F}}'$ we find, $$\label{eq:a1}
{\mathcal{F}}'(x)=\frac{\cosh ({\mathcal{F}}(x))}{2 \sqrt{x} \left(\sqrt{x} \sinh ({\mathcal{F}}(x))-1\right)}\;,$$ while taking of derivative of (\[eq:def:ga\]) gives, $$\label{eq:a2}
{\mathcal{G}}'(x)=-2 \text{sech}^2({\mathcal{F}}(x)) \tanh ({\mathcal{F}}(x)) {\mathcal{F}}'(x)\;.$$ Now using (\[eq:def:F\]) and (\[eq:a1\]) we can rewrite (\[eq:a2\]) as, $$\label{eq:a3}
{\mathcal{G}}'(x)= \frac{\tanh (\mathcal{F})}{\mathcal{F} (\mathcal{F} \tanh (\mathcal{F})-1)}\;,$$ and using (\[eq:branches\]) and taking the limit $x\rightarrow0$, we get, $$\label{eq:a4}
{\cal G}'(0)=\left\{
\begin{array}{ll}
0 & \mbox{(small branch)}\\ -1& \mbox{(large branch)}
\end{array}
\right..$$ Taking another set of derivative, after some algebra we obtain, $$\label{eq:a5}
{\mathcal{G}}''(x)
= \frac{\mathcal{F}\cosh (2 \mathcal{F})-\cosh (\mathcal{F}) \sinh (\mathcal{F})}
{2 \mathcal{F}^3 (\mathcal{F} \tanh (\mathcal{F})-1)^3}\;,$$ and once again taking the $x\rightarrow0$ limit, we get, $$\label{eq:a6}
{\mathcal{G}}''(0)=
\left\{
\begin{array}{ll}
\infty & \mbox{(small branch)}\\ -\tfrac{2}{3}& \mbox{(large branch)}
\end{array}
\right. \;.$$ We found that the second order Taylor expansion of ${\mathcal{G}}_{L}(x)$, (\[taylor:L\]), agrees well with our numerical plot for $x$ small. For example, at $x=w_{0}$, we find that they differ by about $2\%$.
On the other hand, we see that the small branch does not have a nice Taylor expansion about the origin. However, on the small branch, we see from figure \[fig:1\], that when $\xi$ is large, $\gamma$ is also large and consequently, $\lambda/P=\gamma^{-2}$, is small. For $\xi\gg 1$, we can approximate (\[eq:x\]) by, $$\label{eq:a7}
\tfrac{1}{2}e^{\xi}\approx\gamma\xi + {\cal O}(e^{-\xi})\;,$$ which can approximately be solved by, $$\xi \approx - W(-\tfrac{1}{2}\gamma^{-1})\;.$$ where $W$ is the Lambert function defined in (\[lambf\]). For, $-e^{-1}<x<0$, the two real branches of $W$ satisfy $W_{0}(x)\geq -1$ and $W_{-1}(x)\leq-1$, [@lamb], consequently since we are assuming that $\xi\gg 1$, we should take the branch $W_{-1}$. So, using (\[eq:sol:u\],\[eq:def:ga\]), we obtain, $${\mathcal{G}}_{S}(\lambda/P)\approx (\lambda/P){\left[ W_{-1}(-\tfrac{1}{2}\sqrt{\lambda/P})\right]^{-2}}\;,$$ which we found agrees well with our numerical results shown in figure \[fig:2\], for $\lambda/P$ small.
[^1]: The coupling $a=\sqrt{\lambda}$ is commonly used in the literature. With out loss of generality, we can take $\lambda>0$.
[^2]: See [@lamb] for a nice review of the properties of the Lambert function.
|
---
abstract: 'We show the Godbillon-Vey invariant arises as a ‘restricted Casimir’ invariant for three-dimensional ideal fluids associated to a foliation. We compare to a finite-dimensional system, the rattleback, where analogous phenomena occur.'
address: 'H.H. Wills Physics Laboratory, Tyndall Avenue, Bristol BS8 1TL, UK'
author:
- Thomas Machon
title: 'The Godbillon-Vey Invariant as a Restricted Casimir of Three-dimensional Ideal Fluids'
---
Introduction
============
The topological aspect of ideal fluids has its origins in the transport of vorticity. A consequence is the conservation of helicity, which measures the average linking of vortex lines [@moffatt69; @arnold74; @moffatt92]. In the Hamiltonian formulation of ideal fluids as an infinite-dimensional Lie-Poisson system, helicity appears as a Casimir invariant, a degeneracy in the Lie-Poisson bracket [@morrison98].
The state of an ideal fluid on a homology 3-sphere is specified by the vorticity, a divergence-free vector field. A Casimir in an ideal fluid is invariant under all volume-preserving diffeomorphisms of the domain, so can be said to measure a topological property of the vorticity. For a generic vorticity field, helicity is the only topological invariant [@enciso16; @kudryavtseva14; @kudryavtseva16]. However, higher order invariants can be defined in special cases. Here we study the Godbillon-Vey invariant, $GV$, which can be associated to a vorticity field tangent to a codimension-1 foliation [@machon20; @webb14; @webb19; @tur93]. $GV$ originates in the theory of foliations [@gv; @candel]; in ideal fluids it measures topological helical compression of vortex lines [@machon20]. The goal of this paper is to show how $GV$ fits naturally into the Lie-Poisson Hamiltonian formulation of ideal fluids [@morrison98] as a ‘restricted Casimir’ invariant. In particular, we consider a set $S$ of ideal fluids where the Lie-Poisson bracket has an additional degeneracy associated to the Lie subalgebra of volume-preserving vector fields tangent to a foliation, which may vary within $S$. On $S$ we construct a modified Lie-Poisson type bracket, in terms of which the Godbillon-Vey invariant appears as a Casimir.
Imposing this degeneracy also forces the helicity to vanish and in this sense $GV$ is hierarchical, in a manner analogous to that suggested by Arnold and Khesin [@arnold99]. Recent work [@yoshida14; @yoshida16] has studied similar hierarchical structures in Hamiltonian systems, where a singular region in phase space with a Poisson operator of decreased rank can itself be considered as a Poisson submanifold, on which new Casimir invariants appear. What we describe can be considered an example of this phenomenon. We note also that a foliation on $M$ can be defined through an exact sequence of vector bundles $$\begin{CD}
0@>>> T\mathcal{F} @>>> TM @>>>N @>>>0
\end{CD}$$ where $N = T\mathcal{F}/TM$ is the normal bundle of the foliation. This sequence passes to the volume-preserving case, and we note the connection to the classification of Casimir invariants coming from Lie algebra extensions [@thiffeault00].
There is a finite-dimensional example in the Lie-Poisson formulation of the ‘rattleback’ spinning top [@yoshida17], where corresponding phenomena occur: there is a submanifold of phase space where the Poisson operator has an additional degeneracy associated to a Lie subalgebra; on this submanifold the primary Casimir vanishes and a new restricted Casimir appears.
In the finite-dimensional rattleback case, perturbation of the system around the singular manifold leads to interesting dynamical properties [@yoshida17]. Our own analysis of the Godbillon-Vey invariant elsewhere also suggests a strong connection to dynamics; $GV$ provides a global and local obstruction to steady flow and can be used to estimate the rate of change of vorticity [@machon20]. With that in mind, we suggest that flows with $GV\neq 0$ (or perturbations thereof) may prove particularly interesting from a dynamical perspective. Finally, we note our light touch regarding rigour.
Lie-Poisson Systems
===================
See e.g. [@thiffeault00] for a description. Let ${\mathfrak{g}}$ be a Lie algebra associated to a group $G$, with ${\mathfrak{g}}^\ast$ its dual. Given an element $\alpha \in {\mathfrak{g}}^\ast$ and two elements $U,V \in {\mathfrak{g}}$ we form the bracket $$\langle \alpha , [U,V] \rangle,
\label{eq:KKS}$$ where $\langle \cdot, \cdot \rangle : {\mathfrak{g}}^\ast \times {\mathfrak{g}}\to \mathbb{R}$ is the natural pairing between the Lie algebra and its dual, and $[ \cdot, \cdot]$ is the Lie bracket of ${\mathfrak{g}}$. This is then used to define the Lie-Poisson bracket $$\{F,G\}_\pm = \pm \left \langle \alpha , \left [ \frac{\delta F}{\delta \alpha}, \frac{\delta G}{\delta \alpha} \right ] \right \rangle,
\label{eq:pb}$$ where the (functional) derivative $\delta F/ \delta \alpha$ is identified with an element of ${\mathfrak{g}}$ by the relation $$\frac{d}{d \epsilon} F(\alpha + \epsilon \delta \alpha) \big |_{\epsilon=0} =\left \langle \delta \alpha , \frac{\delta F}{\delta \alpha} \right \rangle.$$ The sign in depends on whether we consider right-invariant or left-invariant function(al)s on ${\mathfrak{g}}^\ast$ with respect to the coadjoint representation of $G$, but is irrelevant for our purposes. Coupled with a Hamiltonian function on ${\mathfrak{g}}^\ast$, this specifies the system. The noncanonical nature of the Lie-Poisson bracket allows for the existence of Casimir invariants, $C$, given by the property $\{F,C\}=0$ for any function $F$. We define the coadjoint bracket $[ \cdot, \cdot]^\dagger : {\mathfrak{g}}\times {\mathfrak{g}}^\ast \to {\mathfrak{g}}^\ast$ as $$\langle [U, \alpha]^\dagger , V \rangle = \langle \alpha, [U,V] \rangle.$$ This allows us to give the condition for $C$ to be a Casimir as $$\left [ \frac{\delta C(\alpha)}{\delta \alpha} , \alpha \right ]^\dagger =0.$$
In this paper we will be interested in sets of points $\alpha \in S \subset {\mathfrak{g}}^\ast$ where there is a non-generic degeneracy associated to a subalgebra $\mathfrak{h}_\alpha \subset {\mathfrak{g}}$, such that $$\langle \alpha, U \rangle =0,$$ for $U \in \mathfrak{h}_\alpha$. For a given $\alpha \in {\mathfrak{g}}^\ast$, let $\beta = \textrm{ad}^\ast_g \alpha$, $g \in G$. Then $\beta$ is orthogonal to the subalgebra ${\mathfrak{h}}_\beta = \textrm{ad}_g {\mathfrak{h}}_\alpha $, so that $S$ will, in general, be a set of coadjoint orbits in ${\mathfrak{g}}^\ast$. The precise specification of admissible subalgebras and the subset $S$ in a general formulation is left intentionally vague.
Finite Dimensional Example: the Rattleback
==========================================
An idealised description of the chiral dynamics of the rattleback spinning top [@yoshida17] can be formulated as a Lie-Poisson system based on the three-dimensional Lie algebra with Bianchi classification $\textrm{VI}_{h<-1}$, spanned by three elements, $P$, $R$, $S$ with Lie bracket $$[P,R] =0, \quad [S,P] = h P, \quad [S,R] = R.$$ Physically $P$, $R$, and $S$ are associated to pitching, rolling and spinning motions respectively, and $h$ is a geometric parameter related to the aspect ratio of the top. The dynamical variable is an element of the dual space ${\mathfrak{g}}^\ast$ which we write as a lowercase tuple $(p,r,s)$, in terms of which the dynamics are [@yoshida17; @moffatt08] $$\frac{d}{dt} \begin{pmatrix} p \\ r \\ s \end{pmatrix} = \begin{pmatrix} -h ps \\ - rs \\ r^2 +h
p^2\end{pmatrix}.
\label{eq:rattleback}$$ The Hamiltonian of this system is given by $H = (p^2+r^2+s^2)/2$. At a generic point in ${\mathfrak{g}}^\ast$ the Lie-Poisson bracket has a one-dimensional kernel, associated to the Casimir $$C = pr^{-h},$$ which one can check is conserved by the dynamics .
There is a two-dimensional Abelian subalgebra ${\mathfrak{h}}\subset \textrm{VI}_{h<-1}$, spanned by $P,R$. The set of points $ M \subset {\mathfrak{g}}^\ast$ orthogonal to ${\mathfrak{h}}$ is the singular line $(0,0,s)$, so that on $M$ the Casimir $C=0$. On $M$ the Lie-Poisson bracket is trivial, so the dynamics are trivial (one can see this by setting $p=r=0$ in ). It follows that $s$ is a constant of the motion on $M$ only. Finally, note that $M$ can be thought of as a one-dimensional Poisson manifold with trivial Poisson bracket, and with respect to this bracket $s$ is a Casimir invariant (as is any function of $s$), so that $s$ is a restricted Casimir invariant of the rattleback system. Physically it corresponds to simple spinning motion of the top.
The Godbillon-Vey Invariant as a Restricted Casimir in Three-Dimensional Ideal Fluids
=====================================================================================
Now we see how the same pattern of phenomena is found in three-dimensional ideal fluids on a manifold $M$. We assume throughout that $M$ is a homology 3-sphere (one can take $M=S^3$).
Ideal Hydrodynamics and Helicity {#sec:hyd}
--------------------------------
In the Lie-Poisson formulation of ideal fluids [@morrison98; @arnold99], ${\mathfrak{g}}$ is the Lie algebra of volume-preserving vector fields on $M$ with respect to a volume form $\mu$, so that $\mathcal{L}_U \mu =0$ for $U \in {\mathfrak{g}}$ and the $2$-form $\iota_U \mu$ is closed. The dynamical variable is given by an element of the dual space ${\mathfrak{g}}^\ast$, the smooth part of which can be identified as $\Omega^1(M)/d\Omega^0(M)$, the space of differential 1-forms modulo exact forms, and each element is given by a coset $[\alpha]$, with specific representative $\alpha$. We will suppress the coset notation $[]$. The pairing $\langle \cdot, \cdot \rangle: {\mathfrak{g}}^\ast \times {\mathfrak{g}}\to \mathbb{R}$ is given by $$\langle \alpha, U \rangle = \int_M (\iota_U \alpha) \mu,
\label{eq:pair}$$ which does not depend on the representative 1-form $\alpha$. In this case the Lie-Poisson bracket takes the form $$\langle \alpha, [U,V] \rangle = \int_M \alpha \wedge \iota_{[U,V]} \mu.
\label{eq:pair}$$ The coadjoint bracket is given as $$[U, \alpha]^\dagger = -\iota_U d \alpha = -\iota_U \iota_W \mu.$$ Where the vorticity field $W \in {\mathfrak{g}}$ is given by $d \alpha= \iota_W \mu$. Helicity is defined as $$\mathcal{H} = \int_M \alpha \wedge d \alpha = \langle \alpha, W \rangle.$$ A short calculation gives $$\frac{d}{d \epsilon} \mathcal{H}(\alpha + \epsilon \delta \alpha) \big |_{\epsilon=0} = \int_M \delta \alpha \wedge 2 d \alpha,$$ so $\delta \mathcal{H}/ \delta \alpha = 2 W$ and hence $$\left [ \frac{\delta \mathcal{H}}{\delta \alpha} , \alpha \right ]^\dagger = 0,$$ so that $\mathcal{H}$ is a Casimir.
Foliations and ${\mathfrak{g}}^\ast$
------------------------------------
We now consider a codimension-1 foliation $\mathcal{F}_\alpha$ of $M$ such that $\alpha \in {\mathfrak{g}}^\ast$ satisfies $$\langle \alpha , X \rangle =0,$$ for $X$ in the subalgebra of volume-preserving vector fields $ {\mathfrak{h}}_\alpha \subset {\mathfrak{g}}$ that are tangent to the leaves of $\mathcal{F}_\alpha$. Let $\beta_\alpha$ be a defining form for $\mathcal{F}_\alpha$ and consider the family of closed 2-forms $d(h \beta_\alpha)$, $h$ a function, then the vector field $Y$ defined by $\iota_Y \mu = d(h \beta_\alpha)$ is an element of $\mathfrak{h}_\alpha$. By assumption on $\alpha$, $$0 = \int_M \alpha \wedge d (h \beta_\alpha) = \int_M h \beta_\alpha \wedge d \alpha.$$ As $h$ is arbitrary, $\beta_\alpha \wedge d \alpha =0$. Recall the vorticity field $W$ defined by $d \alpha = \iota_W \mu$, it follows that $$W \in \mathfrak{h}_\alpha.$$ As an immediate consequence, the helicity, $\langle \alpha, W \rangle$, vanishes. Now let $\gamma$ be a closed loop tangent to $\mathcal{F}$. The quantity $$I_\gamma = \int_\gamma \alpha$$ is invariant under leafwise homotopies of $\gamma$, so that $\alpha$ defines a class $[\alpha]_\mathcal{F}$ in the foliated cohomology group $H^1(\mathcal{F}_\alpha)$. In fact $$[\alpha]_\mathcal{F} = 0 \in H^1(\mathcal{F}_\alpha).$$ Consider a family of smooth vector fields $G_\lambda \in {\mathfrak{g}}$ with support in a tubular neighbourhood of $\gamma$ of diameter $\sim \lambda$ (with respect to a metric), tending to the singular vector field with support $\gamma$ and constant flux $\phi$ as $\lambda \to 0$, so that $\int_D \iota_{G_\lambda} \mu \to \phi$ as $\lambda \to 0$, where $D$ is a disk pierced by $\gamma$. Then $\langle \alpha, G_\lambda \rangle \to \phi I_g$ as $\lambda \to 0$. But $|\langle \alpha, G_\lambda \rangle| < \lambda C$ for some constant $C$, so $I_g=0$. As $\gamma$ was arbitary, $[\alpha]_\mathcal{F} = 0 \in H^1(\mathcal{F}_\alpha)$. As an immediate consequence, any representative of the coset $[\alpha] \in \Omega^1(M) / d \Omega^0(M)$ can be written as $ f \beta_\alpha + dg$ for functions $f,g$. There is then a canonical representative form $\alpha_c = f \beta_\alpha$, which we write as $\alpha$ in subsequent sections. With this choice of representative the helicity density vanishes, $\alpha_c \wedge \iota_W \mu =0$.
$\Omega^1_I(M)$ and the Functional Derivative
---------------------------------------------
We may write $\alpha = f \beta_\alpha$. For simplicity we will assume that $f \neq 0$, and we choose $\beta_\alpha$ so that $\alpha = \beta_\alpha$ is a defining form for $\mathcal{F}_\alpha$. Then, with this canonical choice, the subset of ${\mathfrak{g}}^\ast$ we are considering can be identified with the space of nonvanishing integrable 1-forms on $M$, which we write as $\Omega^1_I(M)$ (this space is not connected, we consider a single arbitrary connected component). We note that $\Omega^1_I(M)$ is no longer a vector space.
We would like to define functional derivatives on $\Omega^1_I(M)$. For a one-parameter family $\alpha_t \in \Omega^1_I(M)$, $t \geq 0$ with time derivative $\dot{\alpha}_t$, we write $\dot{\alpha} = \dot{\alpha}_0$ and $\alpha = \alpha_0$. Now for a functional $F$ on $\Omega^1_I(M)$ the functional derivative is defined as $$\frac{d}{d t} F(\alpha_t ) \big |_{t=0} =\left \langle \dot{\alpha} , \frac{\delta F}{\delta \alpha} \right \rangle,$$ and we identify ${\delta F}/{\delta \alpha}$ with a vector field as in Section \[sec:hyd\]. Because we have a canonical choice of $\alpha$, we no longer require invariance under gauge transformations and so are not restricted to volume-preserving vector fields. We suppose instead ${\delta F}/{\delta \alpha} \in \mathfrak{X}(M) / \Xi_\alpha $ where $\mathfrak{X}(M)$ is the space of smooth vector fields on $M$ and $\Xi_\alpha \subset \mathfrak{X}(M)$ is an $\alpha$ dependent subset satisfying $\langle \dot{\alpha}, U \rangle =0$ for $U \in \Xi_\alpha$.
Our characterisation of $\Xi_\alpha$ below is not complete, but is sufficient for our purposes. First, we will show that it is non-empty. As $\alpha_t$ is integrable we have $$\alpha_t \wedge d \alpha_t =0.$$ In particular this gives $$0 = \frac{d}{d t} \left( \int_M f \alpha_t \wedge d \alpha_t \right )\Big |_{t=0}= \int_M \dot{\alpha} \wedge (f d \alpha + d (f \alpha))
\label{eq:degeneracy}$$ for any function $f$, so that fields $V$ satisfying $$\iota_V \mu = f d \alpha + d (f \alpha)
\label{eq:what}$$ are elements of $\Xi_\alpha$. Now we give two properties of general elements of $\Xi_\alpha$.
Firstly we note that any field in $\Xi_\alpha$ must be tangent to $\mathcal{F}_\alpha$. We can choose $\alpha_t = \exp(g t) \alpha$, so that $\dot{\alpha} = g \alpha$ for an arbitrary function $g$. Now suppose $U$ is not tangent to $\mathcal{F}_\alpha$, then by an appropriate choice of $g$ we can force $\langle g \alpha, U \rangle \neq 0$, so $U \notin \Xi_\alpha$.
Secondly we note that any element $V$ of $\Xi_\alpha$ must satisfy $d (\iota_V \mu) = \eta \wedge (\iota_V \mu)$, where $\eta$ is a 1-form defined by the relation $d \alpha = \alpha \wedge \eta$. We can choose $
\alpha_t$ to be generated by a family of diffeomorphisms, so that $\dot{\alpha}= \mathcal{L}_U \alpha$ for $U \in \mathfrak{X}(M)$. Then, writing $\nu = \alpha \wedge \sigma = \iota_V \mu$, $V \in \Xi_\alpha$ we require $$0 = \int_M \mathcal{L}_U \alpha \wedge \alpha \wedge \sigma = \int (\iota_U \alpha) \left (d \alpha \wedge \sigma + d(\alpha \wedge \sigma) \right ),$$ and since $\iota_U \alpha$ is arbitrary we find $d \alpha \wedge \sigma + d(\alpha \wedge \sigma)=0$, or $$d \nu = \eta \wedge \nu.
\label{eq:condon}$$ Any element of $\Xi_\alpha$ must then be tangent to $\mathcal{F}_\alpha$ and satisfy . This is not a complete characterisation, there are vector fields satisfying these two conditions which are not elements of $\Xi_\alpha$. This is demonstrated by example in section \[sec:cas\]. We speculate that vector fields of the form fully characterise $\Xi_\alpha$.
The Poisson Bracket on $\Omega^1_I(M)$
--------------------------------------
We define the Poisson bracket on $\Omega^1_I(M)$ which continues to take the standard form $$\{F,G\}_I = \left \langle \alpha , \left [ \frac{\delta F}{\delta \alpha}, \frac{\delta G}{\delta \alpha} \right ] \right \rangle,
\label{eq:brack}$$ where now $\alpha \in \Omega^1_I(M)$ and the functional derivatives are cosets in $\mathfrak{X}(M)/\Xi_\alpha$. The bracket must not depend on the choice of representative vector field for the functional derivative. Consider a vector field $A$ on $M$ such that $\iota_A \alpha =0$ and $d(\iota_A \mu) = \eta \wedge \iota_A \mu $. From the previous section we know all elements of $\Xi_\alpha$ have these properties. Then $$\langle \alpha , [ A,V ] \rangle = 0
\label{eq:shower}$$ where $V \in \mathfrak{X}(M)$. We compute $$\langle \alpha , [A,V ] \rangle = \int_M \alpha \wedge \iota_{[A,V]} \mu = - \int_M \alpha \wedge \mathcal{L}_V \iota_A \mu.$$ Now $\iota_A \mu = \alpha \wedge \sigma$ for some 1-form $\sigma$. Then we have $$\langle \alpha , [A,V ] \rangle = \int_M \alpha \wedge \sigma \wedge \mathcal{L}_V \alpha = -\int_M \iota_V \alpha (2d \alpha \wedge \sigma - \alpha \wedge d \sigma ) =0,$$ it follows that the bracket $\{F,G\}_I$ does not depend on the choice of representative vector field for the functional derivatives and becomes a Poisson bracket on $\Omega^1_I(M)$. Finally, we note that if $F$ is the restriction to $\Omega^1_I(M)$ of a functional on ${\mathfrak{g}}^\ast$, then its functional derivative is still an element of ${\mathfrak{g}}$, all elements of which are representative vector fields of a coset in $\mathfrak{X}(M)/\Xi_\alpha$, and the bracket reproduces the Lie-Poisson bracket of the original ideal fluid formulation. In particular, we can recover Euler’s equations by choosing the appropriate Hamiltonian.
The Godbillon-Vey Invariant
---------------------------
For a codimension-1 foliation $\mathcal{F}$ on a closed manifold $M$, the Godbillon-Vey class [@gv; @candel] is an element $GV \in H^3(M ; \mathbb{R})$, if $M$ is a closed 3-manifold $H^3(M ; \mathbb{R}) = \mathbb{R}$ and $GV \in \mathbb{R}$ is a diffeomorphism invariant of the foliation. Let $\beta$ be a defining 1-form for $\mathcal{F}$, then the integrability condition $\beta \wedge d \beta =0$ implies there is a 1-form $\eta$ such that $$d \beta = \beta \wedge \eta.$$ The 3-form $\eta \wedge d \eta$ is closed and $GV$ is defined as $$GV = \int_M \eta \wedge d \eta,$$ $\beta$ is only defined up to multiplication by a non-zero function, and $\eta$ is only defined up to addition of a multiple of $\beta$, but under these transformations $\eta \wedge d \eta$ changes by an exact 3-form, so $GV$ is well-defined. By construction $GV$ is a diffeomorphism invariant of $\mathcal{F}$.
The Godbillon-Vey Invariant as a Restricted Casimir {#sec:cas}
---------------------------------------------------
Our goal is to show that the Godbillon-Vey invariant is a Casimir with respect to the Poisson bracket $\{,\}_I$ defined above. Consider the variation of $GV$, $$\frac{d }{dt} GV \big|_{t=0}= 2 \int_M \dot{\eta} \wedge d \eta = 2 \int_M \dot{\eta} \wedge \beta \wedge \gamma.$$ Where we use the fact that $d \eta = \alpha \wedge \gamma$. Using $d \alpha = \alpha \wedge \eta$, we find $d \dot{\alpha} = \dot{\alpha} \wedge \eta+{\alpha} \wedge \dot{\eta}$, so that $$\frac{d }{dt} GV \big|_{t=0}= 2 \int_M (\dot{\alpha} \wedge \eta - d \dot{\alpha})\wedge \gamma = \int_M \dot{\alpha} \wedge 2( \eta \wedge \gamma - d \gamma).$$ Now we consider the 2-form $\chi = 2 (\eta \wedge \gamma - d \gamma)$, there is a freedom in $\chi$ arising from freedom in $\eta$ and $\gamma$. We may make the transformations $$\eta \to \eta + f \alpha, \quad \gamma \to \gamma + f \eta - df + g \alpha,$$ for functions $f,g$. Under these transformations one finds $$\chi \to \chi - 2 ( g d \alpha + d(g \alpha)),$$ which does not affect the value of $d GV / dt$ as per . Now observe that $\alpha\wedge \chi =0$ and $d\chi = \eta \wedge \chi $. Writing $\chi = \iota_T \mu$, we find $T$ is tangent to $\mathcal{F}_\alpha$ and satisfies . Using we find $$\{F, GV \}_I =0$$ for any functional $F$ on $\Omega^1_I(M)$ so that GV is a restricted Casimir of three-dimensional ideal fluids.
[^1]
[99]{} Moffatt HK. 1969 [*The degree of knottedness of tangled vortex lines*]{}, J. Fluid. Mech. [**35**]{}, 117-129. ([doi:10.1017/S0022112069000991](https://doi.org/10.1017/S0022112069000991)) Arnold VI. 1986 [*The asymptotic Hopf invariant and its applications*]{}, Selecta Math. Soviet. [**5**]{}, 327-345. ([doi:10.1007/978-3-642-31031-7\_32](https://doi.org/10.1007/978-3-642-31031-7_32)) Moffatt HK, Ricca RL. 1992 [*Helicity and the Călugăreanu invariant*]{}, Proc. R. Soc. Lond. A [**439**]{}, 411-429. ([doi:10.1098/rspa.1992.0159](https://doi.org/10.1098/rspa.1992.0159))
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Enciso A, Peralta-Salas D, Torres de Lizaur F. 2016 [*Helicity is the only integral invariant of volume-preserving transformations*]{}, Proc. Natl. Acad. Sci. U.S.A. [**113**]{}, 2035-2040. ([doi:10.1073/pnas.1516213113](https://doi.org/10.1073/pnas.1516213113)) Kudryavtseva EA. 2014 [*Conjugation invariants on the group of area-preserving diffeomorphisms of the disk*]{}, Math. Notes [**95**]{}, 877-880. ([doi:10.1134/S0001434614050332](https://doi.org/10.1134/S0001434614050332)) Kudryavtseva EA. 2016 [*Helicity is the only invariant of incompressible flows whose derivative is continuous in $C^1$-topology*]{}, Math. Notes [**99**]{}, 626-630. ([doi:10.1134/S0001434616030366](https://doi.org/10.1134/S0001434616030366))
Machon T. [*Topological Vorticity Compression and Obstruction to Steady Flow in Ideal Fluids*]{}, [*Submitted*]{}. Webb GM, Dasgupta B, McKenzie JF, Hu Q, Zank GP. 2014 [*Local and nonlocal advected invariants and helicities in magnetohydrodynamics and gas dynamics I: Lie dragging approach*]{}, J. Phys. A [**47**]{}, 095501. ([doi:10.1088/1751-8113/47/9/095501](https://doi.org/10.1088/1751-8113/47/9/095501))
Webb GM, Prasad A, Anco SC, Hu Q. 2019 [*Godbillon-Vey helicity and magnetic helicity in magnetohydrodynamics*]{}, J. Plasma Phys. [**85**]{}, 775850502. ([doi:10.1017/S0022377819000679](https://doi.org/10.1017/S0022377819000679)) Tur AV, Yanovsky VV. 1993 [*Invariants in dissipationless hydrodynamic media*]{}, J. Fluid Mech. [**248**]{}, 67-106. ([doi:10.1017/S0022112093000692](https://doi.org/10.1017/S0022112093000692))
Godbillon C, Vey J. 1971. [*Un invariant des feuilletages de codimension 1*]{}, C. R. Acad. Sci. Paris, Série A [**273**]{}, 92-95. Candel A, Conlon L. 1999 *Foliations I*. Providence, Rhode Island: AMS.
Arnold VI, Khesin B. 1999 *Topological methods in hydrodynamics*. New York: Springer.
Yoshida Z, Morrison PJ. 2014 [*A hierarchy of noncanonical Hamiltonian system: circulation laws in an extended phase space*]{}, Fluid Dyn. Res. [**46**]{}, 031412. ([doi:10.1088/0169-5983/46/3/031412](https://doi.org/10.1088/0169-5983/46/3/031412))
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Yoshida Z, Tokieda T, Morrison PJ. 2017 [*Rattleback: a model of how geometric singularities induce dynamic chirality*]{}, Phys. Lett. A [**381**]{}, 2772-2777. ([doi:10.1016/j.physleta.2017.06.039](https://doi.org/10.1016/j.physleta.2017.06.039))
Moffatt HK, Tokieda T. 2008 [*Celt reversals: a prototype of chiral dynamics*]{}, Proc. Royal Soc. Edinburgh [**138A**]{} 361–368. ([doi:10.1017/S0308210506000679](https://doi.org/10.1017/S0308210506000679))
[^1]: I am extremely grateful to PJ Morrison for a hugely enlightening discussion. I would also like to acknowledge many useful conversations with JH Hannay.
|
---
abstract: 'This paper describes a novel approach to software engineering derived from the [*SP Theory of Intelligence*]{} and its realisation in the [*SP Computer Model*]{}. Despite superficial appearances, it is shown that many of the key ideas in software engineering have counterparts in the structure and workings of the SP system. Potential benefits of this new approach to software engineering include: the automation or semi-automation of software development, with support for programming of the SP system where necessary; allowing programmers to concentrate on ‘world-oriented’ parallelism, without worries about parallelism to speed up processing; support for the long-term goal of programming the SP system via written or spoken natural language; reducing or eliminating the distinction between ‘design’ and ‘implementation’; reducing or eliminating operations like compiling or interpretation; reducing or eliminating the need for verification of software; reducing the need for validation of software; no formal distinction between program and database; the potential for substantial reductions in the number of types of data file and the number of computer languages; benefits for version control; and reducing technical debt.'
author:
- 'J Gerard Wolff[^1]'
title: Software engineering and the SP Theory of Intelligence
---
[*Keywords:*]{} SP Theory of Intelligence; software engineering; automatic programming; natural language processing; compiling; interpretation; verification; validation; parallel processing; version control; technical debt.
Introduction
============
This paper is about a novel approach to software engineering with potential advantages over standard approaches. It is a considerable revision, expansion and development of preliminary ideas in @sp_benefits_apps [Section 6.6]. There is an outline description of the SP system in Appendix \[outline\_of\_sp\_system\_appendix\] with pointers to where fuller information may be found.
The novelty of the approach is because it derives from the [*SP Theory of Intelligence*]{} and its realisation in the [*SP Computer Model*]{}. Despite superficial appearances, it is shown that many of the key ideas in software engineering have counterparts in the structure and workings of the SP system.
It is envisaged that the SP theory and its realisation in the SP computer model will be the basis for an industrial-strength [*SP Machine*]{} (Appendix \[sp\_machine\_appendix\]) which would be the vehicle for software engineering as described in this paper.
The workings of the SP system may be classified as natural for three main reasons:
- [*Information compression in human learning, perception, and cognition*]{}. The SP system incorporates the principle that much of human learning, perception, and cognition may be understood as information compression. Relevant evidence derives from: research by Fred Attneave [-@attneave_1954], Horace Barlow [-@barlow_1959; -@barlow_1969], and others, exploring the role of information compression in human perception and cognition. Much additional evidence is described in @sp_compression.
- [*Information compression in language learning*]{}. A programme of research developing computer models of language learning (summarised in @wolff_1988) which demonstrates the importance of information compression in learning artificial analogues of natural language.
- [*Modelling aspects of human intelligence*]{}. Although the SP system is little more than the essentially simple concept of SP-multiple-alignment (Appendix \[sp\_multiple\_alignment\_appendix\]), it has proved to be remarkably versatile in modelling several aspects of human learning, perception, and cognition, as summarised in Appendix \[uai\_appendix\]. In general, the SP system is strongly oriented towards human and thus natural forms of computing.
As a preparation for the main body of the paper, the next section relates concepts in ordinary computers to concepts in the SP system. Sections that follow describe several potential advantages of the SP system in software engineering. All sections presuppose some understanding of the structure and workings of the SP system, as outlined in Appendix \[outline\_of\_sp\_system\_appendix\].
How concepts that are familiar in ordinary computer programming may be seen in the workings of the SP system {#relating_conventional_concepts_to_sp_concepts}
============================================================================================================
Superficially, the workings of an ordinary computer is quite different from the workings of the SP system. Ordinary computers are normally seen to work via the ‘execution’ of ‘procedures’ or ‘functions’ but the SP system works entirely via the compression of information. That the two kinds of processing may be seen to be equivalent is an important insight from the SP programme of research.
This section demonstrates how several of the concepts that are familiar in the programming of ordinary computers may be seen in the workings of the SP system.
‘Function’, ‘calling of a function’, ‘parameter’, and ‘conditional statement’ {#program_etc_section}
-----------------------------------------------------------------------------
We begin with a simple example: the kinds of things that need to be done in preparing a meal in a restaurant in response to an order from a customer, excluding any advance preparation of the ingredients.
### An outline of C code for preparing meals in a restaurant {#c_code_section}
In the C programming language, relevant functions for preparing meals in a restaurant are shown in outline in Figure \[prepare\_meal\_function\_figure\]. Here, the highest level structure is the ‘`prepare_meal()`’ function at the bottom of the figure, with subordinate functions above it, in accordance with convention.
The top-level function may be called like this: ‘`prepare_meal(0, 4, 1)`’. This has the effect of calling the ‘`starter()`’ function with the parameter ‘`0`’, the ‘`main_course(`’ function with the parameter ‘`4`’, and the ‘`pudding()`’ function with the parameter ‘`1`’. As can be seen in the subordinate functions, the parameters have the effect, via conditional statements, of calling the functions ‘`mussels()`’, ‘`salad()`’, and ‘`apple_crumble()`’. Each of these may, in an intelligent robot, prepare the corresponding dish, or may at least instruct a person to prepare that dish.
### An outline of an SP grammar for preparing meals in a restaurant
Figure \[prepare\_meal\_grammar\_figure\_1\] shows an SP [*grammar*]{}, comprising a collection of SP-patterns, which may be seen as a function for the preparation of meals corresponding to the example in Figure \[prepare\_meal\_function\_figure\].
The first SP-pattern in the figure, ‘`PM ST #ST MC #MC PD #PD #PM`’, describes the overall structure of the operation of preparing a meal. It is identified by the pair of SP-symbols ‘`PM ... #PM`’ which are mnemonic for “prepare meal”.
As with the example shown in Figure \[prepare\_meal\_function\_figure\], the main steps are the preparation of a starter (‘`ST ... #ST`’), the preparation of the main course (‘`MC ... #MC`’), and the preparation of a pudding (‘`PD ... #PD`’). Corresponding SP-patterns are shown in the second and subsequent rows in the figure.
### Building SP-multiple-alignments via information compression
To see how this grammar functions in practice, consider the SP-multiple-alignment shown in Figure \[prepare\_meal\_ma\_figure\_1\].[^2]
This SP-multiple-alignment is the best one created by the SP Computer Model with the New SP-pattern, ‘`PM 0 4 1 #PM`’, processed in conjunction with Old SP-patterns shown in Figure \[prepare\_meal\_grammar\_figure\_1\]. Here, the New SP-pattern may be seen as an economical description of what the customer ordered: a starter comprising a dish of mussels, represented by the code ‘`0`’; a main course chosen to be a salad, represented by the code ‘`4`’; and a pudding which in this case is apple crumble, represented by the code ‘`1`’.
Assuming that each of the SP-symbols ‘`mussels`’, ‘`salad`’, and ‘`apple_crumble`’, represents the execution of instructions for preparing the corresponding dish, or is at least an instruction to a person to prepare that dish, the whole SP-multiple-alignment may be seen to achieve the effect of preparing what the customer has ordered, much as with the example discussed in Section \[c\_code\_section\].
### How the concepts of ‘function’, ‘calling of a function’, ‘parameter’, and ‘conditional statement’ may be seen in the workings of the SP system
This example shows how the concepts ‘function’, ‘calling of a function’, ‘parameter’, and ‘conditional statement’ may be seen in the workings of the SP system:
- As mentioned earlier, the whole grammar in Figure \[prepare\_meal\_grammar\_figure\_1\] may be seen as a [*function*]{} for preparing a meal to meet a given order, like the outline code shown in Figure \[prepare\_meal\_function\_figure\].
- Of the remaining SP-patterns in Figure \[prepare\_meal\_grammar\_figure\_1\], each group of SP-patterns that begin with the same SP-symbol, such as ‘`ST 0 mussels #ST`’, ‘`ST 1 soup #ST`’, and ‘`ST 2 avocado #ST`’, may be seen as a subordinate [*function*]{} that is [*called*]{} from the higher-level function ‘`PM ST #ST MC #MC PD #PD #PM`’.
Although the example does not illustrate the point, it should be clear that each subordinate function may itself call one or more lower-level functions, and so on through as many levels as may be required. As we shall see in Section \[recursion\_section\], recursion is also possible.
- Each of the code SP-symbols ‘`0`’, ‘`4`’, and ‘`1`’, in the New SP-pattern ‘`PM 0 4 1 #PM`’, may be seen as a [*parameter*]{} to the top-level function.
- Since the code SP-symbol ‘`0`’ has the effect of selecting the SP-pattern ‘`ST 0 mussels #ST`’ from the set of SP-patterns ‘`ST 0 mussels #ST`’, ‘`ST 1 soup #ST`’, and ‘`ST 2 avocado #ST`’, the process of selection may be seen to achieve the effect of a [*conditional statement*]{} or [*if-then rule*]{} in an ordinary computer program, something like the C statement ‘`if (ST == 0) mussels() ;`’ in Figure \[prepare\_meal\_function\_figure\], meaning “If the value of ‘`ST`’ is ‘0’, perform the subordinate function ‘`mussels()`’, which itself means “prepare a dish of mussels”. Much the same may be said, [*mutatis mutandis*]{}, about the code SP-symbols ‘`4`’, and ‘`1`’.
Variables, values, and types {#variables_values_types_section}
----------------------------
In addition to the programming concepts already considered, the concepts ‘variable’, ‘value’, and ‘type’ may be seen in the workings of the SP system.
Consider, for example, the SP grammar shown in Figure \[salad\_grammar\_figure\]. This is an expansion of the “salad” main course entry in the grammar shown in Figure \[prepare\_meal\_grammar\_figure\_1\]. Instead of simply giving the name of the dish, this grammar provides for choices of ingredients in four categories: salad leaves (‘`L ... #L`’), root vegetables (‘`R ... #R`’), garnish (‘`G ... #G`’), and dressing (‘`D ... #D`’).
When the SP Computer Model is run with the New SP-pattern ‘`MC 2 1 0 1 #MC`’ and Old SP-patterns comprising the SP-patterns shown in Figure \[salad\_grammar\_figure\], the best SP-multiple-alignment created by the SP Computer Model is the one shown in Figure \[salad\_ma\_figure\].
In this example:
- Within the SP-pattern ‘`MC salad L #L R #R G #G D #D #MC`’ (column 1), each of the pair of SP-symbols ‘`L #L`’, ‘`R #R`’, ‘`G #G`’, and ‘`D #D`’, may be seen to represent the concept [*variable*]{} because they are slots where values may be inserted.
- The effect of the SP-multiple-alignment is to assign ‘`water_cress`’ to the first slot, ‘`potato`’ to the second slot, ‘`nuts`’ to the third slot, and ‘`vinaigrette`’ to the fourth slot. Those four things may be seen to be [*values*]{}, each one assigned to an appropriate variable.
- For each of the four variables, its [*type*]{}—meaning the range of values that it may take—may be seen to be defined by the grammar shown in Figure \[salad\_grammar\_figure\]. For example, possible values for the variable ‘`L #L`’ may be seen to be ‘`lettuce`’, ‘`beetroot_leaves`’, ‘`water_cress`’, and ‘`spinach`’. Likewise for the other three variables.
Structured programming {#structured_programming_section}
----------------------
An established feature of software engineering today, which is now partly but not entirely subsumed by object-oriented programming (next), is ‘structured programming’ [@jackson_1975] in which the central idea is that programs should comprise well-defined structures which should reflect the structure of the data that is to be processed and should [*never*]{} use the ‘goto’ statement of an earlier era.
To a large extent, the SP system incorporates the principles of structured programming, since unsupervised learning in the SP system creates structures that reflect the structure of incoming data. And other kinds of processing in the SP system, such as pattern recognition or reasoning, is achieved by recognising and processing similar structures in new data, without the use of anything like a ‘goto’ statement.
Object-oriented design or programming {#oo_design_programming_section}
-------------------------------------
From its introduction in the [*Simula*]{} computer language [@birtwistle_etal_1973], ‘object-oriented programming’ (OOP) and the closely-related ‘object-oriented design’ (OOD) have become central in software engineering and in such widely-used programming languages as C++ and Java.
Key ideas in OOP/OOD are that the structure of each computer program should reflect the objects to which it relates—people, packages, fork-lift trucks, and so on—and the classes and subclasses in which each object belongs. This not only helps to make computer programs easy to understand but it means that the features of any specific object may be ‘inherited’ from the classes and subclasses to which it belongs.
Inheritance applies to all the objects in a given class, meaning that there is an overall saving or compression of information compared with what would be needed without inheritance. In this respect, OOP/OOD is very much in keeping with the central importance of information compression in the SP system.
In Figure \[class\_hierarchy\_figure\], the SP-multiple-alignment produced by the SP Computer Model, with the New SP-pattern ‘`white-bib eats furry purrs`’ and a set of Old SP-patterns representing different categories of animal and their attributes, shows how a previously-unknown entity with features shown in the New SP-pattern in column 1 may be recognised at several levels of abstraction: as an animal (column 1), as a mammal (column 2), as a cat (column 3) and as the specific cat “Tibs” (column 4). These are the kinds of classes used in ordinary systems for OOP/OOD.
From this SP-multiple-alignment, we can see how the entity that has now been recognised [*inherits*]{} unseen characteristics from each of the levels in the class hierarchy: as an animal (column 1) the creature ‘`breathes`’ and ‘`has-senses`’, as a mammal it is ‘`warm-blooded`’, as a cat it has ‘`carnassial-teeth`’ and ‘`retractile-claws`’, and as the individual cat Tibs it has a ‘`white-bib`’ and is ‘`tabby`’.
Recursion {#recursion_section}
---------
The SP system does not provide for the repetition of procedures via these kinds of statement: [*while ...*]{}, [*do ... while ...*]{}, [*for ...*]{}, or [*repeat ... until ...*]{}. But the same effect may be achieved via recursion, as illustrated in Figure \[recursion\_ma\_figure\].
In the figure, the SP-symbols ‘`a6 b1 b1 b1 c4 d3`’ in the New SP-pattern in row 0 (‘`pg a6 b1 b1 b1 c4 d3 #pg`’) may be seen as parameters for the SP ‘program’ or grammar for this example (not shown on this occasion).
One point of interest here is that the SP-pattern ‘`ri ri1 ri #ri b #b #ri`’ (which appears in columns 5, 7, and 9) is recursive because it is self-referential—because the pair of SP-symbols ‘`ri #ri`’ within the larger SP-pattern ‘`ri ri1 ri #ri b #b #ri`’ may be matched and unified with the same two SP-symbols at the beginning and end of that larger SP-pattern. Hence, the larger SP-pattern contains a reference to itself.
Another point of interest is that the recursive SP-pattern ‘`ri ri1 ri #ri b #b #ri`’, and its connected SP-pattern ‘`b b1 procedure_B #b`’, each occur 3 times in the SP-multiple-alignment in Figure \[recursion\_ma\_figure\], although each of them only occurs once in the grammar for this SP-multiple-alignment. Hence, the grammar is relatively compressed compared with what would be needed if all possible repetitions were stored explicitly.
With any kind of recursion, something is needed to tell the system when to stop the repetition. In our example, the number of repetitions is specified explicitly by the three instances of the SP-symbol ‘`b1`’ within the New SP-pattern. Other devices may also be used.
The full or partial automation of software development {#full_partial_automation_se_section}
======================================================
This and the following main sections describe potential advantages of the SP system in software engineering compared with software engineering with conventional computers.
This section considers the full or partial automation of software and the associated issue of generalisation in software, and how to avoid under- and over-generalisaton.
Automation of software development
----------------------------------
Assuming that the SP Machine (Appendix \[sp\_machine\_appendix\]) has been developed to the stage where it has robust abilities for unsupervised learning with both one-dimensional and two-dimensional SP-patterns, and assuming that residual problems in that area have been solved (Appendix \[unsupervised\_learning\_appendix\]), the SP Machine is likely to prove useful in both the automatic and semi-automatic creation of software, discussed in this and the following subsection.
At least two things suggest that such possibilities are credible:
- As noted in Section \[structured\_programming\_section\] and Appendix \[sp\_cc\_differences\_in\_rk\_appendix\], it has been recognised for some time, in connection with the concept of “structured programming”, that the structure of software should mirror the structure of the data that it is designed to process [@jackson_1975]. This fits well with the observation that in forms of unsupervised learning such as grammatical inference, the structure of the resulting grammar reflects the structure of the data from which it was derived.
- In the same vein, in connection with “object-oriented design” and “object-oriented programming” (Section \[oo\_design\_programming\_section\]), it is well-established that a well-structured program should reflect the structure of entities and classes of entities that are significant in the workings of the program. This fits well with the observation that unsupervised learning in the SP system appears to conform to the ‘DONSIC’ principle [@sp_extended_overview Section 5.2]: [*the discovery of natural structures via information compression*]{}—where ‘natural’ means aspects of our environment such as ‘objects’ which we perceive to be natural.
### Example: learning in an autonomous robot {#learning_in_autonomous_robot_section}
Perhaps the best example of how the SP system may facilitate automatic programming is in autonomous robots that learn continually via their senses, much as people do [@sp_autonomous_robots]. Here, the robot’s ever-increasing store of knowledge, together with any in-built motivations, provide the basis for many potential inferences (@wolff_2006 [Chapter 7], @sp_extended_overview [Section 10]) and, perhaps more important in the present context, the creation of one or more plans (@wolff_2006 [Chapter 8], @sp_extended_overview [Section 12]), each one of which may be regarded as a program to guide the robot’s actions.
The potential for this kind of development raises important issues about how much autonomy should be granted to any robot and how external controls may be applied. Pending the resolution of such issues, there is potential in the SP system for more humdrum kinds of automatic programming, as described in the next two subsections.
### Example: processing data received by the SKA {#ska_example_section}
The fully automatic creation of software should be possible in situations where there is a body of data that represents the entire problem or a realistically large sample of it. An example is the large volumes of data that will be gathered by the Square Kilometre Array (SKA)[^3] when it is completed.
With data like this, unsupervised learning by the SP Machine should build grammars that represent entities and classes of entity—such as stars and galaxies—in two dimensions at least, and possibly in three dimensions. And its grammars should also embrace ‘procedural’ or ‘process’ regularities in the time dimension.
Any such grammar may be seen as a ‘program’ for the analysis of similar kinds of data in the future. A neat feature of the SP system is that the SP-multiple-alignment construct serves not only in unsupervised learning but also, without modification, in such operations as SP-pattern recognition, reasoning, and more (Appendix \[versatility\_in\_intelligence\_appendix\]).
With an area of application like the processing of data received by the SKA, it may of course happen that significant structures or events—such as supernovas or gamma-ray bursts—do not appear in any one sample of data. For that reason, unsupervised learning should be an ongoing process, much as in people, so that the system may gain progressively more knowledge of its target environment as time goes by.
Some more observations relating to this example are described in Section \[possible\_augmentations\_section\].
### Example: programming by demonstration {#programming_by_demonstration_example}
Another situation where the SP Machine may achieve fully-automatic creation of software is with a technique for programming robots called “programming by demonstration”.[^4]
As an example, a person who is skilled at some operation in the building of a car (such as paint-spraying the front of the car) may take the ‘hand’ of a robot and guide it through the sequence of actions needed to complete the given operation. Here, signals from sensors in various parts of the robot’s arm, including the robot’s actuators or ‘muscles’, would be recorded and the record would constitute a preliminary kind of ‘program’ of the several positions of the arm and actuators that are needed to complete the operation.
Any such preliminary program may be processed by the SP system to convert it into something that more closely resembles an ordinary program, with the equivalent of subroutines, repetition of operations, and conditional statements. To allow for acceptable variations in the task, there should also be appropriate generalisation from the raw data, as described in Section \[generalisation\_section\].
Some more observations relating to this example are described next.
### Possible augmentations {#possible_augmentations_section}
An assumption behind the two examples just described is that the grammar or program created via unsupervised learning would do everything that is needed.
In many cases, this would probably be true. This is because of a neat feature of the SP system: that the SP-multiple-alignment subsystem is not only an important part of unsupervised learning but is also the key to such operations as pattern recognition, several kinds of reasoning, retrieval of information, and problem solving (Appendix \[versatility\_in\_intelligence\_appendix\]). With the SKA example (Section \[ska\_example\_section\], these kinds of operations may be all that is required. With the programming-by-demonstration example (Section \[programming\_by\_demonstration\_example\]), the program created via unsupervised learning may function directly in controlling the robot arm.
But the user of the SKA system might want to do such things as showing stars in red, galaxies in green, and so on. And the user of the programming-by-demonstration system might want to add some bells and whistles such as playing musical sounds as the robot works.
Clearly, such augmentations fall outside what could be created automatically via unsupervised learning. They take us into to the realm of semi-automatic creation of software, discussed next.
Semi-automatic creation of software {#semi-full_partial_automation_se_section}
-----------------------------------
With some kinds of application, it seems unlikely that the creation of relevant software could be fully automated in the foreseeable future. One example is the kinds of augmentation to an automatically-created program described in Section \[possible\_augmentations\_section\]. Another example is the kind of software that is needed to manage a business—with knowledge of people, vehicles, furniture, packages, warehouses, relevant rules and regulations, and so on.
With the latter kind of problem, there appears to be potential for the system to assist in the refinement of human-created software by detecting redundancies in any draft design, and inconsistencies from one part of the design to another. On the assumption that the software is developed using SP-patterns and is hosted on an SP Machine (as outlined in Section \[non-automatic\_programming\_software\_section\]), then the SP Machine may be a vehicle for verification and validation of the software as described in Sections \[verification\_section\] and \[validation\_section\].
At some point in the future, it is conceivable that knowledge about how a business operates may, at some stage, be built up by an intelligent autonomous robot of the kind described in @sp_autonomous_robots that is allowed to explore different areas of the business, observing the kinds entity and operation that are involved, asking questions, and so on. But for the foreseeable future, it seems likely that any software that may have been created by such a robot would need to be augmented and refined by people.
Generalisation and the avoidance of under- and over-generalisation {#generalisation_section}
------------------------------------------------------------------
As a rule, any given computer program is more general than any set of examples that it may process. For example, an ordinary spreadsheet program can work with millions or perhaps billions of different sets of data, far more than it would ever be used for in practice.
Since we have been considering the possibility that software may be created automatically or semi-automatically in the manner of unsupervised learning (Sections \[full\_partial\_automation\_se\_section\] and \[semi-full\_partial\_automation\_se\_section\]), we need to consider how the system would generalise correctly from the examples it has been given, without either under-generalisation (sometimes called ‘overfitting’) or over-generalisation (sometimes called ‘underfitting’).
The SP system provides an answer outlined in @sp_extended_overview [Section 5.3],[^5] with some supporting evidence. In brief, it appears that correct generalisation may be achieved, without either under- or over-generalisation, like this:
1. Given a body of raw data, [**I**]{}, compress it as much as possible with the program for unsupervised learning.
2. Divide the resulting compressed version of [**I**]{} into two parts: a [*grammar*]{}, [**G**]{}, which represents the recurring features of [**I**]{}, and an [*encoding*]{}, [**E**]{}, of [**I**]{} in terms of [**G**]{}.
3. Discard [**E**]{} and retain [**G**]{}.
Here, [**G**]{} may be seen to be a program for processing [**I**]{} and for processing many other bodies of data with the same general characteristics as [**I**]{}, without either under- or over-generalisation.
Non-automatic programming of the SP system {#non-automatic_programming_software_section}
==========================================
If or when the automatic creation of software is not feasible, or if something more than small revisions are needed with software that has been created semi-automatically, then something like ordinary programming will be needed.
In principle, this can be done using SP-patterns directly. But, mainly for reasons of human psychology, some kind of ‘syntactic sugar’ or other aids may be helpful for programmers. Here are four possibilities:
- With an SP-pattern like ‘`NP D #D N #N #NP`’ in row 4 of Figure \[parsing\_figure\], it may he helpful if, when the first SP-symbol (‘`NP`’) has been typed in, the programming environment would automatically insert the balancing last SP-symbol (‘`#NP`’).
- Unless or until programmers become used to how things are done in the SP system, it may be helpful to create a programming environment in which SP concepts are presented in a manner that resembles ordinary programming concepts, as described in Section \[relating\_conventional\_concepts\_to\_sp\_concepts\].
- Instances of the object-oriented concept of a class-inclusion hierarchy (Section \[oo\_design\_programming\_section\]), and instances of any part-whole hierarchy (dividing an object into its parts and subparts) may be represented graphically and implemented with equivalent sets of SP-patterns.
There will also be a need for programmers to specify aspects of parallel and sequential processing, as described in Section \[sp\_advantage\_with\_parallel\_processing\_section\], next.
A potential advantage of the SP Machine in the application of parallel processing {#sp_advantage_with_parallel_processing_section}
=================================================================================
In the application of parallel processing in the SP Machine, it is important to distinguish between two kinds of parallelism:
- [*World-oriented parallelism*]{}. World-oriented parallelism means the kind of parallelism that one might observe in the world, including the activities of people: it may rain at the same time as the wind blows; the players in a game of football are all doing different things at the same time; a cook may prepare the icing for a cake at the same time as the cake is baking; and so on.
- [*Machine-oriented parallelism*]{}. In the workings of a computer, there may be parallel processing in the MapReduce model, in pipelining, in SIMD parallelism, in MIMD parallelism, and so on.
In the programming of an ordinary high-parallel supercomputer or high-parallel computing cluster, both kinds of parallelism may be applied, with little or no distinction between them. For example, a programmer who is developing a flight simulator may adopt machine-oriented parallelism for such processing as matrix multiplication and apply world-oriented parallelism in modelling the many processes that are involved, largely in parallel, in a plane’s flight.
A potential advantage of the SP machine is that programmers of such a machine may be largely relieved of the need to worry about machine-oriented parallelism and may concentrate on the programming of world-oriented parallelims.
This potential advantage arises because of evidence that the SP system has potential to serve as a [*universal framework for the representation and processing of diverse kinds of knowledge*]{} (UFK) [@sp_big_data Section III].
The evidence is that, already, one relatively simple conceptual and computational framework—SP-multiple-alignment—demonstrates versatility in aspects of intelligence, versatility in the representation of diverse forms of knowledge, with clear potential for the seamless integration of diverse aspects of intelligence and diverse forms of knowledge, in any combination (see [@sp_intro_2018 Sections 4, 5, and 6] and pointers from there). And there are reasons to believe that that versatility may be extended.
If or when the SP system can be developed with full human-like versatility, taking full advantage of machine-oriented parallel processing, then it seems likely that programmers can largely relieved of concerns about that kind of parallelism and, for any given area of application, they may concentrate world-oriented parallelism in that domain.
Programming via natural language {#programming_via_nl_section}
================================
One of the strengths of the SP system is in the processing of natural language, mentioned in Appendices \[versatility\_in\_rk\_appendix\] and \[versatility\_in\_intelligence\_appendix\], and described in more detail in @sp_extended_overview [Section 8] and @wolff_2006 [Chapter 5].
There is clear potential in the SP system for developing human-level processing of writing, and ultimately speech, but there will be some difficult hurdles to overcome, probably requiring a two-pronged attack: working on problems in the processing of natural language together with problems in the unsupervised learning of syntactic knowledge, semantic knowledge, and syntactic/semantic associations [@devt_sp_machine Sections 9 and 10].
If or when these problems are solved, there is potential for programming the SP system using written or spoken natural language, in much the same way that people can be given written or spoken instructions. However, achieving human levels of understanding is an ambitious goal and is not likely to be realised soon.
Bringing ‘design’ closer to ‘implementation’ {#design_implementation_closer_section}
============================================
It has been established for some time that, in conventional development of software, one should begin with a relatively abstract high-level design (which is often represented graphically) and then translate that into a working program. There seem to be three main reasons for this approach:
- With any kind of design, it is often useful to establish a relatively abstract “big picture” before filling in details.
- For the kinds of reasons described in Section \[non-automatic\_programming\_software\_section\], it may be useful to disguise the details of a program behind syntactic sugar that is more congenial for programmers.
- Even with ‘high’ level programming languages such as C++, Python, or Java, or ‘declarative’ systems such as Prolog, it is often necessary to pay attention to the details of how the underlying machine will run a program, details that are not relevant to the more abstract ‘design’ of the software, with its focus on entities and processes that are significant for the user.
The SP system probably makes no difference to the first and second of the above points, but it is likely to be helpful with the third. This is because, in the manner of declarative programming systems, it will probably allow programmers to specify ‘what’ computations are to be achieved, and to reduce or eliminate the need to consider ‘how’ the computations should be done.
Possible reductions in the need for operations like compiling or interpretation {#no_compiling_or_interpretation_section}
===============================================================================
At first sight, the SP system eliminates the need for anything like compiling or interpretation. This is because it works entirely via searches for full or partial matches between SP-patterns, or parts of SP-patterns, with corresponding unifications.
But it is likely that, in the development of the SP Machine, indexing will be introduced to record the first match between a given SP-symbol and any other SP-symbol, and thus speed up the later retrieval of the zero or more matching partners of the given SP-symbol [@devt_sp_machine Section 3.4]. And it is likely that similar measures will be introduced into the computer model for SP-neural [@devt_sp_machine Section 13.2], a version of the SP Theory expressed in terms of neurons and their interconnections.
Indexing of that kind is similar in some respects to the use of compiling or interpretation in a conventional computing system. Hence it would be misleading to suggest that the SP system would eliminate the need for such operations. But there are potential gains in this area, especially if, at some later stage, it became feasible to introduce very fast and highly-parallel searching for matches between SP-patterns which may reduce or eliminate the need for indexing.
Verification {#verification_section}
============
The SP system has potential to reduce the need for ‘verification’ of software—meaning the process of checking that a software system meets its specifications—and there is corresponding potential for improvements in the quality of software. The main reasons for these potential benefits are:
- [*The potential of the system for automatic or semi-automatic creation of software*]{} (Sections \[full\_partial\_automation\_se\_section\] and \[semi-full\_partial\_automation\_se\_section\]). To the extent that automatic or semi-automatic creation of software is possible, it should reduce or eliminate human-induced errors in software.
- [*Potential reductions in the sizes of software systems*]{}. The potential of the system for reductions in the overall sizes of software systems (Section \[overall\_simplification\_of\_applications\_section\]) means that there are likely to be fewer opportunities to introduce bugs into software, and, probably, less searching would be required in the detection of bugs via static analysis of software.
- [*Bringing ‘design’ closer to ‘implementation’*]{}. To the extent that ‘design’ and ‘implementation’ may be merged (Section \[design\_implementation\_closer\_section\]), and in particular to the extent that SP software may concentrate on ‘what’ the user needs and reduce or eliminate details of ‘how’ the underlying machine may meet those needs, there is potential to reduce the numbers of bugs in programs.
Validation {#validation_section}
==========
In addition to its potential with verification, the SP system has potential to strengthen the process of “validation” in software development—meaning the process of checking that a software system fulfills its intended purpose.
As with verification, the potential of the SP system for the automatic or semi-automatic creation of software means elimination or reduction of the kinds of human error that may send a program off track.
Also, the potential of the SP system to bring ‘design’ and ‘implementation’ closer together (Section \[design\_implementation\_closer\_section\]) can mean fewer opportunities for a program to drift away from its original conception.
Seamless integration of ‘software’ with ‘database’ {#integration_of_software_with_database_section}
==================================================
In the SP system, [*all*]{} kinds of knowledge are represented with arrays of atomic [*SP-symbols*]{} in one or two dimensions (Appendix \[overview\_appendix\]), and [*all*]{} kinds of processing is achieved via the matching and unification of SP-patterns. For these two reasons, and because of the system’s potential for [*universal artificial intelligence*]{} (UAI) (Appendix \[uai\_appendix\]), there would be no distinction in the SP system between ‘software’ and ‘database’, as there is normally in conventional software engineering projects.
A potential benefit of this kind seamless integration of software and database is elimination of awkward incompatibilities between different kinds of knowledge and elimination of the need for translations where incompatibilities exist.
An overall simplification of computing applications {#overall_simplification_of_applications_section}
===================================================
With the SP system, there is potential for an overall simplification of applications compared with what is required with ordinary computers [@sp_benefits_apps Section 5].
In broad terms, this potential arises because of the way in which conventional software contains often-repeated procedures for searching amongst data, and similar ‘low level’ operations needed to overcome shortcomings in conventional CPUs. In an SP system, the ‘CPU’ is relatively complex but with fewer of the shortcomings of conventional CPUs, so that that relative complexity is, probably, more than offset by simplifications in software with data, as shown schematically in Figure \[two\_schematic\_computers\_figure\]. That relative advantage is likely to grow, roughly in proportion to the numbers of applications and their sizes.
![Schematic representations of a conventional computer and the proposed [*SP machine*]{}, showing potential benefits in terms of simplification, as discussed in the text. Adapted from Figure 4.7 in [@wolff_2006], with permission.[]{data-label="two_schematic_computers_figure"}](computers_schematic4.pdf){width="90.00000%"}
This kind of idea is not new. In the early days of databases, each database had its own procedures for searching and for retrieval of information, and it had its own user interface and procedures for printing, and so on. People soon realised that it would make better sense to develop a general-purpose system for the management of data, with a user interface and system for retrieval of data, and to load it with different bodies of data according to need. There was a similar evolution in expert systems, from bespoke systems to general-purpose ‘shells’.
Reducing the variety of formats and formalisms in computing {#reducing_formats_and_formalisms_section}
===========================================================
As noted in Section \[integration\_of\_software\_with\_database\_section\], the SP system has potential for [*universal artificial intelligence*]{} (UAI). What this means in the SP programme of research, and how the concept of a UAI differs from alternatives such as the concept of a universal Turing machine, is discussed in Appendix \[uai\_appendix\].
If indeed this expectation is born out, and the evidence is strong, there is clear potential for use of the SP Machine to clean up the curse of variety in the thousands of different formats and formalisms which exist for the representation of data, and the hundreds of different computer languages for describing how data may be processed (Appendix \[the\_curse\_of\_variety\_appendix\]).
Version control {#version_control_section}
===============
In a typical software engineering project, there is a need to keep track of the parts and sub-parts of the developing program. At the same time, there is a need to keep track of a hierarchy of versions and subversions. And, associated with each part or version, there may be several different kinds of document, including a statement of requirements, a high-level design, a low-level design, and notes. To avoid awkward inconsistencies, these things should be smoothly integrated.[^6]
The SP system provides a neat solution to the problem of integrating a class-inclusion hierarchy with a part-whole hierarchy, as described in @sp_extended_overview [Section 9.1] and @wolff_2006 [Section 6.4].[^7] Although these sources do not demonstrate the point, it appears that the SP system also provides for each version or part to have one or more associated documents, as outlined above.
Also relevant to these issues is a brief discussion of how to maintain multiple versions and parts of a document or web page in @sp_benefits_apps [Section 6.10.3].
Technical debt {#technical_debt_section}
==============
As noted in @sp_benefits_apps [Section 6.6.6], the SP system has potential to reduce or eliminate the problem of ‘technical debt’, meaning the way in which software systems can become progressively more unmanageable with the passage of time, owing to an accumulation of postponed or abandoned maintenance tasks, or a progressive deterioration in the design quality or maintainability of the software via the repeated application of ‘fixes’ in response to short-terms concerns, without sufficient attention to their global and long-term effects.
The SP system may reduce or eliminate the problem of technical debt by streamlining the process of software development via automatic or semi-automatic automation of software development, by reducing the gap between design and implementation, by streamlining processes of verification and validation, and other facilitations described in preceding sections.
Conclusion
==========
This paper describes a novel approach to software engineering derived from the [*SP Theory of Intelligence*]{} and its realisation in the [*SP Computer Model*]{}. It is anticipated that the SP Theory and the SP Computer Model, together, will be the basis for the development of an industrial-strength [*SP Machine*]{}. And a mature version of the SP Machine is seen as the likely vehicle for software engineering as described in this paper.
Although concepts associated with software engineering may seem far removed from the structure and workings of the SP system, many of those concepts map quite neatly into elements of the SP system (Section \[relating\_conventional\_concepts\_to\_sp\_concepts\]).
Potential benefits of this new approach to software engineering include:
- [*The automation of semi-automation of software development*]{}. Taking advantage of the SP system’s strengths and potential in unsupervised learning, there is clear potential for the automation or semi-automation of software development.
- [*Non-automatic programming of the SP system*]{}. Where it is not possible to create software automatically, or when human assistance is needed, there is clear potential for programming the SP system in much the same way as a conventional system.
- [*Programming via natural language*]{}. An ambitious goal, which is not likely to be realised soon, is to bring the SP system to a point where it has human levels of understanding and production of natural language, so that the SP system may be ‘programmed’ in much the same way that people can be given written or spoken instructions.
- [*Reducing or eliminating the distinction between ‘design’ and ‘implementation’*]{}. By contrast with conventional systems, there is potential in the SP system to reduce or eliminate the distinction between ‘design’ and ‘implementation’. This is because aspects of software design, such as structured programming and object-oriented programming, may be expressed directly with SP-patterns.
- [*Reducing or eliminating operations like compiling or interpretation*]{}. The SP system has potential to reduce or eliminate operations like compiling or interpretation. This is because the system works directly on ‘source’ code by searching for patterns or parts of patterns that match each other. But it seems likely that indexing of matches between SP-symbols will speed up the system, and the compiling of such an index may be seen to be similar to what is entailed in conventional compiling or interpretation.
- [*Reducing or eliminating the need for verification of software*]{}. The need for verification of SP software may be reduced or eliminated: via the potential of the SP system for the automatic or semi-automatic creation of software; because compression of software is likely to reduce the opportunities for bugs to be introduced; and because there is likely to be a reduced need to bridge the divide between design and implementation.
- [*Reducing or eliminating the need for validation of software*]{}. The SP system also has potential to help ensure that software fulfills its intended purpose. This is because of the system’s potential for automatic or semi-automatic creation of software and because of the way in which design and implementation may be brought closer together or merged.
- [*No formal distinction between program and database*]{}. Unlike conventional systems, where ‘programs’ and ‘databases’ are distinguished quite sharply, there is no formal distinction of that kind in the SP system because all kinds of knowledge are expressed with SP-patterns. This can mean useful simplifications on occasion, and it can reduce or remove awkward incompatibilities.
- [*Potential for an overall simplification of computing applications*]{}. Despite the fact that the processing ‘core’ of the SP system is, almost certainly, more complex than the CPU of a conventional computer, there is potential with the SP system for an overall simplification of computing applications, when hardware and software are considered together.
- [*Potential for substantial reductions in the number of types of data file and the number of computer languages*]{}. Because of the SP system’s potential as a [*universal artificial intelligence*]{} (UAI), there is potential to reduce the many thousands of types of data file to one, and to reduce the hundreds of different computer languages to one.
- [*Allowing programmers to concentrate on ‘world-oriented’ parallelism, without worries about parallelism to speed up processing*]{}. With a mature version of the SP Machine, it is intended that parallelism that is designed only for the purpose of speeding up processing will be built into the system, so that programmers need not worry about it. They would be free to concentrate on parallelism in the real world, perhaps with assistance from unsupervised learning.
- [*Benefits for version control*]{}. The SP system has potential to help organise all the knowledge associated with any given software development project, with provision for: the representation of versions of the software; the representation of parts and sub-parts of the software; the seamless integration of version hierarchies with part-whole hierarchies; and, for any given version or part, the representation of the one or more kinds of information associated with that element. It provides for cross-classification where that is required.
- [*Reducing technical debt*]{}. The potential of the SP system to increase the efficiency of software development can mean reductions in ‘technical debt’, meaning the way in which software systems can become progressively more unmanageable with the passage of time, owing to short-term fixes and the postponement or abandonment of maintenance tasks.
Appendices {#appendices .unnumbered}
==========
Outline of the SP system {#outline_of_sp_system_appendix}
========================
To help ensure that this paper is free standing, the SP system is described here in outline with enough detail to make the rest of the paper intelligible.
The [*SP Theory of Intelligence*]{} and its realisation in the [*SP Computer Model*]{} is the product of a unique extended programme of research aiming to simplify and integrate observations and concepts across artificial intelligence, mainstream computing, mathematics, and human learning, perception, and cognition, with information compression as a unifying theme.[^8]
The latest version of the SP Computer Model is SP71. Details of where the source code and associated files may be obtained are here: [www.cognitionresearch.org/sp.htm\#ARCHIVING](http://www.cognitionresearch.org/sp.htm#ARCHIVING).
It is envisaged that the SP Computer Model will provide the basis for the development of an industrial-strength [*SP Machine*]{}, described briefly in Appendix \[sp\_machine\_appendix\], below.
The SP system is described most fully in @wolff_2006 and quite fully but more briefly in @sp_extended_overview. Other publications from this programme of research are detailed, many with download links, on [www.cognitionresearch.org/sp.htm](http://www.cognitionresearch.org/sp.htm).
Overview {#overview_appendix}
--------
The SP Theory is conceived as a brain-like system which receives [*New*]{} information via its senses and stores some or all of it in compressed form as [*Old*]{} information, as shown schematically in Figure \[sp\_input\_perspective\_figure\].
![Schematic representation of the SP system from an ‘input’ perspective. Reproduced with permission from Figure 1 in @sp_extended_overview.[]{data-label="sp_input_perspective_figure"}](sp_abstract_colour.pdf){width="60.00000%"}
Both New and Old information are expressed as arrays of atomic [*SP-symbols*]{} in one or two dimensions called [*SP-patterns*]{}. To date, the SP Computer Model works only with one-dimensional SP-patterns but it is envisaged that it will be generalised to work with two-dimensional SP-patterns.
In this context, an ‘SP-symbol’ is simply a mark that can be matched with any other SP-symbol to determine whether they are the same or different. No other result is permitted. Apart from some distinctions needed for the internal workings of the SP system, SP-symbols do not have meanings such as ‘plus’ (’$+$’), ‘multiply’ (‘$*$’), and so on. Any meaning associated with an SP-symbol derives entirely from other SP-symbols with which it is associated.
Multiple sequemce alignments in bioinformatics {#multiple_alignments_in_bioinformatics_appendix}
----------------------------------------------
At the heart of the SP system is information compression via the matching and unification of patterns (ICMUP). More specifically, a central part of the SP system is a concept of [*SP-multiple-alignment*]{}, borrowed and adapted from the concept of ‘multiple sequence alignment’ in bioinformatics.
The original concept is an arrangement of two or more DNA sequences or sequences of amino acid residues, in rows or columns, with judicious ‘stretching’ of selected sequences in a computer to bring symbols that match each other from row to row, as many as possible, into line. An example of such a multiple sequennce alignment of five DNA sequences is shown in Figure \[DNA\_figure\].
SP-multiple-alignments in the SP system {#sp_multiple_alignment_appendix}
---------------------------------------
In the SP system, multiple alignments are sufficiently different from those in bioinformatics for them to be given a different name: [*SP-multiple-alignments*]{}.[^9] The distinctive features of an SP-multiple-alignment are:
- One New SP-pattern is shown in row 0 (or column 0 when SP-patterns are arranged in columns).[^10]
- The Old SP-patterns are shown in the remaining rows (or columns), one SP-pattern per row (or column).
- As with the original concept of multiple alignment, the aim in building multiple alignments is to bring matching SP-symbols into alignment. More specifically in SP-multiple-alignments, the aim is to create or discover one or more ‘good’ SP-multiple-alignments that allow the New SP-pattern to be encoded economically in terms of the Old SP-patterns. How this encoding is done is described in @wolff_2006 [Section 2.5] and in @sp_extended_overview [Section 4.1].
An example of an SP-multiple-alignment is shown in Figure \[parsing\_figure\].
In this SP-multiple-alignment, a sentence is shown as a New SP-pattern in row 0. The remaining rows show Old SP-patterns, one per row, representing grammatical structures including words. The overall effect is to analyse (parse) the sentence into its parts and subparts. The SP-pattern in row 8 shows the association between the plural subject of the sentence, marked with the SP-symbol ‘`Np`’, and the plural main verb, marked with the SP-symbol ‘`Vp`’.
Because, with most ordinary multiple sequence alignments or with SP-multiple-alignments, there is an astronomically large number of ways in which sequences may be aligned, discovering good multiple alignments means the use of heuristic methods: building each multiple alignment in stages and discarding all but the best few multiple alignment at the end of each stage. With this kind of technique it is normally possible to find multiple alignments that are reasonably good but it is not normally possible to guarantee that the best possible multiple alignment has been found.
The concept of SP-multiple-alignment has proved to be extraordinarily powerful: in aspects of intelligence (Appendix \[versatility\_in\_intelligence\_appendix\]), in the representation of knowledge (Appendix \[versatility\_in\_rk\_appendix\]), and in the seamless integration of diverse aspects of intelligence and diverse kinds of knowledge in any combination (Appendix \[seamless\_integration\_appendix\]). It could prove to be as significant for an understanding of intelligence as is DNA for biological sciences: it could be the ‘double helix’ of intelligence.
Unsupervised learning {#unsupervised_learning_appendix}
---------------------
Unsupervised learning in the SP system is described quite fully in @wolff_2006 [Sections 3.9 and 9.2]. The aim with unsupervised learning in the SP system is, for a given set of New SP-patterns, to create one or two [*grammars*]{}—meaning collections of Old SP-patterns—that are effective at encoding the given set of New SP-patterns in an economical manner.
The process of creating good grammars entails the creation of Old SP-patterns, partly by the direct assimilation of New SP-patterns and partly via the building of SP-multiple-alignments—which provides a means of creating Old SP-patterns and via the splitting of New SP-patterns and the splitting of pre-existing Old SP-patterns.
The building of SP-multiple-alignments also provides a means of evaluating candidate grammars in terms of their effectiveness at encoding the given set of New SP-patterns in an economical manner.
As with the building of SP-multiple-alignments, the creation of good grammars requires heuristic search through the space of alternative grammars: creating grammars in stages and discarding low-scoring grammars at the end of each stage.
The SP Computer Model can discover plausible grammars from samples of English-like artificial languages. This includes the discovery of segmental structures, classes of structure, and abstract SP-patterns.
At present, the program has two main weaknesses outlined in @sp_extended_overview [Section 3.3]: it does not learn intermediate levels of abstraction or discontinuous dependencies in data. However, it appears that these problems are soluble, and it seems likely that their solution would greatly enhance the performance of the system in the learning of diverse kinds of knowledge.
To ensure that unsupervised learning in the SP system is robust and useful across a wide range of different kinds of data, it will be necessary for the system, including its procedures for unsupervised learning, to have been generalised for two-dimensional SP-patterns as well as one-dimensional SP-patterns (Appendix \[overview\_appendix\]).
The SP Machine {#sp_machine_appendix}
--------------
As mentioned earlier, it is envisaged that an industrial-strength [*SP Machine*]{} will be developed from the SP Theory and the SP Computer Model [@devt_sp_machine]. Initially, this will be created as a high-parallel software virtual machine, hosted on an existing high-performance computer. An interesting possibility is to develop the SP Machine as a software virtual machine that is driven by the high-parallel search processes in any of the leading internet search engines.
Later, there may be a case for developing new hardware for the SP Machine, to take advantage of optimisations that may be achieved by tailoring the hardware to the characteristics of the SP system. In particular, there is potential for substantial gains in efficiency and savings in energy compared with conventional computers by taking advantage of statistical information that is gathered by the SP system as a by-product of how it works (@sp_big_data [Section IX], @sp_autonomous_robots [Section III], @devt_sp_machine [Section 14]).
A schematic representation of how the SP Machine may be developed and applied is shown in Figure \[sp\_machine\_figure\].
![Schematic representation of the development and application of the SP machine. Reproduced from Figure 2 in [@sp_extended_overview], with permission.[]{data-label="sp_machine_figure"}](sp_machine_colour1.pdf){width="90.00000%"}
Distinctive features and advantages of the SP system {#distinctive_features_advantages_appendix}
----------------------------------------------------
Distinctive features of the SP system and its main advantages compared with AI-related alternatives are described in @sp_alternatives. In particular, Section V of that paper describes thirteen problems with deep learning in artificial neural networks and how, with the SP system, those problems may be overcome. The SP system also provides a comprehensive solution to a fourteenth problem with deep learning—“catastrophic forgetting”—meaning the way in which new learning in a deep learning system wipes out old memories.[^11]
The main strengths of the SP system are in its versatility in the representation of several kinds of knowledge (Appendix \[versatility\_in\_rk\_appendix\]), its versatility in several aspects of intelligence (Appendix \[versatility\_in\_intelligence\_appendix\]), and because these things all flow from one relatively simple framework—the SP-multiple-alignment concept—they may work together seamlessly in any combination (Appendix \[seamless\_integration\_appendix\]). That kind of seamless integration appears to be essential in any system that aspires to general human-level artificial intelligence.
Potential benefits and applications of the SP system
----------------------------------------------------
Potential benefits and applications of the SP system are described in several peer-reviewed papers, copies of which may be obtained via links from [www.cognitionresearch.org/sp.htm](http://www.cognitionresearch.org/sp.htm): the SP system may help to solve nine problems with big data [@sp_big_data]; it may help in the development of human-like intelligence in autonomous robots [@sp_autonomous_robots]; the SP system may help in the understanding of human vision and in the development of computer vision [@sp_vision]; it may function as a database system with intelligence [@wolff_sp_intelligent_database]; it may assist medical practitioners in medical diagnosis [@wolff_medical_diagnosis]; it provides insights into commonsense reasoning [@sp_csrk]; and it has several other potential benefits and applications described in @sp_benefits_apps. And, of course, this paper describes how the SP system may be applied in software engineering.
Towards universal artificial intelligence (UAI) {#uai_appendix}
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In Sections \[integration\_of\_software\_with\_database\_section\] and \[reducing\_formats\_and\_formalisms\_section\], it has been noted that the SP system has potential for [*universal artificial intelligence*]{} (UAI). The purpose of this appendix is to describe what this means and to distinguish the concept from alternatives such as a ‘universal Turing machine’ (UTM) [@turing_1936].
The idea that something may have UAI or be a UAI derives from the concept of a [*universal framework for the representation and processing of diverse kinds of knowledge*]{} (UFK) [@sp_big_data Section III] but gives weight to the concept of (human-like) ‘intelligence’.
The idea that the SP system has potential for UAI may at first sight seem to be redundant since it has been recognised for some time that all kinds of computing may be understood in terms of the workings of a UTM or ideas which are recognised as equivalent such as Post’s ‘canonical system’ [@post_1943], or Church’s ‘lambda calculus’ [@church_1941], or indeed the many conventional computers that are in use today. For the sake of brevity these will be referred to collectively as CCs, short for ‘conventional computers’.
The suggestion here is that, by definition: 1) a UAI should demonstrate human-like intelligence, 2) it should be able to represent any kind of knowledge, 3) it should provide for any kind of processing within the limits set by computational complexity, 4) it should facilitate the seamless integration of diverse kinds of knowledge and diverse kinds of processing in any combination, and 5) it should do these things efficiently. In short, a UAI is a Turing-equivalent device with human-like intelligence.
The potential of the SP system in areas 1), 2), 4), and 5), and how it differs from a CC, is described in the following four subsections.
Versatility in aspects of intelligence via the powerful concept of SP-multiple-alignment {#versatility_in_intelligence_appendix}
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As noted in Appendix \[sp\_multiple\_alignment\_appendix\], the concept of SP-multiple-alignment has the potential to be the ‘double helix’ of intelligence, the key to the versatility of the SP system in aspects of intelligence, summarised here:
- [*Unsupervised learning via the processing of SP-multiple-alignments*]{}. The SP system has strengths and potential in ‘unsupervised’ learning of new knowledge, meaning learning without the assistance of a ‘teacher’ or anything equivalent. As outlined in Appendix \[unsupervised\_learning\_appendix\], unsupervised learning is achieved in the SP system via the processing of SP-multiple-alignments to create Old SP-patterns, directly and indirectly, from New SP-patterns, and to build collections of Old SP-patterns, called [*grammars*]{} which are relatively effective in the compression of New SP-patterns (@wolff_2006 [Chapter 9], @sp_extended_overview [Section 5]).
Unsupervised learning appears to be the most fundamental form of learning, with potential as a foundation for other forms of learning such as reinforcement learning, supervised learning, learning by imitation, and learning by being told.
- [*How other aspects of intelligence flow from the building of SP-multiple-alignments*]{}. By contrast with the way in which the SP system models unsupervised learning via the processing of already-constructed ‘good’ SP-multiple-alignments, other aspects of intelligence derive from the building of SP-multiple-alignments (Appendix \[sp\_multiple\_alignment\_appendix\]). These other aspects of intelligence include: analysis and production of natural language; pattern recognition that is robust in the face of errors in data; pattern recognition at multiple levels of abstraction; computer vision [@sp_vision]; best-match and semantic kinds of information retrieval; several kinds of reasoning (more under the next bullet point); planning; and problem solving (@wolff_2006 [Chapters 5 to 8], @sp_extended_overview [Sections 5 to 14]).
- [*How several kinds of reasoning flow from the building of SP-multiple-alignments*]{}. In scientific research and in the applications of science, what is potentially one of the most useful attributes of the SP system is its versatility in reasoning, described in @wolff_2006 [Chapter 7] and @sp_extended_overview [Section 10]. Strengths of the SP system in reasoning, derived from the building of SP-multiple-alignments, include: one-step ‘deductive’ reasoning; chains of reasoning; abductive reasoning; reasoning with probabilistic networks and trees; reasoning with ‘rules’; nonmonotonic reasoning and reasoning with default values; Bayesian reasoning with ‘explaining away’; causal reasoning; reasoning that is not supported by evidence; the already-mentioned inheritance of attributes in class hierarchies; and inheritance of contexts in part-whole hierarchies. There is also potential for spatial reasoning [@sp_autonomous_robots Section IV-F.1], and for what-if reasoning [@sp_autonomous_robots Section IV-F.2].
### Generality in artificial intelligence {#generality_in_ai_appendix}
The close connection that is known to exist between information compression and concepts of prediction and probability [@solomonoff_1964; @solomonoff_1997; @li_vitanyi_2014], the central role of information compression in the SP-multiple-alignment framework (Appendix \[sp\_multiple\_alignment\_appendix\]), and the versatility of the SP-multiple-alignment framework in the representation of knowledge (Appendix \[versatility\_in\_rk\_appendix\]) and aspects of intelligence (Appendix \[versatility\_in\_intelligence\_appendix\]), suggest that SP-multiple-alignment may prove to be the key to the development of general, human-like artificial intelligence.
### What about things that the SP system can’t do, except with some kind of ‘programming’ or ‘training’? {#sp_programming_training_appendix}
In considering the possibility that the SP system might be developed into a UAI is that, while the mechanisms for the building and processing of SP-multiple-alignments, yield several different AI-related capabilities, described above, there are lots of things that a newly-created system, without any ‘experience’, would not be able to do. It would not, for example, have any knowledge of how to hold a pencil, how to climb a ladder, how to negotiate an international treaty, and so on.
Is it reasonable to suggest that such a system might be a UAI when there there are so many shortcomings in what it can do? The answer, of course, is “Yes, such a system can be ‘universal’ in exactly the same way that a universal Turing machine, or a newborn baby, is universal”, because in all three cases there is the potential to do a wide variety of different things, provided that it has appropriate knowledge, acquired via learning (babies and AI systems) or programming (computers).
Since procedures or processes are forms of knowledge, and since we have reason to believe that the SP system may accommodate any kind of knowledge (Appendix \[versatility\_in\_rk\_appendix\]), it is reasonable to believe that the SP system may in principle, with the right knowledge, do any kind of computation that is not ruled out by over-large computational complexity.
### How the SP system differs from a CC in aspects of intelligence {#sp_cc_differences_in_intelligence_appendix}
With regard to the modelling of human-like intelligence, the main attraction of the SP system compared with CCs, is its versatility in diverse aspects of intelligence (Appendix \[versatility\_in\_intelligence\_appendix\]) and its potential for the seamless integration of diverse aspects of intelligence and diverse kinds of knowledge, in any combination (Appendix \[seamless\_integration\_appendix\]).
Unless a CC has been specifically programmed with SP capabilities—in which case it would be an SP system, not a CC—it would be lacking in the above-mentioned capabilities, and, arguably, for that reason, is likely to fall short of general human-like artificial intelligence.
Versatility in the representation of knowledge via the powerful concept of SP-multiple-alignment {#versatility_in_rk_appendix}
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The SP system has the potential for UAI because, although SP-patterns are not very expressive in themselves, they come to life in the SP-multiple-alignment framework. Within that framework, they may serve in the representation of several different kinds of knowledge, including: the syntax of natural languages; class-inclusion hierarchies (with or without cross classification); part-whole hierarchies; discrimination networks and trees; if-then rules; entity-relationship structures [@wolff_sp_intelligent_database Sections 3 and 4]; relational tuples ([*ibid*]{}., Section 3), and concepts in mathematics, logic, and computing, such as ‘function’, ‘variable’, ‘value’, ‘set’, and ‘type definition’ (@wolff_2006 [Chapter 10], @sp_benefits_apps [Section 6.6.1]).
With the addition of two-dimensional SP-patterns to the SP Computer Model, there is potential for the SP system to represent such things as: photographs; diagrams; structures in three dimensions [@sp_vision Section 6.1 and 6,2]; and procedures that work in parallel [@sp_autonomous_robots Sections V-G, V-H, and V-I, and Appendix C].
### Generality in the representation of knowledge {#generality_in_rk_appendix}
The generality of information compression as a means of representing knowledge in a succinct manner, the central role of information compression in the SP-multiple-alignment framework, and the versatility of that framework in the representation of knowledge, suggest that SP-multiple-alignment may prove to be a means of representing [*any*]{} kind of knowledge, as would be needed if the SP system were to be a UAI.
### How the SP system differs from a CC in the representation of knowledge {#sp_cc_differences_in_rk_appendix}
With regard to the representation of knowledge, attractions of the SP system compared with CCs are:
- The SP system provides for the succinct representation of knowledge via ICMUP and the powerful concept of SP-multiple-alignment. By contrast, information compression, ICMUP, and SP-multiple-alignments are barely recognised as guides or principles for the representation of knowledge in CCs.[^12]
- The versatility of the SP system in the representation of knowledge is combined with some constraint—knowledge must be represented with SP-patterns and processed via the building and manipulation of SP-multiple-alignments (Appendix \[versatility\_in\_rk\_appendix\])—and that constraint seems to be largely responsible for how the system facilitates the seamless integration of different kinds of knowledge (Appendix \[seamless\_integration\_appendix\]).
By contrast, the representation of knowledge in a CC is a free-for-all: any kind of structure that may be represented with arrays $0$s and $1$s is accepted. This relative lack of discipline seems to be largely responsible for the excessive number of formats and formalisms in computing today (Appendix \[the\_curse\_of\_variety\_appendix\]) and the many incompatibilities that exist amongst computer applications today.
The need for some discipline in how computing is done is not a new idea. In the early days of computing by machine, there was much ‘spaghetti programming’ with the infamous “goto” statement, leading to the creation of programs that were difficult to understand and to maintain. This problem was largely solved by the introduction of ‘structured programming’ (see, for example, @jackson_1975). Later, it became apparent that there could be more gains in the comprehensibility and maintainability of software via the introduction of ‘object-oriented’ programming and design, modelling software on world-oriented objects and classes of object.
Seamless integration of diverse kinds of knowledge and diverse aspects of intelligence {#seamless_integration_appendix}
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In connection with the potential of the SP system as a UAI, an important third feature of the system, alongside its versatility in aspects of intelligence and its versatility in the representation of knowledge, is that [*there is clear potential for the SP system to provide seamless integration of diverse kinds of knowledge and diverse aspects of intelligence, in any combination.*]{} This is because diverse kinds of knowledge and diverse aspects of intelligence all flow from a single coherent and relatively simple source: SP-patterns within the SP-multiple-alignment framework.
In this respect, there is a sharp contrast between the SP system and the majority of other AI systems, which are either narrowly specialised for one or two functions or, if they aspire to be more general, are collections or kluges of different functions, with little or no integration.[^13]
This point is important because it appears that seamless integration of diverse kinds of knowledge and diverse aspects of intelligence, in any combination, are essential pre-requisites for human levels of fluidity, versatility and adaptability in intelligence.
Figure \[versatility\_integration\_figure\] shows schematically how the SP system, with SP-multiple-alignment centre stage, exhibits versatility and integration.
![A schematic representation of versatility and integration in the SP system, with SP-multiple-alignment centre stage.[]{data-label="versatility_integration_figure"}](sp_ma_diagram.pdf){width="90.00000%"}
Efficiency {#efficiency_appendix}
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As noted near the beginning of Appendix \[uai\_appendix\], the fifth suggested feature of a UAI is that it should in some sense be relatively ‘efficient’ in its ability to represent diverse kinds of knowledge, to support diverse aspects of intelligence, and to provide for seamless integration of diverse kinds of knowledge and diverse aspects of intelligence, in any combination. This section expands on that idea.
It is anticipated that, when the SP system is more fully developed, it is likely to be more ‘efficient’ than a CC, largely because it contains well-developed mechanisms for compression of information via the matching and unification of patterns (ICMUP), expressed via the powerful concept of SP-multiple-alignment. This provides the key to the SP system’s versatility in the representation of diverse kinds of knowledge (Appendix \[versatility\_in\_rk\_appendix\]), its versatility in aspects of intelligence (Appendix \[versatility\_in\_intelligence\_appendix\]), and its potential for the seamless integration of diverse kinds of knowledge and diverse aspects of intelligence in any combination (Appendix \[seamless\_integration\_appendix\]).
Although the computational ‘core’ of a CC is likely to be smaller and simpler than in the SP Machine, the SP system has potential for relative advantages like these:
- [*More intelligence*]{}. A CC is likely to fall short of the SP system in modelling the fluidity, versatility, and adaptability of human intelligence—unless the CC has been programmed with all the features of the SP system, in which case it would be an SP system and not a CC.
- [*Economies in software*]{}. Because of the pervasive influence of information compression in the SP system, its ‘software’ is likely to be relatively compact. By contrast, the absence of well-developed mechanisms for ICMUP in the core of the CC is likely to mean the need for such mechanisms to be repeatedly recreated in different guises and in different applications. This can mean software with a bloating that more than offsets the small size of the central processor. See also Section \[overall\_simplification\_of\_applications\_section\].
- [*Economies in data*]{}. Unlike a CC, the SP system is designed to compress its data via unsupervised learning. This would normally mean that data for the SP Machine would, after compression, be substantially smaller than data for a CC.
- [*Dramatic reductions in the variety of formats and formalisms*]{}. As described in Appendix \[the\_curse\_of\_variety\_appendix\], an enormous variety of formats and formalisms is associated with conventional systems. The SP Machine has potential for dramatic simplifications in this area.
- [*Efficiency in processing*]{}. Although CCs, compared with human brains, are extraordinarily effective in such arithmetic tasks as adding up numbers or finding square roots, the advent of big data is creating demands that exceed the capabilities of the most powerful supercomputers [@kelly_hamm_2013 p. 9]. But by exploiting statistical information that the SP system gathers as a by-product of how it works, there is potential in the system for substantial gains in the energy efficiency of its computations [@sp_big_data Sections VIII and IX].
With regard to the second and third bullet points, all knowledge in the SP system reflects the world outside the system. This may include knowledge of entities and their interrelations—the kind of knowledge that would conventionally be called ‘data’—and knowledge of world-oriented processes or procedures—the kind of knowledge that might conventionally be called ‘software’.
All such knowledge is stored as SP-patterns without any formal distinctions amongst them. But in a CC, stored knowledge may be seen to comprise two components:
- Knowledge of the system’s environment, as in the SP Machine. This knowledge may be contained in external ‘databases’ and also in ‘software’.
- Knowledge of processes or procedures, contained largely in ‘software’, needed to overcome the deficiencies of the core model. This kind of knowledge, such as knowledge of how to search for matching SP-patterns, may be recreated many times in many different guises and in many different applications.
The curse of variety in computing and what can be done about it {#the_curse_of_variety_appendix}
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Wikipedia lists nearly 4,000 different ‘extensions’ for computer files, representing a distinct type of file.[^14] A scan of the list suggests that most of these types of file are designed as input for this or that application. Each application is severely restricted in what kinds of file it can process—it is often only one—and incompatibilities are rife, even within one area of application such as word processing or the processing of images. And a program that will run on one operating system will typically not run on any other, so normally a separate version of each program is needed for each operating system, and, with some exceptions, each version needs its own kind of data file.
This kind of variety may also be found within individual files. In a Microsoft Word file, for example, there may be text in several different fonts and sizes, information generated by the “track changes” system, equations, WordArt, hyperlinks, bookmarks, cross-references, Clip Art, pre-defined shapes, SmartArt graphics, headers and footers, embedded Flash videos, images created by drawing tools, tables, and imported images in any of several formats including JPEG, PNG, Windows Metafile, and many more.
Excess variety is also alive and well amongst computer languages. Several hundred high-level programming languages are listed by Wikipedia, plus large numbers of assembly languages, machine languages, mark-up languages, style-sheet languages, query languages, modelling languages, and more.[^15]
Problems arising from excessive variety in computing {#problems_from_variety_appendix}
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Excessive variety in computing is so familiar that we think of it as normal—part of the ‘wallpaper’ of computing. But, although some may see that variety as evidence of vitality in computing, it is probably more accurate to see it as a symptom of a deep malaise in computing as it is today.
Much of this excessive variety is quite arbitrary, without any real justification, and the source of significant problems in computing such as:
- [*Bit rot*]{}. The first of these, bit rot, is when software or data or both become unusable because technologies have moved on. Vint Cerf of Google has warned that the 21st century could become a second “Dark Age” because so much data is now kept in digital format, and that future generations would struggle to understand our society because technology is advancing so quickly that old files will be inaccessible. See, for example, “Google’s Vint Cerf warns of ‘digital Dark Age’”, [*BBC News*]{}, 2015-02-13, [bbc.in/1D3pemp](http://bbc.in/1D3pemp).
- [*Difficulties in extracting value from big data*]{}. With big data—the humongous quantities of information that now flow from industry, commerce, science, and so on—excessive variety in formalisms and formats for knowledge and in how knowledge may be processed is one of several problems that make it difficult or impossible to obtain more than a small fraction of the value in those floods of data [@kelly_hamm_2013; @national_research_council_2013]. Most kinds of processing—reasoning, pattern recognition, planning, and so on—will be more complex and less efficient than it needs to be [@sp_big_data Section III]. In particular, excess variety is likely to be a major handicap for data mining—the discovery of significant SP-patterns and structures in big data [@sp_big_data Section IV-B].
- [*Inefficiencies in the development of software*]{}. Excessive variety in computing also means inefficiencies in the labour-intensive and correspondingly expensive process of developing software and the difficulty of reducing or eliminating bugs in software.
- [*Safety and security*]{}. And excess variety in computing means potentially serious consequences for such things as the safety of systems that depend on computers and software, and the security of computer systems. With regard to cybersecurity, Mike Walker, head of the Cyber Grand Challenge at DARPA, has said that it counts as a grand challenge because of, [*inter alia*]{}, the sheer complexity of modern software. A relevant news report is “Can machines keep us safe from cyber-attack?”, [*BBC News*]{}, 2016-08-02, [bbc.in/2aLGwOu](http://bbc.in/2aLGwOu).
A potential solution to the problem of excessive variety {#a_potential_solution_to_problems_of_variety_appendix}
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The SP system provides a potential solution to the kinds of problems described in Appendix \[problems\_from\_variety\_appendix\]. It arises from the following three features of the SP system:
- [*Versatility of the SP system in the representation of knowledge*]{}. The SP system has already-demonstrated versatility in the representation of diverse kinds of knowledge (Appendix \[versatility\_in\_rk\_appendix\]), with reasons to think that it may serve in the representation of [*any*]{} kind of knowledge (Appendix \[generality\_in\_rk\_appendix\]).
- [*Versatility of the SP system in aspects of intelligence*]{}. The SP system has already-demonstrated versatility in aspects of intelligence (Appendix \[versatility\_in\_intelligence\_appendix\]), with reasons to think that it provides a relatively firm foundation for the development of general, human-like artificial intelligence (Appendix \[generality\_in\_ai\_appendix\]).
- [*Potential of the SP system to perform any kind of computable process or procedure*]{}. As described in Appendix \[sp\_programming\_training\_appendix\], the SP system has potential, via learning or programming, for any kind of computation that is not ruled out by problems with computational complexity.
An implication of the foregoing is that, instead of the great variety of kinds of input file for programs that prevails in computing today, we need only one: a type of computing file that contains SP-patterns, as described in Appendix \[overview\_appendix\].
In a similar way, there is potential to replace all the many different computer languages with one language composed entirely of SP-patterns to be processed by the SP Machine.
A possible qualification to the idea that there might be only type of data file and one type of computer language is that, in both cases, users may wish to create sub-types of data file and sub-types of computer language by incorporating domain-specific knowledge in any given sub-type. For example, information about physics might be incorporated in a special-purpose language for use by physicists, and information about finance might be incorporated in a special-purpose language for that domain.
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[^1]: Dr Gerry Wolff, BA (Cantab), PhD (Wales), CEng, MBCS, MIEEE; CognitionResearch.org, Menai Bridge, UK; <jgw@cognitionresearch.org>; +44 (0) 1248 712962; +44 (0) 7746 290775; [*Skype*]{}: gerry.wolff; [*Web*]{}: [www.cognitionresearch.org](http://www.cognitionresearch.org).
[^2]: Just to confuse matters, this SP-multiple-alignment has been rotated by $90\degree$ compared with the SP-multiple-alignment shown in Figure \[parsing\_figure\]. These two versions of an SP-multiple-alignment are entirely equivalent. The choice between them depends entirely on what fits best on the page.
[^3]: See, for example, “Square Kilometre Array”, [*Wikipedia*]{}, [bit.ly/2t16xxW](http://bit.ly/2t16xxW), retrieved 2017-07-15.
[^4]: See, for example, “Programming by demonstration”, [*Wikipedia*]{}, [bit.ly/2v3phy8](http://bit.ly/2v3phy8), retrieved 2017-07-15.
[^5]: That accountcl only mentions over-generalisation but it appears that the same procedure will apply to the avoidance of under-generalisation.
[^6]: The problem of integrating a class-inclusion hierarchy with a part-whole hierarchy—a problem that arose in connection with the development of an “Integrated Project Support Environment” (IPSE) when I was working as a software engineer with Praxis Systems plc—was one of the main sources of inspiration for the development of the SP system.
[^7]: The solution also applies to class-inclusion heterarchies, meaning a class-inclusion hierarchy with cross-classification.
[^8]: This ambitious objective is in keeping with Occam’s Razor. And as a means of solving the exceptionally difficult problem of developing general, human-level artificial intelligence, it is in keeping with “If a problem cannot be solved, enlarge it”, attributed to President Eisenhower; it chimes with Allen Newell’s exhortation that psychologists should work to understand “a genuine slab of human behaviour” [@newell_1973 p. 303] and his work on [*Unified Theories of Cognition*]{} [@newell_1990]; and it is in keeping with the quest for “Artificial General Intelligence” ([*Wikipedia*]{}, [bit.ly/1ZxCQPo](http://bit.ly/1ZxCQPo), retrieved 2017-08-15).
[^9]: This name has been introduced fairly recently to make clear that there are important differences between the two kinds of multiple alignment.
[^10]: Sometimes there is more than one New SP-pattern in row 0 or column 0.
[^11]: A solution has been proposed in @kirkpatrick_2017 but it appears to be partial, and it is unlikely to be satisfactory in the long run.
[^12]: For example, none of these ideas is mentioned in “Knowledge representation and reasoning”, [*Wikipedia*]{}, [bit.ly/2fmKVtP](http://bit.ly/2fmKVtP), retrieved 2017-08-07.
[^13]: Although Allen Newell called for the development of [*Unified Theories of Cognition*]{} [@newell_1992; @newell_1990], and researchers in ‘Artificial General Intelligence’ are aiming for a similar kind of integration in AI, it appears that none of the resulting systems are fully integrated: “We have not discovered any one algorithm or approach capable of yielding the emergence of \[general intelligence\].” [@goertzel_2012 p. 1].
[^14]: Details may be seen in “List of filename extensions”, [*Wikipedia*]{}, [bit.ly/28LaT4v](http://bit.ly/28LaT4v), retrieved 2016-08-16.
[^15]: There is more information in “List of programming languages”, [*Wikipedia*]{}, [bit.ly/1GTW05W](http://bit.ly/1GTW05W), retrieved 2016-08-16; and also in “Computer language” and links from there, [*Wikipedia*]{}, [bit.ly/2aZ2kag](http://bit.ly/2aZ2kag), retrieved 2016-08-17.
|
---
abstract: 'In this molecular dynamics study, we examine the local surface geometric effects of the normal impact force between two approximately spherical nanoparticles that collide in a vacuum. Three types of surface geometries, facets, sharp crystal edges, and amorphous surfaces of nanoparticles with radii $R < \unit[10]{nm}$ are considered, and the impact force is compared with its macroscopic counterpart described by a nonlinear contact force, $F_{N} \propto \delta^{n}$ with $n = 3/2$ derived by Hertz (1881), where $\delta$ is the overlap induced by elastic compression. We study the surface geometry-dependent impact force. For facet-facet impact, the mutual contact surface area does not expand due to the large facet surface, and this in turn leads to a non-Hertz impact force, $n < 3/2$. A Hertz-like contact force, $n \approx 3/2$, is recovered in the edge contact and in the amorphous surface contact, allowing expansion of the mutual contact surface area. The results suggest that collisions of amorphous nanoparticles or nanoparticles with sharp edges may maintain dynamic phenomena, such as breathers and solitary waves, originating from the nonlinear contact force.'
address:
- |
Department of Physics\
The State University of New York at Buffalo, Buffalo, New York 14260-1500, USA
- |
Mathematical Soft Matter Unit\
Okinawa Institute of Science and Technology Graduate University, Onna-son, Okinawa, 904-0495 Japan
author:
- Yoichi Takato
- 'Michael E. Benson'
- Surajit Sen
bibliography:
- 'The\_effect\_of\_surface\_geometry\_on\_collisions\_between\_nanoparticles.bib'
title: The effect of surface geometry on collisions between nanoparticles
---
Contact force ,Nanoparticle ,Collision ,Surface roughness
46.55.+d ,45.50.Tn ,61.46.Df ,05.45.-a
Introduction
============
The discrete nature of nanoscale materials has often revealed surprising phenomena. Nanoscale normal contact force and friction laws depend strongly on the contact. A molecular dynamics (MD) study by Luan and Robbins [@luan2005] demonstrated the breakdown of continuum theory in a system of nanoscale non-adhesive cylinders made to model contact between substrates and atomic force microscope tips. The normal contact force for rough surfaces of solids with amorphous and crystal structures revealed significant departures from the Hertz contact force for contacting elastic perfect spheres [@hertz1881]. Atomically rough surfaces on contacting objects influence the contact considerably, and local surface geometry in the vicinity of the contact region promotes variations of the contact surface area [@luan2005; @pastewka2016].
Atomic surface asperities also dictate impact phenomena in nanoparticles. Gold nanoparticles can take a near-spherical shape, one of the thermodynamically stable shapes [@barnard2009], and the surface of such a nanoparticle is comprised of crystal steps and terraces. Facets and crystal steps in small nanoparticles can make them “rough”, which in turn affects the nature of their interactions. The lack of spherical nanoparticle symmetry often leads to more complex dynamics than the Hertz contact theory for perfect spheres. Roll, slide, and deflection of colliding nanoparticles arising from shape asymmetry have been observed elsewhere [@dominik1996; @awasthi2007].
In dynamics, the coefficient of restitution often suffices to predict overall behavior of a dissipative many-body system such as clustering of dissipative particles [@luding1999]. Precise measurement of impact force, however, is critical for certain dynamic systems. A recent MD simulation demonstrated that a one-dimensional chain of nanoscale buckyballs can permit the propagation of a solitary wave [@xu2016], which is a non-dispersive propagating wave in a macroscopic granular system discovered by Nesterenko [@nesterenko1983]. Realization of the solitary wave at nanoscale implies that the interaction between buckyballs is described by a nonlinear force. That is, the power of overlap must satisfy $n > 1$ so that a sharp propagating pulse of energy can be formed [@nesterenko2001; @sen2008].
Direct observation of the impact force of repulsive nanoparticles made by MD simulations, on the other hand, has been reported in a few papers [@kuninaka2009a; @kim2010jkps; @zeng2010]. The nature of impact forces between rough surfaces of colliding nanoparticles is not fully understood yet. Our MD study presents precise details of contact forces for collisions between nanoparticles having three different contact surface geometries, in order to investigate the influence of surface roughness in a systematic fashion. The surface of an amorphous nanoparticle and two surface geometries, facets and sharp crystal edges of a face-centered cubic (fcc) crystal nanoparticle, are considered as contact surfaces.
Many MD studies on nanoparticle collisions indicate that nanoparticles are highly elastic if the impact velocity is kept below their material yield point, although a small amount of the initial kinetic energy admittedly dissipates during the impact process [@kuninaka2009a; @kuninaka2009b; @han2010; @takato2014]. In addition, a recent study showed that the effect of nanoparticle adhesive property is negligibly small [@takato2015] if the adhesive nanoparticles collide beyond a critical velocity determined by the balance between adhesion and elastic energies [@thornton1998; @awasthi2006; @awasthi2007; @jung2010]. Therefore, for simplicity, we consider only repulsive nanoparticles in our simulation, adopting the Weeks-Chandler-Andersen (WCA) potential [@weeks1971; @luan2005; @takato2014], which is a type of Lennard-Jones (LJ) potential modified to yield a purely repulsive interaction.
In our work, a non-Hertzian contact force exhibiting velocity dependence is found in monocrystalline nanoparticles impacting on facets. This type of impact makes the dynamics of a collision considerably different from Hertzian contact mechanics. For instance, the mutual contact surface created between two nanoparticles in contact remains unchanged, as opposed to the evolving contact surface area of a Hertzian sphere. In contrast, the Hertzian contact force is a good approximation for crystalline nanoparticles impacting on sharp crystal step edges. Furthermore, amorphous nanoparticles with relatively smooth surfaces and substantially spherical shapes yield an impact force that resembles the Hertzian contact force.
The present work is organized as follows: the Hertz contact theory is briefly reviewed in Sec. \[hertz-contact-theory\]. Sec. \[numerical-simulations\] discusses our nanoparticle models and computational methods. Contact forces for colliding nanoparticles made from amorphous and crystal structures are displayed in Sec. \[impact-force\]. Furthermore, the maximum contact area and maximum compression for facet contact for monocrystalline nanoparticles are compared with those for Hertzian spheres in Sec. \[dynamic-behaviors-for-facet-contact\]. In Sec. \[discussion\], a departure is discussed. The conclusions are presented in Sec. \[conclusions\].
Hertz contact theory
====================
Hertz derived a normal compressive force $F_\text{H}$ between two statically contacting elastic spheres that have smooth surfaces [@hertz1881; @johnson1987]. The force is expressed in terms of overlap $\delta = (2R - d)/2$ for two identical spheres of diameter $2R$ and center-to-center intersphere distance $d$ under compression, $$F_\text{H} = \kappa_\text{H}\delta^{n}, \label{eq:fh}$$ where $n = 3/2$ and $\kappa_\text{H} = (4/3)E^{*}R^{*1/2}$. The reduced Young’s modulus is $E^{*} = E/\lbrack 2(1 - \nu^{2})\rbrack$ with Young’s modulus $E$ and Poisson ratio $\nu$. The reduced radius is $R^{*} = R/2$ for identical spheres of radius $R$. The contact force grows nonlinearly with increasing overlap. The underlying mechanism that yields the nonlinearity is the varying mutual contact surface area between the spheres as a function of compression. The shape of the contact surface in the theory is assumed to be a circle with radius $a_\text{H}$, from which its area is $\pi a_\text{H}^{2}$. General contact surface shapes and their resultant contact forces including Hertzian force are discussed in Ref. [@sun2011]. The geometrical relation between the displacement and the contact area under compression gives $$a_\text{H} = \sqrt{R^{*}\delta}.$$ Hence, the contact area expands in proportion to the square root $\sqrt{\delta}$ of the overlap.
The equations are derived under static compression. For colliding elastic spheres that are assumed to follow the Hertz contact theory by ignoring energy dissipation via surface vibrations, there are several important quantities that characterize the spheres. A maximum overlap $\delta_\text{H}^{\max}$ is the overlap when the colliding spheres instantaneously come to rest during a collision. Maximum overlap takes the highest value when the initial kinetic energy associated with the relative velocity $v_{\text{imp}}$ is fully converted into elastic energy to deform the spheres. The maximum overlap $\delta_\text{H}^{\max}$ is therefore expressed by Eq. (3), taking into account the energy balance between kinetic energy and elastic energy. $$\delta_\text{H}^{\max} = \left( \frac{15}{16}\frac{M^{*}}{E^{*}R^{*1/2}}{v_{\text{imp}}}^{2} \right)^{2/5}.$$
The contact radius for the spheres at maximum compression is then obtained from Eqs. (2) and (3), $$a_\text{H}^{\max} = \sqrt{R^{*}\delta_\text{H}^{\max}} = \left( \frac{15}{16}\frac{M^{*}R^{*2}}{E^{*}}{v_{\text{imp}}}^{2} \right)^{1/5}.$$
This reveals that the maximum contact radius scales with impact velocity to the $2/5$th power.
Numerical simulations
=====================
Models {#a.-models}
------
For an investigation of the dynamic contact force between colliding nanoparticles, nearly spherical argon nanoparticles $\xi$ and $\eta$ of equal radius $R$ are prepared. To study the influence of surface roughness, we employ crystal and amorphous structures as base materials for making our nanoparticles.
Crystalline nanoparticles are carved out of an fcc single crystal of solid argon. The resultant nanoparticles have atomic roughness such as crystal facets and steps on their exterior surfaces due to their crystal structures (See the insets of Figs. –.). We take advantage of the presence of the surface roughness to obtain contact forces at particular points on the surfaces. The {001} facets are chosen for studying *facet collisions* and some sharp crystal edges are chosen for studying the *edge collision* problem.
The amorphous structure is obtained by quenching a molten argon block from temperature $T = \unit[70]{K}$ to $\unit[0.02]{K}$ at a rate $\unit[8 \times 10^{10}]{K/s}$ [@kristensen1976; @nose1985], followed by equilibrations at $T_\text{eq} = \unit[0.02]{K}$. A nearly spherical nanoparticle is then created by cutting the block (See the insets of Figs. –). Radial distribution functions computed from our equilibrated nanoparticles confirm the amorphous structure [@rahman1976].
A pure repulsion between two contacting nanoparticles is considered by employing the WCA potential in Eq. as used for our previous study [@takato2014] and for the Luan and Robbins’s study [@luan2005].
Interatomic potentials {#b.-interatomic-potentials}
----------------------
Our nonequilibrium MD simulations here use two interatomic potentials for argon nanoparticles. The shifted 12-6 LJ potential $V^{\text{LJ}}$ in Eq. (5) describes an interatomic interaction between a pair of atoms $i$ and $j$ in either nanoparticle $\xi$ or nanoparticle $\eta$, i.e., $i,j \in \xi$ or $i,j \in \eta$ as follows, $$V^{\text{LJ}} = \left\{ 4\epsilon\left\lbrack \left( \frac{\sigma}{r_{ij}} \right)^{12} - \left( \frac{\sigma}{r_{ij}} \right)^{6} \right\rbrack + V(r_{c}^{\text{LJ}}) \right\} H(r_{c}^{\text{LJ}} - r_{ij}). \label{eq:lj}$$ The $r_{ij}$ denotes an interatomic distance between $i$-th and $j$-th atoms, $\sigma$ is the distance at which the potential is zero, and $\epsilon$ is the depth of the potential. The Heaviside step function $H$ describes a cutoff of the potential at $r_{c}^{\text{LJ}} = \unit[2.5]{\sigma}$. The potential is shifted by $V(r_{c}^{\text{LJ}})$ to get rid of a discontinuity at $r_{ij} = r_{c}^\text{LJ}$ that stems from the adoption of the cutoff.
In addition, the WCA potential $V^{\text{WCA}}$ in Eq. (6) is adopted to attain purely repulsive nanoparticles by setting its cutoff radius at $r_{c}^{\text{WCA}} = 2^{1/6}\sigma$ where the potential is minimal. The use of this cutoff value makes the potential purely repulsive [@weeks1971]. This potential applies only to a pair of atoms $p$ and $q$ belonging to nanoparticle $\xi$ or nanoparticle $\eta$, respectively, $$V^{\text{WCA}} = \left\{ 4\epsilon\left\lbrack \left( \frac{\sigma}{r_{pq}} \right)^{12} - \left( \frac{\sigma}{r_{pq}} \right)^{6} \right\rbrack + \epsilon \right\} H(r_{c}^{\text{WCA}} - r_{pq}). \label{eq:wca}$$ For argon atoms, $\sigma = \unit[0.3405]{nm}$ and $\epsilon = \unit[1.654 \times 10^{-21}]{J}$ are set in both potentials.
Computation {#c.-computation}
-----------
Equations of motion for argon atoms are solved by the velocity Verlet algorithm with integration time $\Delta t = \unit[1.08 \times 10^{- 14}]{s}$ for crystalline nanoparticles and $\Delta t = \unit[4.3 \times 10^{- 15}]{s}$ for amorphous nanoparticles.
All the nanoparticles prepared are initially relaxed over sufficient time steps in the canonical (constant-temperature) ensemble at temperature $T_\text{eq} = \unit[0.02]{K}$. After the relaxation, the nanoparticles are brought into head-on collision at a relative impact velocity $v_{\text{imp}} = v_\xi - v_\eta$ in the microcanonical (constant-energy) ensemble. The $v_\xi$ and $v_\eta$ denote the center-of-mass velocities in $z$-direction for the nanoparticles $\xi$ and $\eta$, respectively.
Our MD simulations presented hereafter are carried out by <span style="font-variant:small-caps;">lammps</span> [@plimpton1995].
Calculation of impact force {#d.-calculation-of-contact-force}
---------------------------
An impact force $\mathbf{F}_{\xi\eta}$ acting on the mutual contact surfaces formed between two nanoparticles $\xi$ and $\eta$ in contact is computed by summing the individual interatomic forces $\mathbf{f}_{pq}$ for a pair of atoms $p$ and $q$ that are positioned in separate nanoparticles $\xi$ and $\eta$, respectively. The expression for the force $\mathbf{F}_{\xi\eta}$ is given by $$\mathbf{F}_{\xi\eta} = \sum_{p \in \xi}^{}{\sum_{q \in \eta}^{}\mathbf{f}_{pq}}.$$
The impact force $\mathbf{f}_{pq}$ determined by the WCA potential in Eq. (6) leads to a purely compressive impact force $\mathbf{F}_{\xi\eta}$ that causes deformation of the nanoparticles during a head-on collision.
The normal component $F_{N}$ of the impact force $\mathbf{F}_{\xi\eta} $ is obtained in such a way that $F_{N} = \mathbf{F}_{\xi\eta} \cdot \mathbf{d}_{\xi\eta}/|\mathbf{d}_{\xi\eta}|$, where $\mathbf{d}_{\xi\eta}$ is the instantaneous center-of-mass distance of the colliding nanoparticles $\xi$ and $\eta$ in a direction parallel to a line segment between the centers of the colliding nanoparticles. Although the direction of $\mathbf{d}_{\xi\eta}$ is initially aligned with the $z$ axis, thermal vibrations, slip, and rotation that break the reflectional symmetry of the system frequently result in a small deflection of the contacting nanoparticles. The direction of $\mathbf{d}_{\xi\eta}$ during the collision does not necessarily match the $z$ axis, accordingly. The normal force $F_{N}$ computed by the above-stated definition is used to describe the nanoparticle interactions.
Simulation Results
==================
We present simulation results in this section for impact phenomena found in our repulsive nanoparticles. Impact forces for amorphous and monocrystalline nanoparticles in facet and edge contacts are shown in Sec. \[impact-force\]. Additionally, dynamic behaviors of deformation in the facet contact case are compared in Sec. \[dynamic-behaviors-for-facet-contact\] with theoretical predictions for corresponding Hertzian spheres. For visualization of nanoparticles <span style="font-variant:small-caps;">vmd</span> [@humphrey1996] is utilized.
Impact force
------------
-- --
-- --
-- --
-- --
-- --
-- --
Fig. \[fig:force\] shows normal contact forces for amorphous nanoparticles in (a) and (b), for fcc nanoparticles in edge contact in (c) and (d) and for fcc nanoparticles in facet contact in (e) and (f). Each curve represents a dimensionless normal contact force $\widetilde{F}_{N} \coloneqq F_{N}/E^{*}R^{2}$ between contacting nanoparticles at an impact velocity $v_{\text{imp}}$ plotted as a function of a dimensionless overlap $\widetilde{\delta} \coloneqq \delta/R$. The dimensionless normal force and the dimensionless overlap are henceforth termed a force and an overlap, respectively, for the sake of simplicity. Two different radii $R$ are chosen to see the size dependence of the force. Furthermore, four different impact velocities $v_{\text{imp}} =$ 10, 21, 31, and $\unit[52]{m/s}$ are selected to analyze a dynamic effect on the force. To see the similarity between the nanoparticles and corresponding elastic perfect spheres, the Hertzian contact force described by Eq. is depicted as a dashed line in each plot.
The force is exhibited only in the loading stage of the collision, which is determined by a point where the overlap $\widetilde{\delta}$ takes its maximum value. Although nanoparticles are highly elastic, unlike elastic bodies in contact, the force in the unloading stage mostly does not follow back the path of the force-overlap curve taken in the loading stage. This is because of irreversible processes. A part of the initial kinetic energy is almost always converted to thermal energy during the collision.
All plots in Fig. \[fig:force\] have two compression regimes where the rates of change differ noticeably in impact force. In a curve for $R = \unit[2.5]{nm}$ and $v_{\text{imp}} = \unit[52]{m/s}$ in Fig. \[fig:force\](e), for instance, its critical overlap ${\widetilde{\delta}}_{c}$ is identified as a kink observed at around $\widetilde{\delta} \sim 10^{-2}$, which varies depending on the contact type, size, and impact speed. The force below the critical overlap increases at some constant rate as the overlap increases, and the force above the critical overlap increases further at another rate.
In the high compression regime, $\widetilde{\delta} \gtrsim {\widetilde{\delta}}_{c}$, the edge contact in Fig. \[fig:force\] (c) and (d) shows good agreement with the Hertz contact theory, and the amorphous nanoparticles in Fig. \[fig:force\] (a) and (b) recover the slope of the Hertz contact force. The facet contact, however, turns out to be distinct from the Hertz contact force in both slope and magnitude.
In the low compression regime, $\widetilde{\delta} \lesssim {\widetilde{\delta}}_{c}$, all the forces appear to increase at a rate similar to the Hertz contact force but show large departures in magnitude. The edge contact and amorphous nanoparticles in the same figures possess a gentler mechanical response to impact loads, but the facet contact shows a rapid increase in force in (e) and (f).
Impact forces for the aforementioned surface roughnesses and compression levels are summarized in Table \[table:force\], and we will show more details in the contact types separately.
------------------------------------------------------------------ ----------- -------------- ---------------
Edge contact Facet contact
Large compression ${\widetilde{\delta}}> {\widetilde{\delta}}_c$ Hertz Hertz Non-Hertz
Small compression ${\widetilde{\delta}}< {\widetilde{\delta}}_c$ Non-Hertz Non-Hertz Non-Hertz
------------------------------------------------------------------ ----------- -------------- ---------------
: Impact force for contact surface types and overlap ranges.[]{data-label="table:force"}
### Impact between amorphous nanoparticles {#i.-impact-between-amorphous-nanoparticles}
The exteriors of the amorphous nanoparticles in Figs. \[fig:force\](a) and (b) look more spherical than those of the crystalline nanoparticles in Figs. 1(a) and (b). Although individual atoms randomly stacked on the surface form a small asperity, the arrays of atomic steps and associated sharp edges seen in the crystalline nanoparticles are absent due to the disordered arrangement. The facet-free nanoparticle is therefore expected to recover a Hertzian-like force curve with the $3/2$ power of overlap.
Impact forces ${\widetilde{F}}_{N}$ of the large amorphous nanoparticles in Fig. \[fig:force\] (b) at high compression $\widetilde{\delta} \gtrsim {\widetilde{\delta}}_{c}$ resemble the Hertz contact force in the sense that ${\widetilde{F}}_{N}$ grows with overlap nonlinearly at a rate $\sim 1.5$ though there is a dynamic effect, that is, the forces increase at a certain overlap fixed as impact velocity increases. The nanoparticles in the inset of Fig. \[fig:force\](b) have a flat contact surface developed during the collision. The expanding contact surface area seems consistent with the result in Ref. [@kim2010], which reported an expanding contact surface area of colliding polymer nanoparticles having a relatively smooth surface like our amorphous nanoparticles. This area is in agreement with that of corresponding Hertz spheres in Eq. (2) when the compression is high enough. This contact surface expansion induced by compression as expressed in Eq. (2) is critical to recover the Hertzian contact force.
The smaller amorphous nanoparticles in Fig. \[fig:force\](a) show an impact response similar to that of the large ones, but some noise is present. The contact surface of the small nanoparticles in compression involves only a small number of atoms due to their size, $R = \unit[2.3]{nm}$, the smallest among our nanoparticles.
### Impact on sharp crystal edges {#ii.-impact-on-sharp-crystal-edges}
We consider cases where two monocrystalline nanoparticles collide such that they first come into contact on at least one sharp edge on either of the nanoparticles. Fig. \[fig:force\](c) and (d) illustrate the collisions for fully compressed nanoparticles that are randomly oriented before collision. As noticed, the edge collision case obviously has a contact area smaller than the {001} facet displayed in Fig. \[fig:force\](e). This allows the contact surface area in edge contact to expand as the overlap increases.
This contact force is in excellent agreement with the Hertz contact force, as shown in Figs. \[fig:force\](c) and (d). The impact force is well aligned with the Hertzian force when both the nanoparticles are compressed sufficiently, $\widetilde{\delta} > {\widetilde{\delta}}_{c}$. In this regime, the magnitude of the impact force has almost negligible velocity dependence. Regardless of the impact velocity variation between 10 and $\unit[52]{m/s}$, the forces for the given velocities are all the same up to their maximum overlaps, which now depend on the impact velocity.
### Impact on crystal facets {#iii.-impact-on-crystal-facets}
The contact surface for the facet contact in the inset of Fig. \[fig:force\](e) is obviously distinct from those in the edge contact and amorphous nanoparticle cases. The presence of large surfaces on crystalline nanoparticles and the impact on the facets completely alter the dynamic response of the colliding nanoparticles. Consequently, the impact force between the nanoparticles has a behavior unlike the Hertz contact force in Eq. .
When compression of the nanoparticles is low $\widetilde{\delta} < {\widetilde{\delta}}_{c}$, all impact forces ${\widetilde{F}}_{N}$ at various velocities in Fig. \[fig:force\](e) are the same, and are considerably higher than their corresponding Hertzian forces. Each force then increases at a lower rate as the compression increases further. The slope of the force-overlap curve in the large overlap region, $\widetilde{\delta} > {\widetilde{\delta}}_{c}$, is notably lower than $n = 3/2$ for the Hertz contact force, shown as a dashed line in Figs. \[fig:force\](a–e). Furthermore, the force in the same overlap range has a relatively strong velocity dependence, and faster nanoparticles experience stronger repulsive forces.
The larger nanoparticles in Fig. \[fig:force\](f) behave qualitatively in the same way as seen in the smaller nanoparticles, that is, the impact force has a higher rate of change at $\widetilde{\delta} < {\widetilde{\delta}}_{c}$ and a lower rate at $\widetilde{\delta} > {\widetilde{\delta}}_{c}$. Large nanoparticles are expected to approach Hertz spheres since as nanoparticle size is increased, the facet area progressively becomes smaller and the role of the facet would vanish in the limiting case of $R \rightarrow \infty$. However, these nanoparticles of $R = \unit[7.3]{nm}$ bear no semblance of Hertzian spheres.
Dynamic behaviors for facet contact
-----------------------------------
### Contact surface area {#i.-contact-surface-area}
The contact surface area developed for Hertzian spheres under compression is described by Eq. (2) and it gives rise to nonlinearity in its mechanical response. In the case of the facet collision, the {001} facets are chosen for surfaces to be contacted, and the facet contact resulted in the non-Hertzian force, as already seen. Thus, it is interesting to see how the mutual contact surface evolves as the nanoparticles are compressed. The contact surface area for nanoparticles of several sizes is examined and compared with the Hertzian sphere case.
Fig. \[fig:force\](e) shows a representative snapshot for faceted nanoparticles with their {001} facets in contact at maximum compression, and their colliding surfaces possess relatively flat square-like areas. We adopt a new measure to quantify a contact radius $a$ for our nanoparticles in place of a contact surface area. The contact radius $a$ is an effective radius for an actual contact surface area that equals a circular surface area $\pi a^{2}$. The actual contact surface of the contacting nanoparticles is not a circle; however, it is convenient to regard the square-like contact surface as a circular surface of radius $a$ when compared to the radius of the mutual contact surface area formed between the Hertzian spheres. The contact radius $a$ for nanoparticles is computed by [@vergeles1997] $$\quad a^2 = \frac{1}{N_{s}^{2}}\sum_{i = 1}^{N_{s}}{\sum_{j = 1}^{N_{s}}\left\lbrack \left( x_{i} - x_{j} \right)^{2} + \left( y_{i} - y_{j} \right)^{2} \right\rbrack},$$ where $x_{i}$ ($y_{i}$) and $x_{j}$ ($y_{j}$) are the $x$ ($y$) component of $i$-th and $j$-th atom coordinates on the same contact surface, respectively. $N_{s}$ denotes the total number of atoms involved in contact.
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The nondimensionalized contact radii $\tilde{a} \coloneqq a/R$ of three different sized nanoparticles computed by Eq. (8) at their maximum compression are plotted against the impact velocity in Fig. 2.
The plateaus in the figure clearly indicate that the maximum contact radii ${\widetilde{a}}^{\max}$ remain unchanged. In other words, the facet contact does not allow a contact surface expansion no matter how forcefully the nanoparticles collide, provided that the impact occurs within the range of the quasi-elastic collision regime. Facet sizes are nearly comparable to the radii of small nanoparticles. In particular, the contact radius for the $R = \unit[2.5]{nm}$ nanoparticle reaches $50\%$ of its radius. By contrast, the surface area for Hertzian spheres expands with increasing impact velocity depicted by a dashed line based on Eq. (2), as seen in Fig. 2.
### Compression {#ii.-compression}
Another mechanical property examined here is how much the nanoparticles are compressed at maximal contact. Colliding elastic spheres show a velocity dependence on the dimensionless maximum overlap ${\widetilde{\delta}}^{\max} \propto {v_{\text{imp}}}^{\alpha}$, where $\alpha = 4/5$ for the Hertz, but no size dependence according to Eq. (3). The exponent $\alpha = 4/5$ of the impact velocity $v_{\text{imp}}$ equals the ratio of the exponent, 2, of the impact velocity in kinetic energy $(1/2)M^{*}{v_{\text{imp}}}^{2}$ to the exponent $n + 1 = 5/2$ of the overlap in elastic energy $\kappa_{H}\delta^{n + 1}$ stored in the deformed Hertzian spheres, i.e., $\alpha = 2/(n + 1) = 4/5$. A different value of the exponent $\alpha$ in the facet contact case would be anticipated since its force does not obey the Hertzian force in Figs. \[fig:force\](e) and (f) and the power of the force law, $n$, is likely different from 3/2.
[figs/fig3.eps]{} (61,23)[1]{} (68,29)[1]{} (16.3,52)
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Fig. \[fig:maxComp\] presents the maximum overlap ${\widetilde{\delta}}^{\max}$ of the nanoparticles. The obtained maximum overlap for three nanoparticle sizes has a size dependence, and the smaller nanoparticles are less deformable. The maximum overlap of the nanoparticles exhibits a velocity dependence and appears to be proportional to the impact velocity. This result differs from the $4/5$ power of the velocity that the Hertz contact theory yields. This deviation in the exponent demonstrates further evidence of the non-Hertzian contact force observed in the facet contact.
Discussion
==========
Critical overlap and contact surface area at low compression {#a.-critical-overlap-and-contact-surface-area-at-low-compression}
------------------------------------------------------------
We defined the critical overlap ${\widetilde{\delta}}_{c}$ as an overlap where the dynamic response of the colliding nanoparticles undergoes a sudden change in slope of a force-overlap curve. This value can reduce the somewhat complex force behavior into two regimes in which the forces grow at different rates in a comparatively smooth manner. It then raises questions about what makes such a change in force at a particular overlap and what influences force behaviors above and below the critical overlap.
Fig. 4(a) is a force-overlap plot for facet contact. In addition, the number $N_s$ of atoms involved in contact is also included in the plot in order to see how evolution of the contact surface influences the force. The dimensionless impact force ${\widetilde{F}_N}$ and number $N_{s}$ in the plot are obtained by averaging over different initial conditions.
The number $N_{s}$ of atoms in contact rises and then saturates as the overlap increases. At saturation, all atoms residing on the very first atomic layer of the facet participate in contact. The atomic layers are stacked in the [\[]{}001[\]]{} direction, and the atoms in the second layer are clearly visible from the exteriors of the nanoparticles in Fig. 1(e). Nevertheless, the second layers never come into contact with the other nanoparticle; hence, further growth of the contact area does not occur.
The kink in the number where contact area growth ends, coincides with the transition of the force behavior at the critical overlap ${\widetilde{\delta}}_{c}$. For the small nanoparticle of $R = \unit[2.5]{nm}$ the kink is identified at $\widetilde{\delta} \sim 5 \times 10^{- 3}$ in Fig. 4(a). Thus, we conclude that the critical overlap ${\widetilde{\delta}}_{c}$ is the onset of completion of contact of the first atomic layer of the facet, and that no further growth of the area occurs at $\widetilde{\delta} > {\widetilde{\delta}}_{c}$.
The constant contact areas presented at $\widetilde{\delta} > {\widetilde{\delta}}_{c}$ differ from the results for nanoparticles having a facet-free surface in Refs. [@vergeles1997; @yi2005; @valentini2007; @jung2012] and in our amorphous nanoparticles and edge contacts. The nanoparticles reported in the articles are crystalline nanoparticles with nearly spherical shapes. They have relatively smooth surfaces, although atomic asperity exists. Contact surface areas expand as the nanoparticles get compressed and follow either the Hertz contact theory or the Johnson-Kendall-Roberts theory [@johnson1971], in which surface adhesion is incorporated.
At low compression $\widetilde{\delta} < {\widetilde{\delta}}_{c}$, where partial contact of the facet occurs, the rising rate of the contact force with increasing overlap is higher. The number $N_{s}$ of contacting atoms in this compression domain rises steadily and quickly in a very narrow range of overlap. The dynamic response triggered by the impact load then becomes sharper than that at the higher compression, $\widetilde{\delta} > {\widetilde{\delta}}_{c}$.
In the edge contact case, on the other hand, Fig. 4(b), the very low $N_{s}$ at $\widetilde{\delta} < {\widetilde{\delta}}_c$ shows that only one atom is involved in contact. The $N_{s}$ then increases gradually, which commences in the vicinity of the critical overlap. This means that the nanoparticles are in contact via only a single atom at the very early stage of contact, and for a longer period of contact compared to the facet contact case. The slower compression process happening in a very limited area on the surface is presumably a primary reason that the contact force at $\widetilde{\delta} < {\widetilde{\delta}}_{c}$ increases more slowly by comparison to that at ${\widetilde{\delta}}> {\widetilde{\delta}}_c$.
Hertzian or non-Hertzian? {#b.-hertzian-or-non-hertzian}
-------------------------
The failure of the Hertz contact law for facet contact in our crystalline nanoparticles at high compression seems to be attributable to the occurrence of an impact on a large contact area. Since the forces for edge contact and amorphous surface contact shown earlier recover at least the slope of the Hertzian contact force, for sufficient compression. Two past papers may give us another perspective on repulsive nanoparticles in contact.
In an MD-based nanoindentation study [@bian2013], the contact force of a small copper nanoparticle of radius $R = \unit[10]{nm}$ under uniaxial compression in its {001} facets by two opposing rigid plates behaves according to the Hertzian model. The copper atoms in the simulation were modeled by the Embedded Atom Model (EAM) [@baskes1992] and the pure repulsion between the plates and the copper nanoparticle adopted a harmonic potential.
Moreover, in Ref. [@kuninaka2009a] a facet collision for two LJ nanoparticles with a pure repulsion acting between the nanoparticles described by a soft potential $4\epsilon(\sigma/r_{\text{ij}})^{12}$ with a cutoff $\unit[2.5]{\sigma}$, which is the repulsive term in the 12-6 LJ potential, was reported. Its force on the contact surface turned out to be consistent with Hertzian contact theory.
In both papers, the (100) facet on fcc nanoparticles was chosen for contact. The results are inconsistent with our results for facet contact. This discrepancy perhaps arises from the choices in surface interactions, the softer interactions employed in the above papers and the stiffer interaction, i.e. the WCA potential, used in our simulation. Further investigation using different potentials is needed to identify the disagreement.
Conclusions
===========
We have presented the quasi-elastic interactions of two approximate spherical nanoparticles that undergo a head-on collision obtained by means of non-equilibrium MD simulation. The pure repulsive impact force between two nanoparticles in contact was achieved by adopting the Weeks-Chandler-Andersen potential. To study the effect of atomic scale surface roughness and structure on the impact force, monocrystalline nanoparticles produced from a face-centered cubic crystal and amorphous nanoparticles were prepared. Crystalline nanoparticles possess crystal facets, steps, and sharp edges on their surfaces, while amorphous nanoparticles have comparatively smooth surfaces, although some atomic roughness still remains. We have compared our numerical results with macroscopic counterparts described by Hertzian contact theory.
For monocrystalline nanoparticles, two different contact types, a facet contact and an edge contact, arising from the surface geometry of the faceted nanoparticles were taken into account. The impact on the facets causes dynamic properties to deviate considerably from predictions of the Hertz contact theory. The collision on facets of both particles reveals a non-Hertzian contact force $F_{N} \propto \delta^{n}$, where $\delta$ denotes the overlap and $n < 3/2$ when the nanoparticles are sufficiently compressed. The mutual contact surface area of the colliding nanoparticles does not expand in the quasi-elastic collision regime.
In contrast, during the edge collision, the mutual contact surface area expands, as opposed to the facet collision case. The impact force for the edge collision shows excellent quantitative agreement with the Hertzian contact force when compression is sufficiently high, $\delta/R > 10^{- 2}$.
For a system in which orientation of the nanoparticles cannot be controlled, the possibility of precise facet contact is not high. Overall collisional behavior of the nanoparticles is thought to be dominated by edge contact; hence, Hertzian contact theory is valid in this regard. However, a crystallographic orientation-controlled system, for instance the oriented attachment technique [@halder2007], may have to consider the non-Hertzian interactions we have demonstrated, if nanoparticles are compelled to impact on facets together.
In addition to the crystalline nanoparticle case, collisions of amorphous nanoparticles, which have a rather spherical shape, were simulated in the same manner to examine the influence of facet-free surface structure. The impact force of amorphous nanoparticles agrees qualitatively, $F_{N} \propto \delta^{n}$, where $n \approx 1.5$. However, a dynamic effect on the impact force is observed.
In both crystalline and amorphous nanoparticles, a departure from the Hertzian contact force is commonly seen when nanoparticles are weakly compressed, $\delta/R < 10^{- 2}$. In this compression regime, we found that only a few atoms on each contact surface are involved in contact and surface discreteness influences the impact force.
Acknowledgments {#acknowledgments .unnumbered}
===============
We thank the U.S. Army Research Office for partial support of the present research.
References {#references .unnumbered}
==========
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bibliography:
- 'all.bib'
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[ **Constructing a Stochastic Model of Bumblebee Flights from Experimental Data** ]{}\
Friedrich Lenz$^{1,\ast}$, Aleksei V. Chechkin$^{2,3}$, Rainer Klages$^{1}$\
**[1]{} School of Mathematical Sciences, Queen Mary University of London, Mile End Road, London E1 4NS, UK\
[ **[2]{} Max-Planck Institute for Physics of Complex Systems, Noethnitzer Str. 38, 01187 Dresden, Germany** ]{}\
**[3]{} Institute for Theoretical Physics, NSC KIPT, ul. Akademicheskaya 1,UA-61108 Kharkov, Ukraine\
$\ast$ E-mail: f.lenz@qmul.ac.uk****
Abstract {#abstract .unnumbered}
========
The movement of organisms is subject to a multitude of influences of widely varying character: from the bio-mechanics of the individual, over the interaction with the complex environment many animals live in, to evolutionary pressure and energy constraints. As the number of factors is large, it is very hard to build comprehensive movement models. Even when movement patterns in simple environments are analysed, the organisms can display very complex behaviours. While for largely undirected motion or long observation times the dynamics can sometimes be described by isotropic random walks, usually the directional persistence due to a preference to move forward has to be accounted for, e.g., by a correlated random walk. In this paper we generalise these descriptions to a model in terms of stochastic differential equations of Langevin type, which we use to analyse experimental search flight data of foraging bumblebees. Using parameter estimates we discuss the differences and similarities to correlated random walks. From simulations we generate artificial bumblebee trajectories which we use as a validation by comparing the generated ones to the experimental data.
Introduction {#introduction .unnumbered}
============
Foraging Animals {#foraging-animals .unnumbered}
----------------
The characteristics of the movement of animals play a key role in a variety of ecologically relevant processes, from foraging and group behaviour of animals[@Santos2009] to dispersal[@Petrovskii2009; @Hawkes2009] and territoriality[@Giuggioli2012]. Studying the behaviour of animals, simple random walk models have been proven effective in describing irregular paths[@Codling2008]. While the first studies on random paths of organisms focused on uncorrelated step sequences[@Pearson1905], in many cases of studies of animal behaviour the directional persistence of the animals suggested a modelling in terms of correlated random walks (CRWs) [@Kareiva1983; @Bovet1988]. In many complex environments an intermittent behaviour of animals is observed. In these cases an animal switches either randomly or in reaction to its environment between different movement patterns. The mechanisms which generate, and the factors which influence this switching behaviour have been shown to be important in understanding and modelling complicated animal paths[@Benichou2006; @Plank2009; @Mashanova2010; @Benichou2011review]. While there is a source of switching between free flight and food inspections in the experiment we analyse[@ourPRL], here we concentrate on the former as detailed below. With no clear indication of additional intermittency, we will focus on non-intermittent models in the following.
CRW/Reorientation Model {#crwreorientation-model .unnumbered}
-----------------------
The planar horizontal movement of an animal is often approximated by a sequence of steps: an angle $\alpha(t)$ describes the current direction of movement in a fixed coordinate frame, while the step length $l(t)$ determines the distance travelled during a time step. The direction $\alpha(t)$, often determined by a specific front direction of the animal, changes each time step by a random turning angle $\beta(t)$. The description of the dynamics in a co-moving frame, i.e., via the turning angle, turned out to be most useful for analysis of persistent animal movement [@Kareiva1983; @Bovet1988]. In many cases $\beta(t)$ is drawn independently and identically distributed (i.i.d.) from a wrapped normal distribution or a von Mises distribution [@Codling2005; @Batschelet1981] for each time step, giving rise to a persistence in direction depending on how strongly the distribution is concentrated around $0$. Usually the step length is taken to be either constant or it is drawn i.i.d. from some distribution. The step length can either be the result of a constant speed and a variable time step or (as in our case below) of a constant time step $\Delta t$ and a variable speed $s(t)$.
This class of models can generate a variety of different dynamics. Two special cases with a uniform distribution for $\beta$ and a fixed time step $\Delta t$ are the standard Gaussian random walk for step lengths $l(t)=|z(t)|$ where $z$ is normally distributed and Lévy flights for power-law tails in the step lengths distributions ($l(t)\sim l^{-\mu}$ for $1<\mu\leq3$ and $l>l_0$). Related to Lévy Flights, but using a time step proportional to the step length, are Lévy Walks, which have been of interest as candidates for optimal search behaviour of foraging animals. They have been studied analytically[@ViswanathanStanley1999], by simulations[@pitchford-efficient-or-inaccurate; @Plank2009], and many experimental data sets have been statistically analysed to determine whether Lévy Walks are suitable to describe the movement of animals (see, e.g., [@Viswanathan1996; @Edwards2007; @Benhamou2007; @Sims2010; @overturnFishingEdwards2011]).
As Lévy-type models show anomalous diffusive behaviour, in contrast to models with a finite variance of the step length distribution and a fixed time step $\Delta t$, only the latter are included in the definition of *correlated random walks* which are also called *reorientation models* in the context of animal movement. Apart from pathological cases, CRWs are diffusive in the long time limit according to the central limit theorem.
The estimation of the tortuosity of a trajectory is intimately connected to the distributions of the turning angle and speed [@Bovet1988; @Codling2005; @Benhamou2004]. The relevance of the turning angle distribution for foraging efficiencies when searching in random environments has been analysed, e.g., in [@Bartumeus2008].
Generalisation of the Model {#generalisation-of-the-model .unnumbered}
---------------------------
In the following we will present a generalisation of the CRW above, which we then use to analyse bumblebee flight data. Given movement data with a constant time step $\Delta t$, the step length is determined by the speed $s(t)=|v(t)|$ of the animal. As we will be looking at a flying insect in a data recording using a small time step, we may expect to have a deterministic persistence due to the animals momentum. Additionally, the above CRW model assumes that $s$ and $\beta$ are drawn i.i.d. which is sensible if $\Delta t$ is large enough. However, for small time steps it cannot be excluded that the decision of the animal to turn left or right takes longer than the time step, which can correlate the turning angles $\beta(t)$ over a number of time steps. To allow for these possibilities we therefore model the changes in speed and turning angle via two coupled generalized Langevin equations, $$\begin{aligned}
\frac{d\beta}{dt}(t) &= h(\beta(t),s(t)) + \tilde{\xi}_s(t) \label{eq:generalmodel-beta}\\
\frac{ds}{dt}(t)&= g(\beta(t),s(t)) + \psi(t) , \label{eq:generalmodel-s}
\end{aligned}$$ where we distinguish between the deterministic parts $h$ and $g$ and stochastic terms $\psi$ and $\tilde{\xi}_s$ (whose speed dependency will be discussed in the Results section). We assume that the noise processes are stationary with auto-correlation functions which may be non-trivial, and we make no further assumptions for the shape of their stationary distributions. While Eqs. (\[eq:generalmodel-beta\],\[eq:generalmodel-s\]) represent a time-continuous description, the turning angle $\beta$ still yields the change of $\alpha$ according to our fixed time resolution $\Delta t$. That is, $\beta(t)$ relates to a time-continuous angular velocity $\gamma$ of $\alpha$ via $\beta(t) = \int_{t-\Delta t}^t \gamma(\tau)d\tau$. The animals’ position $r(t)=(x(t),y(t))$ is then given by $dx/dt = s \cos(\alpha(t))$, $dy/dt = s \sin(\alpha(t))$ and $d\alpha/dt = \gamma(t)$. [Not having experimental access to $\gamma$,]{} the numerical analysis is done with time-discrete data where the measured turning angle is given by $\beta(t) ={\alpha(t)-\alpha(t-\Delta t) =} \measuredangle(v(t),v(t-\Delta t))$, where $v(t)=(r(t+\Delta t)-r(t))/\Delta t$ at times $t=n\Delta t$, $n\in \mathbb{N}$.
Application to Experimental Data {#application-to-experimental-data .unnumbered}
--------------------------------
Analysing measured movement data of animals in their natural habitat is intricate due to a variety of factors which may influence the animal’s behaviour, ranging from heterogeneous food source distributions [@Boyer2012; @Kai2009; @Sims2012] and predation threats[@Reynolds2010; @ourPRL] to [individual differences in behaviour within a population]{} [@Petrovskii2009; @Hawkes2009]. Here we analyse data obtained from a small scale laboratory experiment in which single bumblebees forage in an artificial flight arena[@Ings2008]. The set-up is shown in Fig. \[fig:cage\] together with part of a typical trajectory of a bumblebee on its search for food. Each bumblebee can forage on an artificial flower carpet which is positioned on one of the walls of the arena. In this paper we are not interested in the behaviour resulting from the interaction with the flowers which has been studied in detail in [@ourPRL]. Instead we only examine the search flights away from the flower carpet. (See section Materials and Methods for details.) We use our generalised stochastic model (Eqs. (\[eq:generalmodel-beta\],\[eq:generalmodel-s\])) to describe these flights and to examine in which ways the behaviour deviates from a simple CRW model. Here we will focus on the horizontal movements. By neglecting the slower vertical movements, which are of more interest when analysing the starting and landing behaviour near flowers, we thus restrict ourselves to a two-dimensional model.
Results and Discussion {#results-and-discussion .unnumbered}
======================
Estimation of Drift Terms {#estimation-of-drift-terms .unnumbered}
-------------------------
Given the experimental data, we start determining the unknown parameters in our model by first estimating the deterministic parts $h(\beta,s)$ and $g(\beta,s)$ of the Langevin equation. This is done by numerical estimation [@Risken; @FriedrichPeinke1997; @Ragwitz2001; @Lenz2009] of the components of the drift vector field (drift coefficients) $D^1(\beta,s)=(g(\beta,s),h(\beta,s))^\top$ of the corresponding Fokker-Planck equation via $$\label{eq:drift}
D^1(X) = \lim_{\tau\rightarrow 0}\frac{1}{\tau} \left.\left<\widetilde X(t+\tau)-X\right>\right|_{\widetilde{X}(t)\approx X}$$ where $X=(\beta,s)^\top$ and $\left<.\right>$ is the time average over the time series $\widetilde X$ conditioned on $\widetilde{X}(t)\approx X$, where $\widetilde{X}$ is assumed to be stationary (for a detailed discussion see [@Ragwitz2001]). The estimation of the drift terms is based on a Markov approximation: only those parts of the dynamics which match to a Markovian description in the state space variables $\beta$ and $s$ have their deterministic terms reflected in $D^1(X)$. Any other parts of the flight dynamics – stochastic as well as deterministic but not Markovian in $\beta$ and $s$ – are captured by the stochastic terms of Eqs. (\[eq:generalmodel-beta\],\[eq:generalmodel-s\]). Figure \[fig:fpc\] shows the drift vector field, with normalised lengths of the vectors for better visibility. The nearly horizontal vectors show, that the drift quickly pushes the turning angle $\beta$ towards $0$, while the dynamics in the speed $s$ is much slower. As the cross-dependencies of $h(\beta,s)$ on $s$ and of $g(\beta,s)$ on $\beta$ are weak, we can neglect them in our model. Since vector fields are hard to interpret, we will look at the projections in the following.
Examining the drift $h(\beta)$ of the turning angle in Fig. \[fig:anglefpc\] reveals that the drift term seems linear in $\beta$ — indeed we find numerically that its slope $-k$ matches exactly to a decay of the turning angle to $0$ in a single observation time step $\Delta t$ by $k \approx 1/\Delta t$, disregarding the noise term.
This means that by integrating Eq. (\[eq:generalmodel-beta\]) over a time $\Delta t$ and approximating the drift $h(\beta)$ for small $\Delta t$ by $ \int_t^{t+\Delta t} h(\beta(\tau)) d\tau \approx h(\beta(t))\Delta t $, we have $$\label{eq:simpl}
\beta(t+\Delta t)-\beta(t)= -k \beta(t)\Delta t + \int_t^{t+\Delta t}\tilde{\xi}_s(\tau)d\tau
= - \beta(t) + \int_t^{t+\Delta t}\tilde{\xi}_s(\tau)d\tau \; .$$ With $ \xi_s(t) := \int_{t-\Delta t}^t \tilde{\xi}_s(\tau) d\tau $ and Eq. (\[eq:simpl\]), the time scale separation in the $\beta$-Langevin equation due to the very fast relaxation means that we can simplify Eqs. (\[eq:generalmodel-beta\],\[eq:generalmodel-s\]) to: $$\begin{aligned}
\beta(t) &= \xi_s(t) \label{eq:completemodel-beta}\\
\frac{ds}{dt}(t)&= g(s(t)) + \psi(t) \label{eq:completemodel-s} .
\end{aligned}$$ [While this reduction of the turning angle dynamics from $d\beta/dt$ to $\beta$ bears similarity to a simple reorientation model, the turning angles are still correlated and speed-dependent, as we will see below.]{}
The speed drift $g(s)$ displayed in Fig. \[fig:speedfpc\] shows that the deterministic part of the speed-Langevin equation alone would have a stable fixed point around $s_0=\unit[0.27]{m/s}$. Comparing the slopes above and below $s_0$ reveals that for $s<s_0$ the force towards $s_0$ is stronger than for $s>s_0$. This is biologically plausible if one interprets $s_0$ as a preferred speed: if the bumblebee is slower it accelerates, but if it is faster it does not rush to decelerate as it would give up the energy spent to reach a high velocity. For very high velocities (over ) the slope of $g(s)$ increases again. This might be caused by the limited space available to the bumblebee in the flight arena. For our model we approximated $g(s)$ by a piecewise linear function: $$\label{eq:g-piecewise}
g(s)\approx (s-s_0)\times \left\{ \begin{array}{rcl}
-d_1 & \mbox{for}& s<s_0 \\
-d_2 & \mbox{for} & s\geq s_0
\end{array}\right. ,$$ where $d_1>d_2>0$. As the very high velocities are rare, it made no difference in our model whether we used Eq. (\[eq:g-piecewise\]) or a piecewise linear function with three pieces.
Velocity-Dependent Angle-Noise and Noise Auto-Correlations {#velocity-dependent-angle-noise-and-noise-auto-correlations .unnumbered}
----------------------------------------------------------
What we did not specify before was that the turning angle distribution may depend on the speed of the bumblebees. Given that the force a bumblebee can use to change directions is finite, the largest turning angles have to be smaller when flying with high speeds (see Fig. \[fig:schematic\]). This is consistent with the absence of simultaneously having high speed and large turning angle in the data - as is evident, e.g., from the data gaps in Fig. \[fig:fpc\]. However, animals can counteract this geometric dependence by varying the forces used for changing direction with the speed. We approximated the distribution for the turning angles for each speed $s$ by a normal distribution. This approximation works best for low speeds. While there are some deviations for high speeds, it was not possible to reliably fit a better model due to the limited amount of data available. Figure \[fig:beta-s-loglog\] shows how its standard deviation $\sigma_{\beta}$ depends on the current speed. $\sigma_{\beta}$ decreases with increasing speed, however it does not decay to $0$ as a simple geometric model would predict (see Materials and Methods below).
Instead $\sigma_{\beta}(s)$ decays roughly exponentially to a constant offset. We therefore model the turning angles as speed-dependent noise [with a wrapped normal distribution [@Codling2005; @Batschelet1981]]{}: $\xi_s(t)\sim\mathcal{N}(0,\sigma_\xi(s))$ with $\sigma_\xi(s)= c_1 e^{-c_2 s}+c_3$. This offset could either be an effect of the boundedness of the flight arena, since the bumblebee has to turn more often to avoid walls when flying fast. Or it could be that the bumblebees use stronger forces for turning during fast flights to maintain their manoeuvrability. It would be interesting to examine free-flight data to check for the cause. In other models in which the momentum of the animal is not important for the observed directional persistence, this cross-dependence is often neglected [@Kareiva1983].
\[fig:model-angle-acf\]
For the two stochastic parts of the Langevin equations, we estimated the normalised auto-correlation functions from the data. The turning angle auto-correlation is approximated by a steep power-law as seen in Fig. \[fig:angle-acf-log\], which in this case is preferable to the alternative fit by a simple exponential decay. By subtraction of our approximation for the deterministic term $g(s)$ from the observed speed changes $ds/dt$ in Eq. (\[eq:completemodel-s\]) we estimated the distribution and auto-correlation of the noise term $\psi(t)=ds(t)/dt-g(s(t))$. In order not to overestimate the noise term, additive discretization errors of an approximate size of $\sigma_\mathrm{error}=\Delta x/{\Delta t}^2$ due to the finite resolution $\Delta x=\unit[10^{-3}]{m}$ of the cameras have been accounted for, giving the variance $\sigma_\psi^2=\sigma_{\psi^\mathrm{noisy}}^2-\sigma_\mathrm{error}^2$. The noise term $\psi(t)$ is well approximated by Gaussian noise with an auto-correlation function $\mathrm{acf}^{e-e}_\psi(\tau)= a e^{-\lambda_1 \tau} + (1-a) e^{-\lambda_2 \tau}$ (see Fig. \[fig:acc-acf\]).
While an auto-correlation function of the shape of $\mathrm{acf}^{p-p}_\psi(\tau)= b (\tau+1)^{-p_1} + (1-b) (\tau+1)^{-p_2}$ can be exluded, a difference between an exponential and a power-law $\mathrm{acf}^{e-p}_\psi(\tau)= c e^{-\lambda_3 \tau} + (1-c) (\tau+1)^{-p_2}$ is not significantly worse than $\mathrm{acf}^{e-e}_\psi$. For our model we chose the simple difference of exponentials $\mathrm{acf}^{e-e}$.
As the observed anti-correlation between delays of $\unit[0.1]{s}>\tau>\unit[0.3]{s}$ happens on a time scale which is too short to be an effect of the boundedness of the experiment or of residual effects of the presence of the foraging wall[@ourPRL], it is unclear where the anti-correlation comes from. One could speculate that it might be the result of a stabilising mechanism in the bumblebee dynamics.
Validation {#validation .unnumbered}
----------
Given all the parameters of the full model (see Materials and Methods) estimated by minimizing the mean squared errors, we used them to generate artificial bumblebee trajectories, as follows: We simulated the dynamics using an Euler-Maruyama scheme with noise terms $\xi_s(t)$,$\psi(t)$. In rare cases where the Gaussian noise $\psi(t)$ would lead to a negative speed despite the positive drift $g(s)$ for $s<s_0$, we enforce a non-negative speed by setting $s(t)=0$. We correlated the noise terms in advance by modifying their power spectral density in the following way: we take uncorrelated noise of the wanted distribution, multiply its Fourier transform with the root of the desired power spectral density corresponding to our approximate auto-correlation function and then transform back[@NumericalRecipes]. To deal with the speed dependence of the turning angle noise $\xi_s(t)$ we first correlate Gaussian noise and afterwards scale with $\sigma_{\beta}(s)$ at each time step in the integration scheme. While this does not reproduce the auto-correlation of the turning angle exactly, the error made is less than the errors from the estimation of $\mathrm{acf}_{\beta}$.
A sample trajectory of a bumblebee simulated for using $10^5$ time steps is shown in Fig. \[fig:model-traj\]. Using the generated data we checked the validity of the model by comparison to the experimental data of all bumblebees.
Figure \[fig:model-speed-pdf\] compares the probability density function $\mathrm{pdf}(s)$ of the speed extracted from the simulated data with the experimental data. The auto-correlation functions of the speed and turning angle are shown in Figures \[fig:model-speed-acf\] and \[fig:model-angle-acf\]. Considering the number of rough approximations we have made for constructing our model, the agreement between simulation results and experimental data is very good.
Summary {#summary .unnumbered}
-------
We generalised a reorientation model which is often used to describe the correlated random walk of animals by explicitly modelling accelerations via Langevin equations. Analysing movement data from bumblebees, we extracted information on the deterministic and stochastic terms of Eqs. (\[eq:generalmodel-beta\],\[eq:generalmodel-s\]). Simulations of our model and comparison to the data have shown that the resulting model agrees very well with the experimental data despite the approximations we made for the model. With the estimation of the turning angle drift $h(\beta)$ we found that while the usual assumption of i.i.d. turning angles is not valid in our case, the lack of a non-trivial drift and the weak auto-correlation of $\xi_s$ are consistent with the usual reorientation model. However, our generalised model exhibits significant differences in the non-trivial deterministic part $g(s)$ of the speed change $ds/dt$ [and]{} the speed dependence of the turning angles. In terms of active Brownian particle models[@Geier2012; @Geier2011] we described the two-dimensional bumblebee movement by a particle with a non-linear friction term $g(s)$ depending and acting only on the speed, driven by multiplicative coloured noise with different correlations for the angle component and the speed component of the velocity. While this combination of complications might make it difficult to treat the system analytically, progress into this direction has been made[@Peruani2007; @Lindner2010]. [We remark that one could ignore the fast decaying auto-correlations of $\xi_s$ and $\psi(t)$ if one is not interested in the dynamics for short times, thus simplifying the model by using uncorrelated noise terms, since the effect of the noise autocorrelations on the long time dynamics is negligible.]{}
Given that the experiment which yielded our data is rather small and provided the bumblebees with an artificial environment, it would be interesting to apply our new model to free-flying bumblebees to reveal how much the results depend on the specific set-up. This would clarify whether the flight behaviour seen in the laboratory experiment survives as a flight mode for foraging in a patch of flowers in an intermittent model, with an additional flight mode for long flights between flower patches. The analysis of data from other flying insects and birds by using our model could be interesting in order to examine whether the piecewise linear nature of the speed drift and the trivial drift of the turning angle are a common feature. In view of understanding the small-scale bio-mechanical origin of flight dynamics, our model might serve as a reference point for any more detailed dynamical modelling. That is, we would expect that any more microscopic model should reproduce our dynamics after a suitable coarse graining over relevant degrees of freedom.
Materials and Methods {#materials-and-methods .unnumbered}
=====================
Experimental Data {#experimental-data .unnumbered}
-----------------
In this experiment 30 bumblebees ([*Bombus terrestris*]{}) were trained to forage individually in a roughly cubical flight arena with an approximate side length of . Figure \[fig:cage\] shows a diagram of the arena together with data from a typical flight path of a bumblebee. The flight arena included a $4\times4$ grid of artificial flowers on one of the walls. Each of the 16 flowers consisted of a landing platform, a yellow square floral marker and a replenishing food source where syrup was offered. For the analysis presented in this paper all data in zones ($\unit[7]{cm}\times\unit[9]{cm}\times\unit[9]{cm}$) around the flowers has been removed in order to analyse the search behaviour while foraging excluding the interaction with food sources. The 3D flight trajectories of the bumblebees were tracked by two cameras with a temporal resolution of $\Delta t=\unit[0.02]{s}$. Each bumblebee was approximated as a point mass with a spatial resolution of $\unit[0.1]{cm}$: its position was estimated by the arithmetic mean of all image pixels corresponding to the bumblebee via background subtraction. In total $\approx49000$ data points were used for the analysis. For individual bumblebees an average of $51$ search trajectories between flower zones have been sampled and analysed. The thorax widths of the bumblebees have a mean of $\unit[5.6]{mm}$ and a standard deviation of $\unit[0.4]{mm}$.
For calculating auto-correlations small gaps in the time series have been interpolated linearly. As the number of gaps was small the correlations for short times were not affected, however, the interpolation increased the usable data for long time delays. Trajectories were split at larger gaps, e.g., when entering a flower zone, to exclude correlations induced by flower visits.
For a discussion of the influence of the boundedness of the flight arena and for the analysis of the foraging dynamics under varying environmental conditions see [@ourPRL]. More details on the experimental setup can be found in [@Ings2008; @Ings2009].
Estimated Model Parameters {#estimated-model-parameters .unnumbered}
--------------------------
The full set of parameters estimated from the data set which was used for the simulation is given here. For the deterministic drift of the speed the change of slope is at $s_0=\unit[0.275]{m/s}$ while the slopes are $d_1=0.16$ and $d_2=0.06$. The parameters for the standard deviation $\sigma_\xi(s)$ of the angle noise are $c_1=126^\circ$, $c_2=\unit[12]{s/m}$, $c_3=12.5^\circ$ and its auto-correlation is given by $\mathrm{acf}_\beta(\tau)=(\tau+1)^{-1.5476}$. The non-deterministic changes $\psi(t)$ of the speed are assumed to be normally distributed with standard deviation $\sigma_\psi=\unit[3.52]{m/s^2}$ and auto-correlated according to $\mathrm{acf}^{e-e}_\psi(\tau)$ where $a=1.44$, $\lambda_1=25.5$ and $\lambda_2=10.7$.
Speed Dependence of Turning Angles {#speed-dependence-of-turning-angles .unnumbered}
----------------------------------
A simple model showing a dependence of the turning angles on the speed (see Fig. \[fig:schematic\]) is given in the following. Assume that the velocity of an animal changes at each time step $\Delta t$ by an acceleration vector which is given by a binormal i.i.d. random vector with variance $\sigma^2$ in both directions. The turning angle $\beta$ between $v_t$ and $v_{t+\Delta t}$ then depends on the quotient $\eta_t:=s_t/(\sqrt{2}\sigma)$ between the former speed $s_t=|v_{t}|$ and the noise strength $\sigma$. By changing to the comoving frame of the animal and integrating out $s_{t+\Delta t}$ the distribution $\rho(\beta)$ of the turning angle is given by: $$\rho(\beta)=\frac{e^{-\eta^2}}{2\pi}
+ \frac{e^{-\eta^2\sin^2(\beta)}}{2\sqrt{\pi}}\eta\cos(\beta)(1+\mathrm{erf}(\eta\cos(\beta)))$$ for $-\pi\leq\beta\leq\pi$. With vanishing speed $s(t)=\eta(t)=0$ the first term gives a uniform distribution as expected, and for $\eta(t)\rightarrow\infty$ the distribution sharply peaks at $\beta=0$ with its variance $\sigma_\beta$ approaching $0$, similar to the behaviour of the simpler von Mises distribution. As the experimental bumblebee data does not show a decay to $\sigma_\beta=0$ but to a finite value (see Fig. \[fig:beta-s-loglog\]), this simple model does not hold: therefore the accelerations have to be modelled as speed-dependent.
Acknowledgements {#acknowledgements .unnumbered}
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We thank Thomas C. Ings and Lars Chittka for providing us with their experimental data and for their helpful comments.
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